Credit Risk Assessment Using Fuzzy Inhomogeneous Markov Chains Within a Fuzzy Market

Abstract

In the present study, we model the migration process and the changes in the market environment. The migration process is being modeled as an $\mathcal{F}$-inhomogeneous semi-Markov process with fuzzy states. The evolution of the migration process takes place within a stochastic market environment with fuzzy states, the transitions of which are being modeled as an $\mathcal{F}$-inhomogeneous semi-Markov process. We prove a recursive relation from which we could find the survival probabilities of the bonds or debts as functions of the basic parameters of the two $\mathcal{F}$-inhomogeneous semi-Markov processes. The asymptotic behavior of the survival probabilities is being found under certain easily met conditions in closed analytic form. Finally, we provide maximum likelihood estimators for the basic parameters of the proposed models.

1. Introductory Notes

Credit risk pertains to the possibility of financial losses arising from a counterparty’s failure to meet contractual debt obligations. Conceptually, it is straightforward: a firm defaults when it is unable to fulfill significant financial commitments stipulated in a debt contract.

Since the mid-1990s, credit-rating models have gained prominence in the field of risk management. These models utilize the credit-rating of a corporate or sovereign entity as a fundamental input in assessing the probability of default associated with a loan or bond. A key focus within this framework is the credit migration process, which describes the dynamic evolution of an entity’s credit quality over time. This includes transitions across various rating categories for instruments such as sovereign debt, corporate bonds, and other credit exposures.

The modeling of credit migration has become increasingly important for both risk quantification and financial instrument pricing. Its widespread adoption is attributed not only to its methodological accessibility but also to regulatory developments—most notably the introduction of the Basel II Capital Accord by the Basel Committee on Banking Supervision in 2001. This framework, established under the auspices of the Bank for International Settlements (BIS), permits financial institutions to determine capital requirements using both internal and external credit-rating systems, thereby incentivizing the development and use of robust migration models.

The academic modeling of credit migration began in earnest in the mid-1990s, with the introduction of homogeneous Markov chain models in discrete time and state spaces. A majority of contemporary models remain grounded in this discrete-time framework, primarily because leading institutions—such as J.P. Morgan’s CreditMetrics and McKinsey’s CreditPortfolioView—report and estimate rating transition probabilities using discrete-time methodologies.

Accurate modeling of credit migration is critical for assessing portfolio credit risk and for the valuation of structured credit products. Systematic variations in transition matrices can exert significant influence on the credit value-at-risk (VaR) of a portfolio, as well as on the pricing of complex derivatives such as collateralized debt obligations (CDOs). Consequently, rating transition matrices are central to the estimation of economic capital metrics, including expected loss and VaR. Moreover, they provide essential inputs for the pricing of sophisticated credit instruments across the financial sector.

In the study by Carty and Fons (1994), based on Moody’s Investors Service proprietary database covering the period from 1976 to 1993, it was shown that the duration of an entity’s stay within a given credit-rating class follows a Weibull distribution. This finding challenges the assumptions underlying simple homogeneous Markov chain models, particularly their implication of exponentially distributed holding times, and thus highlights their limitations in accurately modeling credit-rating dynamics.

In response to these findings, a substantial body of literature emerged, focusing on duration-based models. Among these, the work of Duffie et al. (2007) is particularly noteworthy and serves as a valuable entry point for further exploration of duration-dependent credit-risk modeling.

1.1. The Importance of the Present Study

This study explores an advanced and novel methodological framework for credit-risk modeling by integrating fuzzy logic with non-homogeneous semi-Markov processes. The proposed approach represents a significant theoretical advancement and addresses critical limitations in existing models.

The importance of this research is accentuated by the contemporary financial landscape, which is characterized by elevated volatility, recurrent economic crises, evolving regulatory frameworks, and the increasing frequency of global disruptions. Under such conditions, conventional credit-risk models, which typically assume data accuracy and time-invariant transition probabilities, have shown decreasing reliability and predictive power.

Fuzzy logic provides a means of incorporating imprecise and uncertain information into risk assessments. This is particularly valuable when dealing with qualitative or subjective inputs—for example, assessments of a borrower’s financial stability or creditworthiness are often ambiguous and not easily quantifiable. In parallel, non-homogeneous semi-Markov processes capture the temporal evolution of credit dynamics, allowing for state transitions whose probabilities depend on both time and duration in a given state. This reflects more accurately the reality of changing borrower behavior and shifting market conditions over time.

By synthesizing these two methodologies, the present study introduces a flexible and robust modeling framework that more effectively accommodates the uncertainty and temporal heterogeneity inherent in real-world credit risk assessment. As such, it contributes to the ongoing development of more resilient and adaptable risk management tools in the face of an increasingly complex financial environment.

This study addresses the asymptotic behavior of survival probabilities in a non-homogeneous semi-Markov process characterized by both fuzzy and absorbing states, evolving within a fuzzy stochastic environment. All non-absorbing states are considered fuzzy, reflecting the necessity of such representations in many practical contexts. The external stochastic environment is likewise modeled as a non-homogeneous semi-Markov process with fuzzy states. Consequently, the problem under investigation corresponds to the quasi-stationarity problem (Seneta and Vere-Jones 1996) for a non-homogeneous semi-Markov process with fuzzy states embedded in a fuzzy stochastic framework.

1.2. A Short Related Literature Review

In the early 2000s, inhomogeneous semi-Markov processes began to be utilized as more realistic models for capturing the variability in credit-rating transition matrices, notably by Vasileiou and Vassiliou (2006). Around the same time, D’Amico et al. (2005) introduced a homogeneous Markov renewal model to describe the dynamics of credit migration processes. While both approaches belong to the broader family of semi-Markov models, they differ fundamentally in the parameters used to construct them. This distinction is rooted in the classical formulation of homogeneous semi-Markov processes, as defined by Howard (1971), in contrast to the Markov renewal framework.

A detailed comparison between inhomogeneous semi-Markov chains and inhomogeneous Markov renewal processes is provided in Vassiliou (2021), highlighting their structural and behavioral differences. Building on this, D’Amico et al. (2007, 2011) proposed inhomogeneous semi-Markov models inspired by reliability theory to model credit migration. Further extending this line of research, D’Amico (2009) developed a semi-Markov maintenance model incorporating imperfect repairs occurring at random times to describe the credit-rating evolution process.

Study Vassiliou and Vasileiou (2013) later introduced an inhomogeneous semi-Markov process for modeling state transitions in the migration process, with particular focus on analyzing the asymptotic behavior of survival probabilities. In another extension, Vassiliou (2013) proposed the use of fuzzy states in modeling migration processes, addressing the significant disagreements among rating agencies. This work also tackles the quasi-stationary distribution problem for non-homogeneous semi-Markov processes with fuzzy states, providing closed-form analytical solutions.

The notion of a stochastic market environment was introduced by Vassiliou (2014) to model the broader economic conditions that influence industry dynamics through varying degrees of turbulence. This concept draws on earlier work in demography, where Cohen (1976, 1977) showed that fertility rates could change dramatically in response to shifting social and economic conditions. Since then, the idea of a stochastic environment has been extended to a wide range of stochastic systems (see Vassiliou 2023).

In parallel, the application of fuzzy logic to stochastic modeling in credit-risk analysis was pioneered by Agliardi and Agliardi (2009, 2011), who proposed a structural model for defaultable bonds within a fuzzy framework. This work led to the development of a fuzzy-form pricing model for these bonds, offering a novel approach to account for uncertainty and imprecision in credit-risk assessments. Building upon these ideas, Vassiliou (2013) demonstrated that a fuzzy market is viable if, and only if, an equivalent martingale measure exists. He further modeled the credit migration of a defaultable bond as an inhomogeneous semi-Markov process with fuzzy states, investigating the asymptotic behavior of survival probabilities within these fuzzy states under the condition of no default. This foundational work has spurred substantial research in the field, as reflected in its extensive citations.

Studies Wu and Zhuang (2015); Wu et al. (2015, 2016) expanded on these concepts by introducing a reduced-form, intensity-based model under fuzzy environments. They applied this methodology to the pricing of defaultable bonds and credit default swaps (CDS), providing practical applications of fuzzy logic in financial markets. In a subsequent study, Wu et al. (2017) incorporated both fuzziness and randomness to examine the pricing of total return swaps in fuzzy random environments, further extending the analytical tools available for modeling credit risk in uncertain conditions.

1.3. The Structure of the Paper

Section 2 provides a brief overview of the credit-rating systems employed by major rating agencies, along with a discussion of the possible states of a stochastic market. This section also motivates the use of fuzzy set theory in modeling both credit-rating transitions and the stochastic market dynamics. The limitations of crisp state classifications in such contexts justify the adoption of fuzzy representations. Additionally, we introduce Zadeh’s membership functions, along with their basic operations and interaction with probability theory.

Section 3 presents the novel contributions of this paper. We model the evolution of credit-rating migrations for defaultable bonds as a non-homogeneous semi-Markov process with fuzzy states, embedded within a fuzzy stochastic market—where the market itself is represented as a non-homogeneous semi-Markov process with fuzzy states.

In Section 3.1, we develop the model for the credit migration process. We derive closed-form matrix expressions for the non-default probabilities, which are central to the analysis of the model’s asymptotic behavior. Given the technical complexity and notation involved, we include a detailed illustrative example to clarify the mathematical structure and the interpretation of parameters.

In Section 3.2, we present analogous results for the evolution of the stochastic market among its fuzzy states, following a similar methodological framework.

Section 4 investigates the transition probabilities of defaultable bonds among fuzzy non-default states, specifically, the probability that a bond occupies a fuzzy non-default state while the stochastic market is in a given fuzzy state at time t. We formulate a theorem to compute the joint transition probabilities of the bond’s credit rating and the market’s state over the interval (t, t + 1]. As in earlier sections, we express these probabilities in closed-form matrix expressions, which are critical for analyzing the long-term behavior of the model. Due to the technical intricacy of the derivations, we again provide an illustrative example to assist with understanding the notation and the underlying economic interpretation.

In Section 5, we study the survival probabilities of defaultable bonds, that is, the probabilities of being in non-default states up to time t, given that if $\tau$ is the time to default then $\tau \ge t$. In this section, we establish a stochastic difference equation for the fuzzy survival probabilities of the defaultable bonds in a fuzzy stochastic market, which will enables us in the next section to study their asymptotic behavior. This is done in a closed matrix form, where the fuzzy survival probabilities of the defaultable bonds in a fuzzy stochastic market in a form of a theorem are found as functions of the basic parameters of the system.

In Section 6, we study the problem of finding the asymptotic behavior of the survival probabilities of defaultable bonds, which is very important for many reasons in credit risk. It certainly influences market stability by creating turbulences, when the survival probabilities are low or providing flows of loans when the survival probabilities are high. It also influences the behavior of bond holders, market liquidity, and the spreads, which influence state and global economies and other.

The study of the asymptotic behavior in the present section is a much deeper result though, as we discussed earlier, since it is the quasi-stationarity problem for an $\mathcal{F}$-inhomogeneous semi- Markov chain with fuzzy states in a fuzzy stochastic environment.

In the form of a theorem, we establish the existence of the limiting distribution of the fuzzy survival probabilities in a fuzzy environment under conditions easily met in practice. Finally, we evaluate these probabilities uniquely.

In Section 7, we discuss the estimation of the the survival probabilities in the inhomogeneous semi-Markov chain with fuzzy states and within a fuzzy stochastic Market, which depends entirely on the estimation of the basic parameters of the system. That is, the elements of the matrices ${\mathbf{Q}}_X\left(t,j\right)$, the transition probabilities among the states of the migration process given the state of the market, ${\mathbf{Q}}_Y\left(t\right)$, the transition probabilities among the states of the market, ${\mathbf{M}}_X^{\left(F\right)}\left(t\right)$, the membership functions of the fuzzy states of the migration process, and ${\mathbf{M}}_Y^{\left(F\right)}\left(t\right)$, the membership functions of the fuzzy states of the market.

Appendix A presents a comprehensive list of the symbols and notation used throughout the paper, with special attention to the fuzzy, non-homogeneous semi-Markov framework modeling of both credit risk and market evolution. For interpretive clarity, readers are advised to consult this appendix when encountering the technical symbols.

2. Credit Rating Classes, States of the Stochastic Market, and Fuzzy Sets

The classification of an issued bond into a specific credit-rating category is typically carried out by independent credit-rating agencies, such as Moody’s and Standard & Poor’s (S&P). S&P is one of the largest rating agencies globally, operating in over 50 countries, while Moody’s, though primarily based in the United States, maintains a strong international presence. Together, Moody’s and S&P dominate the credit-rating industry.

As detailed in Crouhy et al. (2001), the long-term credit-rating scales of Moody’s and S&P exhibit notable similarities in structure and definition. However, the inherent vagueness in these definitions often leads to discrepancies between agencies when evaluating the same debt instrument. Indeed, Vassiliou (2013) found that disagreement between agencies is more probable than agreement, thereby providing a compelling rationale for modeling rating categories using fuzzy state representations.

In practice, credit ratings are initially assigned across 21 distinct categories. However, due to the sparsity of observed transitions in empirical datasets, these categories are commonly aggregated into seven broader classes for analytical tractability. The theoretical foundation for Markov chains with fuzzy states has been explored in various contexts, including the works of Bhattacharyya (1998); Symeonaki and Stamou (2004); Symeonaki et al. (2004).

A common approach to modeling the stochastic market involves classifying its states into four qualitative categories: “Excellent” (state 1), “Good” (state 2), “Medium” (state 3), and “Crisis” (state 4), as seen in the works of Kijima and Yoshida (1993); Silvestrov and Stenberg (2004). These classifications are inherently linguistic, and as such, are best represented using fuzzy set theory, which is particularly well-suited for handling imprecise or subjective categorizations (see Symeonaki 2014). Furthermore, market conditions are influenced by a vast number of variables, which are typically aggregated into a small number of states. This dimensionality reduction introduces significant ambiguity, further reinforcing the appropriateness of fuzzy set theory for modeling market states.

The theoretical justification for such an approach can be traced to the notion of vague propositions, as articulated by Black (1939). A vague proposition is characterized by a lack of clearly defined boundaries of inclusion. For example, the statement “a person is young” lacks a universally accepted age range, as the term “young” is interpreted differently across individuals. This ambiguity prevents a definitive determination of the precise ages at which an individual is considered young or not. Zadeh (1965) elaborates on this distinction by noting that while vague propositions are often fuzzy, not all fuzzy concepts are necessarily vague. Fuzziness pertains to the imprecision or gradual nature of concept boundaries, whereas randomness (or chance) relates to the uncertainty of an event’s occurrence.

To formalize fuzzy concepts, Zadeh (1968) introduced membership functions, which represent the degree of belief that a given element belongs to a fuzzy set. This conceptual framework builds upon earlier philosophical treatments of vagueness, including the foundational ideas explored by Black (1939).

In this section, we consider ${\mu }_A\left(x\right)$ to be a normalized membership function, meaning that for all xx, it satisfies $0\le {\mu }_A\left(x\right)\le 1$. By definition, a membership function quantifies the degree of belief that an element x belongs to a set A, where ${\mu }_A\left(x\right)=1$ indicates full membership, and ${\mu }_A\left(x\right)=0$ indicates complete non-membership. If ${\mu }_A\left(x\right)\in \{0,1\}$ for all x, then A is a crisp set, with sharply defined boundaries. In contrast, if ${\mu }_A\left(x\right)$ takes values in the open interval $(0,1)$ for some x, then A is a fuzzy set, representing uncertainty or gradation in membership.

Ross et al. (2002) introduced a formal framework for operations on fuzzy sets via their membership functions. These operations include fuzzy union, intersection, and complement, defined pointwise on the basis of the corresponding membership values. Such operations provide a foundational mechanism for extending classical set-theoretic constructs to accommodate imprecision and vagueness inherent in many real-world applications (Carty and Fons 1994; D’Amico et al. 2005; Duffie et al. 2007; Howard 1971; Vasileiou and Vassiliou 2006).

$${\mu }_{A\cup B}\left(x\right)=max\left({\mu }_A\left(x\right),{\mu }_B\left(x\right)\right),{\mu }_{A\cap B}\left(x\right)=min\left({\mu }_A\left(x\right),{\mu }_B\left(x\right)\right),$$

(1)

$${\mu }_{A^C}\left(x\right)=1-{\mu }_A\left(x\right),A\subseteq B\iff {\mu }_A\left(x\right)\le {\mu }_B\left(x\right),$$

(2)

$$A=B\iff {\mu }_A\left(x\right)={\mu }_B\left(x\right).$$

(3)

As suggested by Ross et al. (2002), the membership function ${\mu }_A\left(x\right)$ can be interpreted as a likelihood that an element xx belongs to a fixed set A. This interpretation aligns philosophically with the concept of a likelihood function in probability theory, in that both are subjective constructs reflecting the assessor’s belief—here, regarding the degree of membership in AA. While this perspective may appear unconventional from a traditional probabilistic standpoint, it is consistent with broader philosophical interpretations of likelihood and allows for a unified treatment of uncertainty arising from both randomness and imprecision.

This interpretation opens the door to combining probabilistic and fuzzy uncertainties within a coherent framework. In this direction, Zadeh (1965) proposed a methodology for integrating fuzzy set theory with probability theory, enabling these two distinct forms of uncertainty to work in concert. The construction proposed by Zadeh forms the basis for hybrid models that account for both stochastic variability and vagueness in the description of real-world systems.

Let $\left(\mathsf{\Omega },\mathcal{F},\mathbb{P}\right)$ be a probability space with $x\in \mathsf{\Omega }$. Let $C\in$ $\mathcal{F}$ be a crisp set. Then, it is true if ${\mathbf{1}}_C\left(X\right)$ is the indicator function of the set $C\in$ $\mathcal{F}$, that

$$\mathbb{P}\left(C\right)={\int }_{\mathsf{\Omega }}{\mathbf{1}}_C\left(x\right)d\mathbb{P}.$$

(4)

Motivated by the above, Zadeh (1968) defines the $\mathbb{P}$-measure of a fuzzy subset $A\in \mathsf{\Omega }$ that is a fuzzy event, with membership function ${\mu }_A\left(x\right)$ as

$$\mathbb{P}\left(A\right)={\int }_{\mathsf{\Omega }}{\mu }_A\left(x\right)d\mathbb{P}={\mathbb{E}}_{\mathbb{P}}\left[{\mu }_A\left(x\right)\right].$$

(5)

The product of two fuzzy events (sets) A and B is defined

$$A.B\leftrightarrow {\mu }_{A.B}={\mu }_A.{\mu }_B.$$

(6)

Having defined the above, Zadeh (1995) proves that the following rules of probability hold for two sets A and B (Carty and Fons 1994):

$$A\subseteq B\Rightarrow \mathbb{P}\left(A\right)\le \mathbb{P}\left(B\right),$$

Having defined the above, Zadeh (1995) proves that the following rules of probability hold for two sets A and B (Carty and Fons 1994; Duffie et al. 2007; Vasileiou and Vassiliou 2006):

$$A\subseteq B\Rightarrow \mathbb{P}\left(A\right)\le \mathbb{P}\left(B\right),$$

$$\mathbb{P}\left(A\cup B\right)=\mathbb{P}\left(A\right)+\mathbb{P}\left(B\right)-\mathbb{P}\left(A\cap B\right),$$

(7)

$$\mathbb{P}\left(A+B\right)=\mathbb{P}\left(A\right)+\mathbb{P}\left(B\right)-\mathbb{P}\left(A•B\right);$$

Note that in Bhattacharyya (1998), A + B is meant the union of the two fuzzy sets. Extensions of the above to the cases of finite and countable additivity follow by induction. Finally, A and B are declared independent if $\mathbb{P}\left(A•B\right)=\mathbb{P}\left(A\right)\mathbb{P}\left(B\right)$. Furthermore,

$$\mathbb{P}\left(A\mid B\right)={\displaystyle \frac{\mathbb{P}\left(A•B\right)}{\mathbb{P}\left(B\right)}}\mathrm{if}\mathbb{P}\left(B\right)>0,$$

(8)

is the conditional probability of fuzzy event A given a fuzzy event B. Note that, the conditional probability is defined in terms of the $\mathbb{P}\left(A•B\right)$ instead of $\mathbb{P}\left(A\cap B\right)$. Study Zadeh (1968) demonstrated that this is the correct way to define the conditional probability of two fuzzy sets. As a result, in  $\mathbb{P}\left(A\cup B\right)=\mathbb{P}\left(A\right)+\mathbb{P}\left(B\right)-\mathbb{P}\left(A\cap B\right)$, the assessment of $\mathbb{P}\left(A\cap B\right)$ is never established . Instead, $\mathbb{P}\left(A+B\right)=\mathbb{P}\left(A\right)+\mathbb{P}\left(B\right)-\mathbb{P}\left(A•B\right)$ is addressed with $\mathbb{P}\left(A•B\right)=\mathbb{P}\left(A\right)\mathbb{P}\left(B\right)$ if A and B are independent or with $\mathbb{P}\left(A•B\right)=\mathbb{P}\left(A\mid B\right)\mathbb{P}\left(B\right)$. Furthermore, given that we are assessing probabilities of fuzzy events, there is no law of probability that leads to $\mathbb{P}\left(A\right)$. Since ${\mu }_A\left(x\right)$ is not a probability, the law of total probability does not apply when declaring  $\mathbb{P}\left(A\right)={\int }_{\mathsf{\Omega }}{\mu }_A\left(x\right)d\mathbb{P}$.

3. Fuzzy Semi-Markov Models Within a Fuzzy Stochastic Environment

In most practical settings, financial markets are modeled as finite markets, meaning they are described in discrete time and all relevant economic variables take on a finite number of values. Let the probability space $(\mathsf{\Omega },\mathcal{F},P)$ represent the market under consideration. Although this finite framework is common, it can be generalized by allowing $\mathsf{\Omega }$ to be a general state space, provided that the $\sigma$-algebra $\mathcal{F}$ is finitely generated. That is, $\mathcal{F}$ is generated by a finite partition of $\mathsf{\Omega }$ into mutually disjoint sets $A_1,A_2,\dots ,A_n$ such that ${\cup }_{i=1}^n{A}_i=$ $\mathsf{\Omega }$ and $\mathbb{P}\left(A_i\right)>0$ for all i. We fix the time set $T=\{0,1,\dots ,T\}$, where T denotes the terminal time of the trading horizon and each point in T corresponds to an admissible trading date.

In this section, we model the credit migration process of a defaultable bond as an inhomogeneous semi-Markov process with fuzzy states, evolving within a fuzzy stochastic market. The foundational theory of homogeneous semi-Markov chains is comprehensively treated in Howard (1971) and Hunter (1983). The extension to inhomogeneous semi-Markov processes was introduced and analyzed in Vassiliou and Papadopoulou (1992), with their asymptotic behavior studied further in Papadopoulou and Vassiliou (1994).

Various methodologies for estimating transition probabilities in inhomogeneous semi-Markov processes—particularly when data are sparse or irregular—have been developed in McClean and Gribbin (1987, 1991), and McClean and Montgomery (1999). Specifically, applications to credit-risk data, which often involve such challenges, were studied by Vasileiou and Vassiliou (2006).

On the fuzzy side, Bhattacharyya (1998) first introduced the concept of a fuzzy homogeneous Markov decision process, offering a framework for decision making under uncertainty and vagueness. This approach was later extended to non-homogeneous semi-Markov processes by Papadopoulou and Tsaklidis (2007), thus laying the groundwork for the present integration of fuzzy set theory into semi-Markov credit-risk models.

3.1. The Fuzzy Non-Homogeneous Semi-Markov for the Migration Process

Let the pair ${\left\{X_t,S_{t+1}\right\}}_{t=0}^{\infty }$ be a discrete $\mathcal{F}$-inhomogeneous semi-Markov process, the formal definition of which could be found in Vassiliou and Vasileiou (2013). The family of random variables ${\left\{X_t\right\}}_{t=0}^{\infty }$ has state space the set $\mathbb{K}=\left\{1,2,\dots ,k,k+1\right\}$ and represents the state the defaultable bond enters at time t, with state $k+1$ representing the default state. Also, let ${\mathcal{F}}^X$ be the natural filtration generated by the process ${\left\{X_t\right\}}_{t=0}^{\infty }$. We assume ${\mathcal{F}}^X$ is a subfiltration of $\mathcal{F}$, i.e., ${\mathcal{F}}^X\subseteq \mathcal{F}$. The family of random variables ${\left\{S_{t+1}\right\}}_{t=0}^{\infty }$ represent the selection at time interval $\left(t,t+1\right]$ of the next transition of the bond, given that its last entrance to a grade was at time t. Let ${\mathcal{F}}^S$ be the natural filtration generated by the process ${\left\{S_{t+1}\right\}}_{t=0}^{\infty }$ and assume that ${\mathcal{F}}^X\vee {\mathcal{F}}^S\subseteq \mathcal{F}$. Now, let

$$p_{X,ij}\left(t\right)=\mathbb{P}\left\{S_{t+1}=j\mid X_t=i\right\},$$

which in fact is the probability of a defaultable bond eventually moving in its next transition to state j, given that it entered state i at time t or equivalently the probability of a defaultable bond to select at time interval $\left(t,t+1\right]$ the state j given that it entered state i at time t. Denote by ${\tilde{\mathbf{P}}}_X\left(t\right)$ the matrix of transition probabilities, which will be of the form

$${\tilde{\mathbf{P}}}_X\left(t\right)=\left(\begin{array}{ccccc}{p}_{X,11}\left(t\right)& p_{X,12}\left(t\right)& \dots & p_{X,1k}\left(t\right)& p_{X,1,k+1}\left(t\right)\\ p_{X,21}\left(t\right)& p_{X,22}\left(t\right)& \dots & p_{X,2k}\left(t\right)& p_{X,2,k+1}\left(t\right)\\ \dots & \dots & \dots & \dots & \dots \\ p_{X,k1}\left(t\right)& p_{X,k2}\left(t\right)& \dots & p_{X,kk}\left(t\right)& p_{X,k,k+1}\left(t\right)\\ 0& 0& \dots & 0& 1\end{array}\right)=$$

(9)

$$\left(\begin{array}{cc}{\mathbf{P}}_X\left(t\right)& {\mathbf{p}}_{X,k+1}^{\top }\\ \mathbf{0}& 1\end{array}\right).$$

However, after j has been selected, but before making the transition from class i to class j, the process “holds” for a time ${\varpi }_{ij}\left(t\right)$ in state i. Recall that the holding times ${\varpi }_{ij}\left(t\right)$ are positive, integer-valued random variables each governed by a probability density function $h_{ij}\left(t,m\right)$. That is,

$$h_{X,ij}\left(t,m\right)=\mathbb{P}\left\{{\varpi }_{ij}\left(t\right)=m\mid X_t=i,S_{t+1}=j\right\},$$

(10)

and let ${\tilde{\mathbf{H}}}_X\left(t,m\right)={\left\{h_{X,ij}\left(t,m\right)\right\}}_{i,j\in \mathbb{K}}$, where

$${\tilde{\mathbf{H}}}_X\left(t,m\right)=\left(\begin{array}{cc}{\mathbf{H}}_X\left(t,m\right)& {\mathbf{h}}_X^{\top }\left(t,m\right)\\ \mathbf{0}& {\mathbf{1}}_{\left\{m=t+1\right\}}\end{array}\right).$$

We assume that the means of all holding time distributions are finite and that all holding times are nonzero, i.e., $h_{X,ij}\left(t,0\right)=0$ for every t. The parameters ${\tilde{\mathbf{P}}}_X\left(t\right)$ and ${\tilde{\mathbf{H}}}_X\left(t,m\right)$ are called the basic parameters of the $\mathcal{F}$-inhomogeneous semi-Markov process ${\left\{X_t,S_{t+1}\right\}}_{t=0}^{\infty }$. We now make the following assumption

Assumption 1.

For the stochastic process ${\left\{X_t,S_{t+1}\right\}}_{t=0}^{\infty }$, there is a finite duration in each state. Let $b_i$ for $i=1,2,\dots ,k$ be the maximum duration in state i for the $\mathcal{F}$-inhomogeneous semi-Markov process ${\left\{X_t,S_{t+1}\right\}}_{t=0}^{\infty }$.

From the data available from the rating agencies, it is easily apparent that the above assumption is realistic. Consider now the corresponding inhomogeneous Markov duration chain, let say ${\left\{D_X\left(t\right)\right\}}_{t=0}^{\infty }$, of the stochastic process ${\left\{X_t,S_{t+1}\right\}}_{t=0}^{\infty }$, (see Vassiliou and Vasileiou 2013). The random variable $D_X\left(t\right)$ represents the state the defaultable bond is at the end of the interval $\left(t-1,t\right]$, that is, at time t. The state space of the stochastic process ${\left\{D_X\left(t\right)\right\}}_{t=0}^{\infty }$ is

$$S{D}_X=\left\{\left(1,0\right),\left(1,1\right),\dots ,\left(1,b_1\right),\left(2,0\right),\dots ,\left(2,b_2\right),\dots ,\left(k,0\right),\dots ,\left(k,b_k\right)\right\}.$$

(11)

For example, $D_X\left(t\right)=\left(i,s\right)$ means that the defaultable bond is in state i at time t with duration s in this state and apparently that it did not default at any time before t. Let ${\widehat{\mathbf{Q}}}_X\left(t\right)$ be the matrix of transition probabilities among the states of (11), the elements of which are defined as

$$q_{X\left(i,s\right)\left(j,m\right)}\left(t\right)=\mathbb{P}\left\{D_X\left(t+1\right)=\left(j,m\right)\mid D_X\left(t\right)=\left(i,s\right)\right\}.$$

(12)

The matrix ${\widehat{\mathbf{Q}}}_X\left(t\right)$ is of the form

$${\widehat{\mathbf{Q}}}_X\left(t\right)=\left(\begin{array}{cc}{\mathbf{Q}}_X\left(t\right)& {\mathbf{q}}_{X\left(k+1\right)}^{\top }\left(t\right)\\ \mathbf{0}& 1\end{array}\right),$$

(13)

where ${\mathbf{Q}}_X\left(t\right)$ is the matrix of transition probabilities at time t among the non-default states and ${\mathbf{q}}_{X\left(k+1\right)}^{\top }$ is the vector of default probabilities at time t from the non-default states of $D_X\left(t\right)$. The transition probabilities $q_{X\left(is\right)\left(jm\right)}\left(t\right)$ for every $\left(i,s\right),\left(j,m\right)ϵS{D}_X$ are uniquely determined by the basic parameters ${\left\{{\tilde{\mathbf{P}}}_X\left(t\right)\right\}}_{t=0}^{\infty }$ and ${\left\{{\tilde{\mathbf{H}}}_X\left(t,m\right)\right\}}_{t,m=0}^{\infty }$ of the inhomogeneous semi-Markov process ${\left\{X_t,S{X}_t\right\}}_{t=0}^{\infty },$ see (Vassiliou and Vasileiou 2013, p. 2888). It is not difficult to see that

$$\sum _{j=1}^k\sum _{m=0}^{b_j}{q}_{X\left(is\right)\left(jm\right)}\left(t\right)=1.$$

(14)

Let

$$S{D}_X^{\left(F\right)}=\left\{\begin{array}{c}\left(F_{X1,}0\right),\left(F_{X1,}1\right),\dots ,\left(F_{X1,}{f}_{X1}\right),\left(F_{X2,}0\right),\left(F_{X2,}1\right),\dots ,\left(F_{X2,}{f}_{X2}\right),\dots ,\\ \left(F_{Xn,}0\right),\left(F_{Xn,}1\right),\dots ,\left(F_{Xn,}{f}_{Xn}\right).\end{array}\right\},$$

(15)

be the fuzzy state space, that is, the set of fuzzy states of the state space $S{D}_X,$ the original non-fuzzy state space for the system, the elements of which may or may not be observed. Let ${\mu }_{X\left(F_{Xr},m\right)}\left[\left(i,s\right),t\right]:S{D}_X\to \left[0,1\right]$ for $r=1,2,\dots ,n$ denotes the membership function of the fuzzy state $\left(F_{Xr,}.\right)$ for $r=1,2,\dots ,n$. The memberships functions are assumed to be nonhomogeneous in time, expressing the fact that the state of the global economy influences the values of the likelihood functions ${\mu }_{X\left(F_{Xr},m\right)}\left[\left(i,s\right),t\right]$. It is assumed that $S{D}_X^{\left(F\right)}$ defines a pseudopartition of fuzzy sets on $S{D}_X$ such that

$$\sum _{r=1}^n\sum _{m=0}^{f_{Xr}}{\mu }_{X\left(F_{Xr},m\right)}\left[\left(i,s\right),t\right]=1,{\mu }_{X\left(F_{Xr},m\right)}\left[\left(i,s\right),t\right]\ge 0,$$

(16)

$$\mathrm{for}\mathrm{all}i=1,2,\dots ,k,s=0,1,\dots ,b_i\mathrm{and}\mathrm{all}t.$$

We define by ${\widehat{\mathbf{M}}}_X^{\left(F\right)}\left(t\right)$ the matrix with elements ${\mu }_{X\left(F_{Xr},m\right)}\left[\left(i,s\right),t\right]$ for $i=1,2,\dots ,k+1,r=1,2,\dots ,n$ and $m=0,1,\dots ,f_{Xr}$. Then, ${\widehat{\mathbf{M}}}_X^{\left(F\right)}\left(t\right)$ has the elements of $S{D}_X^{\left(F\right)}$ as column and the elements of the state space $S{D}_X$ as rows, that is, it takes the form

$${\widehat{\mathbf{M}}}_X^{\left(F\right)}\left(t\right)=\left(\begin{array}{cc}{\mathbf{M}}_X^{\left(F\right)}\left(t\right)& \mathbf{0}\\ \mathbf{0}& 1\end{array}\right).$$

(17)

We define by ${\left\{D_X^{\left(F\right)}\left(t\right)\right\}}_{t=0}^{\infty }$ the stochastic process with state space given by (15). We now consider the transition probabilities

$$q_{X\left(\left(F_{Xr},m\right),\left(F_{Xl},z\right)\right)}^{\left(F\right)}\left(t\right)=\mathbb{P}\left\{D_X^{\left(F\right)}\left(t+1\right)=\left(F_{Xl},z\right)\mid D_X^{\left(F\right)}\left(t\right)=\left(F_{Xr},m\right)\right\}.$$

(18)

The matrix ${\widehat{\mathbf{Q}}}_X^{\left(F\right)}\left(t\right)$ is of the form

$${\widehat{\mathbf{Q}}}_X^{\left(F\right)}\left(t\right)=\left(\begin{array}{cc}{\mathbf{Q}}_X^{\left(F\right)}\left(t\right)& {\mathbf{q}}_{X\left(k+1\right)}^{\left(F\right)\top }\left(t\right)\\ \mathbf{0}& 1\end{array}\right),$$

where ${\mathbf{Q}}_X^{\left(F\right)}\left(t\right)$ is the matrix of transition probabilities at time t among the fuzzy non-default states and ${\mathbf{q}}_{X\left(k+1\right)}^{\left(F\right)\top }$ is the vector of default probabilities at time t from the fuzzy non-default states of ${\left\{D_X^{\left(F\right)}\left(t\right)\right\}}_{t=0}^{\infty }$. It is not difficult to see that

$$\sum _{l=1}^n\sum _{z=0}^{f_{Xl}}{q}_{X\left(\left(F_{Xr},m\right),\left(F_{Xl},z\right)\right)}^{\left(F\right)}\left(t\right)=1.$$

(19)

Now, we have that

$$\begin{array}{ccc}\hfill q_{X\left(\left(F_{Xr},m\right),\left(F_{Xl},z\right)\right)}^{\left(F\right)}\left(t\right)& =\hfill & \mathbb{P}\left\{D_X^{\left(F\right)}\left(t+1\right)=\left(F_{Xl},z\right)\mid D_X^{\left(F\right)}\left(t\right)=\left(F_{Xr},m\right)\right\}\hfill \\ \hfill & =\hfill & \left[\mathrm{due}\mathrm{to}\mathrm{relation}\left(8\right)\right]=\hfill \\ \hfill & \hfill & {\displaystyle \frac{\mathbb{P}\left\{D_X^{\left(F\right)}\left(t+1\right)=\left(F_{Xl},z\right)·D_X^{\left(F\right)}\left(t\right)=\left(F_{Xr},m\right)\right\}}{\mathbb{P}\left\{D_X^{\left(F\right)}\left(t\right)=\left(F_{Xr},m\right)\right\}}}.\hfill \end{array}$$

(20)

Using from (5) the definition of the probability of a fuzzy event and applying it for the discrete time, case we get

$$\mathbb{P}\left\{D_X^{\left(F\right)}\left(t\right)=\left(F_{Xr},m\right)\right\}=\sum _{i=1}^k\sum _{s=0}^{b_i}{\mu }_{\left(F_{Xr},m\right)}\left[\left(i,s\right),t\right]\mathbb{P}\left\{D_X\left(t\right)=\left(i,s\right)\right\}$$

$$=\sum _{i=1}^k\sum _{s=0}^{b_i}{\mu }_{\left(F_{Xr},m\right)}\left[\left(i,s\right),t\right]q_X\left(i,s,t\right).$$

(21)

Now, combining (5) and (6) we get that

$$\mathbb{P}\left\{D_X^{\left(F\right)}\left(t+1\right)=\left(F_{Xl},z\right)·D_X^{\left(F\right)}\left(t\right)=\left(F_{Xr},m\right)\right\}=$$

$$\begin{array}{ccc}& & \sum _{i=1}^k\sum _{s=0}^{b_i}\sum _{j=1}^k\sum _{m=0}^{b_j}\mathbb{P}\left\{D_X\left(t+1\right)=\left(j,m\right),D_X\left(t\right)=\left(i,s\right)\right\}\times \hfill \\ & & {\mu }_{\left(F_{Xr},m\right)}\left[\left(i,s\right),t\right]{\mu }_{\left(F_{Xl},m\right)}\left[\left(j,m\right),t+1\right]\hfill \end{array}$$

$$=\sum _{i=1}^k\sum _{s=0}^{b_i}\sum _{j=1}^k\sum _{m=0}^{b_j}{q}_{X\left(is\right)\left(jm\right)}\left(t\right)q_X\left(i,s,t\right)$$

(22)

To evaluate the fuzzy probabilities defined in Equation (20), we previously computed the denominator in Equation (21) and the numerator in Equation (22) as functions of the membership functions and the transition probabilities of the non-homogeneous semi-Markov chain ${\mathbf{Q}}_X\left(t\right)$. However, it is evident that the resulting expressions are not only cumbersome but also impractical for further analytical use in their current form.

Therefore, in what follows, we aim to express the matrix of fuzzy probabilities in a more concise, closed-form analytic matrix representation. It should be noted, however, that no general method exists for deriving such forms—each case may require a tailored approach, and success is not always guaranteed. Nevertheless, the need to resolve such challenges will arise repeatedly throughout our analysis. To aid comprehension, we will also provide a numerical example to illustrate the present analysis.

Define

$${\mathbf{q}}_X\left(.,.,t\right)=\left[q\left(1,0,t\right),q\left(1,1,t\right),\dots ,q\left(1,b_1,t\right),\dots ,q\left(k,1,t\right),\dots ,q\left(k,b_k,t\right)\right],$$

(23)

as a stochastic vector of size $1\times {\sum }_{i=0}^k\left(b_i+1\right)$.

$${\mathit{\mu }}_{X\left(F_{Xr},m\right)}\left[\left(i,s\right),t\right]=[{\mu }_{X\left(F_{Xr},m\right)}\left[\left(1,0\right),t\right],{\mu }_{X\left(F_{Xr},m\right)}\left[\left(1,1\right),t\right],\dots ,$$

$${\mu }_{X\left(F_{Xr},m\right)}\left[\left(1,b_1\right),t\right],\dots ,{\mu }_{X\left(F_{Xr},m\right)}\left[\left(k,0\right),t\right],\dots ,{\mu }_{X\left(F_{Xr},m\right)}\left[\left(k,b_k\right),t\right]{]}^{\top },$$

(24)

is a ${\sum }_{i=0}^k\left(b_i+1\right)\times 1$ stochastic vector. We also define

$${\mathbf{M}}_{dg}^{\left(F\right)}\left(t\right)=diag\left[{\mathbf{q}}_X\left(.,.,t\right){\mathit{\mu }}_{X\left(F_{X1},0\right)}\left[\left(.,.\right),t\right],\dots ,{\mathbf{q}}_X\left(.,.,t\right){\mathit{\mu }}_{X\left(F_{Xn},f_n\right)}\left[\left(.,.\right),t\right]\right],$$

(25)

where it is easy from (23) and (24) to see that ${\mathbf{q}}_X\left(.,.,t\right){\mathit{\mu }}_{X\left(F_{Xr},f_r\right)}\left[\left(.,.\right),t\right]$ is a scalar for r = 1, 2, …, n and therefore ${\mathbf{M}}_{dg}^{\left(F\right)}\left(t\right)$ is a ${\sum }_{i=1}^n\left(f_i+1\right)\times {\sum }_{i=1}^n\left(f_i+1\right)$ matrix. Next let

$${\mathbf{D}}_{Xq}\left(t\right)=diag\left[q\left(1,0,t\right),q\left(1,1,t\right),\dots ,q\left(1,b_1,t\right),\dots ,q\left(k,1,t\right),\dots ,q\left(k,b_k,t\right)\right],$$

(26)

where ${\mathbf{D}}_{Xq}\left(t\right)$ is apparently a ${\sum }_{i=1}^k\left(b_i+1\right)\times {\sum }_{i=1}^k\left(b_i+1\right)$ matrix. Now, the reader may verify from (18)–(26) that the following is true

$${\mathbf{Q}}_X^{\left(F\right)}\left(t\right)={\left[{\mathbf{M}}_{dg}^{\left(F\right)}\left(t\right)\right]}^{-1}{\left[{\mathbf{M}}_X^{\left(F\right)}\left(t\right)\right]}^{\top }{\mathbf{D}}_{Xq}\left(t\right){\mathbf{Q}}_X\left(t\right){\mathbf{M}}_X^{\left(F\right)}\left(t+1\right).$$

(27)

A first step towards the verification of relation (27) is that the sizes of the matrices in the two sides of the equation should be equal. In this respect, we must have

$$\left(\sum _{i=1}^n\left(f_i+1\right)\times \sum _{i=1}^n\left(f_i+1\right)\right)=\left(\sum _{i=1}^n\left(f_i+1\right)\times \sum _{i=1}^n\left(f_i+1\right)\right)\times$$

$$\left(\sum _{i=1}^n\left(f_i+1\right)\times \sum _{i=1}^k\left(b_i+1\right)\right)\times \left(\sum _{i=1}^k\left(b_i+1\right)\times \sum _{i=1}^k\left(b_i+1\right)\right)\times$$

(28)

$$\left(\sum _{i=1}^k\left(b_i+1\right)\times \sum _{i=1}^k\left(b_i+1\right)\right)\times \left(\sum _{i=1}^k\left(b_i+1\right)\times \sum _{i=1}^n\left(f_i+1\right)\right),$$

which is true.

An illustrative Example 1.

In this section, we present an illustrative example to support and clarify the theoretical developments established thus far. A detailed exposition of the credit- rating systems employed by major rating agencies can be found in Crouhy et al. (2001). In the context of our non-homogeneous semi-Markov model, the states correspond to credit-rating categories. For brevity and consistency with existing literature, we adopt the commonly used abbreviated notation for these credit ratings. Specifically, the state space is defined as follows:

$$S=\left\{Aaa,Aa,A,Baa,Ba,B,Caa,\mathrm{default}\right\}$$

$$=\left\{1,2,3,4,5,6,7,8\right\}.$$

(29)

Consider now the corresponding inhomogeneous Markov duration chain, lets say ${\left\{D_X\left(t\right)\right\}}_{t=0}^{\infty }$, of the stochastic process ${\left\{X_t,S_{t+1}\right\}}_{t=0}^{\infty }$ with state space being

$$S{D}_X=\left\{\left(1,0\right),\left(1,1\right),\left(1,2\right),\left(2,0\right),\left(2,1\right),\left(3,0\right),\left(3,1\right)\right\}.$$

(30)

Then, we will have

$${\mathbf{D}}_{Xq}\left(t\right)=diag\left[\begin{array}{c}q\left(1,0,t\right),q\left(1,1,t\right),q\left(1,2,t\right),q\left(2,0,t\right),q\left(2,1,t\right),\\ q\left(3,0,t\right),q\left(3,1,t\right)\end{array}\right].$$

(31)

In the example that follows, the numerical values are artificially generated. However, they are constructed to realistically simulate actual market conditions in the sense that the relationships among them preserve the underlying structure and interpretive meaning observed in empirical data.

$${\mathbf{D}}_{Xq}\left(t\right)=\left(\begin{array}{ccccccc}{\displaystyle \frac{1}{12}}& 0& 0& 0& 0& 0& 0\\ 0& {\displaystyle \frac{1}{12}}& 0& 0& 0& 0& 0\\ 0& 0& {\displaystyle \frac{1}{12}}& 0& 0& 0& 0\\ 0& 0& 0& {\displaystyle \frac{1}{4}}& 0& 0& 0\\ 0& 0& 0& 0& {\displaystyle \frac{1}{6}}& 0& 0\\ 0& 0& 0& 0& 0& {\displaystyle \frac{1}{4}}& 0\\ 0& 0& 0& 0& 0& 0& {\displaystyle \frac{1}{12}}\end{array}\right).$$

(32)

Let

$$S{D}_X^{\left(F\right)}=\left\{\left(F_{X1,}0\right),\left(F_{X1,}1\right),\left(F_{X2,}0\right),\left(F_{X2,}1\right)\right\},$$

be the fuzzy state space, that is, the set of fuzzy states of the state space $S{D}_X,$ the original non-fuzzy state space for the system. These elements may or may not be observed. Let the memberships at time t be

$${\mathbf{M}}_X^{\left(F\right)}\left(t\right)=\left(\begin{array}{ccccc}1& 0& 0& 0& 0\\ 0.5& 0.2& 0.1& 0.1& 0.1\\ 0.3& 0.2& 0.2& 0.1& 0.2\\ 0.3& 0.2& 0.2& 0.1& 0.2\\ 0.1& 0.2& 0.2& 0.2& 0.3\\ 0.1& 0.1& 0.2& 0.2& 0.4\\ 0& 0& 0& 0& 1\end{array}\right).$$

(33)

Observe that the sum of rows of matrix (33) are equal to one as in (16). Then, from the above data we get that

$${\left[{\mathbf{M}}_{Xdg}^{\left(F\right)}\left(t\right)\right]}^{-1}=\left(\begin{array}{ccccc}3.7037& 0.0& 0.0& 0.0& 0.0\\ 0.0& 8.3333& 0.0& 0.0& 0.0\\ 0.0& 0.0& 7.6923& 0.0& 0.0\\ 0.0& 0.0& 0.0& 10.0& 0.0\\ 0.0& 0.0& 0.0& 0.0& 2.6316\end{array}\right).$$

(34)

Now let

$${\mathbf{M}}_X^{\left(F\right)}\left(t+1\right)=\left(\begin{array}{ccccc}1& 0& 0& 0& 0\\ 0.5& 0.2& 0.1& 0.1& 0.1\\ 0.3& 0.3& 0.1& 0.1& 0.2\\ 0.3& 0.1& 0.3& 0.1& 0.2\\ 0.1& 0.2& 0.2& 0.2& 0.3\\ 0.1& 0.1& 0.2& 0.2& 0.4\\ 0& 0& 0& 0& 1\end{array}\right),$$

(35)

and

$${\mathbf{Q}}_X\left(t\right)=\left(\begin{array}{ccccccc}0.4& 0.2& 0& 0.2& 0& 0.2& 0\\ 0.3& 0& 0.2& 0.3& 0& 0.2& 0\\ 0.2& 0& 0& 0.4& 0& 0.4& 0\\ 0.1& 0& 0& 0.3& 0.3& 0.3& 0\\ 0.1& 0& 0& 0.3& 0& 0.6& 0\\ 0& 0& 0& 0.4& 0& 0.3& 0.3\\ 0.1& 0& 0& 0.4& 0& 0.5& 0\end{array}\right).$$

(36)

Using the above data and relation (27), we get that

$${\mathbf{Q}}_X^{\left(F\right)}\left(t\right)=\left(\begin{array}{ccccc}0.41736& 0.09398& 0.16206& 0.10582& 0.21978\\ 0.28876& 0.09846& 0.19354& 0.12682& 0.29342\\ 0.25333& 0.09317& 0.19481& 0.12551& 0.33418\\ 0.243& 0.089& 0.191& 0.121& 0.356\\ 0.25107& 0.08897& 0.20687& 0.13109& 0.322\end{array}\right),$$

which is a stochastic matrix as expected.

3.2. The Fuzzy Non-Homogeneous Semi-Markov for the Market Evolution

Let the stochastic process ${\left\{Y_t,S{Y}_{t+1}\right\}}_{t=0}^{\infty }$ be a discrete $\mathcal{F}$-inhomogeneous semi-Markov process, where ${\left\{Y_t\right\}}_{t=0}^{\infty }$ represents the state of the market that the defaultable bond enters at time t. We assume that there are $\nu$ possible states of the economy, which from now on it is realistic to take as a finite number. However, note that mathematically in what follows, $\nu$ could be countable as well. Now, $S{Y}_{t+1}$ represents the selection at time interval $\left(t,t+1\right]$ of the next transition of the market. Define $\mathbf{C}\left(t\right)={\left\{c_{ij}\left(t\right)\right\}}_{ij}$ and $\mathbf{HC}\left(t,m\right)={\left\{h{c}_{ij}\left(t,m\right)\right\}}_{ij}$ as the basic parameters of the discrete $\mathcal{F}$-inhomogeneous semi-Markov process ${\left\{Y_t,S{Y}_{t+1}\right\}}_{t=0}^{\infty }$. Also, let ${\mathcal{F}}^Y$ be the natural filtration generated by the process ${\left\{Y_t\right\}}_{t=0}^{\infty }$. We assume ${\mathcal{F}}^Y$ is a subfiltration of $\mathcal{F}$, i.e., ${\mathcal{F}}^Y\subseteq \mathcal{F}$. In addition, let ${\mathcal{F}}^{SY}$ be the natural filtration generated by the process ${\left\{S{Y}_{t+1}\right\}}_{t=0}^{\infty }$ and assume that ${\mathcal{F}}^Y\vee {\mathcal{F}}^{SY}\subseteq \mathcal{F}$.

Consider now the corresponding inhomogeneous Markov duration chain, ${\left\{D_Y\left(t\right)\right\}}_{t=0}^{\infty }$, of the stochastic process ${\left\{Y_t,S{Y}_{t+1}\right\}}_{t=0}^{\infty }$, assuming that there is a maximum duration ${\gamma }_{\iota }$ for $i=1,2,\dots ,\nu$ in each state. Then, the state space will be

$$S{D}_Y=\left\{\left(1,0\right),\left(1,1\right),\dots ,\left(1,{\gamma }_1\right),\left(2,0\right),\dots ,\left(2,{\gamma }_2\right),\dots ,\left(\nu ,0\right),\dots ,\left(\nu ,{\gamma }_{\nu }\right)\right\}.$$

(37)

We now make the following basic assumption:

Assumption 2.

(i) At time t , suppose the defaultable bond is in state $D_X\left(t\right)=\left(i,s\right)$, and the market is in state $D_Y\left(t\right)=\left(j,m\right)$. We assume that during the interval $[t,t+1]$, the transition of the market state occurs first, resulting in a new market state $(l,0)$ at some time $tϵ[t,t+1]$. Following this market transition, the transition of the defaultable bond takes place. Consequently, the bond transition is governed by the parameters ${\tilde{\mathbf{P}}}_l\left(t\right)$ and ${\tilde{\mathbf{H}}}_l\left(t,m\right)$, which are associated with the new market state l and the elapsed time m.

(ii) The evolution of the market is assumed to be conditionally independent of the evolution of the bond, given the past information on the market process.

Remark 1.

Note that we distinguish ν states of the market and not ${\sum }_{i=1}^{\nu }\left(1+{\gamma }_i\right)$ as one might expect from the state space $S{D}_Y$ in (37). The reason for this is that when the market changes state, for example, from $\left(j,m\right)$ to the new state $\left(l,0\right)$, this has in general a positive probability and all other transition probabilities from $\left(j,m\right)\to \left(l,s\right)$ for $s=1,2,\dots ,{\gamma }_s$ have zero probability. Hence, the set of parameters with which transition of the bonds will take place will be $\left\{{\tilde{\mathbf{P}}}_l\left(t\right),{\tilde{\mathbf{H}}}_l\left(t,m\right)\right\}$. When the state of the market remains the same, for example, from state $\left(j,m\right)$, we move to state $\left(j,m+1\right)$, and the set of parameters with which transition of the bonds will take place will be again $\left\{{\tilde{\mathbf{P}}}_j\left(t\right),{\tilde{\mathbf{H}}}_j\left(t,m\right)\right\}$.

Define the transition probabilities of the inhomogeneous Markov duration chain that corresponds to the inhomogeneous semi-Markov process ${\left\{Y_t,S{Y}_{t+1}\right\}}_{t=0}^{\infty }$ as

$$\begin{array}{ccc}\hfill q_{Y\left(i,s\right)\left(j,m\right)}\left(t\right)& =& \mathbb{P}\left\{D_Y\left(t+1\right)=\left(j,m\right)\mid D_Y\left(t\right)=\left(i,s\right)\right\},\mathrm{and}\hfill \\ \hfill {\mathbf{Q}}_Y\left(t\right)& =& {\left\{q_{Y\left(i,s\right)\left(j,m\right)}\left(t\right)\right\}}_{i,s,j,m}.\hfill \end{array}$$

The transition probabilities $q_{Y\left(i,s\right)\left(j,m\right)}\left(t\right)$ for every $\left(i,s\right),\left(j,m\right)\in S{D}_y$ are uniquely determined by the basic parameters ${\left\{\mathbf{C}\left(t\right)\right\}}_{t=0}^{\infty }$ and ${\left\{\mathbf{HC}\left(t,m\right)\right\}}_{t,m=0}^{\infty }$ of the inhomogeneous semi-Markov process ${\left\{Y_t,S{Y}_{t+1}\right\}}_{t=0}^{\infty },$ see (Vassiliou and Vasileiou 2013, p. 2888). Now, let

$$S{D}_Y^{\left(F\right)}=\left\{\begin{array}{c}\left(F_{Y1,}0\right),\left(F_{Y1,}1\right),\dots ,\left(F_{Y1,}{f}_{Y1}\right),\left(F_{Y2,}0\right),\left(F_{Y2,}1\right),\dots ,\left(F_{Y2,}{f}_{Y2}\right),\dots ,\\ \left(F_{Yr,}0\right),\left(F_{Yr,}1\right),\dots ,\left(F_{Yr,}{f}_{Yr}\right)\end{array}\right\},$$

(38)

be the fuzzy state space, that is, the set of fuzzy states of the state space $S{D}_Y,$ the original non-fuzzy state space for the market. Let ${\mu }_{Y\left(F_{Yj},m\right)}\left[\left(i,s\right),t\right]:S{D}_Y\to \left[0,1\right]$ for $j=1,2,\dots ,r$ denote the membership function of the fuzzy state $\left(F_{Yj,}.\right)$ for $j=1,2,\dots ,r$. The membership functions are assumed to be nonhomogeneous in time. That is, expressing the fact that the state of the global economy influences the values of the likelihood functions ${\mu }_{Y\left(F_{Yj},m\right)}\left[\left(i,s\right),t\right]$. It is assumed that $S{D}_Y^{\left(F\right)}$ defines a pseudopartition of fuzzy sets on $S{D}_X$ such that

$$\sum _{j=1}^r\sum _{m=0}^{f_{Yj}}{\mu }_{Y\left(F_{Yj},m\right)}\left[\left(i,s\right),t\right]=1,{\mu }_{Y\left(F_{Yj},m\right)}\left[\left(i,s\right),t\right]\ge 0,$$

(39)

$$\mathrm{for}\mathrm{all}i=1,2,\dots ,k+1,s=0,1,\dots ,b_i\mathrm{and}\mathrm{all}t.$$

We define ${\widehat{\mathbf{M}}}_Y^{\left(F\right)}\left(t\right)$ as the matrix with elements ${\mu }_{Y\left(F_{Yj},m\right)}\left[\left(i,s\right),t\right]$ for $i=1,2,\dots ,k+1,j=1,2,\dots ,r$ and $m=0,1,\dots ,f_{Yj}$. Then, ${\widehat{\mathbf{M}}}_Y^{\left(F\right)}\left(t\right)$ has the elements of $S{D}_Y^{\left(F\right)}$ as rows and the elements of the state space $S{D}_Y$ as columns, that is, it takes the form

$${\widehat{\mathbf{M}}}_Y^{\left(F\right)}\left(t\right)=\left(\begin{array}{cc}{\mathbf{M}}_Y^{\left(F\right)}\left(t\right)& \mathbf{0}\\ \mathbf{0}& 1\end{array}\right).$$

(40)

We define by ${\left\{D_Y^{\left(F\right)}\left(t\right)\right\}}_{t=0}^{\infty }$ the stochastic process with state space given by (38). We now consider the transition probabilities

$$q_{Y\left(\left(F_{Yj},m\right),\left(F_{Yl},z\right)\right)}^{\left(F\right)}\left(t\right)=\mathbb{P}\left\{D_Y^{\left(F\right)}\left(t+1\right)=\left(F_{Yl},z\right)\mid D_Y^{\left(F\right)}\left(t\right)=\left(F_{Yj},m\right)\right\}.$$

(41)

where ${\mathbf{Q}}_Y^{\left(F\right)}\left(t\right)$ is the matrix of transition probabilities at time t among the fuzzy non-default states given their survival up to time t. Then, following similar steps as the ones with which we founded relation (27), we get that

$${\mathbf{Q}}_Y^{\left(F\right)}\left(t\right)={\left[{\mathbf{M}}_{Ydg}^{\left(F\right)}\left(t\right)\right]}^{-1}{\left[{\mathbf{M}}_Y^{\left(F\right)}\left(t\right)\right]}^{\top }{\mathbf{D}}_{Yq}\left(t\right){\mathbf{Q}}_Y\left(t\right){\mathbf{M}}_Y^{\left(F\right)}\left(t+1\right).$$

(42)

An illustrative Example 2.

In the present, we illustrate the theoretical results founded in Section 3.2. In this respect, let $\nu =3,{\gamma }_1=2,{\gamma }_2={\gamma }_3=1$. Then, we get

$$S{D}_Y\left(t\right)=\left\{\left(1,0\right),\left(1,1\right),\left(1,2\right),\left(2,0\right),\left(2,1\right),\left(3,0\right),\left(3,1\right)\right\}.$$

(43)

Now, let

$$S{D}_Y^{\left(F\right)}=\left\{\left(F_{Y1,}0\right),\left(F_{Y1,}1\right),\left(F_{Y2,}0\right),\left(F_{Y2,}1\right)\right\},$$

(44)

be the fuzzy state space, that is, the set of fuzzy states of the state space $S{D}_Y,$ the original non-fuzzy state space for the system. In the example that follows, the numerical values are artificially generated. However, they are constructed to realistically simulate actual market conditions in the sense that the relationships among them preserve the underlying structure and interpretive meaning observed in empirical data. Let the memberships at time t be

$${\left[{\mathbf{M}}_Y^{\left(F\right)}\left(t\right)\right]}^{\top }=\left(\begin{array}{ccccccc}1& 0.4& 0.3& 0.2& 0.1& 0.1& 0\\ 0& 0.2& 0.2& 0.2& 0.2& 0.1& 0\\ 0& 0.2& 0.2& 0.2& 0.2& 0.2& 0\\ 0& 0.1& 0.1& 0.2& 0.2& 0.2& 0\\ 0& 0.1& 0.2& 0.2& 0.3& 0.4& 1\end{array}\right).$$

(45)

Now, let

$${\mathbf{D}}_{Yq}\left(t\right)=\left(\begin{array}{ccccccc}{\displaystyle \frac{1}{14}}& 0& 0& 0& 0& 0& 0\\ 0& {\displaystyle \frac{1}{14}}& 0& 0& 0& 0& 0\\ 0& 0& {\displaystyle \frac{1}{14}}& 0& 0& 0& 0\\ 0& 0& 0& {\displaystyle \frac{2}{7}}& 0& 0& 0\\ 0& 0& 0& 0& {\displaystyle \frac{1}{7}}& 0& 0\\ 0& 0& 0& 0& 0& {\displaystyle \frac{2}{7}}& 0\\ 0& 0& 0& 0& 0& 0& {\displaystyle \frac{1}{14}}\end{array}\right).$$

(46)

Then, from the above data we get that

$${\left[{\mathbf{M}}_{Ydg}^{\left(F\right)}\left(t\right)\right]}^{-1}={\left(\begin{array}{ccccc}0.22& 0& 0& 0& 0\\ 0& 0.14& 0& 0& 0\\ 0& 0& 0.17& 0& 0\\ 0& 0& 0& 0.16& 0\\ 0& 0& 0& 0& 0.31\end{array}\right)}^{-1}.$$

(47)

Now let

$${\mathbf{M}}_Y^{\left(F\right)}\left(t+1\right)=\left(\begin{array}{ccccc}1& 0& 0& 0& 0\\ 0.3& 0.2& 0.2& 0.1& 0.1\\ 0.3& 0.2& 0.1& 0.2& 0.2\\ 0.3& 0.2& 0.2& 0.1& 0.2\\ 0.1& 0.2& 0.2& 0.2& 0.3\\ 0.1& 0.2& 0.2& 0.2& 0.3\\ 0& 0& 0& 0& 1\end{array}\right),$$

(48)

also

$${\mathbf{Q}}_Y\left(t\right)=\left(\begin{array}{ccccccc}0& 0.5& 0& 0.3& 0& 0.2& 0\\ 0& 0& 0.4& 0.3& 0& 0.3& 0\\ 0.2& 0& 0& 0.4& 0& 0.4& 0\\ 0.3& 0& 0& 0& 0.5& 0.2& 0\\ 0.3& 0& 0& 0.4& 0& 0.3& 0\\ 0.1& 0& 0& 0.4& 0& 0& 0.5\\ 0.2& 0& 0& 0.4& 0& 0.4& 0\end{array}\right).$$

(49)

Using the above data in (42), we get that

$${\mathbf{Q}}_Y^{\left(F\right)}\left(t\right)=\left(\begin{array}{ccccc}0.3048& 0.1632& 0.158& 0.1213& 0.2462\\ 0.3457& 0.1374& 0.1334& 0.1144& 0.2791\\ 0.3339& 0.1306& 0.1274& 0.103& 0.3551\\ 0.3234& 0.1224& 0.1204& 0.0973& 0.3365\\ 0.3251& 0.1286& 0.127& 0.0958& 0.3435\end{array}\right).$$

(50)

3.3. Complete Transitions of the Inhomogeneous Markov Duration Chain

At each state of the economy (market), there corresponds a different set of basic parameters for the evolution of the defaultable bond, that is, ${\mathbf{P}}_l\left(t\right)$, ${\mathbf{H}}_l\left(t,m\right)$ for $l=1,2,\dots ,\nu$. Equivalently, the basic parameters of the discrete $\mathcal{F}$-inhomogeneous semi-Markov process that represents the evolution of a defaultable bond are being selected from the set

$$\mathcal{PH}\left(t\right)={\left[\left\{{\mathbf{P}}_l\left(t\right),{\mathbf{H}}_l\left(t,m\right)\right\}\right]}_{l=1}^{\nu }.$$

(51)

The selection is realized through the stochastic process ${\left\{Y_t,S{Y}_{t+1}\right\}}_{t=0}^{\infty }$, for which we assume that there is a finite duration in each state. According to Assumptions 1 and 2, during the interval $\left(t,t+1\right]$, the transition of the market state occurs first at some time $t^{+}ϵ(t,t+1]$. This transition determines which specific set of basic parameters, from the collection $\mathcal{PH}\left(t\right)$ as defined in Equation (51), will govern the subsequent transitions of the defaultable bond. Consequently, we define the probabilities of the complete transitions of the inhomogeneous Markov duration chain over the interval $(t,t+1]$ as follows:

$$q_{X\left(i,s\right)\left(j,m\right)}\left(t,l\right)=\mathbb{P}\left\{D_X\left(t+1\right)=\left(j,m\right)\mid D_Y\left(t+1\right)=\left(l,d\right),D_X\left(t\right)=\left(i,s\right)\right\},$$

(52)

That is, (52) are the transition probabilities of the inhomogeneous Markov duration chain ${\left\{D_X\left(t\right)\right\}}_{t=0}^{\infty }$ in the interval $\left(t,t+1\right]$, given that the Market has moved into state l. Note that these transition probabilities could be found as functions of the basic parameters ${\mathbf{P}}_l\left(t\right)$ and ${\mathbf{H}}_l\left(t,m\right)$ (see Vassiliou and Vasileiou 2013, p. 2888). We denote with ${\mathbf{Q}}_X\left(t,l\right)$ the matrix with the probabilities (52) as elements.

An illustrative Example 3.

For each $l=1,2,\dots ,\nu$, which represents the state of the market in the interval $\left(t,t+1\right]$, we get a different sequence of matrices of the type (42). We call ${\mathbf{Q}}_X\left(t,l\right)$ the complete transition probabilities of the inhomogeneous Markov chain.

4. The Transition Probabilities of the Defaultable Bonds Among the Non-Default Fuzzy States in a Fuzzy Stochastic Market

In the present section, we will study the transition probabilities of the defaultable bonds among the non-default fuzzy states. That is, the probabilities of a defaultable bond to be in a fuzzy non–default state, while the stochastic market is in a given fuzzy state at time t.

Our goal will be to find the joint probabilities of the total movements among the fuzzy grades of a defaultable bond and the fuzzy states of the stochastic market during the time interval $\left(t,t+1\right]$. We are interested in finding the probabilities:

$$\mathbb{P}\left\{\begin{array}{c}{D}_X^{\left(F\right)}\left(t+1\right)=\left(F_{Xi},l\right)•D_Y^{\left(F\right)}\left(t+1\right)=\left(F_{Yj},m\right)\mid \\ D_X^{\left(F\right)}\left(t\right)=\left(F_{Xa},s\right)•D_Y^{\left(F\right)}\left(t\right)=\left(F_{Yz},s\right)\end{array}\right\}=$$

$$\mathbb{P}\left\{D_Y^{\left(F\right)}\left(t+1\right)=\left(F_{Yj},m\right)\mid D_X^{\left(F\right)}\left(t\right)=\left(F_{Xa},s\right)•D_Y^{\left(F\right)}\left(t\right)=\left(F_{Yz},s\right)\right\}\times$$

$$\mathbb{P}\left\{\begin{array}{c}{D}_X^{\left(F\right)}\left(t+1\right)=\left(F_{Xi},l\right)\mid D_Y^{\left(F\right)}\left(t+1\right)=\left(F_{Yj},m\right)•\\ D_X^{\left(F\right)}\left(t\right)=\left(F_{Xa},s\right)•D_Y^{\left(F\right)}\left(t\right)=\left(F_{Yz},s\right)\end{array}\right\}=$$

$$\left(\mathrm{due}\mathrm{to}\mathrm{Assumption}2\left(ii\right)\right)=$$

$$\mathbb{P}\left\{D_Y^{\left(F\right)}\left(t+1\right)=\left(F_{Yj},m\right)\mid D_Y^{\left(F\right)}\left(t\right)=\left(F_{Yz},s\right)\right\}\times$$

$$\mathbb{P}\left\{D_X^{\left(F\right)}\left(t+1\right)=\left(F_{Xi},l\right)\mid D_Y^{\left(F\right)}\left(t+1\right)=\left(F_{Yj},m\right)•D_X^{\left(F\right)}\left(t\right)=\left(F_{Xa},s\right)\right\}=$$

$$q_{Y\left(\left(F_{Yz},s\right)\left(F_{Yj},m\right)\right)}^{\left(F\right)}\left(t\right)q_{X\left(\left(F_{Xa},s\right)\left(F_{Xi},l\right)\right)}^{\left(F\right)}\left(t;j\right).$$

(53)

We will now prove the following theorem concerning the probabilities of transitions of the defaultable bond among the fuzzy states.

Theorem 1.

Consider the inhomogeneous semi-Markov chain $\left\{X_t,S{X}_{t+1}\right\}$ with fuzzy states, which models the movements of the defaultable bond , as described in Section 2. Also, consider the inhomogeneous semi-Markov chain $\left\{Y_t,S{Y}_{t+1}\right\}$ with fuzzy states, which models the movements of the stochastic Market, as described in Section 2. Denote by $Q_X^{\left(F\right)}\left(t;F_{Y_j}\right)$ the matrix of the transition probabilities $q_{X\left(\left(F_{Xa},s\right)\left(F_{Xi},l\right)\right)}^{\left(F\right)}\left(t;j\right)$. Then

$${\mathbf{Q}}_X^{\left(F\right)}\left(t;F_{Yj}\right)={\left[{\mathbf{D}}_{q_X{q}_Y}\left(t\right)\right]}^{-1}\times$$

$${\mathbf{Q}}_{dg{\mathbf{q}}_Y}\left(t\right){\mathbf{M}}_{dg\left(F_{Yj},m\right)}\left(t\right){\mathfrak{P}}_{X\left(\left(F_{Xa},s\right)\left(F_{Xi},l\right)\right)}\left(t\right).$$

where ${\mathbf{D}}_{q_X{q}_Y}\left(t\right)$ is a diagonal matrix encoding the normalizing denominators based on joint fuzzy membership values and transition probabilities;

$${\mathbf{D}}_{q_X{q}_Y}\left(t\right)=diag\{{\mathbf{q}}_Y\left(.,.,t\right){\mathbf{Q}}_Y\left(t\right){\mathit{\mu }}_{Y\left(F_{Yj},m\right)}^{\top }\left[\left(.,.\right),t+1\right]{\mathbf{q}}_X\left(.,.,t\right){\mathit{\mu }}_{X\left(F_{X1},0\right)}^{\top }\left[\left(.,.\right),t\right],\dots ,$$

$${\mathbf{q}}_Y\left(.,.,t\right){\mathbf{Q}}_Y\left(t\right){\mathit{\mu }}_{Y\left(F_{Yj},m\right)}^{\top }\left[\left(.,.\right),t+1\right]{\mathbf{q}}_X\left(.,.,t\right){\mathit{\mu }}_{X\left(F_{Xr},r\right)}^{\top }\left[\left(.,.\right),t\right]\},$$

and, ${\mathbf{Q}}_{dg{\mathbf{q}}_Y}\left(t\right)$ is a block-diagonal matrix constructed from market transition probabilities

$${\mathbf{Q}}_{dg{\mathbf{q}}_Y}\left(t\right)=diag\left\{{\mathbf{q}}_Y\left(.,.,t\right){\mathbf{Q}}_Y\left(t\right),{\mathbf{q}}_Y\left(.,.,t\right){\mathbf{Q}}_Y\left(t\right),\dots ,{\mathbf{q}}_Y\left(.,.,t\right){\mathbf{Q}}_Y\left(t\right)\right\},$$

and, ${\mathbf{M}}_{dg\left(F_{Yj},m\right)}\left(t\right)$ is a diagonal matrix containing fuzzy membership values for market states:

$${\mathbf{M}}_{dg\left(F_{Yj},m\right)}\left(t\right)=$$

$$\left(\begin{array}{cccc}{\mathit{\mu }}_{dg\left(F_{Yj},m\right)}\left[\left(.,.\right),t+1\right]& \mathbf{0}& \dots & \mathbf{0}\\ \mathbf{0}& {\mathit{\mu }}_{dg\left(F_{Yj},m\right)}\left[\left(.,.\right),t+1\right]& \dots & \mathbf{0}\\ \dots & \dots & \dots & \dots \\ \mathbf{0}& \mathbf{0}& \dots & {\mathit{\mu }}_{dg\left(F_{Yj},m\right)}\left[\left(.,.\right),t+1\right]\end{array}\right)$$

with

$${\mathit{\mu }}_{dg\left(F_{Yj},m\right)}\left[\left(.,.\right),t+1\right]=$$

$$\left(\begin{array}{cccc}{\mu }_{\left(F_{Yj},m\right)}\left[\left(1,0\right),t+1\right]& 0& \dots & 0\\ 0& {\mu }_{\left(F_{Yj},m\right)}\left[\left(1,1\right),t+1\right]& \dots & 0\\ \dots & \dots & \dots & ..\\ 0& 0& \dots & {\mu }_{\left(F_{Yj},m\right)}\left[\left(\nu ,{\gamma }_{\nu }\right),t+1\right]\end{array}\right),$$

and where ${\mathfrak{P}}_{X\left(\left(F_{Xa},s\right)\left(F_{Xi},l\right)\right)}\left(t\right)$ is a matrix of transition probabilities of the defaultable bond, indexed by market state $\left(\lambda ,d_{Y_{\lambda }}\right):$

$${\mathfrak{P}}_{X\left(\left(F_{Xa},s\right)\left(F_{Xi},l\right)\right)}\left(t\right)=$$

$$\left(\begin{array}{ccc}{\mathfrak{p}}_{X\left(F_{X1},0\right)\left(F_{X1},0\right)}\left(t+1;\left(1,0\right)\right)& \dots & {\mathfrak{p}}_{X\left(F_{X1},0\right)\left(F_{Xn},f_{X_n}\right)}\left(t+1;\left(1,0\right)\right)\\ \dots & \dots & \dots \\ {\mathfrak{p}}_{X\left(F_{X1},0\right)\left(F_{X1},0\right)}\left(t+1;\left(1,{\gamma }_1\right)\right)& \dots & {\mathfrak{p}}_{X\left(F_{X1},0\right)\left(F_{Xn},f_{X_n}\right)}\left(t+1;\left(1,{\gamma }_1\right)\right)\\ \dots & \dots & \dots \\ {\mathfrak{p}}_{X\left(F_{Xn},f_{X_n}\right)\left(F_{Xn},0\right)}\left(t+1;\left(1,0\right)\right)& \dots & {\mathfrak{p}}_{X\left(F_{Xn},f_{X_n}\right)\left(F_{Xn},f_{Xn}\right)}\left(t+1;\left(1,0\right)\right)\\ \dots & \dots & \dots \\ {\mathfrak{p}}_{X\left(F_{Xn},f_{X_n}\right)\left(F_{Xn},0\right)}\left(t+1;\left(\nu ,{\gamma }_{\nu }\right)\right)& \dots & {\mathfrak{p}}_{X\left(F_{Xn},f_{X_n}\right)\left(F_{Xn},f_{X_n}\right)}\left(t+1;\left(\nu ,{\gamma }_{\nu }\right)\right)\end{array}\right),$$

$$\begin{array}{ccc}\hfill {\mathfrak{p}}_{X\left(\left(F_{Xa},s\right)\left(F_{Xi},l\right)\right)}\left(t;\left(\lambda ,d_{Y_{\lambda }}\right)\right)& =& {\mathit{\mu }}_{X\left(F_{Xa},s\right)}\left[\left(.,.\right),t\right]{\mathbf{D}}_{X_q}\left(t\right){\mathbf{Q}}_X\left(t;\left(\lambda ,d_{Y_{\lambda }}\right)\right)\times \hfill \\ & & {\mathit{\mu }}_{X\left(F_{Xi},s\right)}^{\top }\left[\left(.,.\right),t+1\right],\hfill \end{array}$$

Proof. 

From (53), we now have

$$\begin{array}{ccc}\hfill & \hfill & q_{X\left(\left(F_{Xa},s\right)\left(F_{Xi},l\right)\right)}^{\left(F\right)}\left(t;F_{Yj}\right)=\hfill \\ \hfill & \hfill & {\displaystyle \frac{\mathbb{P}\left\{D_X^{\left(F\right)}\left(t+1\right)=\left(F_{Xi},l\right)•D_Y^{\left(F\right)}\left(t+1\right)=\left(F_{Yj},m\right)•D_X^{\left(F\right)}\left(t\right)=\left(F_{Xa},s\right)\right\}}{\mathbb{P}\left\{D_Y^{\left(F\right)}\left(t+1\right)=\left(F_{Yj},m\right)•D_X^{\left(F\right)}\left(t\right)=\left(F_{Xa},s\right)\right\}}}.\hfill \end{array}$$

(54)

We start with the denominator in (54)

$$\mathbb{P}\left\{D_Y^{\left(F\right)}\left(t+1\right)=\left(F_{Yj},m\right)•D_X^{\left(F\right)}\left(t\right)=\left(F_{Xa},s\right)\right\}=$$

$$\begin{array}{ccc}\hfill \sum _{l=1}^{\nu }\sum _{d_{Yl}=0}^{{\gamma }_i}\sum _{c=1}^k\sum _{d_{Xc}=0}^{b_c}\mathbb{P}\{D_Y\left(t+1\right)& =& \left(l,d_{Yl}\right),D_X\left(t\right)=\left(c,d_{Xc}\right)\}\hfill \\ & & \times {\mu }_{Y\left(F_{Yj},m\right)}\left[\left(l,d_{Yl}\right),t+1\right]{\mu }_{X\left(F_{Xa},s\right)}\left[\left(c,d_{Xc}\right),t\right]\hfill \end{array}$$

$$={\mathbf{q}}_Y\left(.,.,t+1\right){\mathit{\mu }}_{Y\left(F_{Yj},m\right)}^{\top }\left[\left(.,.\right),t+1\right]{\mathbf{q}}_X\left(.,.,t\right){\mathit{\mu }}_{X\left(F_{Xa},s\right)}^{\top }\left[\left(.,.\right),t\right]$$

$$={\mathbf{q}}_Y\left(.,.,t\right){\mathbf{Q}}_Y\left(t\right){\mathit{\mu }}_{Y\left(F_{Yj},m\right)}^{\top }\left[\left(.,.\right),t+1\right]{\mathbf{q}}_X\left(.,.,t\right){\mathit{\mu }}_{X\left(F_{Xa},s\right)}^{\top }\left[\left(.,.\right),t\right],$$

(55)

where ${\mathbf{q}}_Y\left(.,.,t\right){\mathbf{Q}}_Y\left(t\right){\mathit{\mu }}_{Y\left(F_{Yj},m\right)}^{\top }\left[\left(.,.\right),t+1\right]$ is of size

$$1\times \sum _{i=1}^{\nu }\left(1+{\gamma }_i\right)\times \sum _{i=1}^{\nu }\left(1+{\gamma }_i\right)\times \sum _{i=1}^{\nu }\left(1+{\gamma }_i\right)\times \sum _{i=1}^{\nu }\left(1+{\gamma }_i\right)\times 1,$$

while ${\mathbf{q}}_X\left(.,.,t+1\right){\mathit{\mu }}_{X\left(F_{Ya},s\right)}^{\top }\left[\left(.,.\right),t\right]$ is of size

$$1\times \sum _{i=1}^{b_i}\left(1+b_i\right)\times \sum _{i=1}^{b_i}\left(1+b_i\right)\times 1,$$

Hence, the denominator in (54) is a scalar as it should be. Note that the right hand side of Equation (54) consists of parameters of the two $\mathcal{F}$-inhomogeneous semi-Markov process at time t. When time t is the present, then these parameters, as we have seen, could be found as functions of the two $\mathcal{F}$-inhomogeneous semi-Markov process at time $t$. Consider now the nominator in the right hand side of Equation (53). We have that

$$\mathbb{P}\left\{D_X^{\left(F\right)}\left(t+1\right)=\left(F_{Xi},l\right)•D_Y^{\left(F\right)}\left(t+1\right)=\left(F_{Yj},m\right)•D_X^{\left(F\right)}\left(t\right)=\left(F_{Xa},s\right)\right\}=$$

$$\sum _{u=1}^k\sum _{d_{Xu}=0}^{b_u}\sum _{\lambda =1}^{\nu }\sum _{d_{Y\lambda }=0}^{{\gamma }_{\lambda }}\sum _{c=1}^k\sum _{d_{Xc}=0}^{b_c}\mathbb{P}\{D_X\left(t+1\right)=\left(u,d_{Xu}\right),D_Y\left(t+1\right)=\left(\lambda ,d_{Y\lambda }\right),$$

$$D_X\left(t\right)=\left(c,d_{Xc}\right)\}{\mu }_{X\left(F_{Xi},l\right)}\left[\left(u,d_{Xu}\right),t+1\right]\times$$

$${\mu }_{Y\left(F_{Yj},m\right)}\left[\left(\lambda ,d_{y\lambda }\right),t+1\right]{\mu }_{X\left(F_{Xa},s\right)}\left[\left(c,d_{Xc}\right),t\right]=$$

(56)

$$\sum _{u=1}^k\sum _{d_{Xu}=0}^{b_u}\sum _{\lambda =1}^{\nu }\sum _{d_{Y\lambda }=0}^{{\gamma }_{\lambda }}\sum _{c=1}^k\sum _{d_{Xc}=0}^{b_c}\mathbb{P}\{D_X\left(t+1\right)=\left(u,d_{Xu}\right)\mid D_Y\left(t+1\right)=\left(\lambda ,d_{Y\lambda }\right),$$

$$D_X\left(t\right)=\left(c,d_{Xc}\right)\}\mathbb{P}\left\{D_Y\left(t+1\right)=\left(\lambda ,d_{y\lambda }\right)\right\}\mathbb{P}\left\{D_X\left(t\right)=\left(c,d_{Xc}\right)\right\}\times$$

$${\mu }_{\left(F_{Xi},l\right)}\left[\left(u,d_{Xu}\right),t+1\right]{\mu }_{\left(F_{Yj},m\right)}\left[\left(\lambda ,d_{Y\lambda }\right),t+1\right]{\mu }_{\left(F_{Xa},s\right)}\left[\left(c,d_{Xc}\right),t\right]=$$

$$\sum _{u=1}^k\sum _{d_{Xu}=0}^{b_u}\sum _{\lambda =1}^{\nu }\sum _{d_{Y\lambda }=0}^{{\gamma }_l}\sum _{c=1}^k\sum _{d_{Xc}=0}^{b_c}{q}_{X\left(c,d_{Xc}\right)\left(u,d_{Xu}\right)}\left(t;\lambda ,d_{Y\lambda }\right)q_Y\left(\lambda ,d_{Y\lambda },t+1\right)q_X\left(c,d_{Xc},t\right)\times$$

$${\mu }_{\left(F_{Xi},l\right)}\left[\left(u,d_{Xu}\right),t+1\right]{\mu }_{\left(F_{Yj},m\right)}\left[\left(\lambda ,d_{Y\lambda }\right),t+1\right]{\mu }_{\left(F_{Xa},s\right)}\left[\left(c,d_{Xc}\right),t\right]=$$

$$\sum _{\lambda =1}^{\nu }\sum _{d_{Y\lambda }=0}^{{\gamma }_{\lambda }}\sum _{i=1}^{\nu }\sum _{d_{Yi}=0}^{{\gamma }_i}\sum _{u=1}^k\sum _{d_{Xu}=0}^{b_u}\sum _{c=1}^k\sum _{d_{Xc}=0}^{b_c}{q}_Y\left(i,d_{Yi},t\right)q_{Y\left(i,d_{Yi}\right)\left(\lambda ,d_{Y\lambda }\right)}\left(t\right)\times$$

$${\mu }_{\left(F_{Yj},m\right)}\left[\left(\lambda ,d_{Y\lambda }\right),t+1\right]q_X\left(c,d_{Xc},t\right)q_{X\left(c,d_{Xc}\right)\left(u,d_{Xu}\right)}\left(t;\lambda ,d_{Y\lambda }\right)\times$$

$${\mu }_{\left(F_{Xa},s\right)}\left[\left(c,d_{Xc}\right),t\right]{\mu }_{\left(F_{Xi},l\right)}\left[\left(u,d_{Xu}\right),t+1\right]=$$

$$\sum _{\lambda =1}^{\nu }\sum _{d_{Y\lambda }=0}^{{\gamma }_{\lambda }}\sum _{i=1}^{\nu }\sum _{d_{Yi}=0}^{{\gamma }_i}{q}_Y\left(i,d_{Yi},t\right)q_{Y\left(i,d_{Yi}\right)\left(\lambda ,d_{Y\lambda }\right)}\left(t\right){\mu }_{\left(F_{Yj},m\right)}\left[\left(\lambda ,d_{Y\lambda }\right),t+1\right]\times$$

$${\mathit{\mu }}_{X\left(F_{Xa},s\right)}\left[\left(.,.\right),t\right]{\mathbf{D}}_{q_X}\left(t\right){\mathbf{Q}}_X\left(t;\left(\lambda ,d_{Y_{\lambda }}\right)\right){\mathit{\mu }}_{X\left(F_{Xi},s\right)}^{\top }\left[\left(.,.\right),t+1\right]$$

Define

$${\mathbf{D}}_{q_X}\left(t\right)=diag\left\{q_X\left(1,0,t\right),q_X\left(1,1,t\right)\right\},\dots ,q_X\left(k,b_k,t\right)\}.$$

$${\mathfrak{p}}_{X\left(\left(F_{Xa},s\right)\left(F_{Xi},l\right)\right)}\left(t;\left(\lambda ,d_{Y_{\lambda }}\right)\right)={\mathit{\mu }}_{X\left(F_{Xa},s\right)}\left[\left(.,.\right),t\right]{\mathbf{D}}_{q_X}\left(t\right){\mathbf{Q}}_X\left(t;\left(\lambda ,d_{Y_{\lambda }}\right)\right)\times$$

(57)

$${\mathit{\mu }}_{X\left(F_{Xi},s\right)}^{\top }\left[\left(.,.\right),t+1\right],$$

the size of which, retaining the order of vectors and matrices, is

$$1\times \sum _{i=1}^k\left(1+b_i\right)\times \sum _{i=1}^k\left(1+b_i\right)\times \sum _{i=1}^k\left(1+b_i\right)\times \sum _{i=1}^k\left(1+b_i\right)\times \sum _{i=1}^k\left(1+b_i\right)\times \sum _{i=1}^k\left(1+b_i\right)\times 1,$$

that is, a scalar. □

Note that according to Remark 1, we have that

$${\mathbf{Q}}_X\left(t;\left(\lambda ,d_{Y_{\lambda }}\right)\right)={\mathbf{Q}}_X\left(t;\lambda \right)\mathrm{for}\mathrm{every}{d}_{Y_{\lambda }}=0,1,\dots ,{\gamma }_{\lambda }.$$

(58)

Define

$${\mathbf{Q}}_{dg{\mathbf{q}}_Y}\left(t\right)=dg\left\{{\mathbf{q}}_Y\left(.,.,t\right){\mathbf{Q}}_Y\left(t\right),{\mathbf{q}}_Y\left(.,.,t\right){\mathbf{Q}}_Y\left(t\right),\dots ,{\mathbf{q}}_Y\left(.,.,t\right){\mathbf{Q}}_Y\left(t\right)\right\},$$

(59)

which is chosen to have ${\sum }_{i=1}^n\left(1+f_{Xi}\right)$ rows, and hence the matrix (60) is

$$\sum _{i=1}^n\left(1+f_{Xi}\right)\times \sum _{i=1}^{\nu }\left(1+{\gamma }_i\right).\sum _{i=1}^n\left(1+f_{Xi}\right),$$

Also define

$${\mathbf{M}}_{dg\left(F_{Yj},m\right)}\left(t\right)=$$

(60)

$$\left(\begin{array}{cccc}{\mathit{\mu }}_{dg\left(F_{Yj},m\right)}\left[\left(.,.\right),t+1\right]& \mathbf{0}& \dots & \mathbf{0}\\ \mathbf{0}& {\mathit{\mu }}_{dg\left(F_{Yj},m\right)}\left[\left(.,.\right),t+1\right]& \dots & \mathbf{0}\\ \dots & \dots & \dots & \dots \\ \mathbf{0}& \mathbf{0}& \dots & {\mathit{\mu }}_{dg\left(F_{Yj},m\right)}\left[\left(.,.\right),t+1\right]\end{array}\right)$$

which is chosen to have ${\sum }_{i=1}^n\left(1+f_{Xi}\right)$ blocs of matrices, where each one is

$${\mathit{\mu }}_{dg\left(F_{Yj},m\right)}\left[\left(.,.\right),t+1\right]=$$

(61)

$$\left(\begin{array}{cccc}{\mu }_{\left(F_{Yj},m\right)}\left[\left(1,0\right),t+1\right]& 0& \dots & 0\\ 0& {\mu }_{\left(F_{Yj},m\right)}\left[\left(1,1\right),t+1\right]& \dots & 0\\ \dots & \dots & \dots & ..\\ 0& 0& \dots & {\mu }_{\left(F_{Yj},m\right)}\left[\left(\nu ,{\gamma }_{\nu }\right),t+1\right]\end{array}\right),$$

which is a matrix with size

$$\sum _{i=1}^{\nu }\left(1+{\gamma }_i\right)\times \sum _{i=1}^{\nu }\left(1+{\gamma }_i\right),$$

and as a consequence the matrix (15) is

$$\sum _{i=1}^{\nu }\left(1+{\gamma }_i\right).\sum _{i=1}^n\left(1+f_{Xi}\right)\times \sum _{i=1}^{\nu }\left(1+{\gamma }_i\right).\sum _{i=1}^n\left(1+f_{Xi}\right).$$

Now, define the matrix

$${\mathfrak{P}}_{X\left(\left(F_{Xa},s\right)\left(F_{Xi},l\right)\right)}\left(t\right)=$$

$$\left(\begin{array}{ccc}{\mathfrak{p}}_{X\left(F_{X1},0\right)\left(F_{X1},0\right)}\left(t+1;\left(1,0\right)\right)& \dots & {\mathfrak{p}}_{X\left(F_{X1},0\right)\left(F_{Xn},f_{X_n}\right)}\left(t+1;\left(1,0\right)\right)\\ \dots & \dots & \dots \\ {\mathfrak{p}}_{X\left(F_{X1},0\right)\left(F_{X1},0\right)}\left(t+1;\left(1,{\gamma }_1\right)\right)& \dots & {\mathfrak{p}}_{X\left(F_{X1},0\right)\left(F_{Xn},f_{X_n}\right)}\left(t+1;\left(1,{\gamma }_1\right)\right)\\ \dots & \dots & \dots \\ {\mathfrak{p}}_{X\left(F_{Xn},f_{X_n}\right)\left(F_{Xn},0\right)}\left(t+1;\left(1,0\right)\right)& \dots & {\mathfrak{p}}_{X\left(F_{Xn},f_{X_n}\right)\left(F_{Xn},f_{Xn}\right)}\left(t+1;\left(1,0\right)\right)\\ \dots & \dots & \dots \\ {\mathfrak{p}}_{X\left(F_{Xn},f_{X_n}\right)\left(F_{Xn},0\right)}\left(t+1;\left(\nu ,{\gamma }_{\nu }\right)\right)& \dots & {\mathfrak{p}}_{X\left(F_{Xn},f_{X_n}\right)\left(F_{Xn},f_{X_n}\right)}\left(t+1;\left(\nu ,{\gamma }_{\nu }\right)\right)\end{array}\right),$$

(62)

which is a matrix of size

$$\sum _{i=1}^n\left(1+f_{Xi}\right).\sum _{j=1}^{\nu }\left(1+{\gamma }_j\right)\times \sum _{i=1}^n\left(1+f_{Xi}\right).$$

From (54) and up to (61), we could prove that the matrix of the following probabilities for all possible values of $\left(F_{Xa},s\right)$, $\left(F_{Xi},l\right)$ and for a specific value of $\left(F_{Yj},m\right)$ is given by

$${\left\{\mathbb{P}\left[D_X^{\left(F\right)}\left(t+1\right)=\left(F_{Xi},l\right)•D_Y^{\left(F\right)}\left(t+1\right)=\left(F_{Yj},m\right)•D_X^{\left(F\right)}\left(t\right)=\left(F_{Xa},s\right)\right]\right\}}_{\left(.,.\right)\left(.,.\right)}=$$

$${\mathbf{Q}}_{dg{\mathbf{q}}_Y}\left(t\right){\mathbf{M}}_{dg\left(F_{Yj},m\right)}\left(t\right){\mathfrak{P}}_{X\left(\left(F_{Xa},s\right)\left(F_{Xi},l\right)\right)}\left(t\right),$$

(63)

which is, retaining the order of vectors and matrices, of size

$$\begin{array}{ccc}& & \sum _{i=1}^n\left(1+f_{Xi}\right)\times \sum _{i=1}^{\nu }\left(1+{\gamma }_i\right).\sum _{i=1}^n\left(1+f_{Xi}\right)\sum _{i=1}^{\nu }\left(1+{\gamma }_i\right).\sum _{i=1}^n\left(1+f_{Xi}\right)\times \hfill \\ & & \sum _{i=1}^{\nu }\left(1+{\gamma }_i\right).\sum _{i=1}^n\left(1+f_{Xi}\right)\sum _{i=1}^n\left(1+f_{Xi}\right).\sum _{j=1}^{\nu }\left(1+{\gamma }_j\right)\times \sum _{i=1}^n\left(1+f_{Xi}\right)\hfill \end{array}$$

$$=\sum _{i=1}^n\left(1+f_{Xi}\right)\times \sum _{i=1}^n\left(1+f_{Xi}\right).$$

Define the diagonal matrix

$${\mathbf{D}}_{q_X{q}_Y}\left(t\right)=diag\{{\mathbf{q}}_Y\left(.,.,t\right){\mathbf{Q}}_Y\left(t\right){\mathit{\mu }}_{Y\left(F_{Yj},m\right)}^{\top }\left[\left(.,.\right),t+1\right]{\mathbf{q}}_X\left(.,.,t\right){\mathit{\mu }}_{X\left(F_{X1},0\right)}^{\top }\left[\left(.,.\right),t\right],\dots ,$$

$${\mathbf{q}}_Y\left(.,.,t\right){\mathbf{Q}}_Y\left(t\right){\mathit{\mu }}_{Y\left(F_{Yj},m\right)}^{\top }\left[\left(.,.\right),t+1\right]{\mathbf{q}}_X\left(.,.,t\right){\mathit{\mu }}_{X\left(F_{Xr},r\right)}^{\top }\left[\left(.,.\right),t\right]\}$$

Now, from (9), (10), and (18), we get that

$${\mathbf{Q}}_X^{\left(F\right)}\left(t;F_{Yj}\right)={\left[{\mathbf{D}}_{q_X{q}_Y}\left(t\right)\right]}^{-1}\times$$

$${\mathbf{Q}}_{dg{\mathbf{q}}_Y}\left(t\right){\mathbf{M}}_{dg\left(F_{Yj},m\right)}\left(t\right){\mathfrak{P}}_{X\left(\left(F_{Xa},s\right)\left(F_{Xi},l\right)\right)}\left(t\right).$$

(64)

Intuitive Expanation 1

This construction reflects a two-stage process in line with Assumption 2:

Market Evolves First:

At each time step t, the market transitions to a new fuzzy state. This transition influences which set of parameters governs the bond evolution. The matrix ${\mathbf{Q}}_Y\left(t\right)$, along with fuzzy memberships ${\mu }_Y\left(t\right)$ , captures this uncertainty in market movement.

Bond Evolution Depends on Market:

Once the market state updates, the defaultable bond transitions according to parameters that are conditional on the new market state. The matrix ${\mathfrak{P}}_{X\left(\left(F_{Xa},s\right)\left(F_{Xi},l\right)\right)}\left(t\right)$ encodes these transition probabilities, modulated by the fuzzy information about both the origin and destination states.

Fuzzy Information Integration:

The membership functions ${\mu }_X^{\left(F\right)}\left(t\right)$ and ${\mu }_Y^{\left(F\right)}\left(t\right)$ allow partial or imprecise knowledge about states to be incorporated. The diagonal matrices ${\mathbf{D}}_{q_X{q}_Y}\left(t\right)$ and ${\mathbf{M}}_{dg\left(F_{Yj},m\right)}\left(t\right)$ help weigh the transitions based on this fuzzy knowledge.

Normalization:

Since the transitions are defined over fuzzy states, the resulting probability matrix is normalized using ${\mathbf{D}}_{q_X{q}_Y}\left(t\right)$ and ${\mathbf{Q}}_X^{\left(F\right)}$ $\left(t\right)$, ensuring that each row of the final matrix corresponds to a valid fuzzy probability distribution.

Remark 2.

In (64), we found the probabilities $q_{X\left(\left(F_{Xa},s\right)\left(F_{Xi},l\right)\right)}^{\left(F\right)}\left(t;F_j\right)$ of transitions among the fuzzy states of the migration process: $\left(1\right)$ as functions of the known parameters at time t, of the $\mathcal{F}$-inhomogeneous semi-Markov process that describes the movements among the states $S{D}_X$ (that is, of the migration process and their memberships functions with the fuzzy states); and $\left(2\right)$ as functions of the known parameters at time $t,$ of the $\mathcal{F}$-inhomogeneous semi-Markov process that describes the movements among the states $S{D}_Y$ (that is, of the stochastic market and their memberships functions with the fuzzy states). Hence, we will treat these probabilities as functions, which could be estimated from the basic parameters of the two $\mathcal{F}$-inhomogeneous semi-Markov processes.

An illustrative Example 4.

It is apparent that we cannot illustrate Theorem 2, due to the large space needed to continue in detail with the data given in examples 1, 2, and 3. In such a case, we will have to write matrices of size $49\times 49$ and $7\times 49$ which makes no sense. In Theorem 2, the new elements we did not illustrate in the previous examples are the matrices ${\mathfrak{P}}_{X\left(\left(F_{Xa},s\right)\left(F_{Xi},l\right)\right)}\left(t\right)$. In order to constrain the size of the matrices, we will switch to the following $4\times 4$ matrices:

Let the state space of migration process be

$$S{D}_X=\left\{\left(1,0\right),\left(1,1\right),\left(2,0\right),\left(2,1\right)\right\},$$

and let the probability distribution in the various states of $S{D}_X$ at time t be

$${\mathbf{q}}_X\left(.,.,t\right)=\left(\begin{array}{cccc}{\displaystyle \frac{1}{6}}& {\displaystyle \frac{1}{3}}& {\displaystyle \frac{1}{4}}& {\displaystyle \frac{1}{4}}\end{array}\right),$$

Then, we have that

$${\mathbf{D}}_{X_q}\left(t\right)=\left(\begin{array}{cccc}{\displaystyle \frac{1}{6}}& 0& 0& 0\\ 0& {\displaystyle \frac{1}{3}}& 0& 0\\ 0& 0& {\displaystyle \frac{1}{4}}& 0\\ 0& 0& 0& {\displaystyle \frac{1}{4}}\end{array}\right).$$

Assume now that the membership functions at time t are given by the matrix:

$${\mathbf{M}}_X^{\left(F\right)}\left(t\right)=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0.2& 0.5& 0.2& 0.1\\ 0.1& 0.1& 0.6& 0.2\\ 0& 0& 0& 1\end{array}\right),$$

where the fuzzy state space of the migration process is given by

$$S{D}_X^{\left(F\right)}=\left\{\left(F_{X_1},0\right),\left(F_{X_1},1\right),\left(F_{X_2},0\right),\left(F_{X_2},1\right)\right\},$$

and let

$${\left[{\mathbf{M}}_X^{\left(F\right)}\left(t+1\right)\right]}^{\top }=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 0.9& 0.1& 0\\ 0& 0.1& 0.9& 0\\ 0& 0& 0& 1\end{array}\right).$$

Now, the matrices  ${\mathfrak{P}}_{X\left(\left(F_{Xa},s\right)\left(F_{Xi},l\right)\right)}\left(t\right)$ are given in Equation (62) and it is apparent that in the present set up, they will be matrices of size $16\times 4$. However, each element of the matrix will be given by the relation

$$\begin{array}{ccc}\hfill {\mathfrak{p}}_{X\left(\left(F_{Xa},s\right)\left(F_{Xi},l\right)\right)}\left(t;\left(\lambda ,d_{Y_{\lambda }}\right)\right)& =& {\mathit{\mu }}_{X\left(F_{Xa},s\right)}\left[\left(.,.\right),t\right]{\mathbf{D}}_{X_q}\left(t\right){\mathbf{Q}}_X\left(t;\left(\lambda ,d_{Y_{\lambda }}\right)\right)\times \hfill \\ & & {\mathit{\mu }}_{X\left(F_{Xi},s\right)}^{\top }\left[\left(.,.\right),t+1\right],\hfill \end{array}$$

which, as we have seen, is a scalar function. We will illustrate the evaluation of one of these elements. In this respect, let

$${\mathbf{Q}}_X\left(t,\left(1,0\right)\right)=\left(\begin{array}{cccc}0.4& 0.2& 0.4& 0\\ 0.4& 0& 0.6& 0\\ 0.3& 0& 0.3& 0.4\\ 0.2& 0& 0.8& 0\end{array}\right),$$

Then, from our given data we have that

$${\mathit{\mu }}_{X\left(F_{X_2},0\right)}\left[\left(.,.\right),t+1\right]=\left(\begin{array}{cccc}0.1& 0.1& 0.6& 0.2\end{array}\right),$$

$${\mathit{\mu }}_{X\left(F_{X_1},1\right)}^{\top }\left[\left(.,.\right),t+1\right]=\left(\begin{array}{c}0\\ 0.9\\ 0.1\\ 0\end{array}\right),$$

and consequently

$${\mathfrak{p}}_{X\left(\left(F_{X_2},0\right)\left(F_{X_1},1\right)\right)}\left(t;\left(1,0\right)\right)=$$

$$\left(\begin{array}{cccc}0.1& 0.1& 0.6& 0.2\end{array}\right)\left(\begin{array}{cccc}{\displaystyle \frac{1}{6}}& 0& 0& 0\\ 0& {\displaystyle \frac{1}{3}}& 0& 0\\ 0& 0& {\displaystyle \frac{1}{4}}& 0\\ 0& 0& 0& {\displaystyle \frac{1}{4}}\end{array}\right)\times$$

$$\left(\begin{array}{cccc}0.4& 0.2& 0.4& 0\\ 0.4& 0& 0.6& 0\\ 0.3& 0& 0.3& 0.4\\ 0.2& 0& 0.8& 0\end{array}\right)\left(\begin{array}{c}0\\ 0.9\\ 0.1\\ 0\end{array}\right)=0.014.$$

5. The Fuzzy Survival Probabilities of the Defaultable Bonds in a Fuzzy Stochastic Market

We define the survival probabilities of defaultable bonds as the probabilities that the bond remains in non-default states up to time t; that is, if $\tau$ denotes the time of default, then the survival condition is $\tau \ge t$. In this section, we establish a stochastic difference equation governing the fuzzy survival probabilities of defaultable bonds within a fuzzy stochastic market framework. This formulation will serve as the foundation for analyzing their asymptotic behavior in the subsequent section.

Let $\tau$ be the time to default for a bond and define the following probability:

$${\pi }_{\left(F_{Xi},d_{F_{Xi}}\right)}^{\left(F\right)}\left[\left(F_{Yj},d_{F_{Yj}}\right);t\right]=\mathbb{P}\{D_X^{\left(F\right)}\left(t\right)=\left(F_{Xi},d_{F_{Xi}}\right)•$$

$$D_Y^{\left(F\right)}\left(t\right)=\left(F_{Yj},d_{F_{Yj}}\right)\mid \tau \ge t\}.$$

This survival probability corresponds to the event that the defaultable bond has survived up to time t, is in the fuzzy state $F_{X_i}$ with duration $d_{F_{X_i}}$ , and that the stochastic market is simultaneously in the fuzzy state $F_{Y_j}$ with duration $d_{F_{Y_j}}$. To formalize this, we define the following probability vector, which encapsulates all such survival probabilities corresponding to the bond being in a fuzzy migration state while the market resides in the fuzzy state $\left(F_{Y_j},d_{F_{Y_j}}\right)$:

$${\mathbf{\pi }}_D^{\left(F\right)}\left[\left(F_{Yj},d_{F_{Yj}}\right);t\right]=[{\pi }_{\left(F_{X1},0\right)}^{\left(F\right)}\left[\left(F_{Yj},d_{F_{Yj}}\right);t\right],{\pi }_{\left(F_{X1},1\right)}^{\left(F\right)}\left[\left(F_{Yj},d_{F_{Yj}}\right);t\right],\dots ,$$

(65)

$${\pi }_{\left(F_{X1},f_{X1}\right)}^{\left(F\right)}\left[\left(F_{Yj},d_{Fj}\right);t\right],\dots ,{\pi }_{\left(F_{Xn},0\right)}^{\left(F\right)}\left[\left(F_{Yj},d_{F_{Yj}}\right);t\right],\dots ,{\pi }_{\left(F_{Xn},f_{Xn}\right)}^{\left(F\right)}\left[\left(F_{Yj},d_{F_{Yj}}\right);t\right]],$$

which is an $1\times {\sum }_{i=1}^n\left(1+f_{Xi}\right)$ vector. Also,

$${\mathbf{\pi }}_D^{\left(F\right)}\left(t\right)=[{\mathbf{\pi }}_D^{\left(F\right)}\left[\left(F_{Y1},0\right);t\right],{\mathbf{\pi }}_D^{\left(F\right)}\left[\left(F_{Y1},1\right);t\right],\dots ,{\mathbf{\pi }}_D^{\left(F\right)}\left[\left(F_{Y1},f_{Y1}\right);t\right],$$

$$\dots ,{\mathbf{\pi }}_D^{\left(F\right)}\left[\left(F_{Yr},0\right);t\right],\dots ,{\mathbf{\pi }}_D^{\left(F\right)}\left[\left(F_{Yr},f_{Yr}\right);t\right]],$$

(66)

which is a $1\times {\sum }_{i=1}^n\left(1+f_{Xi}\right){\sum }_{j=1}^r\left(1+f_{Yj}\right)$ row vector.

We will now prove the following theorem

Theorem 2.

Consider the inhomogeneous semi-Markov chain $\left\{X_t,S{X}_{t+1}\right\}$ with fuzzy states, which models the movements of the defaultable bond as described in Section 3 and also consider the inhomogeneous semi-Markov chain $\left\{Y_t,S{Y}_{t+1}\right\}$ with fuzzy states, which models the movements of the stochastic market, as described in Section 3. Then

$${\mathbf{\pi }}_D^{\left(F\right)}\left(t+1\right)={\displaystyle \frac{{\mathbf{\pi }}_D^{\left(F\right)}\left(0\right)\mathbf{F}\left(0,t\right)}{{\mathbf{\pi }}_D^{\left(F\right)}\left(0\right)\mathbf{F}\left(0,t\right){\mathbf{1}}^{\top }}},$$

where $\mathbf{F}\left(t\right)={\mathbf{Q}}_{M_Y}^{\left(F\right)}\left(t\right){\mathbf{Q}}_{Xdg}^{\left(F\right)}\left(t,.\right)$ and $\mathbf{F}\left(0,t\right)=\mathbf{F}\left(0\right)\mathbf{F}\left(1\right)\dots \mathbf{F}\left(t\right)$ is the cumulative fuzzy transition operator up to time t, and where ${\mathbf{Q}}_{M_Y}^{\left(F\right)}\left(t\right)$ is the block matrix composed of fuzzy transition probabilities for the market process across fuzzy states and durations,

$${\mathbf{Q}}_{M_Y}^{\left(F\right)}\left(t\right)=$$

$$\left(\begin{array}{ccccc}{\mathbf{Q}}_{M_{Ydg\left(F_{Y1},0\right)\left(F_{Y1},0\right)}}^{\left(F\right)}\left(t\right)& \dots & {\mathbf{Q}}_{M_{Ydg\left(F_{Y1},0\right)\left(F_{Y1},f_{Y1}\right)}}^{\left(F\right)}\left(t\right)& \dots & {\mathbf{Q}}_{M_{Ydg\left(F_{Yr},0\right)\left(F_{Yr},f_{Yr}\right)}}^{\left(F\right)}\left(t\right)\\ {\mathbf{Q}}_{M_{Ydg\left(F_{Y1},1\right)\left(F_{Y1},0\right)}}^{\left(F\right)}\left(t\right)& \dots & {\mathbf{Q}}_{M_{Ydg\left(F_{Y1},1\right)\left(F_{Y1},f_{Y1}\right)}}^{\left(F\right)}\left(t\right)& \dots & {\mathbf{Q}}_{M_{Ydg\left(F_{Yr},1\right)\left(F_{Yr},f_{Yr}\right)}}^{\left(F\right)}\left(t\right)\\ \dots & \dots & \dots & \dots & \dots \\ {\mathbf{Q}}_{M_{Ydg\left(F_{Y1},f_{Y1}\right)\left(F_{Y1},0\right)}}^{\left(F\right)}\left(t\right)& \dots & {\mathbf{Q}}_{M_{Ydg\left(F_{Y1},f_{Y1}\right)\left(F_{Y1},f_{Y1}\right)}}^{\left(F\right)}\left(t\right)& \dots & {\mathbf{Q}}_{M_{Ydg\left(F_{Y1},f_{Y1}\right)\left(F_{Yr},f_{Yr}\right)}}^{\left(F\right)}\left(t\right)\\ \dots & \dots & \dots & \dots & \dots \\ {\mathbf{Q}}_{M_{Ydg\left(F_{Yr},f_{Yr}\right)\left(F_{Y1},0\right)}}^{\left(F\right)}\left(t\right)& \dots & {\mathbf{Q}}_{M_{Ydg\left(F_{Yr},f_{Yr}\right)\left(F_{Y1},f_{Y1}\right)}}^{\left(F\right)}\left(t\right)& \dots & {\mathbf{Q}}_{M_{Ydg\left(F_{Yr},f_{Yr}\right)\left(F_{Yr},f_{Yr}\right)}}^{\left(F\right)}\left(t\right)\end{array}\right),$$

with

$${\mathbf{Q}}_{M_{Ydg\left(F_{Yj},d_{F_{Yj}}\right)\left(F_{Yu},d_{F_{Yu}}\right)}}^{\left(F\right)}\left(t\right)=$$

$$\left(\begin{array}{cccc}{q}_{Y\left(F_{Yj},d_{F_{Yj}}\right)\left(F_{Yu},d_{F_{Yu}}\right)}^{\left(F\right)}\left(t\right)& 0& \dots & 0\\ \dots & q_{Y\left(F_{Yj},d_{F_{Yj}}\right)\left(F_{Yu},d_{F_{Yu}}\right)}^{\left(F\right)}\left(t\right)& \dots & 0\\ \dots & \dots & \dots & \dots \\ 0& 0& \dots & q_{Y\left(F_{Yj},d_{F_{Yj}}\right)\left(F_{Yu},d_{F_{Yu}}\right)}^{\left(F\right)}\left(t\right)\end{array}\right),$$

and where ${\mathbf{Q}}_{Xdg}^{\left(F\right)}\left(t,.\right)$ is the block diagonal matrix capturing the fuzzy transition behavior of the defaultable bond conditional on the market state.

$${\mathbf{Q}}_{Xdg}^{\left(F\right)}\left(t,.\right)=$$

$$diag\{\underbrace{{\mathbf{Q}}_X^{\left(F\right)}\left(t,F_{Y1}\right),{\mathbf{Q}}_X^{\left(F\right)}\left(t,F_{Y1}\right),\dots ,{\mathbf{Q}}_X^{\left(F\right)}\left(t,F_{Y1}\right)}1\times f_{Y1},\underbrace{{\mathbf{Q}}_X^{\left(F\right)}\left(t,F_{Y2}\right),{\mathbf{Q}}_X^{\left(F\right)}\left(t,F_{Y2}\right),\dots ,{\mathbf{Q}}_X^{\left(F\right)}\left(t,F_{Y2}\right)}1\times f_{Y2},$$

$$\dots ,\underbrace{{\mathbf{Q}}_X^{\left(F\right)}\left(t,F_{Yr}\right),{\mathbf{Q}}_X^{\left(F\right)}\left(t,F_{Yr}\right),\dots ,{\mathbf{Q}}_X^{\left(F\right)}\left(t,F_{Yr}\right)}1\times f_{Yr}\}.$$

Proof. 

Consider the row vector of probabilities

$${\widehat{\mathbf{\pi }}}_D^{\left(F\right)}\left[\left(F_{Y\lambda },d_{F_{Y\lambda }}\right);t+1\right]=q_{Y\left(F_{Yj},d_{F_{Yj}}\right)\left(F_{Y\lambda },d_{F_{Y\lambda }}\right)}^{\left(F\right)}\left(t\right){\mathbf{\pi }}_D^{\left(F\right)}\left[\left(F_{Yj},d_{F_{Yj}}\right);t\right]$$

(67)

$$\times {\mathbf{Q}}_X^{\left(F\right)}\left(t,F_{Y\lambda }\right),$$

where ${\mathbf{Q}}_X^{\left(F\right)}\left(t,F_{Y\lambda }\right)$ is

$$\sum _{i=1}^n\left(1+f_{Xi}\right)\times \sum _{i=1}^n\left(1+f_{Xi}\right)$$

a matrix with elements $q_{X\left(F_{Xi},d_{F_{Xi}}\right)\left(F_{Xu},d_{F_u}\right)}^{\left(F\right)}\left(t,F_{Y\lambda }\right)$ for $F_{Xi},F_{Xu}=F_{X1},F_{X2},\dots ,F_{Xn}$, $d_{F_{Xi}}=0,1,\dots ,f_{Xi}$, $d_{F_{Xu}}=0,1,\dots ,f_{Xi}$ and for a specific value of $F_{Y\lambda }$, the possible values of which are $F_{Y\lambda }=F_{Y1},F_{Y2},\dots ,F_{Yr}$.

Also, define ${\widehat{\mathbf{\pi }}}_D^{\left(F\right)}\left(t\right)$ as

$${\widehat{\mathbf{\pi }}}_D^{\left(F\right)}\left(t\right)=[{\widehat{\mathbf{\pi }}}_D^{\left(F\right)}\left[\left(F_{Y1},0\right);t\right],{\widehat{\mathbf{\pi }}}_D^{\left(F\right)}\left[\left(F_{Y1},1\right);t\right],\dots ,{\widehat{\mathbf{\pi }}}_D^{\left(F\right)}\left[\left(F_{Y1},f_{Y1}\right);t\right],$$

$$\dots ,{\widehat{\mathbf{\pi }}}_D^{\left(F\right)}\left[\left(F_{Yr},0\right);t\right],\dots ,{\widehat{\mathbf{\pi }}}_D^{\left(F\right)}\left[\left(F_{Yr},f_{Yr}\right);t\right]],$$

(68)

Now define

$${\mathbf{Q}}_{M_Y}^{\left(F\right)}\left(t\right)=$$

$$\left(\begin{array}{ccccc}{\mathbf{Q}}_{M_{Ydg\left(F_{Y1},0\right)\left(F_{Y1},0\right)}}^{\left(F\right)}\left(t\right)& \dots & {\mathbf{Q}}_{M_{Ydg\left(F_{Y1},0\right)\left(F_{Y1},f_{Y1}\right)}}^{\left(F\right)}\left(t\right)& \dots & {\mathbf{Q}}_{M_{Ydg\left(F_{Yr},0\right)\left(F_{Yr},f_{Yr}\right)}}^{\left(F\right)}\left(t\right)\\ {\mathbf{Q}}_{M_{Ydg\left(F_{Y1},1\right)\left(F_{Y1},0\right)}}^{\left(F\right)}\left(t\right)& \dots & {\mathbf{Q}}_{M_{Ydg\left(F_{Y1},1\right)\left(F_{Y1},f_{Y1}\right)}}^{\left(F\right)}\left(t\right)& \dots & {\mathbf{Q}}_{M_{Ydg\left(F_{Yr},1\right)\left(F_{Yr},f_{Yr}\right)}}^{\left(F\right)}\left(t\right)\\ \dots & \dots & \dots & \dots & \dots \\ {\mathbf{Q}}_{M_{Ydg\left(F_{Y1},f_{Y1}\right)\left(F_{Y1},0\right)}}^{\left(F\right)}\left(t\right)& \dots & {\mathbf{Q}}_{M_{Ydg\left(F_{Y1},f_{Y1}\right)\left(F_{Y1},f_{Y1}\right)}}^{\left(F\right)}\left(t\right)& \dots & {\mathbf{Q}}_{M_{Ydg\left(F_{Y1},f_{Y1}\right)\left(F_{Yr},f_{Yr}\right)}}^{\left(F\right)}\left(t\right)\\ \dots & \dots & \dots & \dots & \dots \\ {\mathbf{Q}}_{M_{Ydg\left(F_{Yr},f_{Yr}\right)\left(F_{Y1},0\right)}}^{\left(F\right)}\left(t\right)& \dots & {\mathbf{Q}}_{M_{Ydg\left(F_{Yr},f_{Yr}\right)\left(F_{Y1},f_{Y1}\right)}}^{\left(F\right)}\left(t\right)& \dots & {\mathbf{Q}}_{M_{Ydg\left(F_{Yr},f_{Yr}\right)\left(F_{Yr},f_{Yr}\right)}}^{\left(F\right)}\left(t\right)\end{array}\right),$$

which in fact is

$$\sum _{i=1}^r\left(1+f_{Yi}\right)\times \sum _{i=1}^r\left(1+f_{Yi}\right),$$

a bloc matrix, and each bloc is of the type

$${\mathbf{Q}}_{M_{Ydg\left(F_{Yj},d_{F_{Yj}}\right)\left(F_{Yu},d_{F_{Yu}}\right)}}^{\left(F\right)}\left(t\right)=$$

$$\left(\begin{array}{cccc}{q}_{Y\left(F_{Yj},d_{F_{Yj}}\right)\left(F_{Yu},d_{F_{Yu}}\right)}^{\left(F\right)}\left(t\right)& 0& \dots & 0\\ \dots & q_{Y\left(F_{Yj},d_{F_{Yj}}\right)\left(F_{Yu},d_{F_{Yu}}\right)}^{\left(F\right)}\left(t\right)& \dots & 0\\ \dots & \dots & \dots & \dots \\ 0& 0& \dots & q_{Y\left(F_{Yj},d_{F_{Yj}}\right)\left(F_{Yu},d_{F_{Yu}}\right)}^{\left(F\right)}\left(t\right)\end{array}\right),$$

(69)

which is defined to be a matrix with size

$$\sum _{i=1}^n\left(1+f_{Xi}\right)\times \sum _{i=1}^n\left(1+f_{Xi}\right),$$

and therefore ${\mathbf{Q}}_{M_Y}^{\left(F\right)}\left(t\right)$ is a matrix

$$\sum _{j=1}^r\left(1+f_{Yj}\right).\sum _{i=1}^n\left(1+f_{Xi}\right)\times \sum _{j=1}^r\left(1+f_{Yj}\right).\sum _{i=1}^n\left(1+f_{Xi}\right),$$

According to Assumption 2, we could prove that

$${\mathbf{\pi }}_D^{\left(F\right)}\left(t^{+}\right)={\mathbf{\pi }}_D^{\left(F\right)}\left(t\right){\mathbf{Q}}_{M_Y}^{\left(F\right)}\left(t\right)=$$

$$[\sum _{j=1}^r\sum _{d_{F_{Yj}}=0}^{f_{Y{F}_{Yj}}}{\mathbf{\pi }}_D^{\left(F\right)}\left(\left(F_{Yj},d_{F_{Yj}}\right);t\right){\mathbf{Q}}_{M_{Ydg\left(F_{Yj},d_{F_{Yj}}\right)\left(F_{Y1},0\right)}}^{\left(F\right)}\left(t\right),\dots ,$$

$$\sum _{j=1}^r\sum _{d_{F_{Yj}}=0}^{f_{Y{F}_{Yj}}}{\mathbf{\pi }}_D^{\left(F\right)}\left(\left(F_{Yj},d_{F_{Yj}}\right);t\right){\mathbf{Q}}_{M_{Ydg\left(F_{Yj},d_{F_{Yj}}\right)\left(F_{Yr},f_{Yr}\right)}}^{\left(F\right)}\left(t\right)].$$

(70)

Consequently, we could prove that

$${\widehat{\mathbf{\pi }}}_D^{\left(F\right)}\left(t+1\right)={\widehat{\mathbf{\pi }}}_D^{\left(F\right)}\left(t^{+}\right)\times$$

$$diag\{\underbrace{{\mathbf{Q}}_X^{\left(F\right)}\left(t,F_{Y1}\right),{\mathbf{Q}}_X^{\left(F\right)}\left(t,F_{Y1}\right),\dots ,{\mathbf{Q}}_X^{\left(F\right)}\left(t,F_{Y1}\right)}1\times f_{Y1},\underbrace{{\mathbf{Q}}_X^{\left(F\right)}\left(t,F_{Y2}\right),{\mathbf{Q}}_X^{\left(F\right)}\left(t,F_{Y2}\right),\dots ,{\mathbf{Q}}_X^{\left(F\right)}\left(t,F_{Y2}\right)}1\times f_{Y2},$$

$$\dots ,\underbrace{{\mathbf{Q}}_X^{\left(F\right)}\left(t,F_{Yr}\right),{\mathbf{Q}}_X^{\left(F\right)}\left(t,F_{Yr}\right),\dots ,{\mathbf{Q}}_X^{\left(F\right)}\left(t,F_{Yr}\right)}1\times f_{Yr}\},$$

from which we write

$${\widehat{\mathbf{\pi }}}_D^{\left(F\right)}\left(t+1\right)={\widehat{\mathbf{\pi }}}_D^{\left(F\right)}\left(t^{+}\right){\mathbf{Q}}_{Xdg}^{\left(F\right)}\left(t,.\right),$$

(71)

where ${\mathbf{Q}}_{Xdg}^{\left(F\right)}\left(t,.\right)$ is a matrix

$$\sum _{j=1}^r\left(1+f_{Yj}\right).\sum _{i=1}^n\left(1+f_{Xi}\right)\times \sum _{j=1}^r\left(1+f_{Yj}\right).\sum _{i=1}^n\left(1+f_{Xi}\right),$$

Finally, from the above we could prove that

$${\widehat{\mathbf{\pi }}}_D^{\left(F\right)}\left(t+1\right)={\mathbf{\pi }}_D^{\left(F\right)}\left(t\right){\mathbf{Q}}_{M_Y}^{\left(F\right)}\left(t\right){\mathbf{Q}}_{Xdg}^{\left(F\right)}\left(t,.\right).$$

(72)

Now, following the analogous reasoning as in Vassiliou and Vasileiou (2013), we arrive easily at the relation

$${\mathbf{\pi }}_D^{\left(F\right)}\left(t+1\right)={\displaystyle \frac{{\mathbf{\pi }}_D^{\left(F\right)}\left(t\right){\mathbf{Q}}_{M_Y}^{\left(F\right)}\left(t\right){\mathbf{Q}}_{Xdg}^{\left(F\right)}\left(t,.\right)}{{\mathbf{\pi }}_D^{\left(F\right)}\left(t\right){\mathbf{Q}}_{M_Y}^{\left(F\right)}\left(t\right){\mathbf{Q}}_{Xdg}^{\left(F\right)}\left(t,.\right){\mathbf{1}}^{\top }}}.$$

(73)

From relation (73), we recursively get that

$${\mathbf{\pi }}_D^{\left(F\right)}\left(t+1\right)={\displaystyle \frac{{\mathbf{\pi }}_D^{\left(F\right)}\left(t-1\right){\mathbf{Q}}_{M_Y}^{\left(F\right)}\left(t-1\right){\mathbf{Q}}_{Xdg}^{\left(F\right)}\left(t-1,.\right){\mathbf{Q}}_{M_Y}^{\left(F\right)}\left(t\right){\mathbf{Q}}_{Xdg}^{\left(F\right)}\left(t,.\right)}{{\mathbf{\pi }}_D^{\left(F\right)}\left(t-1\right){\mathbf{Q}}_{M_Y}^{\left(F\right)}\left(t-1\right){\mathbf{Q}}_{Xdg}^{\left(F\right)}\left(t-1,.\right){\mathbf{Q}}_{M_Y}^{\left(F\right)}\left(t\right){\mathbf{Q}}_{Xdg}^{\left(F\right)}\left(t,.\right){\mathbf{1}}^{\top }}}$$

$$={\displaystyle \frac{{\mathbf{\pi }}_D^{\left(F\right)}\left(t-1\right)\mathbf{F}\left(t-1\right)\mathbf{F}\left(t\right)}{{\mathbf{\pi }}_D^{\left(F\right)}\left(t-1\right)\mathbf{F}\left(t-1\right)\mathbf{F}\left(t\right){\mathbf{1}}^{\top }}}$$

$$={\displaystyle \frac{{\mathbf{\pi }}_D^{\left(F\right)}\left(t-1\right)\mathbf{F}\left(t-1,t\right)}{{\mathbf{\pi }}_D^{\left(F\right)}\left(t-1\right)\mathbf{F}\left(t-1,t\right){\mathbf{1}}^{\top }}},$$

(74)

$$={\displaystyle \frac{{\mathbf{\pi }}_D^{\left(F\right)}\left(0\right)\mathbf{F}\left(0,t\right)}{{\mathbf{\pi }}_D^{\left(F\right)}\left(0\right)\mathbf{F}\left(0,t\right){\mathbf{1}}^{\top }}}$$

where $\mathbf{F}\left(t\right)={\mathbf{Q}}_{M_Y}^{\left(F\right)}\left(t\right){\mathbf{Q}}_{Xdg}^{\left(F\right)}\left(t,.\right)$ and $\mathbf{F}\left(0,t\right)=\mathbf{F}\left(0\right)\mathbf{F}\left(1\right)\dots \mathbf{F}\left(t\right)$.  □

Intuitive Explanation 2

This theorem gives us a recursive formula to compute the fuzzy survival probabilities of a defaultable bond over time in a market that is both stochastic and fuzzy. Here is how to interpret it step by step:

Think of ${\mathbf{\pi }}_D^{\left(F\right)}\left(t+1\right)$ as a comprehensive probability vector: it tells us, at time $t+1$, what the chances are that the bond is still “alive” (i.e., not in default), and in which fuzzy credit state it might be, while simultaneously tracking the fuzzy state of the market.

The formula evolves this survival probability vector by using two main ingredients:

(1).

Market dynamics: captured by the matrix ${\mathbf{Q}}_{M_Y}^{\left(F\right)}\left(t\right)$, which contains the fuzzy probabilities of moving from one market state-duration pair to another.

(2).

Bond dynamics given the market: encoded in ${\mathbf{Q}}_{Xdg}^{\left(F\right)}\left(t,.\right)$, which handles how the bond transitions between fuzzy credit states, conditional on the fuzzy market environment.

The product $\mathbf{F}\left(t\right)={\mathbf{Q}}_{M_Y}^{\left(F\right)}\left(t\right){\mathbf{Q}}_{Xdg}^{\left(F\right)}\left(t,.\right)$ combines the evolution of both the market and the bond into a single matrix that governs how survival probabilities update from one time step to the next.

The full product $\mathbf{F}\left(0,t\right)$ aggregates all transitions from time 0 to t, representing the entire system’s behavior over time.

Finally, the normalization by ${\mathbf{\pi }}_D^{\left(F\right)}\left(0\right)\mathbf{F}\left(0,t\right){\mathbf{1}}^{\top }$ ensures that the resulting vector is a proper probability distribution (i.e., sums to 1).

This result is important because it gives a computable recursive structure for tracking survival in a complex system where both the credit quality of a bond and the market are uncertain and described with fuzzy states. Moreover, this formulation lays the groundwork for examining long-run or asymptotic behavior, such as whether survival probabilities stabilize or decline under particular market conditions.

6. The Asymptotic Behavior of the Survival Probabilities in the Fuzzy States

The problem of determining the asymptotic behavior of survival probabilities for defaultable bonds holds significant importance in the field of credit risk. These probabilities directly impact market stability—low survival probabilities may trigger turbulence and financial distress, while high values encourage lending activity and support confidence in the financial system. Moreover, they affect the behavior of bondholders, market liquidity, credit spreads, and, ultimately, have implications for both national and global economies.

However, the analysis presented in this section goes beyond practical financial concerns. As we will discuss later, it involves solving the quasi-stationarity problem for an $\mathcal{F}$-inhomogeneous semi-Markov chain with fuzzy states evolving within a fuzzy stochastic environment. This adds considerable depth and complexity to the study.

In what follows, we will employ the recursive structure established in Equation (74) to analyze the long-run behavior of the fuzzy survival probabilities. Specifically, we aim to determine the limit

$$\underset{t\to \infty }{lim}{\mathbf{\pi }}_D^{\left(F\right)}\left(t\right).$$

Let us now introduce the norm of a matrix, which we will use in what follows in order to prove convergence of sequence of matrices. Let $\mathbf{A}\in M_n\left(\mathbb{R}\right)$ be the set of all $n\times n$ matrices with elements from the field $\mathbb{R}$ and $\mathbf{P}\in S{M}_n\left(\mathbb{R}\right)$ be a stochastic matrix. We will use the known norm

$${∥\mathbf{A}∥}_r=\underset{i}{max}\sum _{j=1}^n\left|a_{ij}\right|,$$

(75)

which is usually a useful mathematical instrument when dealing with convergence of matrices. Note, however, that under any norm, convergence of finite matrices is equivalent with element wise convergence. Hence, in what follows we will use the symbol $∥.∥$ in the sense that the arguments hold for any norm.

A fundamental assumption for many theorems of ergodicity is that the products of matrices $\mathbf{P}\left(1,t\right)$ is a regular stochastic matrix. This assumption is not easily checked in a practical way and from the point of view of utility we need conditions on individual matrices $\mathbf{P}\left(t\right)$. The difficulty increases by the following facts which can be easily checked:

(a)

the product of two non-regular matrices may be regular and

(b)

the products of regular matrices may not be regular.

In order to overcome partially at least these difficulties, we distinguish the following classes of matrices (Vassiliou 2023, Def. 219).

Definition 1.

(a) A matrix $\mathbf{P}ϵ\mathcal{M}$ is called Markov if at least one column of $\mathbf{P}$ is entirely positive. (b) $\mathbf{P}ϵG_2$ if $\left(i\right)$ $\mathbf{P}ϵG_1$ a regular matrix; $\left(ii\right)$ $\mathbf{QP}ϵG_1$ for any $\mathbf{Q}ϵG_1.$ (c) $\mathbf{P}ϵG_3$ if its ergodicity coefficient is ${\tau }_{{∥.∥}_1}\left(\mathbf{P}\right)<1,$ i.e., $\mathbf{P}$ is scrambling.

We now state a theorem which links these classes of matrices:

Theorem 3

(Vassiliou 2023, p. 143). For all $\mathbf{P}\in S{M}_n\left(\mathbb{R}\right)$ we have $\mathcal{M}\subset G_3\subset G_2\subset G_1$.

We will now state and prove the following fundamental theorem for the asymptotic behavior of the survival probabilities in the fuzzy states within a fuzzy stochastic environment.

Intuitive Explanation

To understand what we are doing in this section, think of a defaultable bond as an entity moving through a landscape of credit states—such as high-grade, medium-grade, or low-grade—where each state is not precisely defined, but “fuzzy”, reflecting the ambiguity and uncertainty of real-world credit assessments.

At the same time, this movement does not happen in isolation. The broader market itself is also shifting between various fuzzy conditions—say, a stable market, a volatile one, or one in crisis. These market states are themselves uncertain and modeled fuzzily, and they influence how likely a bond is to improve, deteriorate, or default.

Now, suppose we observe this interaction over time: the bond moving through fuzzy credit states, influenced by a fuzzy-evolving market. What we want to know is whether there is any long-run pattern or stable behavior that emerges—do the survival probabilities (i.e., the chances that the bond hasn’t defaulted yet) settle down into some equilibrium or predictable form?

This is the idea behind quasi-stationarity in our fuzzy, inhomogeneous semi-Markov setting. It asks: if the bond has not defaulted yet, what is the likelihood it will be in each fuzzy credit state (while the market is also in some fuzzy state), as time goes to infinity?

By framing and solving this asymptotic problem, we can answer deep questions about long-term creditworthiness under uncertainty and how resilient bonds are under prolonged market influences, even when the system does not settle into a traditional steady-state due to inhomogeneities and fuzziness.

The recursive equation from the previous section gives us a way to compute these probabilities step by step, and by studying its behavior as time advances, we aim to uncover the long-term survival structure of defaultable bonds in a realistically uncertain world.

Theorem 4

(Existence and Evaluation of Limiting Fuzzy Survival Probabilities). Let $(\mathsf{\Omega },\mathcal{F},\mathbb{P})$ be a real-world filtered probability space. Consider a zero-coupon defaultable bond whose migration process is governed by an $\mathcal{F}$-inhomogeneous semi-Markov process as described in Equations (9) and (11), and let the associated duration $\mathcal{F}$-inhomogeneous Markov chain be defined as in (12) and (13). Assume the fuzzy states of the duration chain are given by (14) and (15). Furthermore, let the bond be traded in a stochastic market with ν states, the evolution of which is modeled by an $\mathcal{F}$-inhomogeneous semi-Markov process as in (36). Assume the market environment is fuzzy with states as in (38), and that its duration chain is defined in (41). Suppose the following conditions hold:

$$\left(i\right)\underset{t\to \infty }{lim}∥{\mathbf{Q}}_X\left(t;j\right)-{\mathbf{Q}}_X∥=0,\mathit{with}{\mathbf{Q}}_XϵG_2$$

(76)

$$\left(ii\right)\underset{t\to \infty }{lim}∥{\mathbf{Q}}_Y\left(t\right)-{\mathbf{Q}}_Y∥=0,\mathit{with}{\mathbf{Q}}_YϵG_2$$

(77)

$$\left(iii\right)\underset{t\to \infty }{lim}∥{\mathbf{M}}_X^{\left(F\right)}\left(t\right)-{\mathbf{M}}_X^{\left(F\right)}∥=0,\mathit{with}{\mathbf{M}}_XϵG_2$$

(78)

and

$$\left(iv\right)\underset{t\to \infty }{lim}∥{\mathbf{M}}_Y^{\left(F\right)}\left(t\right)-{\mathbf{M}}_Y^{\left(F\right)}∥=0,\mathit{with}{\mathbf{M}}_YϵG_2$$

(79)

and ${\mathbf{M}}_Y^{\left(F\right)}>0$. Then, the limiting distribution of the fuzzy survival probabilities exists and is given by

$$\underset{t\to \infty }{lim}∥{\mathbf{\pi }}_D^{\left(F\right)}\left(t+1\right)-{\mathbf{\pi }}^{*}∥=0,$$

where ${\mathbf{\pi }}^{*}$ is the unique solution of the system

$${\mathbf{\pi }}^{*}={\mathbf{\pi }}^{*}\mathsf{\Pi }$$

where Π is defined in (105).

Proof. 

From (74), we have that $\mathbf{F}\left(t\right)={\mathbf{Q}}_{M_Y}^{\left(F\right)}\left(t\right){\mathbf{Q}}_{Xdg}^{\left(F\right)}\left(t,.\right)$. We will prove that

$$\left(a\right)\underset{t\to \infty }{lim}∥{\mathbf{Q}}_{Xdg}^{\left(F\right)}\left(t,.\right)-{\mathbf{Q}}_{Xdg}^{\left(F\right)}\left(.\right)∥=0\mathrm{where}{\mathbf{Q}}_{Xdg}^{\left(F\right)}\left(.\right)ϵG_2,$$

(80)

and

$$\left(b\right)\underset{t\to \infty }{lim}∥{\mathbf{Q}}_{M_Y}^{\left(F\right)}\left(t\right)-{\mathbf{Q}}_{M_Y}^{\left(F\right)}∥=0\mathrm{where}{\mathbf{Q}}_{M_Y}^{\left(F\right)}ϵG_2.$$

(81)

$\left(a\right)$ Let us define

$${\mathbf{Q}}_{Xdg}^{\left(F\right)}\left(.\right)=$$

$$diag\{\underbrace{{\mathbf{Q}}_X^{\left(F\right)}\left(F_{Y1}\right),{\mathbf{Q}}_X^{\left(F\right)}\left(F_{Y1}\right),\dots ,{\mathbf{Q}}_X^{\left(F\right)}\left(F_{Y1}\right)}1\times f_{Y1},\underbrace{{\mathbf{Q}}_X^{\left(F\right)}\left(F_{Y2}\right),{\mathbf{Q}}_X^{\left(F\right)}\left(F_{Y2}\right),\dots ,{\mathbf{Q}}_X^{\left(F\right)}\left(F_{Y2}\right)}1\times f_{Y2},$$

$$\dots ,\underbrace{{\mathbf{Q}}_X^{\left(F\right)}\left(F_{Yr}\right),{\mathbf{Q}}_X^{\left(F\right)}\left(F_{Yr}\right),\dots ,{\mathbf{Q}}_X^{\left(F\right)}\left(F_{Yr}\right)}1\times f_{Yr}\},$$

(82)

Then, it is sufficient to show that there exists a ${\mathbf{Q}}_X^{\left(F\right)}\left(F_{Yj}\right)$ such that

$$\underset{t\to \infty }{lim}∥{\mathbf{Q}}_X^{\left(F\right)}\left(t;F_{Yj}\right)-{\mathbf{Q}}_X^{\left(F\right)}\left(F_{Yj}\right)∥=0\mathrm{for}\mathrm{all}{F}_{Yj},$$

(83)

where ${\mathbf{Q}}_X^{\left(F\right)}\left(t;F_{Yj}\right)$ is given in (63) and we seek its limit as $t\to \infty$. It is known (77) that ${\mathbf{Q}}_YϵG_2$. It is well known that in such a case

$$\underset{t\to \infty }{lim}∥{\mathbf{q}}_Y\left(.,.t\right){\mathbf{Q}}_Y\left(0,t\right)-{\mathbf{q}}_Y^{*}∥=0,$$

(84)

where ${\mathbf{q}}_Y^{*}$ is the row of the stable matrix $li{m}_{t\to \infty }{\mathbf{Q}}_Y^t={\mathbf{Q}}_Y^{*}$. Hence, if we define by

$${\mathbf{Q}}_{dg{\mathbf{q}}_Y}=dg\left\{{\mathbf{q}}_Y^{*},{\mathbf{q}}_Y^{*},\dots ,{\mathbf{q}}_Y^{*}\right\},$$

(85)

then it could easily be proved that

$$\underset{t\to \infty }{lim}∥{\mathbf{Q}}_{dg{\mathbf{q}}_Y}\left(t\right)-{\mathbf{Q}}_{dg{\mathbf{q}}_Y}∥=0.$$

(86)

From (59), (60), and (79), we get that

$$\underset{t\to \infty }{lim}∥{\mathbf{M}}_{dg\left(F_{Yj}\right)}\left(t\right)-{\mathbf{M}}_{dg\left(F_{Yj}\right)}∥=0,$$

(87)

where ${\mathbf{M}}_{dg\left(F_{Yj}\right)}$ is constructed as in (59) and (60), placing in the positions of the elements ${\mu }_{\left(F_{Yj},m\right)}\left[\left(1,0\right),t+1\right]$ the corresponding elements of the matrix ${\mathbf{M}}_Y^{\left(F\right)}$. From the relation just before (63), we have that the matrix ${\mathbf{D}}_{q_X{q}_Y}\left(t\right)$ is a diagonal matrix with elements

$${\mathbf{q}}_Y\left(.,.,t\right){\mathbf{Q}}_Y\left(t\right){\mathit{\mu }}_{Y\left(F_{Yj},m\right)}^{\top }\left[\left(.,.\right),t+1\right]{\mathbf{q}}_X\left(.,.,t\right){\mathit{\mu }}_{X\left(F_{Xi},m\right)}^{\top }\left[\left(.,.\right),t\right].$$

(88)

From (84) and (87), we have that

$$\underset{t\to \infty }{lim}∥{\mathbf{q}}_Y\left(.,.,t\right){\mathbf{Q}}_Y\left(t\right){\mathit{\mu }}_{Y\left(F_{Yj},m\right)}^{\top }\left[\left(.,.\right),t+1\right]-{\mathbf{q}}_Y^{*}{\mathit{\mu }}_{Y\left(F_{Yj},m\right)}^{\top }\left[\left(.,.\right)\right]∥=0.$$

(89)

From (76) and the fact that ${\mathbf{Q}}_XϵG_2$, we get that

$$\underset{t\to \infty }{lim}∥{\mathbf{q}}_X\left(.,.,t\right)-{\mathbf{q}}_X^{*}∥=0,$$

(90)

where ${\mathbf{q}}_X^{*}$ is the row of the stable matrix ${\lim }_{t\to \infty }{\mathbf{Q}}_Y^t={\mathbf{Q}}_Y^{*}$. From (89), (90), and (78) it is not difficult to show that

$$\underset{t\to \infty }{lim}\parallel {\mathbf{q}}_Y\left(.,.,t\right){\mathbf{Q}}_Y\left(t\right){\mathit{\mu }}_{Y\left(F_{Yj},m\right)}^{\top }\left[\left(.,.\right),t+1\right]{\mathbf{q}}_X\left(.,.,t\right){\mathit{\mu }}_{X\left(F_{Xi},m\right)}^{\top }\left[\left(.,.\right),t\right]$$

$$-{\mathbf{q}}_Y^{*}{\mathit{\mu }}_{Y\left(F_{Yj},m\right)}^{\top }\left[\left(.,.\right)\right]{\mathbf{q}}_X^{*}{\mathit{\mu }}_{X\left(F_{Xi},m\right)}^{\top }\left(.,.\right)\parallel .$$

(91)

Define the matrix

$${\mathbf{D}}_{q_X{q}_Y}=diag{\left\{{\mathbf{q}}_Y^{*}{\mathit{\mu }}_{Y\left(F_{Yj},m\right)}^{\top }\left[\left(.,.\right)\right]{\mathbf{q}}_X^{*}{\mathit{\mu }}_{X\left(F_{Xi},m\right)}^{\top }\left(.,.\right)\right\}}_{\left(F_{Xi},m\right)}.$$

(92)

Then, it is apparent that

$$\underset{t\to \infty }{lim}∥{\mathbf{D}}_{q_X{q}_Y}\left(t\right)-{\mathbf{D}}_{q_X{q}_Y}∥=0.$$

(93)

Now, consider the matrix ${\mathfrak{P}}_{X\left(\left(F_{Xa},s\right)\left(F_{Xi},l\right)\right)}\left(t\right)$, which is defined in (62) and its elements are of the form which is given in (56) and is the following

$${\mathfrak{p}}_{X\left(\left(F_{Xa},s\right)\left(F_{Xi},l\right)\right)}\left(t;\left(\lambda ,d_{Y_{\lambda }}\right)\right)={\mathit{\mu }}_{X\left(F_{Xa},s\right)}\left[\left(.,.\right),t\right]{\mathbf{D}}_{q_X}\left(t\right){\mathbf{Q}}_X\left(t;\left(\lambda ,d_{Y_{\lambda }}\right)\right)\times$$

(94)

$${\mathit{\mu }}_{X\left(F_{Xi},s\right)}^{\top }\left[\left(.,.\right),t+1\right].$$

From (91), we immediately get that

$$\underset{t\to \infty }{lim}∥{\mathbf{D}}_{q_X}\left(t\right){\mathbf{Q}}_X\left(t;\left(\lambda ,d_{Y_{\lambda }}\right)\right)-{\mathbf{q}}_X^{*}∥=0.$$

(95)

Therefore, from (94), (95), and (78), it is not difficult to show that

$$\underset{t\to \infty }{lim}∥{\mathfrak{p}}_{X\left(\left(F_{Xa},s\right)\left(F_{Xi},l\right)\right)}\left(t;\left(\lambda ,d_{Y_{\lambda }}\right)\right)-{\mathfrak{p}}_{X\left(\left(F_{Xa},s\right)\left(F_{Xi},l\right)\right)}∥=$$

$$\parallel {\mathit{\mu }}_{X\left(F_{Xa},s\right)}\left[\left(.,.\right),t\right]{\mathbf{D}}_{q_X}\left(t\right){\mathbf{Q}}_X\left(t;\left(\lambda ,d_{Y_{\lambda }}\right)\right){\mathit{\mu }}_{X\left(F_{Xi},s\right)}^{\top }\left[\left(.,.\right),t+1\right]$$

$$-{\mathit{\mu }}_{X\left(F_{Xa},s\right)}\left[\left(.,.\right)\right]{\mathbf{q}}_X^{*}{\mathit{\mu }}_{X\left(F_{Xi},s\right)}^{\top }\left[\left(.,.\right)\right]\parallel =0.$$

(96)

Therefore, if we define by ${\mathfrak{P}}_{X\left(\left(F_{Xa},s\right)\left(F_{Xi},l\right)\right)}$ the matrix with elements the probabilities ${\mathfrak{p}}_{X\left(\left(F_{Xa},s\right)\left(F_{Xi},l\right)\right)}$ then it is apparent that

$$\underset{t\to \infty }{lim}∥{\mathfrak{P}}_{X\left(\left(F_{Xa},s\right)\left(F_{Xi},l\right)\right)}\left(t\right)-{\mathfrak{P}}_{X\left(\left(F_{Xa},s\right)\left(F_{Xi},l\right)\right)}∥=0.$$

(97)

It is not difficult from (63), (86)–(88), and (97) to show that

$$\underset{t\to \infty }{lim}∥{\mathbf{Q}}_X^{\left(F\right)}\left(t;F_{Yj}\right)-{\mathbf{Q}}_X^{\left(F\right)}\left(F_{Yj}\right)∥=$$

$$\parallel {\left[{\mathbf{D}}_{q_X{q}_Y}\left(t\right)\right]}^{-1}{\mathbf{Q}}_{dg{\mathbf{q}}_Y}\left(t\right){\mathbf{M}}_{dg\left(F_{Yj},m\right)}\left(t\right){\mathfrak{P}}_{X\left(\left(F_{Xa},s\right)\left(F_{Xi},l\right)\right)}\left(t\right)$$

$$-{\left[{\mathbf{D}}_{q_X{q}_Y}\right]}^{-1}{\mathbf{Q}}_{dg{\mathbf{q}}_Y}{\mathbf{M}}_{dg\left(F_{Yj},m\right)}{\mathfrak{P}}_{X\left(\left(F_{Xa},s\right)\left(F_{Xi},l\right)\right)}\parallel =0.$$

(98)

Moreover, from the fact that ${\mathbf{Q}}_X^{\left(F\right)}\left(F_{Yi}\right)ϵG_2$ for every i and (82), we get that ${\mathbf{Q}}_{Xdg}^{\left(F\right)}\left(.\right)ϵG_2$, which concludes the proof of $a)$.

$\left(b\right)$ From (65) and Theorem 2, we observe that ${\mathbf{Q}}_{M_Y}^{\left(F\right)}\left(t\right)$ is constructed exclusively from elements of the matrix ${\mathbf{Q}}_Y^{\left(F\right)}\left(t\right)$ the analytic matrix form of which is given in (42) and Example 2. Therefore, it suffices to find ${lim}_{t\to \infty }{\mathbf{Q}}_Y^{\left(F\right)}\left(t\right)$.

$$\underset{t\to \infty }{lim}∥{\mathbf{Q}}_Y^{\left(F\right)}\left(t\right)-{\mathbf{Q}}_Y^{\left(F\right)}∥=0,$$

(99)

where

$${\mathbf{Q}}_Y^{\left(F\right)}={\left[{\mathbf{M}}_{Ydg}^{\left(F\right)}\right]}^{-1}{\left[{\mathbf{M}}_Y^{\left(F\right)}\right]}^{\top }{\mathbf{D}}_{Y_{{\mathbf{q}}^{*}}}{\mathbf{Q}}_Y{\mathbf{M}}_Y^{\left(F\right)}ϵG_2,$$

(100)

where ${\mathbf{M}}_{Ydg}^{\left(F\right)}$ is the diagonal matrix

$${\mathbf{M}}_{Ydg}^{\left(F\right)}=diag\left\{{\mathbf{q}}_Y^{*}{\mathit{\mu }}_{Y\left(F_{Y1},0\right)}^{\top }\left[.,.\right],\dots ,{\mathbf{q}}_Y^{*}{\mathit{\mu }}_{Y\left(F_{Yn},f_n\right)}^{\top }\left[.,.\right]\right\}.$$

(101)

Now, it is apparent that

$$\underset{t\to \infty }{lim}{∥{\mathbf{Q}}_{M_Y}^{\left(F\right)}\left(t\right)-{\mathbf{Q}}_{M_Y}^{\left(F\right)}∥}_m=0,$$

(102)

where ${\mathbf{Q}}_{M_Y}^{\left(F\right)}$ is being constructed from the elements of the matrix ${\mathbf{Q}}_Y^{\left(F\right)}$ in exactly the same way as ${\mathbf{Q}}_{M_Y}^{\left(F\right)}\left(t\right)$ is constructed from the elements of ${\mathbf{Q}}_Y^{\left(F\right)}\left(t\right)$. Now, from the conditions of the theorem ${\mathbf{Q}}_X$, ${\mathbf{Q}}_Y$, ${\mathbf{M}}_X^{\left(F\right)}$, ${\mathbf{M}}_Y^{\left(F\right)}ϵG_2,$, hence ${\mathbf{Q}}_{M_Y}^{\left(F\right)}ϵG_2$. That concludes the proof of $\left(b\right)$.

From the proofs of parts $\left(a\right)$ and $\left(b\right)$, we get that

$$\mathbf{F}={\mathbf{Q}}_{M_Y}^{\left(F\right)}{\mathbf{Q}}_X^{\left(F\right)}\left(F_{Yj}\right)ϵG_2.$$

(103)

Define by $\rho \left(\mathbf{F}\right)$ the spectral radius of the matrix $\mathbf{F}$ and define

$$\mathsf{\Pi }={\displaystyle \frac{\mathbf{F}}{\rho \left(\mathbf{F}\right)}}.$$

(104)

Then, from (103) and Theorem 2 in Vassiliou and Vasileiou (2013) or in Papadopoulou and Tsaklidis (2007); Wu et al. (2015) we get that there exists a stable stochastic matrix, let say Π* such that

$$\underset{t\to \infty }{lim}∥{\mathsf{\Pi }}^t-{\mathsf{\Pi }}^{*}∥=0.$$

(105)

Define by ${\mathbf{\pi }}^{*}$ the row of the matrix ${\mathsf{\Pi }}^{*}$. Then, following the steps of the proof of Theorem 3 in Vassiliou (2014), we arrive at the conclusion that

$$\underset{t\to \infty }{lim}∥{\mathbf{\pi }}_D^{\left(F\right)}\left(t+1\right)-{\mathbf{\pi }}^{*}∥=0.$$

□

Intuitive Explanation 3

This result establishes that, under natural convergence assumptions on the transition probabilities and membership functions of both the migration and market processes, the system stabilizes in the long run. That is, the fuzzy survival probabilities of the defaultable bond settle into a limiting distribution.

The key idea is that both the bond and the market processes eventually behave like time-homogeneous systems, with their respective transition dynamics converging to fixed matrices. Similarly, the uncertainty (fuzziness) in the classification of the states becomes stable over time. As a result, the overall joint evolution of the system, despite its initial non-stationarity and fuzziness, converges to a stable pattern.

The limiting vector ${\mathbf{\pi }}^{*}$ represents this long-term behavior and satisfies a stationary distribution equation, capturing the quasi-stationary regime of a fuzzy, inhomogeneous environment. This is crucial in applications such as pricing long-term credit derivatives, evaluating the long-run creditworthiness of issuers, or stress-testing market scenarios under prolonged exposure to systemic risk.

7. Parameter Estimation

As established in Section 5, the survival probabilities within the inhomogeneous semi-Markov chain with fuzzy states—operating in a fuzzy stochastic market environment—are entirely determined by the estimation of the fundamental system parameters. These include the elements of the transition probability matrices ${\mathbf{Q}}_X\left(t;j\right)$ and ${\mathbf{Q}}_Y\left(t\right)$, as well as the membership function matrices ${\mathbf{M}}_X^{\left(F\right)}\left(t\right)$ and ${\mathbf{M}}_Y^{\left(F\right)}\left(t\right)$. Consequently, accurate estimation of these quantities from empirical data is crucial, and it necessitates the use of statistical estimators with desirable properties such as consistency and asymptotic normality.

In empirical studies, rating transition histories are often sourced from comprehensive datasets such as Moody’s Corporate Bond Default Database, which includes complete issuer histories dating back to 1970. To manage the sparsity of transitions across an otherwise large number of rating grades, these grades are typically aggregated into seven well-known categories: Aaa, Aa, A, Baa, B, Caa, and Default. These categories are widely recognized by the public and are indexed numerically for modeling purposes: Aaa is indexed as 1 (the highest credit quality), while Caa and Default are indexed as 6 and 7, respectively. We illustrate this structure with two representative empirical examples in

Table 1. Data examples.

Example 1
Date	Raiting
29 May 1988	Ba
27 December 2000	B
1 October 2001	Default
Example 2
Date	Raiting
11 January 1984	A
20 June 1991	Baa
7 February 1993	WR

drawn from (Lando 2004, p. 94) and Carty and Fons (1994).

The entry WR means that the issuers had their rating withdrawn at the date referred to. A senior rating might be withdrawn for any number of reasons, from retirement of all rated debt to completion of an exchange offer of all rated debt. For the state of the stochastic market, there are many statistical indices collected in different databases, with the use of which one could distinguish various states of the market, which could be an entire continent, a group of nations, a nation, an industry, etc. The most common case would be to distinguish four grades such as “excellent” as state 1, “good” as state 2, “medium” as state 3, and “crisis” as state 4. The fuzzy states defined for the various grades of the bonds or debts depend on the physical characteristics of the bond or debts and these define also their membership functions ${\mathbf{M}}_X^{\left(F\right)}\left(t\right)$. These also depend on the definition of the grades and the amount of groupings that will be necessary to perform. Usually a good practice would be not to perform much groupings on the real data and the fuzzy states to be the seven grades referred above. The same applies for the fuzzy states of the market, where the groupings that will be necessary to perform are a lot more. We assume that data are grouped into bond cohorts, belonging to the same grade at a specific time from 1970 onwards at intervals of 3 months, 6 months, or 1 year, etc. The length of the time interval is chosen in such a way as to satisfy the basic assumption that first the transition of the stochastic market takes place and then follows the transitions among the grades of the bond ratings. That is, for every time interval $\left(t,t+1\right]$ and for every cohort of bonds, the state of the market during the interval where their migration will take place, is known. Let $i=1,2,\dots ,7$ be the number of grades of the bonds, $j=1,2,3,4$ be the number of the states of the market, and $t=1,2,\dots ,T^{*}$ be the time span of the available data. Let $N_{is}\left(t,j\right)$ be the number of bonds in grade i with duration $s=0,1,\dots ,b_i$ at time t given the state of the market is $j=1,2,3,4$. Let $N_{is,lm}\left(t,j\right)$ be the number of bonds at time t and the market in state j, which move during the interval $\left(t,t+1\right]$ from state $\left(i,s\right)$ to state $\left(l,m\right)$. Also, let $W_{is}\left(t,j\right)$ be the number of bonds withdrawn from state i with duration s in the time interval $\left(t,t+1\right]$ and the market being in state j. Then, the bond movements in the time interval $\left(t,t+1\right]$ can be tabulated in the typical form shown in

Table 2. Movements of bonds in the time interval $\left(t,t+1\right]$, with the state of the market being j.

$\left(1,0\right)$	$\left(1,1\right)$	…	$\left(1,{\mathit{b}}_1\right)$	…	$\left(\mathit{k},0\right)$	…	$\left(\mathit{k},{\mathit{b}}_{\mathit{k}}\right)$	$\mathit{k}+1$
$N_{10}$	$N_{11}$	…	$N_{1b_1}$	…	$N_{k0}$	…	$N_{k{b}_k}$	-
$N_{10,10}$	$N_{10,11}$	…	$N_{10,1b_1}$	…	$N_{10,k0}$	…	$N_{10,k{b}_k}$	$N_{10,k+1}$
$N_{11,10}$	$N_{11,11}$	…	$N_{11,1b_1}$	…	$N_{11,k1}$	…	$N_{11,k{b}_k}$	$N_{11,k+1}$
…	…	…	…	…	…	…	…	…
$N_{1b_1,10}$	$N_{1b_1,11}$	…	$N_{1b_1,1b_1}$	…	$N_{1b_1,k0}$	…	$N_{1b_1,k{b}_k}$	$N_{1b_1,k+1}$
…	…	…	…	…	…	…	…	…
$N_{k0,10}$	$N_{k0,11}$	…	$N_{k0,1b_1}$	…	$N_{k0,k0}$	…	$N_{k0,k{b}_k}$	$N_{k0,k+1}$
$N_{k1,10}$	$N_{k1,11}$	…	$N_{k1,1b_1}$	…	$N_{k1,k0}$	…	$N_{k1,k{b}_k}$	$N_{k1,k+1}$
…	…	…	…	…	…	…	…	…
$N_{k{b}_k,10}$	$N_{k{b}_k,11}$	…	$N_{k{b}_k,1b_1}$	…	$N_{k{b}_k,k0}$	…	$N_{k{b}_k,k{b}_k}$	$N_{k{b}_k,1b_1}$
$W_{10}$	$W_{11}$	…	$W_{1b_1}$	…	$W_{ko}$	…	$W_{k{b}_k}$	-

, where the arguments of time and the state of the market have been omitted for the sake of space.

The flows from any state $\left(i,s\right)$ during the time interval $\left(t,t+1\right]$ to any other state $\left(j,n\right),$ that is, $N_{is,lm}\left(t,j\right)$ may be considered as a random variable with a multinomial distribution for $N_{is}\left(t,j\right)-W_{is}\left(t,j\right)$ experiments and probability $q_{is,lm}\left(t,j\right)$. The maximum likelihood estimates for $q_{is,lm}\left(t,j\right)$ are given by

$${\widehat{q}}_{Xis,lm}\left(t,j\right)={\displaystyle \frac{N_{is,lm}\left(t,j\right)}{N_{is}\left(t,j\right)-W_{is}\left(t,j\right)}}.$$

(106)

Looking again at $N_{is,lm}\left(t,j\right)$ as a multinomial sample from the population $N_{is}\left(t,j\right)$, with probability $q_{is,lm}\left(t,j\right)$ of moving to state $\left(l,m\right)$ within the interval $\left(t,t+1\right]$, we obtain the standard error $q_{is,lm}\left(t,j\right)$ to be

$$se\left({\widehat{q}}_{Xis,lm}\left(t,j\right)\right)={\left[{\displaystyle \frac{{\widehat{q}}_{is,lm}\left(t,j\right)\left(1-{\widehat{q}}_{is,lm}\left(t,j\right)\right)}{N_{is,lm}\left(t,j\right)}}\right]}^2{R}_{is}\left(t,j\right),$$

(107)

where $R_{Xis}\left(t,j\right)$ is the probability of a bond to be uncensored or, equivalently, is not withdrawn during the interval $\left(t,t+1\right]$, given by

$$R_{Xis}\left(t,j\right)={\displaystyle \frac{N_{is}\left(t,j\right)-W_{is}\left(t,j\right)}{N_{is}\left(t,j\right)}}.$$

(108)

It is evident from the above that we can estimate the probabilities ${\mathbf{Q}}_X\left(t,j\right)$ for the time span available as data. The same applies for the probabilities ${\mathbf{Q}}_Y\left(t\right)$, whose estimation is a well known problem with no particular difficulties.

8. Conclusions

In conclusion, an advanced and novel theoretical methodological framework for credit-risk modeling by integrating fuzzy logic with non-homogeneous semi-Markov processes has been presented and founded . The proposed approach represents a significant theoretical advancement and addresses critical limitations in existing models. Hence, by the construction of the present model the uncertainty and temporal heterogeneity inherent in real-word situations in credit-risk assessment have been more effectively accommodated. Thus, it provides new risk management tools in the face of an increasingly complex financial environment. It also evaluates the asymptotic behavior of fuzzy survival probabilities in a non-homogeneous semi-Markov process evolving within a fuzzy stochastic environment. This is done in a close analytic form easily computable by practitioners. The external stochastic environment is likewise modeled as a non-homogeneous semi-Markov process with fuzzy states. Hence, the general problem of quasi-stationarity for a non-homogeneous semi-Markov process with fuzzy states embedded in a fuzzy stochastic framework has been solved in close analytic form.