Modeling Financial Bubbles with Optional Semimartingales in Nonstandard Probability Spaces

Abstract

Deviation of an asset price from its fundamental value, commonly referred to as a price bubble, is a well-known phenomenon in financial markets. Mathematically, a bubble arises when the deflated price process transitions from a martingale to a strict local martingale. This paper explores price bubbles using the framework of optional semimartingale calculus within nonstandard probability spaces, where the underlying filtration is not necessarily right-continuous or complete. We present two formulations for financial markets with bubbles: one in which asset prices are modeled as càdlàg semimartingales and another where they are modeled as làdlàg semimartingales. In both models, we demonstrate that the formation and re-emergence of price bubbles are intrinsically tied to the lack of right continuity in the underlying filtration. These theoretical findings are illustrated with practical examples, offering novel insights into bubble dynamics that hold significance for both academics and practitioners in the field of mathematical finance.

1. Introduction

Optional semimartingales represent a class of stochastic processes with a well-established calculus that extends to both standard and nonstandard (or unusual) conditions. In this paper, we focus on optional semimartingales with càdlàg (right-continuous with left limits) and làdlàg (left-continuous with right limits) paths under nonstandard conditions. We explore their potential as a framework for modeling financial markets exhibiting price bubbles, extending the traditional approach by accommodating more general filtrations and capturing a wider range of market phenomena and information structures.

The market value of an asset is primarily driven by supply and demand, while its fundamental value is determined by internal factors such as the company’s products, management quality, and brand strength. Common methods for estimating fundamental value include discounted models based on dividends, cash flows, or residual income. Price bubbles occur when a significant gap develops between an asset’s market price and its fundamental value. However, the two prices are intertwined in various ways. It is conceivable that the disparity between them results from differences in the valuation of intrinsic value, as well as differing short- and long-term outlooks—whether positive or negative—on the firm’s market value.

Price bubbles, as the name suggests, are episodes of sharp price increases followed by sudden collapses. They have been extensively studied by economists and mathematicians, leading to the identification of various conditions under which bubbles can form. For instance, bubbles arise in infinite-horizon growing economies with rational traders (O’Connell and Zeldes 1988; Tirole 1985; Weil 1989). Bubbles can also emerge in economies where rational traders hold divergent beliefs or behave myopically (Tirole 1982) or in markets with irrational traders (De-Long et al. 1990). Moreover, bubbles can occur when arbitrageurs are unable to synchronize their trades (Abreu and Brunnermeier 2003) or in markets with constraints on borrowing (Santos and Woodford 1997; Scheinkman and Xiong 2004). Furthermore, price bubbles have been observed on a sector-wide scale, such as the housing bubbles in the U.S. (Case and Shiller 2003) and earlier in Japan (Stone and Ziemba 1993). Notably, in all scenarios where price bubbles arise, arbitrageurs cannot profit from them. That is to say that bubbles appear at and survive for unpredictable times and grow to unpredictable sizes such that consistent arbitrage opportunities are not possible to construct.

Mathematically, a stock price exhibits a bubble when the price process is a positive strict local martingale under the equivalent local martingale measure (Jarrow 2015). A strict local martingale refers to a local martingale that is not a true martingale. The characterization of price bubbles and the pricing of derivatives in finite-horizon economies, under the No Free Lunch with Vanishing Risk (NFLVR) hypothesis, has been extensively explored in several studies (Cox and Hobson 2005; Heston et al. 2007; Loewenstein and Willard 2000a, 2000b). However, bubbles have been shown to violate several classical option pricing theorems—most notably, the put–call parity, which is almost never violated empirically Kamara and Miller (1995); Klemkosky and Resnick (1980); Ofek and Richardson (2003). In (Jarrow 2015), it was demonstrated that these violations arise due to insufficient structural assumptions about the economy within the NFLVR framework. The study further characterized potential price bubbles in incomplete markets under both the NFLVR and No-Dominance (ND) assumptions (Merton 1973) and proposed a theory for bubble birth that involves a market exhibiting different local martingale measures across time.

A study by Kardaras et al. (2015) examined the impact of asset price bubbles on derivative pricing. It provided decomposition formulas for specific classes of European path-dependent options, demonstrating how stock price bubbles affect option values under the NFLVR condition. Furthermore, Biagini et al. (2014) studied the flow in the space of equivalent martingale measures and the corresponding shifts in perception of the fundamental value of a given asset. This framework captured the birth of a bubble as an initial submartingale that transitions into a supermartingale before eventually returning to its initial value of zero. Additionally, Biagini et al. (2023) demonstrated that price bubbles are filtration-dependent, meaning they may emerge in one informational setting while vanishing in another. Their findings also highlight how traders with limited information can misinterpret bubbles as arbitrage opportunities, further emphasizing the role of information asymmetry in financial markets. Carr et al. (2014) utilized the Föllmer measure to construct a pricing operator for market models where the exchange rate is driven by a strict local martingale. This construction allowed them to preserve put–call parity and provided the minimal joint replication price for a contingent claim. Furthermore, Jarrow et al. (2022) introduced invariance theorems to test asset price bubbles in markets where prices evolve as Markov diffusion processes. Their results show that the existence of a bubble can be identified solely through the quadratic variation of the price process, eliminating the need to estimate the drift term under an equivalent local martingale measure. For a comprehensive survey of recent literature on financial bubbles, we refer readers to (Camerer 1989; Hirano and Toda 2024; Protter 2013; Scheinkman and Xiong 2003).

This paper presents a new approach to the theory of price bubbles using the calculus of optional semimartingales in nonstandard probability spaces. Our goal is to expand the mathematical toolkit for addressing problems in mathematical finance. We propose two market models based on optional semimartingales to explore bubble dynamics. The first model characterizes asset prices as càdlàg semimartingales, illustrating how price bubbles arise from the progressive incorporation of external information—beyond that generated by the price process—into market filtration. The second model employs làdlàg optional semimartingales, necessitating a redefinition of market structures to align with nonstandard probability spaces. This framework effectively captures the emergence, evolution, and eventual dissolution of bubbles. To substantiate our theoretical results, we provide illustrative examples that demonstrate practical applications of the proposed models. Through the application of optional semimartingale calculus, this work delivers new insights into the formation and dynamics of price bubbles. In the next section, we offer a concise introduction to the stochastic calculus of optional processes, establishing the foundation for our analysis.

2. Stochastic Calculus of Optional Processes

The study of stochastic processes without the usual conditions was initiated by Dellacherie and Meyer in 1970 (Dellacherie 1975). Subsequent advancements were made by numerous mathematicians, with significant contributions by Galtchouk (1980, 1985). These works developed a stochastic calculus for processes in nonstandard probability spaces. In this section, we describe some aspects of this theory.

Let $\left(\Omega ,\mathcal{F},\mathbf{F}=\left({\mathcal{F}}_t\right),\mathbf{P}\right)$ for $t\in [0,\infty )$ be a complete probability space, where $\mathbf{P}$ is a complete measure and $\mathcal{F}$ contains all $\mathbf{P}$ null sets. However, the family ($\mathbf{F}$) is not assumed to be complete or right- or left-continuous. We introduce $\mathcal{O}\left(\mathbf{F}\right)$ and $\mathcal{P}\left(\mathbf{F}\right)$ as optional and predictable $\sigma$ algebras on $\left(\Omega ,[0,\infty )\right)$, respectively. $\mathcal{O}\left(\mathbf{F}\right)$ is generated by all $\mathbf{F}$-adapted processes whose trajectories are right-continuous and have limits from the left. $\mathcal{P}\left(\mathbf{F}\right)$ is generated by all $\mathbf{F}$-adapted predictable processes whose trajectories are left-continuous and have limits on the right. In addition to the filtration $\mathbf{F}$, we introduce ${\mathbf{F}}_{-}={\left({\mathcal{F}}_{t-}\right)}_{t\ge 0}$, ${\mathbf{F}}^{+}={\left({\mathcal{F}}_t^{+}\right)}_{t\ge 0}$, and ${\mathbf{F}}^{\mathbf{P}}={\left({\mathcal{F}}_t^{\mathbf{P}}\right)}_{t\ge 0}$. ${\mathcal{F}}_t^{+}:={\mathcal{F}}_{t+}={\cap }_{s\ge 0}{\mathcal{F}}_{t+s}$ is right-continuous filtration, and the ${\mathbf{F}}^{\mathbf{P}}$ family satisfies the usual conditions under $\mathbf{P}$.

A random process ($X:={\left(X_t\right)}_{t\ge 0}$) is said to be optional if it is $\mathcal{O}\left(\mathbf{F}\right)$-measurable. An optional process has right and left limits but is not necessarily right- or left-continuous in $\mathbf{F}$. A random process (X) is predictable if $X\in \mathcal{P}\left(\mathbf{F}\right)$, and strongly predictable if $X\in \mathcal{P}\left(\mathbf{F}\right)$ and $X_{+}\in \mathcal{O}\left(\mathbf{F}\right)$. A predictable process has right and left limits but may not necessarily be right- or left-continuous in $\mathbf{F}$. For either optional or predictable processes, we can define the following processes: $X_{-}={\left(X_{t-}\right)}_{t\ge 0}$, $X_{+}={\left(X_{t+}\right)}_{t\ge 0}$, and $△X={(△X_t)}_{t\ge 0}$ such that $△X_t=X_t-X_{t-}$ and ${△}^{+}X={\left({△}^{+}{X}_t\right)}_{t\ge 0}$ such that ${△}^{+}{X}_t=X_{t+}-X_t$.

On an unusual stochastic basis, three canonical types of stopping times exist. Predictable stopping times ($S\in \mathcal{T}\left({\mathbf{F}}_{-}\right)$) are such that $\left(S\le t\right)$ is ${\mathcal{F}}_{t-}$-measurable for all t. Totally inaccessible stopping times ($T\in \mathcal{T}\left(\mathbf{F}\right)$) are such that $\left(T\le t\right)$ is ${\mathcal{F}}_t$-measurable for all t; however, we point out that $\left(T<t\right)$ is not necessarily ${\mathcal{F}}_t$-measurable, since ${\mathcal{F}}_t$ may not be right-continuous. Finally, totally inaccessible stopping times in the broad sense ($U\in \mathcal{T}\left({\mathbf{F}}_{+}\right)$) are such that $\left(U\le t\right)$ is ${\mathcal{F}}_{t+}$-measurable for all t, but since ${\mathcal{F}}_{t+}$ is right-continuous, $\left(U<t\right)$ is also ${\mathcal{F}}_{t+}$-measurable. A process ($X={\left(X_t\right)}_{t\ge 0}$) belongs to the ${\mathcal{J}}_{loc}$ space if there is a localizing sequence of stopping times in the broad sense ($\left(R_n\right)$, $n\in \mathbb{N}$, $R_n\in \mathcal{T}\left({\mathbf{F}}_{+}\right)$, $R_n\uparrow \infty$) such that $X{\mathbf{1}}_{[0,R_n]}\in \mathcal{J}$ for all n, where $\mathcal{J}$ is a space of processes and ${\mathcal{J}}_{loc}$ is an extension of $\mathcal{J}$ by localization.

A process ($A={\left(A_t\right)}_{t\ge 0}$) is increasing if it is non-negative; its trajectories do not decrease; and, for any t, the random variable ($A_t$) is ${\mathcal{F}}_t$-measurable. Let ${\mathcal{V}}^{+}(\mathbf{F},\mathbf{P})$ (${\mathcal{V}}^{+}$ for short) be a collection of increasing processes. An increasing process (A) is integrable if $\mathbf{E}{A}_{\infty }<\infty$ and locally integrable if there is a sequence ($\left(R_n\right)\subset \mathcal{T}\left({\mathbf{F}}_{+}\right)$, $n\in \mathbb{N}$, $R_n\uparrow \infty$) such that $\mathbf{E}{A}_{R_{n+}}<\infty$ for all $n\in \mathbb{N}$. The collection of such processes is denoted by ${\mathcal{A}}^{+}$ (${\mathcal{A}}_{loc}^{+}$). (A process $A=\left(A_t\right)$, $t\in {\mathbb{R}}_{+}$) is a finite variation process if it has finite variation on every segment ($[0,t]$, $t\in {\mathbb{R}}_{+}$), that is, $\mathbf{Var}{\left(A\right)}_t<\infty$, for all $t\in {\mathbb{R}}_{+}$, where

$$\mathbf{Var}{\left(A\right)}_t={\int }_{0+}^t|d{A}_s^{\mathtt{r}}|+{\sum }_{s<t}|{△}^{+}{A}_s|.$$

$\mathcal{V}(\mathbf{F},\mathbf{P})$ ($\mathcal{V}$ for short) denotes a set of $\mathbf{F}$-adapted finite variation processes. A process ($A={\left(A_t\right)}_{t\ge 0}$) of finite variation belongs to the $\mathcal{A}$ space of integrable finite variation processes if $\mathbf{E}\left[\mathbf{Var}{\left(A\right)}_{\infty }\right]<\infty$. A process ($A={\left(A_t\right)}_{t\ge 0}$) belongs to ${\mathcal{A}}_{loc}$ if there is a sequence ($\left(R_n\right)\subset \mathcal{T}\left({\mathbf{F}}_{+}\right)$, $n\in \mathbb{N}$, $R_n\uparrow \infty$) such that $A{\mathbf{1}}_{[0,R_n]}\in \mathcal{A}$ for any $n\in \mathbb{N}$, i.e., for any n, $\mathbf{E}\left[\mathbf{Var}{\left(A\right)}_{R_n}\right]<\infty$. A finite variation or an increasing process (A) can be decomposed as $A=A^{\mathtt{r}}+A^{\mathtt{g}}=A^{\mathtt{c}}+A^{\mathtt{d}}+A^{\mathtt{g}}$, where $A^{\mathtt{c}}$ is continuous, $A^{\mathtt{r}}$ is right-continuous, $A^{\mathtt{d}}$ is discrete right-continuous, and $A^{\mathtt{g}}$ is discrete left-continuous such that

$$A_t^{\mathtt{d}}={\sum }_{s\le t}△A_s\mathrm{and}{A}_t^{\mathtt{g}}={\sum }_{s<t}{△}^{+}{A}_s,$$

where the series converges absolutely (Abdelghani and Melnikov (2019)).

Definition 1.

A process ($M={\left(M_t\right)}_{t\ge 0}$) is an optional martingale (supermartingale or submartingale) if $M\in \mathcal{O}\left(\mathbf{F}\right)$ and there exists an integrable random variable ($\mu \in \mathcal{F}$) such that, for any stopping time ($T<\infty$),

$$M_T=\mathbf{E}[\mu |{\mathcal{F}}_T](\mathrm{respectively},M_T\ge \mathbf{E}[\mu |{\mathcal{F}}_T],M_T\le \mathbf{E}[\mu |{\mathcal{F}}_T])\mathrm{a}.\mathrm{s}.$$

Let $\mathcal{M}(\mathbf{F},\mathbf{P})$ ($\mathcal{M}$ for short) denote a set of optional martingales and ${\mathcal{M}}_{loc}$ denote a set of optional local martingales. If M is a (local) optional martingale, then it can be decomposed as

$$M=M^{\mathtt{r}}+M^{\mathtt{g}}\mathrm{where}{M}^{\mathtt{r}}=M^{\mathtt{c}}+M^{\mathtt{d}},$$

where $M^{\mathtt{c}}$ is a continuous, $M^{\mathtt{d}}$ is a right-continuous, and $M^{\mathtt{g}}$ is a left-continuous (local) optional martingale. $M^{\mathtt{d}}$ and $M^{\mathtt{g}}$ are orthogonal to each other and to any continuous local martingale. Moreover, $M^{\mathtt{d}}$ and $M^{\mathtt{g}}$ can be written as

$$M_t^{\mathtt{d}}={\sum }_{s\le t}△M_s\mathrm{and}{M}_t^{\mathtt{g}}={\sum }_{s<t}{△}^{+}{M}_s.$$

Definition 1 tells us that all optional martingales are right-closed. Also, for a càdlàg martingale (Y) adapted to ${\mathbf{F}}^{\mathbf{P}}$, the optional projection ($X_t=\mathbf{E}[Y_t|{\mathcal{F}}_t]$) is a làdlàg optional martingale under unusual conditions.

An optional semimartingale ($X={\left(X_t\right)}_{t\ge 0}$) can be decomposed to an optional local martingale and an optional finite variation process:

$$X=X_0+M+A,$$

where $M\in {\mathcal{M}}_{loc}$ and $A\in \mathcal{V}$. A semimartingale (X) is called special if the above decomposition exists with a strongly predictable process ($A\in {\mathcal{A}}_{loc}$). Let $\mathcal{S}(\mathbf{F},\mathbf{P})$ denote a set of optional semimartingales and $\mathcal{S}\mathrm{p}(\mathbf{F},\mathbf{P})$ be a set of special optional semimartingales. If $X\in \mathcal{S}\mathrm{p}(\mathbf{F},\mathbf{P})$, then the semimartingale decomposition is unique. Through optional martingale decomposition and decomposition of predictable processes (see Galtchouk (1977, 1985)), we can decompose a semimartingale further to $X=X_0+X^{\mathtt{r}}+X^{\mathtt{g}}$ with $X^{\mathtt{r}}=A^{\mathtt{r}}+M^{\mathtt{r}}$, $X^{\mathtt{g}}=A^{\mathtt{g}}+M^{\mathtt{g}}$, and $M^{\mathtt{r}}=M^{\mathtt{c}}+M^{\mathtt{d}}$, where $A^{\mathtt{r}}$ and $A^{\mathtt{g}}$ are finite variation processes that are right- and left-continuous, respectively. $M^{\mathtt{r}}\in {\mathcal{M}}_{loc}^{\mathtt{r}}$ are right-continuous local martingales, $M^{\mathtt{d}}\in {\mathcal{M}}_{loc}^{\mathtt{d}}$ are discrete right-continuous local martingales, and $M^{\mathtt{g}}\in {\mathcal{M}}_{loc}^{\mathtt{g}}$ are left-continuous local martingales. This decomposition is useful for defining integration with respect to optional semimartingales.

A stochastic integral with respect to an optional semimartingale was defined in (Galtchouk 1985) as

$$\begin{array}{c}\phi \circ X_t={\int }_0^t{\phi }_{s}d{X}_s={\int }_{0+}^t{\phi }_{s-}d{X}_s^{\mathtt{r}}+{\int }_0^{t-}{\phi }_{s}d{X}_{s+}^{\mathtt{g}}\mathrm{where}\\ {\int }_{0+}^t{\phi }_{s-}d{X}_s^{\mathtt{r}}={\int }_{0+}^t{\phi }_{s-}d{A}_s^{\mathtt{r}}+{\int }_{0+}^t{\phi }_{s-}d{M}_s^{\mathtt{r}}\mathrm{and}\\ {\int }_0^{t-}{\phi }_{s}d{X}_{s+}^{\mathtt{g}}={\int }_0^{t-}{\varphi }_{s}d{A}_{s+}^{\mathtt{g}}+{\int }_0^{t-}{\varphi }_{s}d{M}_{s+}^{\mathtt{g}}.\end{array}$$

The stochastic integral with respect to a finite variation process or a strongly predictable process ($A^{\mathtt{r}}$ over $(0,t]$ and $A^{\mathtt{g}}$ over $[0,t)$) is interpreted as usual in the Lebesgue sense. The integral expressed as ${\int }_{0+}^t{\phi }_{s-}d{M}_s^{\mathtt{r}}$ over $(0,t]$ is our usual stochastic integral with respect to a càdlàg local martingale, whereas ${\int }_0^{t-}{\varphi }_{s}d{M}_{s+}^{\mathtt{g}}$ over $[0,t)$ is a Galtchouk stochastic integral with respect to a left-continuous local martingale. In general, the stochastic integral with respect to an optional semimartingale (X) can be defined as a bilinear form ($(f,g)\circ X_t$) such that

$$\begin{array}{c}{Y}_t=(f,g)\circ X_t=f\circ X_t^{\mathtt{r}}+g\circ X_t^{\mathtt{g}},\\ f\circ X^{\mathtt{r}}:={\int }_{0+}^t{f}_{s-}d{X}_s^{\mathtt{r}},g\circ X^{\mathtt{g}}:={\int }_0^{t-}{g}_{s}d{X}_{s+}^{\mathtt{g}},\end{array}$$

where Y is, again, an optional semimartingale, $f_{-}\in \mathcal{P}\left(\mathbf{F}\right)$, and $g\in \mathcal{O}\left(\mathbf{F}\right)$. Therefore, the stochastic integral over optional semimartingales is defined on a much larger space of integrands, where $\mathcal{P}\left(\mathbf{F}\right)\times \mathcal{O}\left(\mathbf{F}\right)$ is the product space of predictable and optional processes. To understand this stochastic integral, one must keep in mind the definition of integration intervals and the relationship between integrands and integrators. Consider the following integral:

$$g\circ X_t^{\mathtt{g}}:={\int }_0^{t-}{g}_{s}d{X}_{s+}^{\mathtt{g}},$$

which is ${\mathcal{F}}_t$-adapted and well defined according to Galtchouk (1985); the integration is taken over the interval of $[0,t)$ and the integrand ($g_s$) is adapted to ${\mathcal{F}}_s$, while $X_{s+}^{\mathtt{g}}$ is adapted to ${\mathcal{F}}_{s+}\subseteq {\mathcal{F}}_t$ for all $s\in [0,t)$.

The properties of optional stochastic integrals are described as follows: First, isometry is satisfied with

$${\left(f^2\circ \left[X^{\mathtt{r}},X^{\mathtt{r}}\right]\right)}^{1/2}\in {\mathcal{A}}_{loc}\left(\mathbf{F}\right)\mathrm{and}{\left(g^2\circ \left[X^{\mathtt{g}},X^{\mathtt{g}}\right]\right)}^{1/2}\in {\mathcal{A}}_{loc}\left(\mathbf{F}\right).$$

The quadratic variation $\left[X,X\right]$ is defined as

$$\begin{array}{c}\left[X,X\right]=\left[X^{\mathtt{r}},X^{\mathtt{r}}\right]+\left[X^{\mathtt{g}},X^{\mathtt{g}}\right]\mathrm{where}\\ {\left[X^{\mathtt{r}},X^{\mathtt{r}}\right]}_t={〈X^{\mathtt{c}},X^{\mathtt{c}}〉}_t+{\sum }_{s\le t}{\left(△X_s\right)}^2\mathrm{and}\\ {\left[X^{\mathtt{g}},X^{\mathtt{g}}\right]}_t={\sum }_{s<t}{\left({△}^{+}{X}_s\right)}^2.\end{array}$$

Linearity is also satisfied with $(f^1+f^2,g^1+g^2)\circ X_t=(f^1,g^1)\circ X_t+(f^2,g^2)\circ X_t$ for any $(f^1,g^1)$ and $(f^2,g^2)$ in the space expressed as $\mathcal{P}\left(\mathbf{F}\right)\times \mathcal{O}\left(\mathbf{F}\right)$; ${△}^{+}{X}^{\mathtt{g}}$ is $\mathcal{O}\left({\mathbf{F}}_{+}\right)$, with its martingale part satisfying $\mathbf{E}\left[{△}^{+}{M}_T^{\mathtt{g}}{\mathbf{1}}_{\left(T<\infty \right)}|{\mathcal{F}}_T\right]=0$ a.s. for any stopping time (T) in the broad sense, $△X^{\mathtt{r}}$ is $\mathcal{O}\left(\mathbf{F}\right)$, with its martingale part satisfying $\mathbf{E}\left[△M_T^{\mathtt{r}}{\mathbf{1}}_{\left(T<\infty \right)}|{\mathcal{F}}_T\right]=0$ a.s. for any stopping time (T). Moreover, orthogonality is such that $X^{\mathtt{r}}\perp X^{\mathtt{g}}$ are orthogonal in the sense that their product is a local optional martingale. Also, differentials are independent: $△Y=f△X^{\mathtt{r}}$ and ${△}^{+}Y=g{△}^{+}{X}^{\mathtt{g}}$. Lastly, for any semimartingale (Z), the quadratic projection is $\left[Y,Z\right]=f\circ \left[X^{\mathtt{r}},Z^{\mathtt{r}}\right]+g\circ \left[X^{\mathtt{g}},Z^{\mathtt{g}}\right]$.

The stochastic calculus of optional processes and its applications have witnessed significant advancements in recent years. In Abdelghani and Melnikov (2020), a solution to the nonhomogeneous linear stochastic equation of optional semimartingales was provided. The work reported in (Gasparyan 1985) established the existence and uniqueness of solutions under Lipschitz conditions for nonlinear stochastic equations driven by optional semimartingales, while Abdelghani and Melnikov (2020) proved the same under monotonicity conditions. A comparison of solutions for stochastic equations of optional semimartingales was presented in Abdelghani and Melnikov (2020), and Jarni and Ouknine (2021) proved the existence and uniqueness of solutions of reflected stochastic equations driven by optional semimartingales. The study reported in Falkowski (2025) further addressed the existence, uniqueness, and approximation of solutions of stochastic differential equations with two time-dependent reflecting barriers driven by optional semimartingales.

In the context of filtering theory, Gasparyan (1988) derived nonlinear filtering equations for optional semimartingales under a restricted filtration that is neither right-continuous nor complete, while Abdelghani and Melnikov (2019) derived a nonlinear filtering equation for optional supermartingales under various conditions, utilizing a version of the optional decomposition of local optional supermartingales. In the field of statistics, Abdelghani et al. (2021) investigated regression problems where the observed process is an optional semimartingale.

In mathematical finance, Kühn and Stroh (2009) characterized a tractable domain of general integrands for optional semimartingales that possesses desirable properties for modeling dynamic trading gains when price processes follow optional semimartingales. The work reported in (Abdelghani and Melnikov 2017) employed optional semimartingales with random times to model derivative contracts with default. Additionally, the optional decomposition of local optional supermartingales derived in (Abdelghani and Melnikov 2019) aids in describing the capital evolution for corresponding minimal hedging portfolios. A criterion for the existence of local optional martingale deflators in financial markets following optional semimartingales was provided in (Abdelghani and Melnikov 2020). Optional semimartingale markets have found application in energy markets, where prices exhibit spikes (Abdelghani et al. 2022). For a comprehensive review of optional semimartingale calculus and its applications, we recommend referring to (Abdelghani and Melnikov 2020).

3. Càdlàg Semimartingale Market in Unusual Probability Space

Here, we develop a model that describes a financial market with an asset price following a càdlàg semimartingale in an unusual probability space. In this model, we demonstrate that price bubbles emerge as a result of incorporating additional external information, beyond the information generated by the price process, into the market’s filtration. In real-world economies, fluctuations in optional local martingale measures naturally occur, reflecting regime changes influenced by diverse beliefs, risk aversion attitudes, institutional structures, technological advancements, tax policy changes, and fluctuations in endowments.

Let $(\Omega ,\mathcal{F},\mathbf{P})$ be a complete probability space. On this space, we define the following filtrations: $\mathbf{F}={\left({\mathcal{F}}_t\right)}_{t\ge 0}$, ${\mathbf{F}}^{+}={\left({\mathcal{F}}_t^{+}\right)}_{t\ge 0}$, and ${\mathbf{F}}^{\mathbf{P}}={\left({\mathcal{F}}_t^{\mathbf{P}}\right)}_{t\ge 0}$. $\mathbf{F}$ may not be right-continuous nor complete. ${\mathbf{F}}^{+}$ is right-continuous. ${\mathbf{F}}^{\mathbf{P}}$ satisfies the usual hypotheses. Let the market price of the risky asset be given by $S={\left(S_t\right)}_{0\le t\le T}$, where T is a stopping time in ${\mathbf{F}}^{+}$ that represents the termination time of the risky asset. Let $D={\left(D_t\right)}_{0\le t<T}$ be the cumulative dividend process of the risky asset and $r={\left(r_t\right)}_{t\ge 0}$ be the risk-free rate of return of a money market account. The money market account serves as a numeraire to make the spot interest rate zero. We assume that S, D, and r are càdlàg semimartingales that are $\mathbf{F}$-measurable and remain càdlàg semimartingales under ${\mathbf{F}}^{+}$. Furthermore, we assume that $S,D\ge 0$, $▵D>0$, and $r>0$. Let $L_T\in {\mathcal{F}}_T$ be the terminal payoff or the liquidation value of the asset at time $T\in \left[0,\infty \right]$. Therefore, the wealth process (X) associated with the market price of the risky asset is

$$X_t={\mathbf{1}}_{t<T}{S}_t+L_T{\mathbf{1}}_{t\ge T}+{\int }_0^{t\wedge T}d{D}_u.$$

(1)

The market value of wealth is the current position in stock plus the accumulated dividends and the terminal payoff. Since the asset does not exist after T, we stop the wealth at T.

Remark 1.

Since S and D are càdlàg semimartingales with respect to $\mathbf{F}$ and ${\mathbf{F}}^{+}$, the wealth (X) is a càdlàg semimartingale with respect to $\mathbf{F}$ and ${\mathbf{F}}^{+}$. In other words, S, D, and X remain càdlàg semimartingales, even after expansion of $\mathbf{F}$ to ${\mathbf{F}}^{+}$.

Remark 2.

The assumption that S, D, and r are càdlàg semimartingales and that the probability space $(\Omega ,\mathcal{F},\mathbf{P})$ is complete is needed for us to be able to use the results of the theory of no free lunch with vanishing risk.

Next, we establish the following criteria for allowable trading strategies: Firstly, all trading strategies must be self-financing to prevent doubling strategies. Secondly, trading strategies must be admissible, imposing restrictions on traders’ ability to borrow without bounds. Thirdly, our market must satisfy NFLVR to safeguards against the emergence of arbitrage opportunities. Furthermore, we impose the no-dominance assumption, which ensures that no cash flow generated by a trading strategy can result in one portfolio dominating another over any trading time intervals.

3.1. Trading Strategies

A trading strategy is a pair ($\pi :=(\xi ,\eta )$) of processes adapted to $\mathbf{F}$ representing the number of units in the risky asset (S) and money market account (r) held at time t. The corresponding portfolio value process ($V^{\pi }$) of the trading strategy $(\xi ,\eta )$ is

$$V_t^{\pi }={\eta }_t+{\xi }_t{S}_t.$$

(2)

Let us suppose $\xi$ is a semimartingale; then, with $V_0^{\pi }=0$, the value process ($V^{\pi }$) is given by

$$\begin{array}{ccc}\hfill V_t^{\pi }& =& {\int }_0^t{\xi }_{u-}d{X}_u\hfill \\ & =& {\int }_0^t{\xi }_{u-}d{S}_u+{\int }_0^{t\wedge T}{\xi }_{u}d{D}_u+{\xi }_T{L}_T{\mathbf{1}}_{t\ge T}\hfill \\ & =& ({\xi }_t{S}_t-{\int }_0^{t}S-d{\xi }_u-{[{\xi }^c,S^c]}_t)+{\int }_0^{t\wedge T}{\xi }_{u-}d{D}_u+{\xi }_T{L}_T{\mathbf{1}}_{t\ge T}\hfill \\ & =& {\eta }_t+{\xi }_t{S}_t,\hfill \end{array}$$

(3)

where

$${\eta }_t={\int }_0^{t\wedge T}{\xi }_{u-}d{D}_u+{\xi }_T{L}_T{\mathbf{1}}_{T\le t}-{\int }_0^t{S}_{u-}d{\xi }_u-{[{\xi }^c,S^c]}_t.$$

(4)

Discarding the temporary assumption that $\xi$ is a semimartingale, we can define a self-financing trading strategy $(\xi ,\eta )$ to be a pair of processes, with $\xi$ being predictable and $\eta$ being optional such that

$$V_t^{\pi }={\xi }_t{S}_t+{\eta }_t={\int }_0^t{\xi }_{u}d{X}_u=\xi \circ X_t,$$

where $\xi \in \mathrm{L}\left(X\right)$. As noted, a self-financing trading strategy starts with zero dollars ($V_0^{\pi }=0$), and all proceeds from purchases/sales of the risky asset are financed/invested in the money market account because Equation (4) shows that $\eta$ is uniquely determined by $\xi$ if a trading strategy is self-financing.

To avoid doubling strategies by indefinite borrowing of S, we need to restrict the class of self-financing trading strategies further. The notion of admissibility corresponds to a lower bound on the wealth process—an inability to borrow without bound.

Definition 2.

Admissibility. Let $V^{\pi }$ be a wealth process given by Equation (3). We say that the trading strategy (π) is a-admissible if it is self-financing and $V_t^{\pi }\ge -a$ for all $t\ge 0$, almost surely. We say a trading strategy is admissible if it is self-financing and there exists an $a\in {\mathbb{R}}_{+}$ such that $V_t^{\pi }\ge -a$ for all t, almost surely. We denote the collection of admissible strategies as $\mathsf{A}$.

Now, we can introduce the notion of an arbitrage-free market. Let

$$\mathrm{K}=\{X_{\infty }^{\pi }={(\xi \circ X)}_{\infty }:\xi \in \mathsf{A}\}$$

(5)

and

$$\mathrm{C}=(\mathrm{K}-{\mathrm{L}}_0^{+})\cap {\mathrm{L}}_{\infty }$$

(6)

where ${\mathrm{L}}_{\infty }$ is the set of a.s. bounded random variables and ${\mathrm{L}}_0^{+}$ is the set of non-negative, finite-valued random variables.

Definition 3.

We say that a market satisfies NFLVR if

$$\overline{\mathrm{C}}\cap {\mathrm{L}}_{\infty }^{+}=\left\{0\right\}$$

(7)

where $\overline{\mathrm{C}}$ denotes the closure of $\mathrm{C}$ in the sup-norm topology on ${\mathrm{L}}_{\infty }$.

The NFLVR excludes all self-financing trading strategies with zero initial investment that guarantee non-negative cash flows or offer strictly positive cash flows with positive probability. The NFLVR condition also excludes sequences of trading strategies that approach arbitrage opportunities. Therefore, we assume the following:

Assumption 1.

The market satisfies NFLVR.

A key concept to characterize a market that satisfies the NFLVR is the equivalent local martingale measure:

Definition 4.

Equivalent Local Martingale Measure. Let $\mathbf{Q}$ be a probability measure equivalent to $\mathbf{P}$ such that the wealth process (X) is a $\mathbf{Q}$-local martingale. We call $\mathbf{Q}$ an Equivalent Local Martingale Measure (ELMM), and we denote the set of ELMMs as ${\mathfrak{Q}}_{loc}\left(X\right)$.

The first fundamental theorem of asset pricing (Delbaen and Schachermayer 1994, 1995, 1998, 1999) posits that a market is free of arbitrage in the NFLVR sense if and only if there is an equivalent probability measure ($\mathbf{Q}$) under which the asset price process (S) is a $\sigma$ martingale. Any $\sigma$ martingale that is bounded from below is also a local martingale. Since S is non-negative, it is also a local martingale with respect to $\mathbf{Q}$. This leads us to the following theorem:

Theorem 1.

First Fundamental Theorem. A market satisfies NFLVR if and only if there exists an ELMM.

Next, we define what we mean by no dominance. For each admissible trading strategy ($\pi \in \mathsf{A}$), its wealth process ($V^{\pi }$) is given by $V_t^{\pi }=\xi \circ X_t$, where $V_t^{\pi }$ is a local martingale under each $\mathbf{Q}\in {\mathfrak{Q}}_{loc}\left(X\right)$.

Definition 5.

Set of Super-Replicated Cash Flows. Consider a fixed future time denoted by v and let ${\Phi }_0$ be the collection of all payoffs ($\varphi =(D,{\Lambda }^v)$) associated with an asset derived from an admissible trading strategy. Here, $D={\left(D_t\right)}_{0\le t\le v}$ is the asset’s cumulative dividend process. It is a non-negative, non-decreasing càdlàg semimartingale adapted to $\mathbf{F}$. ${\Lambda }^v\in {\mathcal{F}}_v$ is a non-negative random variable representing the asset’s terminal payoff at time v. Now, let us define the set of super-replicating trading strategies (Φ):

$$\Phi :=\{\varphi \in {\Phi }_0:\exists \xi \in \mathsf{A},a\in {\mathbb{R}}_{+}\mathrm{such}\mathrm{that}{D}_v+{\Lambda }^v\le a+V_v^{\pi }\}.$$

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Set $\Phi$ represents the cash flows of assets that can be super-replicated through trading in the risky asset and the money market account. As we illustrate later, this particular set of cash flows is relevant for our assumption of no dominance. Our initial demonstration involves establishing that this subset of asset cash flows comprises a convex cone.

Lemma 1.

Φ is closed under addition and multiplication by positive scalars, i.e., it is a convex cone.

Proof. 

See Jarrow et al. (2010). □

If $\varphi \in \Phi$, then for each $\mathbf{Q}\in {\mathfrak{Q}}_{loc}\left(X\right)$,

$${\mathbf{E}}_{\mathbf{Q}}[D_v+{\Lambda }^v]\le a+{\mathbf{E}}_{\mathbf{Q}}\left[V_v^{\pi }\right]\le a.$$

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The first inequality follows because $V^{\pi }$ is a wealth process of admissible trading strategies. The second inequality follows because $V_v^{\pi }$ is a non-negative (because both $D_v$ and ${\Lambda }_v$ are non-negative) $\mathbf{Q}$-local martingale bounded below and, hence, a $\mathbf{Q}$ supermartingale such that ${\Phi }_0=0$. Therefore, each asset ($\varphi \in \Phi$) is integrable under any ELMM. This is the reason for restricting our attention to the set of cash flows ($\Phi \subset {\Phi }_0$). This set ($\Phi$) is large enough to contain many of the assets of interest in derivatives pricing.

We do not expect to see in a well-functioning market or any dominated assets or portfolios. To formalize this idea, let us denote the market price of $\varphi$ at time t as ${\Gamma }_t\left(\varphi \right)$ and fix $\varphi =(D,{\Lambda }^v)\in \Phi$. Consider a pair of stopping times ($\sigma <\mu \le v$) and define the net gain ($G_{\sigma ,\mu }\left(\varphi \right)$) by purchasing $\sigma$ and selling at $\mu \le v$ as follows:

$$G_{\sigma ,\mu }\left(\varphi \right)={\Gamma }_{\mu }\left(\varphi \right)+{\int }_{\sigma }^{\mu }d{D}_s+{\Lambda }^v{\mathbf{1}}_{v=\mu }-{\Gamma }_{\sigma }\left(\varphi \right).$$

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Definition 6.

Dominance. Let ${\varphi }^1$ and ${\varphi }^2\in \Phi$ be two assets. If a pair of stopping times ($\sigma <\mu \le T$) exists such that

$$G_{\sigma ,\mu }\left({\varphi }^2\right)\ge G_{\sigma ,\mu }\left({\varphi }^1\right),\mathrm{a}.\mathrm{s}.$$

and $\mathbf{E}\left[{\mathbf{1}}_{G_{\sigma ,\mu }^2>G_{\sigma ,\mu }^1}|{\mathcal{F}}_{\sigma }\right]>0$ a.s., then we say that asset 2 dominates asset 1 at time σ.

Therefore, we impose the following condition:

Assumption 2.

(No Dominance): Let the market price be represented by a function (${\Gamma }_t:\Phi \to {\mathbb{R}}_{+}$) such that there are no dominated assets in the market.

The addition of a no-dominance assumption prevents the occurrence of price bubbles in a complete market. Therefore, studying bubbles necessitates looking at an incomplete market, which, according to the second fundamental theorem of asset pricing, is characterized by numerous local martingale measures in the absence of arbitrage. The choice of one of these measures defines the fundamental price and can lead to a bubble. Notably, the theory of bubbles in incomplete markets suggests that bubbles can burst but cannot re-emerge, contradicting economic observations. Accommodating the possibility of bubble rebirth requires a significant adaptation of the martingale pricing framework to allow the market to exhibit different local martingale measures over time. These shifts correspond to changes in underlying economic fundamentals and can create new bubbles. Implementing this change, which contradicts the NFLVR theory’s assumption of a consistent local martingale measure, demands a noteworthy extension of this theory, as carried out in Jarrow et al. (2010) under usual conditions with the method of filtration expansion. In the subsequent section, we outline a comparable extension approach, leveraging the calculus of optional processes within nonstandard probability spaces, where filtration naturally progresses from $\mathbf{F}$ to ${\mathbf{F}}^{+}$.

3.2. Regime-Change Processes

We define regime-shift processes as follows. Let ${\left({\sigma }_i\right)}_{i\ge 0}$ denote an increasing sequence of random times with ${\sigma }_0=0$ such that ${\mathbf{1}}_{{\sigma }_i\le t}$ are ${\mathcal{F}}_{t+}$-measurable for any i and t. In other words, ${\sigma }_i$ values are stopping times in the wide sense whenever ${\mathcal{F}}_{{\sigma }_i+}\supset {\mathcal{F}}_{{\sigma }_i}$; otherwise, they are regular stopping times. Let ${\left(Y^i\right)}_{i\ge 0}$ be a sequence of ${\mathcal{F}}_{{\sigma }_i+}$-measurable random variables that characterize the state of the economy at those times. Furthermore, we define the two stochastic processes (${\left(N_t\right)}_{t\ge 0}$ and ${\left(Y_t\right)}_{t\ge 0}$) as

$$N_t=\sum _{i\ge 0}{\mathbf{1}}_{t\ge {\sigma }_i}\mathrm{and}{Y}_t=\sum _{i\ge 0}{Y}^i{\mathbf{1}}_{{\sigma }_i\le t<{\sigma }_{i+1}}.$$

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$N_t$ counts the number of regime shifts up to and including time t, whereas $Y_t$ identifies the characteristics of the regime at time t. Obviously, N and Y are ${\mathbf{F}}^{+}$-measurable. Let $\mathcal{H}$ be a natural filtration generated by N and Y and the filtration be defined as ${\mathcal{G}}_t={\mathcal{F}}_t\vee {\mathcal{H}}_t\subseteq {\mathcal{F}}_{t+}$. According to the definition of $\mathcal{G}$, ${\left({\sigma }_i\right)}_{i\ge 0}$ is an increasing sequence of $\mathcal{G}$-stopping times. We just take ${\mathcal{G}}_t={\mathcal{F}}_{t+}$ so that ${\mathcal{F}}_t$ is extended to ${\mathcal{F}}_{t+}$ only by N and Y. To avoid confusion that may arise from “$t+$” in the ${\mathcal{F}}_{t+}$ symbol, we set ${\mathcal{F}}_t^{+}:={\mathcal{F}}_{t+}$.

Remark 3.

We need not suppose that Y and N are independent of the filtration ($\mathbf{F}$) to which the price process (S) and wealth process (X) are adapted. Allowing for dependence would enable the birth of bubbles to be influenced by intrinsic uncertainty (see Froot and Obstfeld (1991)). However, if Y and N are independent of $\mathbf{F}$, an extrinsic source of randomness is introduced into our economic framework.

3.3. Fundamental Price

In classical mathematical finance, the fundamental price is equivalent to the market price. But in finite-horizon markets, the difference between the market price and fundamental price is nil. This is because the market price is a $\mathbf{Q}$ martingale for any EMM’s $\mathbf{Q}$ and is equal to the arbitrage-free price, which is equal to the conditional expectation of the asset’s payoffs under $\mathbf{Q}$. The conditional expectation of the stock’s payoffs is the present value of the asset’s cash flow, which is its fundamental value. In these cases, there are no price bubbles.

In contrast, the local martingale approach to mathematical finance allows for the existence of price bubbles (see, for example, Cox and Hobson (2005); Delbaen and Schachermayer (1994, 1995, 1998, 1999); Loewenstein and Willard (2000a, 2000b)). In a market adhering to the no free lunch with vanishing risk (NFLVR) condition, the arbitrage-free price and the market price for a primary asset are indistinguishable, yet they may differ from the conditional expectation of the asset’s payoffs, which represents the fundamental price. Indeed, for any $\mathbf{Q}\in {\mathfrak{Q}}_{loc}\left(X\right)$, if the asset’s price is a strict local martingale, then a bubble exists.

We take the fundamental price as the asset’s discounted expected cash flows, given a local martingale measure ($\mathbf{Q}$), as is in (Jarrow et al. 2010). The local martingale measure ($\mathbf{Q}$) selected from ${\mathfrak{Q}}_{loc}\left(X\right)$ for valuation is a measure consistent with market prices of traded derivatives. Schweizer and Wissel (2008) and Jacod and Protter (2010) showed that if enough derivative securities of a certain type are traded, then $\mathbf{Q}$ can be determined. Similar to Jarrow et al. (2010), we assume that the selection of the measure can depend on the current economic regime, and as the regime shifts, so does the local martingale measure selected by the market. This selection process determines the fundamental value and the birth of price bubbles.

First, let us consider the ELMMs of the wealth process (X) in ${\mathcal{F}}_{\infty }$. With respect to this restricted set, the Radon–Nikodym derivative is $Z_{\infty }={\displaystyle \frac{d\mathbf{Q}}{d\mathbf{P}}}{|}_{{\mathcal{F}}_{\infty }}$. Its density process is defined as $Z_t=\mathbf{E}[Z_{\infty }|{\mathcal{F}}_t]$. Z is ${\mathcal{M}}_{loc}(X,\mathbf{F})$, an $\mathbf{F}$-adapted process.

Let the local martingale measure in our extended economy depend on the state of the economy at time t, as represented by the filtration (${\left({\mathcal{F}}_t\right)}_{t\ge 0}$), the number of regime shifts be $N_t$, and the state variable be $Y_t$. Suppose at t, $N_t=i$; let ${\mathbf{Q}}^i\in {\mathfrak{Q}}_{loc}\left(X\right)$ be the ELMM selected by the market at time t, given $(N_t,Y_t)$. Given ${\mathbf{Q}}^i$, the fundamental price of an asset or a portfolio is the asset’s expected discounted cash flows at time t:

Definition 7.

Fundamental Price. Let $\varphi \in \Phi$ be an asset with a maturity of v and payoff of $(\alpha ,{\Lambda }^v)$. The fundamental price (${\Gamma }_t\left(\varphi \right)$) of asset ϕ is defined by

$${\Gamma }_t\left(\varphi \right)=\sum _{i=0}^{\infty }{\mathbf{E}}_{{\mathbf{Q}}^i}\left[{\int }_t^{v}d{D}_u+{\Lambda }^v{\mathbf{1}}_{v<\infty }|{\mathcal{F}}_t\right]{\mathbf{1}}_{\{t<v\}\cap \{t\in [{\sigma }_i,{\sigma }_{i+1})\}}$$

(12)

$\forall t\in [0,\infty )$, where ${\Gamma }_{\infty }\left(\varphi \right)=0$.

In particular, the fundamental price of the risky asset ($S_t$) is the discounted future cash flow on ${\mathcal{F}}_t$ under ${\mathbf{Q}}^i$ for all i, given by

$${\widehat{S}}_t=\sum _{i=0}^{\infty }{\mathbf{E}}_{{\mathbf{Q}}^i}\left[{\int }_t^{\tau }d{D}_u+L_{\tau }{\mathbf{1}}_{\tau <\infty }|{\mathcal{F}}_t\right]{\mathbf{1}}_{\{t<\tau \}\cap \{t\in [{\sigma }_i,{\sigma }_{i+1})\}}.$$

(13)

To understand this definition, let us focus on the risky asset’s fundamental price. At any time ($t<\tau$), given that we are in the ith regime ($\{{\sigma }_i\le t<{\sigma }_{i+1}\}$), the right side of expression (13) simplifies to

$${\widehat{S}}_t={\mathbf{E}}_{{\mathbf{Q}}^i}\left[{\int }_t^{\tau }d{D}_u+L_{\tau }{\mathbf{1}}_{\{\tau <\infty \}}|{\mathcal{F}}_t\right].$$

Note that the payoff of the asset at infinity ($L_{\tau }{\mathbf{1}}_{\tau =\infty }$) does not contribute to the fundamental price, since agents cannot consume it. Furthermore, when the time is $\tau =\infty$, the fundamental price is ${\widehat{S}}_{\tau }=0$. We emphasize that under NFLVR and no-dominance, the market price ($S_t$) equals the arbitrage-free price but needs not equal the fundamental price (${\widehat{S}}_t$).

Next, we present an alternative formulation of the fundamental price (Equation (12)) using a conditional expectation with respect to an equivalent probability measure ($\tilde{\mathbf{Q}}$) instead of the sum of conditional expectation with respect to measures ${\left\{{\mathbf{Q}}^i\right\}}_{i\ge 0}$.

Theorem 2.

There exists an equivalent probability measure (${\mathbf{Q}}_t$) such that

$${\Gamma }_t\left(\varphi \right)={\mathbf{E}}_{{\mathbf{Q}}_t}\left[{\int }_t^{v}d{D}_u+{\Lambda }^v{\mathbf{1}}_{v<\infty }|{\mathcal{F}}_t\right]{\mathbf{1}}_{t<v}.$$

(14)

Proof. 

Let $Z^i\in {\mathcal{F}}_{\infty }$ be a Radon–Nikodym derivative of ${\mathbf{Q}}^i$ with respect to $\mathbf{P}$, $Z^i={\displaystyle \frac{d{\mathbf{Q}}^i}{d\mathbf{P}}}$, and $Z_t^i=\mathbf{E}[Z^i|{\mathcal{F}}_t]$. We define

$$Z_t=\sum _{i=0}^{\infty }{Z}^i{\mathbf{1}}_{t\in [{\sigma }_i,{\sigma }_{i+1})}.$$

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Then, $Z_t>0$, almost surely, and

$$\begin{array}{ccc}\hfill \mathbf{E}\left[Z_t^{\infty }\right]& =& \mathbf{E}\left[\sum _{i=0}^{\infty }{Z}^i{\mathbf{1}}_{t\in [{\sigma }_i,{\sigma }_{i+1})}\right]=\sum _{i=0}^{\infty }\mathbf{E}\left[Z^i{\mathbf{1}}_{t\in [{\sigma }_i,{\sigma }_{i+1})}\right]\hfill \\ & =& \sum _{i=0}^{\infty }\mathbf{E}\left[Z^i\right]\mathbf{E}\left[{\mathbf{1}}_{t\in [{\sigma }_i,{\sigma }_{i+1})}\right],\mathrm{since}\mathbf{E}\left[Z^i\right]=1\hfill \\ & =& \sum _{i=0}^{\infty }\mathbf{P}({\sigma }_i\le t<{\sigma }_{i+1})=1,\hfill \end{array}$$

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at any moment of time (t). Therefore, we can define an equivalent measure (${\mathbf{Q}}_t^{\infty }$) on ${\mathcal{F}}_{\infty }$ as $d{\mathbf{Q}}_t^{\infty }=Z_t^{\infty }d\mathbf{P}$. The Radon–Nikodym density ($Z_t$) on ${\mathcal{F}}_t^{+}$ is

$$\begin{array}{ccc}\hfill Z_t& =& {\left.{\displaystyle \frac{d{\mathbf{Q}}_t^{\infty }}{d\mathbf{P}}}\right|}_{{\mathcal{F}}_t^{+}}=\mathbf{E}[Z_t^{\infty }|{\mathcal{F}}_t^{+}]=\sum _{i=0}^{\infty }\mathbf{E}\left[Z^i{\mathbf{1}}_{t\in [{\sigma }_i,{\sigma }_{i+1})}|{\mathcal{F}}_t^{+}\right]\hfill \\ & =& \sum _{i=0}^{\infty }\mathbf{E}[Z^i|{\mathcal{F}}_t^{+}]{\mathbf{1}}_{t\in [{\sigma }_i,{\sigma }_{i+1})}.\hfill \end{array}$$

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Then,

$$\begin{array}{ccc}\hfill {\Gamma }_t\left(\varphi \right)& =& \sum _{i=0}^{\infty }{\mathbf{E}}_{{\mathbf{Q}}^i}\left[{\int }_t^{v}d{D}_u+{\Lambda }^v{\mathbf{1}}_{v<\infty }|{\mathcal{F}}_t^{+}\right]{\mathbf{1}}_{\{t<v\}\cap \{t\in [{\sigma }_i,{\sigma }_{i+1})\}}\hfill \\ & =& \sum _{i=0}^{\infty }\mathbf{E}\left[{\displaystyle \frac{Z^i}{Z_t^i}}{\int }_t^{v}d{D}_u+{\Lambda }^v{\mathbf{1}}_{v<\infty }|{\mathcal{F}}_t^{+}\right]{\mathbf{1}}_{\{t<v\}\cap \{t\in [{\sigma }_i,{\sigma }_{i+1})\}}\hfill \\ & =& \mathbf{E}\left[\sum _{i=0}^{\infty }{\displaystyle \frac{Z^i}{Z_t^i}}{\mathbf{1}}_{t\in [{\sigma }_j,{\sigma }_{i+1})}\left({\int }_t^{v}d{D}_u+{\Lambda }^v{\mathbf{1}}_{v<\infty }\right)|{\mathcal{F}}_t^{+}\right]{\mathbf{1}}_{t<v},\hfill \end{array}$$

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and we observe that

$${\displaystyle \frac{Z^i}{Z_t^i}}{\mathbf{1}}_{t\in [{\sigma }_i,{\sigma }_{i+1})}={\displaystyle \frac{Z^i{\mathbf{1}}_{t\in [{\sigma }_i,{\sigma }_{i+1})}}{{\sum }_{i=0}^{\infty }{Z}_t^i{\mathbf{1}}_{t\in [{\sigma }_i,{\sigma }_{i+1})}}},$$

and continue to

$$\begin{array}{ccc}& =& \mathbf{E}\left[\left({\displaystyle \frac{{\sum }_{i=0}^{\infty }{Z}^i{\mathbf{1}}_{t\in [{\sigma }_i,{\sigma }_{i+1})}}{{\sum }_{i=0}^{\infty }{Z}_t^i{\mathbf{1}}_{t\in [{\sigma }_i,{\sigma }_{i+1})}}}\right)\left({\int }_t^{v}d{D}_u+{\Lambda }^v{\mathbf{1}}_{v<\infty }\right)|{\mathcal{F}}_t^{+}\right]{\mathbf{1}}_{t<v}\hfill \\ & =& \mathbf{E}\left[{\displaystyle \frac{Z_t^{\infty }}{Z_t}}\left({\int }_t^{v}d{D}_u+{\Lambda }^v{\mathbf{1}}_{v<\infty }\right)|{\mathcal{F}}_t^{+}\right]{\mathbf{1}}_{t<v}\hfill \\ & =& {\mathbf{E}}_{{\mathbf{Q}}_t}\left[{\int }_t^{v}d{D}_u+{\Lambda }^v{\mathbf{1}}_{v<\infty }|{\mathcal{F}}_t^{+}\right]{\mathbf{1}}_{t<v}\hfill \\ & =& {\mathbf{E}}_{{\mathbf{Q}}_t}\left[{\int }_t^{v}d{D}_u+{\Lambda }^v{\mathbf{1}}_{v<\infty }|{\mathcal{F}}_t\right]{\mathbf{1}}_{t<v}.\hfill \end{array}$$

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□

${\mathbf{Q}}_t$ is known as the valuation measure at t, and collection ${\left\{{\mathbf{Q}}_t\right\}}_{t\ge 0}$ is known as the valuation system. If $N_t=1$ for all t, then the valuation system is static with ${\mathbf{Q}}_t={\mathbf{Q}}_0$, $t\ge 0$, and there are no regime shifts. If the market is not static, then it is dynamic, with the valuation system expressed as ${\left\{{\mathbf{Q}}_t\right\}}_{t\ge 0}$, where, in the ith regime $({\sigma }_j\le t<{\sigma }_{i+1})$, the valuation measure coincides with ${\mathbf{Q}}^i\in {\mathcal{M}}_{loc}\left(X\right)$.

Given the definition of the asset’s fundamental price (13), the fundamental wealth process is expressed as follows:

$${\widehat{X}}_t={\widehat{S}}_t{\mathbf{1}}_{t<\tau }+{\int }_0^{\tau \wedge t}d{D}_u+L_{\tau }{\mathbf{1}}_{t\ge \tau }.$$

(20)

Thus,

$${\widehat{X}}_t=\sum _{i=0}^{\infty }{\mathbf{E}}_{{\mathbf{Q}}^i}\left[{\int }_0^{\tau }d{D}_u+L_{\tau }{\mathbf{1}}_{\tau <\infty }|{\mathcal{F}}_t\right]{\mathbf{1}}_{t\in [{\sigma }_i,{\sigma }_{i+1})}$$

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$\forall t\in [0,\infty )$, where

$$X_{\infty }={\int }_0^{\tau }d{D}_u+L_{\tau }{\mathbf{1}}_{\tau <\infty }.$$

Therefore, we can rewrite ${\widehat{X}}_t$ as

$${\widehat{X}}_t=\sum _{i=0}^{\infty }{\mathbf{E}}_{{\mathbf{Q}}^i}\left[X_{\infty }|{\mathcal{F}}_t\right]{\mathbf{1}}_{t\in [{\sigma }_i,{\sigma }_{i+1})}\forall t\in [0,\infty ).$$

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Then, given the definition of the asset’s fundamental price (13), an asset price bubble (B) is defined as the difference between the market price of the asset (S) and its fundamental value ($\widehat{S}$):

$$B_t=S_t-{\widehat{S}}_t.$$

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Remark 4.

So far, we have presented a model of an economy that exhibits regime changes. In this market model, the random times responsible for creating regime shifts in the economy are stopping times that are in the filtration (${\mathbf{F}}^{+}$). By doing so, we have avoided the requirement that every $(\mathbf{Q},\mathbf{F})$ local martingale should also be a $(\mathbf{Q},\mathbf{G})$ local martingale, where $\mathbf{G}$ represents the extended filtration that incorporates random regime-shift times. However, we have made an important assumption that the market price process remains a càdlàg optional semimartingale under both $\mathbf{F}$ and ${\mathbf{F}}^{+}$. This assumption allows us to utilize the theory of NFLVR. Other results concerning the decomposition of bubbles and derivatives with bubbles, among others, can be easily extended to optional extended financial markets with some modifications.

4. Làdlàg Optional Semimartingale Market with Bubbles

In this section, we introduce alternative financial markets where assets are characterized by làdlàg optional semimartingales in unusual probability spaces. We provide a version of NFLVR specifically tailored for these markets. Additionally, we develop a theory of local martingale deflators for làdlàg optional semimartingales and study the formation of price bubbles in these markets. Finally, we present illustrative examples.

4.1. Absence of Arbitrage

Research on no-arbitrage arguments has culminated in the Fundamental Theorem of Asset Pricing, which states that under the usual conditions, for a real-valued semimartingale (X), there exists a probability measure ($\mathbf{Q}$) equivalent to $\mathbf{P}$ under which X is a $\sigma$ martingale if and only if X does not permit a free lunch with vanishing risk (NFLVR). Given $K=${${(\varphi ·X)}_{\infty }:\varphi$ admissible and ${(\varphi ·X)}_{\infty }={lim}_{t\to \infty }{(\varphi ·X)}_t$ exists a.s.} and $\mathrm{C}=\{g\in {\mathrm{L}}^{\infty }\left(\mathbf{P}\right):g\le f\forall f\in K\}$, X is said to satisfy NFLVR if $\overline{\mathrm{C}}\cap {\mathrm{L}}_{+}^{\infty }\left(\mathbf{P}\right)=\left\{0\right\}$, where $\overline{\mathrm{C}}$ is the closure of $\mathrm{C}$ in the norm topology of ${\mathrm{L}}_{+}^{\infty }\left(\mathbf{P}\right)$ Delbaen and Schachermayer (1994).

While, the comprehensive development of NFLVR or NA₁ (no-arbitrage of the first kind), as well as the equivalence relations—ELMM (equivalent local martingale measure) and ELMD (equivalent local martingale deflator)—for optional semimartingales in unusual probability spaces, are important and achievable, such topics exceed the purview of this paper. Nevertheless, we present a compelling case that demonstrates the plausibility of financial markets in unusual probability spaces and establish that they are free of arbitrage under certain conditions.

Consider a market on the unusual probability space expressed as $\left(\Omega ,\mathcal{F},\mathbf{F},\mathbf{P}\right)$. Let $Y\in \mathcal{O}\left(\mathbf{F}\right)$ be a real-valued optional semimartingale and $Y^{+}={\left(Y_{t+}\right)}_{t\ge 0}$ be a $\mathcal{O}\left({\mathbf{F}}^{\mathbf{P}}\right)$ càdlàg semimartingale. Suppose $Y^{+}$ satisfies the NFLVR condition with the admissible portfolio expressed as $\varphi \in \mathcal{P}\left({\mathbf{F}}^{\mathbf{P}}\right)$ and ${(\varphi ·Y)}_{\infty }<\infty$ a.s.; then there, is ELMM $\mathbf{Q}\sim \mathbf{P}$ such that $\varphi ·Y^{+}$ is a local martingale in $\mathcal{O}\left({\mathbf{F}}^{\mathbf{Q}}\right)$. Given that $\varphi ·Y_s^{+}$ is a local martingale under $\left({\mathbf{F}}^{\mathbf{Q}},\mathbf{Q}\right)$, we aim to recover Y and identify the portfolio ($\varphi$) as a process in $(\mathbf{F},\mathbf{Q})$ using the following optional projection:

$$“\varphi \circ Y_s={\mathbf{E}}_{\mathbf{Q}}[\varphi ·Y_t^{+}|{\mathcal{F}}_s]\mathrm{a}.\mathrm{s}.\mathbf{Q}”,$$

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where “·” is the stochastic integral with respect to the càdlàg semimartingale ($Y_t^{+}$) and predictable integrand and “∘” is the stochastic integral with respect to the làdlàg optional semimartingales (Y) with the optional integrand, “$\varphi$”.

Theorem 3.

Let $\varphi ·Y_t^{+}$ be a $\mathbf{Q}$-local martingale; then, $\varphi \circ Y_s={\mathbf{E}}_{\mathbf{Q}}[\varphi ·Y_t^{+}|{\mathcal{F}}_s]$ a.s. $\mathbf{Q}$.

Proof. 

Consider the sequence ($R_k\uparrow \infty$) of stopping times in ${\mathbf{F}}^{\mathbf{Q}}$, where $\varphi ·Y_{t\wedge R_k}^{+}$ is a martingale for all k and ${\mathbf{E}}_{\mathbf{Q}}\left[\varphi ·Y_{t\wedge R_k}^{+}\right]<\infty$. Since ${\mathcal{F}}_s^{\mathbf{Q}}\supseteq {\mathcal{F}}_s$ for all s, then

$${\mathbf{E}}_{\mathbf{Q}}\left[\varphi ·Y_{t\wedge R_k}^{+}-\varphi ·Y_{s\wedge R_k}^{+}|{\mathcal{F}}_s\right]={\mathbf{E}}_{\mathbf{Q}}\left[{\mathbf{E}}_{\mathbf{Q}}\left[\varphi ·Y_{t\wedge R_k}^{+}-\varphi ·Y_{s\wedge R_k}^{+}|{\mathcal{F}}_s^{\mathbf{Q}}\right]|{\mathcal{F}}_s\right]=0.$$

Observe that

$$d{Y}_t^{+}=d{Y}_t+▵Y_t^{+}=d{Y}_t+{▵}^{+}{Y}_t,$$

and if $Y_u$ is evolving in the interval of $[s,t)$, then $Y_u^{+}$ is evolving in the interval of $(s,t]$. Consequently,

$$\begin{array}{ccc}\hfill {\mathbf{E}}_{\mathbf{Q}}\left[\varphi ·Y_{t\wedge R_k}^{+}|{\mathcal{F}}_s\right]& =& {\mathbf{E}}_{\mathbf{Q}}\left[\varphi ·Y_{s\wedge R_k}^{+}|{\mathcal{F}}_s\right]={\mathbf{E}}_{\mathbf{Q}}\left[{\int }_{(0,s\wedge R_k]}{\varphi }_{u-}d{Y}_u^{+}|{\mathcal{F}}_s\right]\hfill \\ & =& {\mathbf{E}}_{\mathbf{Q}}\left[{\int }_{(0,s\wedge R_k]}{\varphi }_{u-}\left(d{Y}_u+▵Y_u^{+}\right)|{\mathcal{F}}_s\right]\hfill \\ & =& {\mathbf{E}}_{\mathbf{Q}}\left[{\int }_{(0,s\wedge R_k]}{\varphi }_{u-}d{Y}_u|{\mathcal{F}}_s\right]+{\mathbf{E}}_{\mathbf{Q}}\left[{\int }_{[0,s\wedge R_k)}{\varphi }_v{▵}^{+}{Y}_v|{\mathcal{F}}_s\right]\hfill \\ & =& {\mathbf{E}}_{\mathbf{Q}}\left[{\int }_{(0,s\wedge R_k]}{\varphi }_{u-}d{Y}_u+{\int }_{[0,s\wedge R_k)}{\varphi }_{v}d{Y}_{v+}|{\mathcal{F}}_s\right]\hfill \\ & =& {\mathbf{E}}_{\mathbf{Q}}\left[{\int }_{[0,s\wedge R_k]}{\varphi }_{u}d{Y}_u|{\mathcal{F}}_s\right]\hfill \\ & =& \varphi \circ Y_{s\wedge R_k}.\hfill \end{array}$$

□

After establishing that $\varphi \circ Y$ is the optional projection of $\varphi ·Y^{+}$ on $\mathbf{F}$, we proceed to demonstrate that $\varphi \circ Y$ is a local optional martingale with respect to $(\mathbf{F},\mathbf{Q})$.

Lemma 2.

$\varphi \circ Y_t$ is a local optional martingale under $\mathbf{Q}$.

Proof. 

Utilizing the aforementioned theorem, for any $u\le s\le t$, we have the following:

$${\mathbf{E}}_{\mathbf{Q}}\left[\varphi \circ Y_s|{\mathcal{F}}_u\right]={\mathbf{E}}_{\mathbf{Q}}\left[{\mathbf{E}}_{\mathbf{Q}}\left[\varphi ·Y_t^{+}|{\mathcal{F}}_s\right]|{\mathcal{F}}_u\right]={\mathbf{E}}_{\mathbf{Q}}\left[\varphi ·Y_t^{+}|{\mathcal{F}}_u\right]=\varphi \circ Y_u.$$

□

By applying the same approach that led us to establish NFLVR under unusual conditions, we observe that if NA₁ holds for $Y^{+}$ on $\left(\Omega ,\mathcal{F},{\mathbf{F}}^{\mathbf{Q}},\mathbf{Q}\right)$, then it also holds for Y on $\left(\Omega ,\mathcal{F},\mathbf{F},\mathbf{Q}\right)$.

Consequently, we can confidently assert that if NFLVR or NA₁ is satisfied for càdlàg semimartingales on $\left(\Omega ,\mathcal{F},{\mathbf{F}}^{\mathbf{P}},\mathbf{P}\right)$, it must also hold for their optional version through optional projection on $\left(\Omega ,\mathcal{F},\mathbf{F},\mathbf{P}\right)$. Thus, we can conclude that optional markets are free of arbitrage opportunities under the appropriate conditions on $\varphi$ and $Y^{+}$.

4.2. Market and Portfolios

Consider the market price of a risky asset ($S={\left(S_t\right)}_{t\ge 0}$) a non-negative làdlàg optional semimartingale adapted to $\mathbf{F}$. Once more, let $D={\left(D_t\right)}_{t\ge 0}$ be a càdlàg semimartingale process adapted to $\mathbf{F}$ representing the cumulative dividend of the asset. Let $L_T\in {\mathcal{F}}_T$ be a terminal payoff of the asset at some future time ($T\in (0,\infty ]$). We assume that both $S\ge 0$, $S_T=S_{T+}=0$, $D\ge 0$ and $▵D>0$, $D_0=D_T=D_{T+}=0$. As usual, the money market account serves as a numeraire that we assume to be a strictly positive process, and we suppose that its value has already been incorporated in the different components of our market (S, D, and L). The wealth process associated with S, D, and L is

$$X_t=\left(S_t+D_t\right)\left(1-{\mathsf{T}}_t\right)+L_T{\mathsf{T}}_t.$$

where ${\mathsf{T}}_t={\mathbf{1}}_{t\ge T}$. Consider the integral form of X,

$$\begin{array}{ccc}\hfill X_t& =& X_0+{\int }_0^t\left(1-{\mathsf{T}}_u\right)d\left(S_u+D_u\right)-{\int }_0^t\left(S_u+D_u\right)d{\mathsf{T}}_u-{[S+D,\mathsf{T}]}_t+L_T{\mathsf{T}}_t\hfill \\ & =& S_0+\left(1-\mathsf{T}\right)\circ {\left(S+D\right)}_t-\left(S+D\right)\circ {\mathsf{T}}_t-{[S+D,\mathsf{T}]}_t+L_T{\mathsf{T}}_t\hfill \\ & =& S_0+\left(1-\mathsf{T}\right)\circ {\left(S+D\right)}_t+L_T{\mathsf{T}}_t.\hfill \end{array}$$

The integral is $\left(S+D\right)\circ {\mathsf{T}}_t=0$, since $S_T+D_T=S_{T+}+D_{T+}=0$ and ${\mathsf{T}}_T={\mathsf{T}}_{T+}=1$, and zero otherwise. Moreover,

$$[S+D,\mathsf{T}]=▵(S+D)▵\mathsf{T}+{▵}^{+}(S+D){▵}^{+}\mathsf{T}=0.$$

Remark 5.

S can be decomposed to $S^{\mathtt{r}}+S^{\mathtt{g}}$, where $S^{\mathtt{r}}$ is càdlàg semimartingale and $S^{\mathtt{g}}$ is a left-continuous optional semimartingale. Theorem (1.14) in Galtchouk (1980) tells us that there are three sequences of stopping times that absorb all the jumps of S: a predictable sequence of stopping times (${\left({\tau }_i^{\mathtt{p}}\right)}_{i\ge 0}$) measurable in ${\mathbf{F}}^{-}$, an inaccessible sequence of stopping times (${\left({\tau }_i^{\mathtt{d}}\right)}_{i\ge 0}$) that are $\mathbf{F}$-measurable, and wide-sense stopping times (${\left({\tau }_i^{\mathtt{g}}\right)}_{i\ge 0}$) that are ${\mathbf{F}}^{+}$-measurable. As we develop this theory, we see that stopping times in the wide sense (${\left({\tau }_i^{\mathtt{g}}\right)}_{i\ge 0}$), which absorb the left-optional jumps of S, are the ones that can lead to regime shifts in the economy.

Let the portfolio expressed as $\pi =(\eta ,\xi )$ consists of the optional processes ($\eta$ and $\xi$). The value process of the portfolio is given by $V_t^{\pi }={\eta }_t+{\xi }_t{S}_t$. We restrict the portfolio ($\pi$) to be self-financing in X, meaning that there is no inflow or outflow of wealth beyond the initially invested amount, and any dividends gained from S are reinvested in it. In other words, the change in the value process ($V_t^{\pi }$) is only due to change in X – $d{V}_t^{\pi }={\xi }_{t}d{X}_t$ or

$$V_t^{\pi }=\xi \circ X_t,$$

(25)

with $V_0^{\pi }=0$. Hence, it follows that

$$\begin{array}{ccc}\hfill V_t^{\pi }& =& \xi \circ X_t=\xi \left(1-\mathsf{T}\right)\circ {\left(S+D\right)}_t+L_T\xi \circ {\mathsf{T}}_t.\hfill \\ & =& {\xi }_t{S}_t-S\circ {\xi }_t-{[\xi ,S]}_t+\xi \left(1-\mathsf{T}\right)\circ D_t+L_T\xi \circ {\mathsf{T}}_t\hfill \\ & =& {\xi }_t{S}_t+{\eta }_t,\hfill \end{array}$$

where

$${\eta }_t=\xi \left(1-\mathsf{T}\right)\circ D_t+L_T\xi \circ {\mathsf{T}}_t-S\circ {\xi }_t-{[\xi ,S]}_t.$$

Since $\eta$ is uniquely determined by $\xi$, the portfolio is, indeed, self-financing. The process (X) is an optional semimartingale, and $\xi$ is X-integrable, i.e., ${\xi }^2\circ {[X,X]}_{\infty }\in {\mathcal{A}}_{loc}$. Thus, $\xi$ evolves in the space expressed as $\mathcal{P}\left(\mathbf{F}\right)\times \mathcal{O}\left(\mathbf{F}\right)$, which is not our usual predictable portfolio space but contains predictable and optional components. On the other hand, $\eta$ belongs to $\mathcal{O}\left(\mathbf{F}\right)$.

The optional integral ($\xi \circ X_t$) can be written as

$$\xi \circ X_t={\int }_{0+}^t{\xi }_{s-}^{\mathtt{1}}d{X}_s^{\mathtt{r}}+{\int }_0^{t-}{\xi }_s^{\mathtt{2}}d{X}_{s+}^{\mathtt{g}}$$

where the trading strategy ($\xi$) is of two components: a predictable component (${\xi }_{-}^{\mathtt{1}}$) and an optional component (${\xi }^{\mathtt{2}}$). The integral (${\int }_{0+}^t{\xi }_{s-}^{\mathtt{1}}d{X}_s^{\mathtt{r}}$) is the usual stochastic integral over the càdlàg semimartingale ($X_t^{\mathtt{r}}$). Similarly, the stochastic integral,

$${\int }_0^{t-}{\xi }_s^{\mathtt{2}}d{X}_{s+}^{\mathtt{g}},$$

is well-defined (see Galtchouk (1985)) and is ${\mathcal{F}}_t$-measurable, as the integration is taken over the interval of $[0,t)$. Moreover, the integrand (${\xi }_s^{\mathtt{2}}$) is ${\mathcal{F}}_s$-measurable, while $X_{s+}^{\mathtt{g}}$ is ${\mathcal{F}}_{s+}$-measurable for any $s\in [0,t)$.

Now, for convenience, let X be given by

$$X_t=X_0\mathcal{E}{(f\circ H)}_t,$$

(26)

where $\mathcal{E}(f\circ H)$ is the stochastic exponential, $H=H^{\mathtt{r}}+H^{\mathtt{g}}$ is an optional semimartingale adapted to $\mathbf{F}$ and $f\in \mathcal{P}\left(\mathbf{F}\right)\times \mathcal{O}\left(\mathbf{F}\right)$ and is H-integrable. Additionally, suppose that the dividend process is given by $D_t=\mu \circ S_t$, where $\mu \in \mathcal{P}\left(\mathbf{F}\right)$, and is S-integrable. Then, we can write X in terms of S as

$$\begin{array}{ccc}\hfill X_t& =& S_0+\left(1-\mathsf{T}\right)\circ {\left(S+D\right)}_t+L_T{\mathsf{T}}_t\hfill \\ & =& S_0+\left(1-\mathsf{T}\right)\left(1+\mu \right)\circ S_t+L_T{\mathsf{T}}_t.\hfill \end{array}$$

Since $X_t=\left(S_t+D_t\right)\left(1-{\mathsf{T}}_t\right)+L_T{\mathsf{T}}_t$, it follows that

$$X_t=\left\{\begin{array}{cc}{S}_0+\left(1+\mu \right)\circ S_t\hfill & t<T\hfill \\ L_T\hfill & t\ge T\hfill \end{array}\right.\Rightarrow S_t=\left\{\begin{array}{cc}{S}_0+{\displaystyle \frac{1}{1+\mu }}\circ X_t\hfill & t<T\hfill \\ 0\hfill & t\ge T\hfill \end{array}\right..$$

(27)

Next, we proceed to construct a local optional martingale deflator for X.

4.3. Transforming Optional Semimartingales to Local Optional Martingales

Our objective is to find a transformation ($Z>0$ a.s. $\mathbf{P}$) belonging to ${\mathcal{M}}_{loc}(\mathbf{P},\mathbf{F})$ that will render $ZX\in {\mathcal{M}}_{loc}(\mathbf{P},\mathbf{F})$. This Z is known as a local martingale deflator. For Z, we can define a local optional martingale ($N\in {\mathcal{M}}_{loc}(\mathbf{P},\mathbf{F})$) as the optional stochastic integral ($N=Z^{-1}\circ Z$) which allows us to write Z as the stochastic exponential ($Z=\mathcal{E}\left(N\right)$). Next, we present a way to find N.

Given $X=X_0\mathcal{E}(f\circ H)$ and $Z=\mathcal{E}\left(N\right)$, $ZX\in {\mathcal{M}}_{loc}(\mathbf{P},\mathbf{F})$ is a local optional martingale if

$$\begin{array}{ccc}\hfill {\Psi }_t(H,N)& :& =N_t+f\circ H_t+f\circ [N,H]\hfill \\ & =& N_t+f\circ H_t+f\circ {〈N^c,H^c〉}_t+\sum _{0<s\le t}{f}_s▵N_s▵H_s+\sum _{0\le s<t}{f}_s{▵}^{+}{N}_s{▵}^{+}{H}_s\hfill \end{array}$$

belongs to ${\mathcal{M}}_{loc}(\mathbf{P},\mathbf{F})$. Here, ${\Psi }_t(H,N)$ is derived from

$$\begin{array}{ccc}\hfill ZX& =& X_0\mathcal{E}\left(N\right)\mathcal{E}(f\circ H)=X_0\mathcal{E}{(N+f\circ H+f\circ [N,H])}_t\hfill \\ & =& X_0\mathcal{E}\left({\Psi }_t\right).\hfill \end{array}$$

If we also have ${▵}^{+}\Psi \ne -1$ and $▵\Psi \ne -1$, then $\Psi \in {\mathcal{M}}_{loc}(\mathbf{P},\mathbf{F})\iff ZX\in {\mathcal{M}}_{loc}(\mathbf{P},\mathbf{F})$.

Since, H is an optional semimartingale, it can be decomposed as $H=A+M$, where A is a finite-variation process and M is a local martingale. Let $N=g\circ M$ and write $\Psi$ as

$$\Psi =f\circ A+\left(f+g\right)\circ M+fg\circ \left[M,M\right].$$

$\Psi$ is a local optional martingale under $\mathbf{P}$ if

$$f\circ A+fg\circ \widetilde{[M,M]}=0$$

(28)

where $\widetilde{[M,M]}$ represents the compensators of $[M,M]$. By finding all $N\in {\mathcal{M}}_{loc}(\mathbf{P},\mathbf{F})$, i.e., the process (g) such that Equation (28) holds and $\mathcal{E}\left(N\right)>0$, we obtain the set (${\mathcal{Z}}_{loc}(\mathbf{P},\mathbf{F})$) of all appropriate local optional martingale deflators (Z) such that $ZX$ is a local optional martingale.

Lemma 3.

A solution to $f\circ A+fg\circ \widetilde{[M,M]}=0$ is given by

$$g=-{\displaystyle \frac{dA}{d\widetilde{[M,M]}}}$$

if it exists, where $f\ne 0$ and g is M-integrable.

Proof. 

The result follows directly by solving the equation for g, assuming the existence of the Radon–Nikodym derivative ($dA/d\widetilde{[M,M]}$). □

Lemma 4.

If Z is a local martingale deflator of X and π is a self-financing portfolio, then $Z{V}^{\pi }$ is a local optional martingale. Conversely, if $Z{V}^{\pi }$ is a local optional martingale for some $Z>0$, then Z is a local martingale deflator of X.

Proof. 

Suppose that $Z>0$ is a local martingale deflator of X and $\pi =(\eta ,\xi )$ is a self-financing, X-integrable portfolio. Then, $Z{V}^{\pi }$ can be written as

$$\begin{array}{cc}\hfill Z_t{V}_t^{\pi }& =Z\circ V_t^{\pi }+V^{\pi }\circ Z_t+{\left[Z,V^{\pi }\right]}_t\hfill \\ \hfill & =\xi Z\circ X_t+\xi \circ {\left[Z,X\right]}_t+V^{\pi }\circ Z_t\hfill \\ \hfill & =\xi \circ Z_t{X}_t+\left(V-\xi X\right)\circ Z_t,\hfill \end{array}$$

from which it follows that $Z_t{V}_t^{\pi }$ is a local optional martingale. Conversely, if $Z{V}^{\pi }$ is a local optional martingale for some $Z>0$ and $\pi$ is a self-financing, X-integrable portfolio, then

$$\xi \circ Z_t{X}_t=Z_t{V}_t^{\pi }-\left(V-\xi X\right)\circ Z_t$$

is the difference of two local optional martingales and, therefore, is, itself, a local optional martingale for any optional process ($\xi$). □

Lemma 5.

If $ZX$ is a local optional martingale, then $ZS$ is a local optional martingale.

Proof. 

Since $S_t=S_0+{\displaystyle \frac{1}{1+\mu }}\circ X_t$ if $t<T$ and 0 otherwise and $\mu \ge 0$ and finite for that, the dividend cannot explode in finite time; then, if X is a local optional martingale, so is S. Let us now consider a representation of $ZS$ in terms of $ZX$:

$$\begin{array}{ccc}\hfill ZS& =& Z\circ S+S\circ Z+[S,Z]\hfill \\ & =& {\displaystyle \frac{1}{1+\mu }}\circ \left(Z\circ X\right)+\left(S_0+{\displaystyle \frac{1}{1+\mu }}\right)\circ X\circ Z+{\displaystyle \frac{1}{1+\mu }}\circ [X,Z]\hfill \\ & =& {\displaystyle \frac{1}{1+\mu }}\circ \left(Z\circ X+X\circ Z+[X,Z]\right)+S_{0}X\circ Z\hfill \\ & =& {\displaystyle \frac{1}{1+\mu }}\circ ZX+S_{0}X\circ Z.\hfill \end{array}$$

Since $ZX$ and Z are local optional martingales, $ZS$ is also a local optional martingale. □

Remark 6.

If $Z>0$ and $Z\in {\mathcal{M}}_{loc}(\mathbf{P},\mathbf{F})$, then we can define ${\mathbf{Q}}_t={\int }_{\Omega }{Z}_{t}d{\mathbf{P}}_t$ as a new measure equivalent to ${\mathbf{P}}_t$, i.e., $\mathbf{Q}{\sim }_{loc}\mathbf{P}$, and $Z_t=d{\mathbf{Q}}_{t}d{\mathbf{P}}_t^{-1}$. The set of all optional local martingale measures ($\mathbf{Q}$) corresponding to Z in ${\mathcal{Z}}_{loc}(\mathbf{P},\mathbf{F})$ is denoted as as ${\mathfrak{Q}}_{loc}(\mathbf{P},\mathbf{F})$.

4.4. Bubbles and Fundamental Values

For a làdlàg optional semimartingale market, the definition of fundamental price does not require the valuation system of local martingale measures used in extended economies to explicitly define regime shifts. In optional semimartingale markets, regime shifts are implicitly encoded in the set of local optional martingale deflators (${\mathcal{Z}}_{loc}$) of X, in which a deflator (Z) incorporates market-regime shifts in its left-continuous part ($Z^{\mathtt{g}}$).

At time T or thereafter, the total accumulated wealth ($X_T$) of an investor in S is

$$X_T={\int }_0^{T}d{D}_u+L_T{\mathbf{1}}_{T<\infty }.$$

If $\mathbf{E}[|X_T|]<\infty$, then the present value of $X_T$ scaled by the local martingale deflator (Z) is ${\widehat{X}}_t=Z_t^{-1}\mathbf{E}[Z_T{X}_T|{\mathcal{F}}_t]$. The difference between X and $\widehat{X}$ ($Y_t=X_t-{\widehat{X}}_t$) is an optional semimartingale. Deflating the difference by the local martingale deflator (Z), we obtain

$$\begin{array}{ccc}\hfill Z_t{Y}_t& =& Z_t{X}_t-Z_t{\widehat{X}}_t\hfill \\ & =& Z_t{X}_t-\mathbf{E}[Z_T{X}_T|{\mathcal{F}}_t]\hfill \\ & =& \mathbf{E}[Z_t{X}_t-Z_T{X}_T|{\mathcal{F}}_t]\hfill \end{array}$$

where $Z_t{X}_t$ is an optional local martingale and $\mathbf{E}[Z_T{X}_T|{\mathcal{F}}_t]$ is an optional martingale; hence, $Z_t{Y}_t$ is a local optional martingale. Since $Z_t{X}_t$ is a positive local optional martingale, it is an optional supermartingale and $\mathbf{E}[Z_T{X}_T|{\mathcal{F}}_t]\le Z_t{X}_t$. Therefore, $Z_t{Y}_t\ge 0$.

Next, we decompose $Z_t{\widehat{X}}_t$ to attained wealth plus the remaining wealth up to time T. To do this, consider $Z_t{\widehat{X}}_t=Z_t{\widehat{X}}_t{\mathbf{1}}_{t<T}+Z_t{\widehat{X}}_t{\mathbf{1}}_{t\ge T}$, where

$$\begin{array}{ccc}\hfill Z_t{\widehat{X}}_t{\mathbf{1}}_{t<T}& =& \mathbf{E}\left[Z_T\left({\int }_0^{T}d{D}_u+L_T{\mathbf{1}}_{T<\infty }\right)|{\mathcal{F}}_t\right]{\mathbf{1}}_{t<T}\hfill \\ & =& \mathbf{E}\left[Z_T\left({\int }_t^{T}d{D}_u+{\int }_0^{t}d{D}_u+L_T{\mathbf{1}}_{T<\infty }\right)|{\mathcal{F}}_t\right]{\mathbf{1}}_{t<T}\hfill \\ & =& \mathbf{E}\left[Z_T{\int }_t^{T}d{D}_u+L_T{\mathbf{1}}_{T<\infty }|{\mathcal{F}}_t\right]{\mathbf{1}}_{t<T}+\mathbf{E}\left[Z_T{\int }_0^{t}d{D}_u|{\mathcal{F}}_t\right]{\mathbf{1}}_{t<T},\hfill \end{array}$$

and

$$Z_t{\widehat{X}}_t{\mathbf{1}}_{t\ge T}=\mathbf{E}\left[Z_T{\int }_0^{T}d{D}_u+L_T{\mathbf{1}}_{T<\infty }|{\mathcal{F}}_t\right]{\mathbf{1}}_{t\ge T}.$$

Therefore,

$$\begin{array}{ccc}\hfill Z_t{\widehat{X}}_t& =& \mathbf{E}\left[Z_T{\int }_t^{T}d{D}_u+L_T{\mathbf{1}}_{T<\infty }|{\mathcal{F}}_t\right]{\mathbf{1}}_{t<T}+\mathbf{E}\left[Z_T{\int }_0^{t}d{D}_u|{\mathcal{F}}_t\right]{\mathbf{1}}_{t<T}\hfill \\ & & +\mathbf{E}\left[Z_T{\int }_0^{T}d{D}_u+L_T{\mathbf{1}}_{T<\infty }|{\mathcal{F}}_t\right]{\mathbf{1}}_{t\ge T}\hfill \\ & =& \mathbf{E}\left[Z_T{\int }_t^{T}d{D}_u+L_T{\mathbf{1}}_{T<\infty }|{\mathcal{F}}_t\right]{\mathbf{1}}_{t<T}\hfill \\ & & +Z_T\left({\int }_0^{t\wedge T}d{D}_u+L_T{\mathbf{1}}_{T<\infty }{\mathbf{1}}_{t\ge T}\right).\hfill \end{array}$$

Let $Z_t{\widehat{S}}_t$ be

$$Z_t{\widehat{S}}_t=\mathbf{E}\left[Z_T{\int }_t^{T}d{D}_u+L_T{\mathbf{1}}_{T<\infty }|{\mathcal{F}}_t\right]{\mathbf{1}}_{t<T}.$$

${\widehat{S}}_t$ is what we may call the fundamental price, as it is the present value of all unrealized future dividends and the final asset value. Therefore, the fundamental wealth is

$$Z_t{\widehat{X}}_t=Z_t{\widehat{S}}_t{\mathbf{1}}_{t<T}+Z_T\left({\int }_0^{t\wedge T}d{D}_u+L_T{\mathbf{1}}_{T<\infty }{\mathbf{1}}_{t\ge T}\right),$$

where ${\int }_0^{t\wedge T}d{D}_u+L_T{\mathbf{1}}_{T<\infty }{\mathbf{1}}_{t\ge T}$ is the realized wealth.

Granted a local optional martingale deflator ($Z>0$), the above discussion leads us to the following definitions of fundamental price and wealth under the corresponding equivalent local optional martingale measure ($\mathbf{Q}$):

Definition 8.

Optional Fundamental Price. For an asset with a maturity of T and payoff of $(D,L_T)$, the fundamental price (${\Gamma }_t(D,L_T)$) is defined by

$${\Gamma }_t(D,L_T)={\mathbf{E}}_{\mathbf{Q}}\left[{\int }_t^{T}d{D}_u+L_T{\mathbf{1}}_{T<\infty }|{\mathcal{F}}_t\right]$$

$\forall t\in [0,\infty )$ where ${\Gamma }_{\infty }(D,L_T)=0$.

In particular, the fundamental price of the risky asset ($S_t$) is the discounted future cash flow on ${\mathcal{F}}_t$ under $\mathbf{Q}$, as given by

$${\widehat{S}}_t={\mathbf{E}}_{\mathbf{Q}}\left[{\int }_t^{T}d{D}_u+L_T{\mathbf{1}}_{T<\infty }|{\mathcal{F}}_t\right].$$

Its fundamental wealth ($\widehat{X}$) is

$$\begin{array}{ccc}\hfill {\widehat{X}}_t& =& {\mathbf{E}}_{\mathbf{Q}}\left[{\int }_t^{T}d{D}_u+L_T{\mathbf{1}}_{T<\infty }|{\mathcal{F}}_t\right]{\mathbf{1}}_{t<T}+\left({\int }_0^{t\wedge T}d{D}_u+L_T{\mathbf{1}}_{T<\infty }\right){\mathbf{1}}_{t\ge T}\hfill \\ & =& {\mathbf{E}}_{\mathbf{Q}}\left[{\int }_0^{T}d{D}_u+L_T{\mathbf{1}}_{T<\infty }|{\mathcal{F}}_t\right],\hfill \end{array}$$

which is a uniformly integrable optional martingale.

Knowing the fundamental value, we can define the price and wealth bubbles as follows:

Definition 9.

An asset price bubble (B) is $B=S-\widehat{S}$, and the associated wealth bubble is $Y=X-\widehat{X}$.

Moreover, knowing that $Z_t{X}_t$ and $Z_t{S}_t$ are positive local martingales, we have the following lemma:

Lemma 6.

Under ${\mathfrak{Q}}_{loc}$, bubbles Y and B are non-negative local optional martingales.

Bubbles in optional semimartingale markets share similarities with those in càdlàg markets but with a key difference: in optional semimartingale markets, regime shifts naturally arise due to the presence of wide-sense random times. These shifts introduce additional complexity to the market dynamics. Before delving into the properties of bubbles as optional local martingales, let us first revisit some foundational concepts.

A càdlàg stochastic process (M) is a martingale under the usual conditions if for any t with $\mathbf{E}\left[|M_t|\right]<\infty$, and for any t and s with $0<s<t<T$, the $\mathbf{E}\left[M_t|{\mathcal{F}}_s^{\mathbf{P}}\right]=M_s$ a.s., where $T\in (0,\infty ]$ and ${\mathcal{F}}_t^{\mathbf{P}}$ satisfies the usual hypothesis. Furthermore, M is a uniformly integrable martingale if $\mathbf{E}[|M_t|{\mathbf{1}}_{|M_t|>n}]$ converges to zero as $n\to \infty$ uniformly in t – ${lim}_{n\to \infty }{sup}_t\mathbf{E}[|M_t|{\mathbf{1}}_{|M_t|>n}]=0$, where the supremum is over $[0,T]$ for a finite time interval and $[0,\infty )$ if the process is considered on $0<t<\infty$. Also, any martingale (M) on a finite time interval ($0<t<T<\infty$) is uniformly integrable and is closed by $M_T$. The martingale convergence theorem tells us that if M is a martingale on $0\le t<\infty$ and ${sup}_{t\ge 0}\mathbf{E}\left[|M_t|\right]<\infty$, then there exists an almost sure limit (${lim}_{t\to \infty }{M}_t=Y$, where Y is an integrable random variable). The optional sampling theorem allows us to express the martingale property in terms of stopping time. Let $\sigma \le \tau$ represent bounded stopping times belonging to $\mathcal{T}\left({\mathbf{F}}^{\mathbf{P}}\right)$. Then, for any càdlàg martingale, random variables $M_{\sigma }$ and $M_{\tau }$ are integrable and $\mathbf{E}\left[M_{\tau }|{\mathcal{F}}_{\sigma }^{\mathbf{P}}\right]=M_{\sigma }$. M is a càdlàg local martingale if there exists a sequence of stopping times (${\tau }_n\in \mathcal{T}\left({\mathbf{F}}^{\mathbf{P}}\right)$) such that ${\tau }_n\uparrow \infty$, and for each n, the stopped processes ($M_{t\wedge {\tau }_n}$) is a uniformly integrable martingale in t.

In contrast, a process ($M={\left(M_t\right)}_{t\ge 0}$) is defined as a strong optionalmartingale, as per Dellacherie (1975), if $M\in \mathcal{O}\left(\mathbf{F}\right)$, the random variable ($M_T$) is integrable for any bounded stopping time ($T\in \mathcal{T}\left(\mathbf{F}\right)$) and for every pair of bounded stopping times (S and T with $S\le T$), and we have $M_S=\mathbf{E}[M_T|{\mathcal{F}}_S]$. Also, every càdlàg martingale is a strong optional martingale, but not all strong optional martingales are càdlàg. However, optional martingales were also defined in Galtchouk (1980) as follows: a process (M) is an optional martingale if it is optional and there exists an integrable random variable ($\mu \in {\mathcal{F}}_{\infty }$) such that $M_T=\mathbf{E}[\mu |{\mathcal{F}}_T]$ a.s. for any stopping time ($T\in {\mathcal{F}}_{\infty }$ with $(T<\infty )$). With either definition of optional martingale, the definition of local optional martingale coincides. Moreover, every càdlàg local martingale in $\mathbf{F}$ is an optional local martingale in $\mathbf{F}$.

In the following discussion, we adopt the definition of optional martingales from Galtchouk (1980), where a true optional martingale is characterized by the existence of an integrable random variable ($\mu \in {\mathcal{F}}_{\infty }$) such that $M_T=\mathbf{E}[\mu |{\mathcal{F}}_T]$ a.s. for any stopping time ($T\in {\mathcal{F}}_{\infty }$ with $(T<\infty )$). However, our focus remains on local optional martingales, and we explore the conditions under which they qualify as strict local martingales or true martingales.

4.4.1. Fundamental Value Invariance

Let ${\mathcal{Z}}_{\mathrm{UI}}$ denote a set of local optional martingale deflators such that for any $Z\in {\mathcal{Z}}_{\mathrm{UI}}$, the ${\left(Z_{T\wedge {\ell }_n}\right)}_{n\ge 1}$ family is uniformly integrable for some localizing sequence (${\left({\ell }_n\right)}_{n\ge 0}$). In other words,

$$\underset{z\to \infty }{lim}\underset{n}{sup}\mathbf{E}\left[Z_{T\wedge {\ell }_n}{\mathbf{1}}_{\left\{Z_{T\wedge {\ell }_n}>z\right\}}\right]=0,$$

or equivalently, ${lim}_{n\to \infty }\mathbf{E}\left[Z_{T\wedge {\ell }_n}\right]\to 1=\mathbf{E}{Z}_T$ (see Abdelghani and Melnikov (2024)). Additionally, let ${\mathcal{Z}}_{\mathrm{NUI}}$ be a set of local optional martingale deflators that are not uniformly integrable. It is evident that the choice of an equivalent local martingale deflator impacts the fundamental value. However, for the ${\mathcal{Z}}_{\mathrm{UI}}$ class, fundamental values are invariant. We characterize this invariant class with the following lemmas:

Lemma 7.

Let $T\in [0,\infty ]$. Suppose that at T, the fundamental wealth (${\widehat{X}}_T$) is equal to its market value ($X_T$); then, the fundamental wealth process ($\widehat{X}$) and the fundamental price ($\widehat{S}$) do not depend on the choice of $Z\in {\mathcal{Z}}_{\mathrm{UI}}$.

Proof. 

Suppose Z and $Z^{\prime }\in {\mathcal{Z}}_{\mathrm{UI}}$. Let $Z\widehat{X}$ be the fundamental wealth given Z, and let $Z^{\prime }{\widehat{X}}^{\prime }$ be the fundamental wealth given $Z^{\prime }$. Since $\widehat{X}$ and ${\widehat{X}}^{\prime }$ are uniformly integrable local martingales under Z and $Z^{\prime }$, respectively, for any ${\ell }_n$,

$$\begin{array}{ccc}\hfill Z_{t\wedge {\ell }_n}{\widehat{X}}_{t\wedge {\ell }_n}& =& \mathbf{E}[Z_{T\wedge {\ell }_n}{\widehat{X}}_{T\wedge {\ell }_n}|{\mathcal{F}}_{t\wedge {\ell }_n}]=\mathbf{E}[Z_{T\wedge {\ell }_n}{X}_{T\wedge {\ell }_n}|{\mathcal{F}}_{t\wedge {\ell }_n}]\hfill \\ & =& Z_{t\wedge {\ell }_n}{X}_{t\wedge {\ell }_n}=\mathbf{E}[Z_{T\wedge {\ell }_n}^{\prime }{X}_{T\wedge {\ell }_n}|{\mathcal{F}}_{t\wedge {\ell }_n}]\hfill \\ & =& \mathbf{E}[Z_{T\wedge {\ell }_n}^{\prime }{\widehat{X}}_{T\wedge {\ell }_n}|{\mathcal{F}}_{t\wedge {\ell }_n}]=Z_{t\wedge {\ell }_n}^{\prime }{\widehat{X}}_{t\wedge {\ell }_n}^{\prime }\mathrm{a}.\mathrm{s}.\mathrm{for}\mathrm{all}n.\hfill \end{array}$$

(29)

The difference between $\widehat{X}$ and $\widehat{S}$ does not depend on the choice of Z. Therefore, since $Z\widehat{X}=Z^{\prime }{\widehat{X}}^{\prime }$, $Z\widehat{S}=Z^{\prime }{\widehat{S}}^{\prime }$. □

The following lemma describes the relationship between the fundamental price under a deflator ($Z^{\prime }\in {\mathcal{Z}}_{\mathrm{NUI}}$); a non-uniform integrable local optional martingale; and $Z\in {\mathcal{Z}}_{\mathrm{UI}}$, a uniformly integrable martingale.

Lemma 8.

Let $T\in [0,\infty ]$, $Z\in {\mathcal{Z}}_{\mathrm{UI}}$, and $Z^{\prime }\in {\mathcal{Z}}_{\mathrm{NUI}}$. Then, even if the fundamental wealth (${\widehat{X}}_T$) is equal to its market value ($X_T$), we have

$$Z_{t\wedge {\ell }_n}^{\prime }{X}_{t\wedge {\ell }_n}^{\prime }\le Z_{t\wedge {\ell }_n}{X}_{t\wedge {\ell }_n}a.s.\mathit{for}\mathit{all}n\mathit{and}t.$$

(30)

Proof. 

For any ${\ell }_n$, we have that

$$\begin{array}{ccc}\hfill Z_{t\wedge {\ell }_n}{\widehat{X}}_{t\wedge {\ell }_n}-Z_{t\wedge {\ell }_n}^{\prime }{\widehat{X}}_{t\wedge {\ell }_n}^{\prime }& =& \mathbf{E}[Z_T{\widehat{X}}_T|{\mathcal{F}}_{t\wedge {\ell }_n}]-\mathbf{E}[Z_T^{\prime }{\widehat{X}}_T|{\mathcal{F}}_{t\wedge {\ell }_n}]\hfill \\ & =& \mathbf{E}[Z_T{X}_T|{\mathcal{F}}_{t\wedge {\ell }_n}]-\mathbf{E}[Z_T^{\prime }{X}_T|{\mathcal{F}}_{t\wedge {\ell }_n}]\hfill \\ & =& Z_{t\wedge {\ell }_n}{X}_{t\wedge {\ell }_n}-Z_{t\wedge {\ell }_n}^{\prime }{X}_{t\wedge {\ell }_n}^{\prime }\ge 0,\hfill \end{array}$$

(31)

since $Z_{·\wedge {\ell }_n}-Z_{·\wedge {\ell }_n}^{\prime }$ is a submartingale—$\mathbf{E}[(Z_T-Z_T^{\prime })X_T|{\mathcal{F}}_{t\wedge {\ell }_n}]\ge 0$. □

Remark 7.

In (Abdelghani and Melnikov 2024), it was demonstrated that a local optional martingale deflator ($Z_t$, $t\in \left[0,T\right]$, and $\mathbf{E}\left[Z_t\right]\le 1$) can be a true optional martingale if one can find a localizing sequence (${\left({\tau }_n\right)}_{n\ge 1}$) such that $\mathbf{E}\left[Z_{t\wedge {\tau }_n}\right]=1$ for any n and the ${\left(Z_{T\wedge {\tau }_n}\right)}_{n\ge 1}$ family is uniformly integrable on $\left[0,T\right]$. This leads to the conclusion that ${lim}_{n\to \infty }\mathbf{E}\left[Z_{T\wedge {\tau }_n}\right]\to 1=\mathbf{E}\left[Z_T\right]$. To verify the uniform integrability of ${\left(Z_{T\wedge {\tau }_n}\right)}_{n\ge 1}$, it is sufficient to check that $\underset{n}{sup}\mathbf{E}\left[Z_{T\wedge {\tau }_n}log\left(Z_{T\wedge {\tau }_n}\right)\right]<\infty$.

4.4.2. Decomposition Theorems

According to the Riesz decomposition theorem (see Theorem 2.1 in Galtchouk (1982)), any optional supermartingale can be decomposed into two components: an optional martingale, which is uniformly integrable by definition, and a potential, which is a non-negative optional supermartingale whose expectation converges to zero. Applying this decomposition to a price bubble (B), which is an optional supermartingale, we can express it as the sum of a uniformly integrable bubble $(B^{\mathtt{u}}$ with $B_t^{\mathtt{u}}\to X_{\infty }$ a.s.) and a potential bubble ($B^{\mathtt{z}}$ – $B=B^{\mathtt{u}}+B^{\mathtt{z}}$). Therefore, the price (S) admits the following unique decomposition: $S=\widehat{S}+B=\widehat{S}+(B^{\mathtt{u}}+B^{\mathtt{z}})$.

Theorem 4.

Under ${\mathfrak{Q}}_{loc}$, if the price bubble (B) is a local optional martingale, which is also an optional supermartingale, then the price can be decomposed as follows:

$$S=\widehat{S}+B=\widehat{S}+(B^{\mathtt{u}}+B^{\mathtt{z}}),$$

where $B^{\mathtt{u}}$ is a uniformly integrable martingale with $B_t^{\mathtt{u}}\to X_{\infty }$ a.s. and $B^{\mathtt{z}}$ is a potential.

Proof. 

This result follows from the Riesz decomposition theorem for optional supermartingales, which allows for the decomposition of B into a uniformly integrable martingale ($B^{\mathtt{u}}$) and a potential ($B^{\mathtt{z}}$). □

On the other hand, if the deflated price bubble ($ZB$) is a strict local optional supermartingale, it can be decomposed into a uniformly integrable optional martingale, a local optional martingale, and a local optional potential. We demonstrate this in the following section. It is important to note that Kazamaki (1970) showed that a local supermartingale can be decomposed into a local martingale and a potential under standard conditions. The proof we present for the decomposition of a local optional martingale extends Kazamaki’s result to local optional supermartingales under nonstandard conditions.

Let ${\left({\ell }_n\right)}_{n\ge 0}$ be a localizing sequence such that $\mathbf{P}({\ell }_n\le n)=1$ for all n. The localized $Z_{·\wedge {\ell }_n}{B}_{·\wedge {\ell }_n}$ is an optional supermartingale. Using the decomposition of Riesz, we obtain

$$Z_{t\wedge {\ell }_n}{B}_{t\wedge {\ell }_n}=U_t^n+V_t^n$$

where $U_t^n$ is a uniformly integrable optional martingale and $V_t^n$ is an optional potential for every n. Let us further assume that

$$\underset{k}{inf}\mathbf{E}\left[Z_{{\ell }_k}{B}_{{\ell }_k}\right]>-\infty ;$$

(32)

but this assumption is true for price bubbles, as they are always $\ge$0 by construction. Since

$$\begin{array}{ccc}\hfill \mathbf{E}\left[Z_{{\ell }_{k+m+1}}{B}_{{\ell }_{k+m+1}}|{\mathcal{F}}_{t\wedge {\ell }_k}\right]& =& \mathbf{E}\left[\mathbf{E}[Z_{{\ell }_{k+m+1}}{B}_{{\ell }_{k+m+1}}|{\mathcal{F}}_{{\ell }_{k+m}}]|{\mathcal{F}}_{t\wedge {\ell }_k}\right],\hfill \\ & \le & \mathbf{E}\left[Z_{{\ell }_{k+m}}{B}_{{\ell }_{k+m}}|{\mathcal{F}}_{t\wedge {\ell }_k}\right],\hfill \end{array}$$

implying that $\mathbf{E}\left[Z_{{\ell }_{k+m}}{B}_{{\ell }_{k+m}}|{\mathcal{F}}_{t\wedge {\ell }_k}\right]$ is decreasing in m for any k and t. Thus, we are compelled to set

$$U_t^n=\left\{\begin{array}{cc}\underset{m\to \infty }{lim}\mathbf{E}\left[Z_{{\ell }_m}{B}_{{\ell }_m}|{\mathcal{F}}_{t\wedge {\ell }_n}\right]\hfill & {\left({\mathcal{N}}_{t,n}\right)}^{\mathsf{c}}\hfill \\ Z_{t\wedge {\ell }_n}{B}_{t\wedge {\ell }_n}\hfill & {\mathcal{N}}_{t,n}\hfill \end{array}\right.$$

where ${\mathcal{N}}_{t,n}$ is an $\mathcal{F}$ set of $\mathbf{P}$-measure zero, which may depend on t and n, and ${\left({\mathcal{N}}_{t,n}\right)}^{\mathsf{c}}$ is its compliment. From the monotone convergence theorem, we have that for each pair ($s<t$) and each n,

$$\begin{array}{ccc}\hfill \mathbf{E}[U_t^n|{\mathcal{F}}_{s\wedge {\ell }_n}]& =& \mathbf{E}\left[\underset{m\to \infty }{lim}\mathbf{E}\left[Z_{{\ell }_m}{B}_{{\ell }_m}|{\mathcal{F}}_{t\wedge {\ell }_n}\right]|{\mathcal{F}}_{s\wedge {\ell }_n}\right]\hfill \\ & =& \underset{m\to \infty }{lim}\mathbf{E}[Z_{{\ell }_m}{B}_{{\ell }_m}|{\mathcal{F}}_{s\wedge {\ell }_n}]=U_s^n\mathrm{a}.\mathrm{s}.\hfill \end{array}$$

It follows from condition (32) that

$$\mathbf{E}\left[Z_{t\wedge {\ell }_n}{B}_{t\wedge {\ell }_n}-U_t^n\right]=\mathbf{E}\left[Z_{t\wedge {\ell }_n}{B}_{t\wedge {\ell }_n}\right]-\underset{n}{inf}\mathbf{E}\left[Z_{{\ell }_n}{B}_{{\ell }_n}\right]<+\infty$$

which implies that $Z_{t\wedge {\ell }_n}{B}_{t\wedge {\ell }_n}-U_t^n$ is integrable. Hence, $U_t^n$ is integrable as well. Therefore, for each n, $U_t^n$ is an optional martingale. Next, we consider the relationship between $U_t^n$ and $U_t^{n+p}$. We know that

$$\begin{array}{ccc}\hfill U_{t\wedge {\ell }_n}^{n+p}& =& \mathbf{E}\left[U_t^{n+p}|{\mathcal{F}}_{t\wedge {\ell }_n}\right]=\mathbf{E}\left[\underset{m\to \infty }{lim}\mathbf{E}\left[Z_{{\ell }_m}{B}_{{\ell }_m}|{\mathcal{F}}_{t\wedge {\ell }_{n+p}}\right]|{\mathcal{F}}_{t\wedge {\ell }_n}\right]\hfill \\ & =& \underset{m\to \infty }{lim}\mathbf{E}\left[Z_{{\ell }_m}{B}_{{\ell }_m}|{\mathcal{F}}_{t\wedge {\ell }_n}\right]=U_t^n,\hfill \end{array}$$

so $U_{t\wedge {\ell }_n}^{n+p}$ and $U_t^n$ coincide and are ${\mathcal{F}}_{t\wedge {\ell }_n}$-measurable for any p. As a result, we may put forth the definition of $U_t={lim}_j{U}_t^j$ and observe that $U_t^n=U_{t\wedge {\ell }_n}$ is an optional martingale for every n. Therefore, $U_t$ is a local optional martingale, and

$$0\le Z_{t\wedge {\ell }_n}{B}_{t\wedge {\ell }_n}-U_t^n=Z_{t\wedge {\ell }_n}{B}_{t\wedge {\ell }_n}-U_{t\wedge {\ell }_n},$$

for which ${lim}_{t\to \infty }\mathbf{E}\left[Z_{t\wedge {\ell }_n}{B}_{t\wedge {\ell }_n}-U_{t\wedge {\ell }_n}\right]=0$ for all n. Therefore, $V_t=Z_t{B}_t-U_t$ is a local potential. Furthermore, for each $U_{t\wedge {\ell }_n}$, there exist $u^n\in {\mathcal{F}}_{\infty }$ such that $U_{t\wedge {\ell }_n}=\mathbf{E}[u^n|{\mathcal{F}}_t]$. However, note that the ${\left(U_{t\wedge {\ell }_n}\right)}_{n\ge 0}$ family may not be uniformly integrable. Let ${inf}_n{u}^n=\breve{u}$; then, we may write $U_{t\wedge {\ell }_n}$ as

$$\begin{array}{ccc}\hfill U_{t\wedge {\ell }_n}& =& \mathbf{E}[\breve{u}|{\mathcal{F}}_t]+\mathbf{E}[u^n-\breve{u}|{\mathcal{F}}_t]\hfill \\ & =& {\breve{U}}_t+{\tilde{U}}_{t\wedge {\ell }_n}\hfill \end{array}$$

where ${\breve{U}}_t$ is a uniformly integrable optional martingale and the ${\left({\tilde{U}}_{t\wedge {\ell }_n}\right)}_{n\ge 0}$ family comprises local optional martingales. Therefore,

$$Z_t{B}_t={\breve{U}}_t+{\tilde{U}}_t+V_t,$$

and we have the following lemma:

Theorem 5.

If condition (32) is satisfied, then the deflated price bubble has the following decomposition:

$$Z_t{B}_t={\breve{U}}_t+{\tilde{U}}_t+V_t$$

where ${\breve{U}}_t$ is an optional martingale, ${\tilde{U}}_t$ is a local optional martingale, and $V_t$ is a local optional potential. Thus, the deflated price process is

$$ZS=Z\widehat{S}+{\breve{U}}_t+{\tilde{U}}_t+V_t.$$

Proof. 

This follows from the application of the Riesz decomposition of local optional supermartingales, as discussed above. □

5. Illustrative Examples

In an optional semimartingale market, bubble birth is possible, as we show with the following examples.

Example 1.

Regime-shifted geometric Brownian motion.

Consider an asset price described by the following optional semimartingale:

$$S_t={\mathcal{E}}_t\left(W+\eta {\mathbf{1}}_{·>\sigma }\right)$$

where W is Brownian motion, $\sigma \in \mathcal{T}\left({\mathbf{F}}_{+}\right)$ is a random regime-shift time, and $\eta$ is a continuously increasing process that is ${\mathcal{F}}_t$-measurable. The asset (S) pays no dividends and expires at $T\in (0,\infty ]$, with a terminal payoff of $S_T$.

We can write $S_t={\mathcal{E}}_t\left(W\right){\mathbf{1}}_{t\le \sigma }+{\mathcal{E}}_t\left(W+\eta \right){\mathbf{1}}_{t>\sigma }$. This shows that S has two phases; one phase is $\mathcal{E}\left(W\right)$ before $\sigma$, and the other is $\mathcal{E}\left(W+\eta \right)$ after $\sigma$. Before $\sigma$, S is a martingale free of bubbles, whereas after $\sigma$, it is not. Let $V_t={\mathbf{1}}_{t>\sigma }-v_t$, where $v_t$, the compensator of ${\mathbf{1}}_{t>\sigma }$, is continuous and ${\mathcal{F}}_t$-measurable. $V_t$ is a left-continuous, ${\mathcal{F}}_t$-measurable optional martingale.

We can choose a local martingale deflator (Z) in the form of ${\mathcal{E}}_t\left(\alpha \circ W+\beta \circ V\right)$ such that $ZS$ is a local martingale, as follows:

$$\begin{array}{ccc}\hfill ZS& =& \mathcal{E}\left(\alpha \circ W+\beta \circ V\right)\mathcal{E}\left(W+\eta V+\eta v\right)\hfill \\ & =& \mathcal{E}\left(\alpha \circ W+\beta \circ V\right)\mathcal{E}\left(W+\eta \circ V+V\circ \eta +\eta v\right)\hfill \\ & =& \mathcal{E}\left((1+\alpha )\circ W+\left(\beta +\eta +\beta \eta \right)\circ V+\alpha \circ t+\eta v+\beta \eta \circ v+V\circ \eta \right)\hfill \end{array}$$

since ${〈W,W〉}_t=t$ and ${\left[V,V\right]}_t={\mathbf{1}}_{t>\sigma }=V_t+v_t$.

For $ZS$ to be a local martingale, we must have the following:

$$\begin{array}{c}\alpha \circ t+\beta \eta \circ v+\eta v+V\circ \eta =0\Rightarrow \\ \alpha \circ t+\beta \eta \circ v+\eta v-v\circ \eta +{\mathbf{1}}_{·>\sigma }\circ \eta =0\Rightarrow \\ \alpha \circ t+(1+\beta )\eta \circ v+{\mathbf{1}}_{·>\sigma }\circ \eta =0.\end{array}$$

(33)

Ideally, the integral equation (33) admits at least one solution for $\alpha$ and $\beta$ in terms of v, $\eta$, and ${\mathbf{1}}_{t>\sigma }$. However, it is possible that Equation (33) has infinitely many solutions for $(\alpha ,\beta )$, resulting in many possible deflators (Z) with various properties. In addition to satisfying (33), the deflator must also satisfy $Z>0$, imposing further constraints on the permissible values of $(\alpha ,\beta )$.

Let us consider several choices of solutions for Equation (33). Setting $\beta =0$, we find that $Z=\mathcal{E}\left(\alpha \circ W\right)$, and the deflated price becomes $ZS=\mathcal{E}\left((1+\alpha )\circ W+\eta \circ V\right)$, which depends on the regime-shift time ($\sigma$). On the other hand, if $\beta =-\eta {(1+\eta )}^{-1}$, the resulting deflator is $Z=\mathcal{E}\left(\alpha \circ W-\eta {(1+\eta )}^{-1}\circ V\right)$, which subsumes the regime-shift time ($\sigma$). In this case, the deflated price simplifies to $ZS=\mathcal{E}\left((1+\alpha )\circ W\right)$, becoming independent of $\sigma$. Thus, depending on the choice of $\beta$ and the properties of the v, $\eta$, and ${\mathbf{1}}_{t>\sigma }$, the characteristics of Z and $ZS$ are determined.

Since the asset pays no dividends, the fundamental price under $\mathbf{P}$ is given by $Z_t{\widehat{S}}_t=\mathbf{E}\left[Z_T{S}_T|{\mathcal{F}}_t\right]$. A price bubble, defined as $B=S-\widehat{S}$, is an optional semimartingale. However, after deflating by Z, the bubble becomes a local optional martingale:

$$\begin{array}{ccc}\hfill Z_t{B}_t& =& Z_t{S}_t-Z_t{\widehat{S}}_t\hfill \\ & =& {\mathcal{E}}_t\left((1+\alpha )\circ W+\left(\beta +\eta +\beta \eta \right)\circ V\right)-\mathbf{E}\left[{\mathcal{E}}_T\left(\alpha \circ W+\beta \circ V\right)S_T|{\mathcal{F}}_t\right].\hfill \end{array}$$

Depending on the choice of $\beta$ and the properties of the v, $\eta$, and ${\mathbf{1}}_{t>\sigma }$, $ZB$ may be either a true martingale or a strict local optional martingale.

Example 2.

Jump diffusion exhibiting a regime shift.

Here, we present a concrete example of a regime-shifted diffusion process. Suppose the market price of an asset is given by

$$S_t=\mathcal{E}\left(\mu t+\sigma W_t+\gamma N_{t-}\right)$$

where the constants are $\mu \in (-\infty ,\infty )$, $\sigma \in (0,\infty )$, and $\gamma \in (-\infty ,\infty )$; W is Brownian motion; and N is a regime-shift Poisson process adapted to ${\mathbf{F}}^{+}$ with a constant intensity of $\lambda \in [0,\infty )$. The price process (S) reflects the price of the asset in all regimes. Suppose the asset pays no dividends so that the wealth process (X) is equal to S. To find a local martingale deflator (Z) of S, we write

$$\begin{array}{ccc}\hfill Z_t{S}_t& =& \mathcal{E}\left(\zeta \right)\mathcal{E}\left(\mu t+\sigma W_t+N_{t-}\right)\hfill \\ & =& \mathcal{E}\left(\mu t+\sigma W_t+N_{t-}+\zeta +[\sigma W_t+N_{t-},\zeta ]\right).\hfill \end{array}$$

Let $\zeta =a{W}_t+b(N_{t-}-\lambda t)$, where $a,b\in \mathbb{R}$. Then,

$$\begin{array}{ccc}\hfill [\sigma W_t+N_{t-},a{W}_t+b(N_{t-}-\lambda t)]& =& \left[\sigma W_t,a{W}_t\right]+[N_{t-},b{N}_{t-}]\hfill \\ & =& \left(\sigma a+b\lambda \right)t+b\left(N_{t-}-\lambda t\right)\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill Z_t{S}_t& =& \mathcal{E}\left(\mu t+\sigma W_t+N_{t-}+a{W}_t+b(N_{t-}-\lambda t)+[\sigma W_t+N_{t-},a{W}_t+b(N_{t-}-\lambda t)]\right)\hfill \\ & =& \mathcal{E}\left(\left(\mu +\sigma a+\left(1+b\right)\lambda \right)t+(1+a)W_t+\left(1+2b\right)\left(N_{t-}-\lambda t\right)\right).\hfill \end{array}$$

To transform S to a local optional martingale using Z, we must choose a and b such that

$$\mu +\sigma a+\left(1+b\right)\lambda =0.$$

There are many solutions to these equations; therefore, there are many possible local martingale deflators (Z). Let us consider few possible solutions to these equations.

Suppose $b=-{\displaystyle \frac{1}{2}}$; then, $a=-{\sigma }^{-1}(\mu +{\displaystyle \frac{1}{2}}\lambda )$,

$$Z=\mathcal{E}\left(-{\sigma }^{-1}\left(\mu +{\displaystyle \frac{1}{2}}\lambda \right)W-{\displaystyle \frac{1}{2}}(N_{t-}-\lambda t)\right),$$

and

$$Z_t{S}_t=\mathcal{E}\left(\left(1-\mu {\sigma }^{-1}\right)W_t\right).$$

In this case, $Z_t{S}_t$ is a uniformly integrable martingale under the Novikov condition.

On the other hand, if $b=0$, then $a=-{\displaystyle \frac{\left(\mu +\lambda \right)}{\sigma }}$ and $Z=\mathcal{E}\left(-{\displaystyle \frac{\left(\mu +\lambda \right)}{\sigma }}W\right)$. Therefore,

$$Z_t{S}_t=\mathcal{E}\left(\left(1+-{\displaystyle \frac{\left(\mu +\lambda \right)}{\sigma }}\right)W_t+\left(N_{t-}-\lambda t\right)\right)$$

is not a uniformly integrable martingale, depending on N, which jumps at wide-sense random times, causing a regime shift in the market.

Example 3.

Regime change and bubble rebirth.

Suppose the market’s valuation measure shifts at time $\sigma$ from $Z\in {\mathcal{Z}}_{\mathrm{U}\mathrm{I}}\left(X\right)$ to $Z^{\prime }\in {\mathcal{Z}}_{\mathrm{N}\mathrm{U}\mathrm{I}}\left(X\right)$. Then, according to Lemma (8),

$$Z_{\sigma }{\widehat{X}}_{\sigma }-Z_{\sigma }^{\prime }{\widehat{X}}_{\sigma }^{\prime }\ge 0.$$

The fundamental price of the asset is then given by

$${\widehat{X}}_t=Z_t{\widehat{X}}_t{\mathbf{1}}_{t\le \sigma }+Z_t^{\prime }{\widehat{X}}_t^{\prime }{\mathbf{1}}_{t>\sigma },$$

and the bubble is

$${\tilde{B}}_t=Z_t^{\prime }{B}_t{\mathbf{1}}_{t>\sigma }.$$

Thus, a bubble is born at time $\sigma$. As shown in Lemma (7), a switch between uniformly integrable deflators does not alter the value of X, ensuring that no bubble exists. Bubble formation occurs only when the valuation measure transitions from a uniformly integrable martingale to a non-uniformly integrable martingale.

6. Conclusions

This paper introduces a novel framework for modeling price bubbles in financial markets using the calculus of optional semimartingales within nonstandard probability spaces. By challenging the conventional assumption that market filtrations must be right-continuous and complete, we propose two market models that extend classical bubble theory. The first model characterizes asset prices as càdlàg semimartingales, demonstrating how bubbles emerge through the progressive incorporation of external information into the filtration. The second model employs làdlàg optional semimartingales, illustrating how market structures can be redefined to accommodate discontinuities in price evolution and the reemergence of bubbles over time.

A key insight from this work is that price bubbles are not static phenomena but dynamic occurrences that arise as a consequence of market filtrations lacking right continuity. Unlike traditional approaches, which treat bubbles as binary states—either present or absent throughout an asset’s lifetime—our models reveal that bubbles can emerge, dissipate, and reappear depending on how external events are integrated into the available market information. This reformulation provides a more flexible and realistic framework for understanding financial instability, liquidity fluctuations, and systemic risks.

Beyond its theoretical contributions, this work has practical implications for both investors and regulatory frameworks. Market participants can use this framework to better identify the formation and persistence of bubbles, while regulators can apply these insights to assess systemic risks and enhance financial stability monitoring. The application of optional semimartingales to financial modeling broadens the mathematical toolkit available for analyzing asset price dynamics under incomplete information and discontinuous market processes.