An Optional Semimartingales Approach to Risk Theory

Abstract

This paper aims to develop optional semimartingale methods in risk theory to allow for a larger class of risk models. Optional semimartingales are left-continuous with right-limit stochastic processes defined on a probability space where the usual conditions—completeness and right-continuity of the filtration—are not assumed. Three risk models are formulated, accounting for inflation, interest rates, and claim occurrences. The first model extends the martingale approach to calculate ruin probabilities, the second employs the Gerber–Shiu function to evaluate the expected discounted penalty from financial oscillations or jumps, and the third introduces a Gaussian risk model using counting processes to capture premium and claim cash flow jumps in insurance companies.

1. Introduction

Mathematical risk theory is concerned with the study of stochastic models of risk in finance and insurance. In a basic risk model, the value of a risk portfolio is the sum of opposing cash flows: premium payments that increase the value of the portfolio and claim payouts that decrease it. Premium payments are received to cover liabilities such as expected losses from claim payouts and other costs. Claims result from risk events that occur at random times. An important problem in risk theory is calculating the probability of ruin and the expected discounted penalty at the time of ruin. These serve as metrics for assessing the likelihood of a negative portfolio value within an insurance company. “Ruin probability” refers to the likelihood that the risk portfolio will ever become negative, while the “time of ruin” refers to the moment when the portfolio value becomes negative.

Classical risk models typically rely on semimartingales with cadlag (right-continuous with left limits) sample paths. However, in many realistic financial and insurance settings, sudden jumps in cash flows—such as multiple claim arrivals in quick succession or abrupt premium adjustments—may violate the left-continuity assumption inherent in cadlag processes. Such events pose challenges when modeled in traditional probability spaces, which assume the completeness and right-continuity of the filtration. In contrast, our approach leverages the calculus of optional semimartingales, allowing us to construct probability spaces where these non-cadlag jumps are well-defined. This generalization enables us to extend and reformulate classical results, such as ruin probabilities and expected discounted penalties, within the broader framework of optional semimartingales.

The development of risk models has historically relied on martingale methods, which were first introduced in risk theory by Gerber in Gerber (1973). While these methods have led to significant advancements, they still operate within the classical probability space framework. Building on this foundation, we explore how optional semimartingales can further extend the scope of risk modeling by addressing the limitations of cadlag assumptions. Since then, these methods have become a standard technique, and numerous papers have been published using martingale methods to analyze increasingly complex risk models.

It is well known that risk models are often generalized in two key ways. The first type includes models that account for inflation and interest. In the second type, the occurrence of claims may be described by a more general point process than the Poisson process. While papers such as Delbaen and Haezendonck (1989), Dassios and Embrechts (1989), and Embrechts and Schmidli (1994) consider general risk models of the first type, the works of Grandell (2012) and Jacobsen (2012) focus primarily on generalizations of the second type. A recent comprehensive review of the literature can be found in Asmussen and Albrecher (2010).

Building on these foundations, more recent research in risk theory has aimed to refine ruin probability estimation and explore alternative modeling approaches. For example, Ieosanurak and Moumeesri (2025) proposed an approach for estimating ruin probability using claim simulation enhanced by the Wang-PH transform, allowing for better fitting of various right-skewed loss distributions. Their work provides a tool for assessing and mitigating financial risks in the non-life insurance sector, focusing on right-continuous processes. Similarly, Korzeniowski (2023) derived a closed-form expression for ruin probability in a discrete-time risk model with random premiums, leveraging the invariance principle for Brownian motion in a complete probability space. Another study in recent years is Huang et al. (2021), where ruin probability was estimated in a risk model with stochastic premium income using the complex Fourier series (CFS) expansion method. They employed the Gerber–Shiu function within a right-continuous filtration in a complete probability space. Additionally, further studies have been conducted in the Cramér–Lundberg model, such as Frihi et al. (2022), which examined asymptotic ruin probabilities in the presence of extreme values. Using the peaks-over-threshold method, they developed a new estimator for ruin probability in a risk process with heavy-tailed claim amounts of infinite variance, particularly for stationary claim arrivals over an infinite time horizon in a standard probability space.

Despite these advancements, existing models largely rely on standard probability spaces, where the completeness and right-continuity of the filtration are assumed. Our objective is to further develop new formulations in risk theory grounded in the framework of optional semimartingales within a probability space where the usual conditions—the completeness and right-continuity of the filtration—are not assumed, referred to as unusual probability spaces. We aim to derive optional local martingale representations that are not restricted to one-sided jumps. To our knowledge, no existing research has explored risk models in these probability spaces or investigated general optional risk models. By adopting this approach and building on the methods of Sorensen (1996), we seek to establish bounds for ruin probability, compute the expected discounted penalty at ruin, and analyze the characteristic equations of these models.

Given a probability space $\left(\Omega ,\mathcal{F},\mathbf{F}={\left({\mathcal{F}}_t\right)}_{t\ge 0},\mathbf{P}\right)$, the usual conditions refer to a non-decreasing family of $\sigma$-algebras ${\left({\mathcal{F}}_t\right)}_{t\ge 0}$ that is complete and right-continuous. Stochastic processes adapted to $\mathbf{F}$ are right-continuous with left limits (RCLL) semimartingales, with a well-defined stochastic calculus. In the mid-1970s, Dellacherie (1975) began studying stochastic processes without the usual conditions and referred to this framework as the unusual conditions. Further developments were carried out by many mathematicians, most notably by Galchuk in Galchuk (1980, 1985), who developed a theory of stochastic calculus for processes on unusual probability spaces. For a review of the stochastic calculus of processes on unusual probability spaces, along with current developments and applications, see the publications by Abdelghani and Melnikov (see, e.g., Abdelghani and Melnikov 2020).

The remaining sections of this paper are organized as follows. In Section 2, we present further developments of risk models based on the general theory of optional semimartingales on unusual probability spaces. First, we derive an optional local martingale representation that, unlike the approach in Sorensen (1996), is not limited to risk models with only negative jumps. Our formulation considers two-sided jumps, encompassing both negative and positive jumps. This leads to an optional risk model that, in fact, incorporates both types of generalizations mentioned earlier.

Optional semimartingales, on which our surplus process is based, admit trajectories that are not right-continuous and arise when usual conditions are not assumed on a filtered probability space. This generalization allows, for example, for the case where two consecutive jumps occur, with the second jump immediately following the first. Such behavior is not possible with cadlag semimartingales, as they cannot, by definition, have two consecutive jumps. To our knowledge, no current work explores the unusual conditions or investigates such general optional risk models. Essential concepts required to understand these discussions are provided in Appendix A.

We continue our exploration of semimartingale risk models in Section 2, focusing on the work of Gerber and Landry (1998). Gerber and his collaborators computed the expected discounted penalty at ruin for a risk model with three components: a linear function describing premiums, a jump-diffusion process characterizing claim processes, and an independent Wiener process modeling the random environment. In this paper, we extend their model by considering a case where premiums are described by a predictable process, claim processes are modeled as compound Poisson processes, and the random environment is modeled by a Gaussian optional martingale. This section concludes by developing a more general and complex structure of risk models, where both the premium and claim processes are counting processes, allowing the model to account for each jump in the calculation of risk measures.

In Section 3, we primarily focus on the work of Sorensen (1996), who first used classical semimartingale theory to establish bounds for ruin probability. In this section, we leverage the optional local martingale representation to compute the probability of ruin. To derive the results, we rely on the auxiliary results presented in Appendix B.

In Section 4, we analyze Lundberg’s equation and the renewal equation for a developed Gaussian risk model, ultimately calculating the expected discounted penalty at ruin. In Section 5, we explore the characteristic equation for the counting risk model and demonstrate that, given that the claim and premium processes have finite changes, it is an optional semimartingale. Therefore, we can apply the ruin probability bound derived for the optional risk model in Section 3. We conclude the paper by employing the developed risk models to provide examples, including cases where two consecutive jumps occur, with the second jump immediately following the first.

2. Optional Semimartingales Risk Models

As mentioned, an important problem in risk theory is the calculation of the probability of ruin and the time to ruin. “Ruin probability” is the likelihood that the risk portfolio, denoted by $U\left(t\right)$, will ever become negative, and “time to ruin”, denoted by $\tau$, is the time when $U\left(\tau \right)<0$. In a basic risk model, the portfolio’s capital value is

$$U\left(t\right)=u+B_t+Z_t.$$

where $u>0$ is the initial wealth, $B_t$ represents premium payments, and $Z_t$ represents claim payouts. In 1998, a diffusion risk model was studied by Gerber and Landry (1998), which generalizes the basic risk model by adding an independent diffusion process.

$$U\left(t\right)=u+B_t+Z_t+\sigma W_t$$

$Z_t$ is a compound Poisson process representing claims payouts, and $W_t$ is a standard Wiener process representing random environment with $\sigma >0$. In our study, we generalize these models and construct three risk models: the Optional Risk Model, the Gaussian Risk Model, and the Counting Risk Model.

2.1. Optional Risk Model

Let $\left(\Omega ,\mathcal{F},\mathbf{F}={\left({\mathcal{F}}_t\right)}_{t\ge 0},\mathbf{P}\right)$ be an unusual probability space on which surplus processes lie. Elements of a risk portfolio include, but are not limited to, the following components: premium payments, dividend payments, capital injections, liability payments, costs, and claims. These elements are inherently random. For example, some premium payments may not be made, costs may increase, and claims and liabilities may exacerbate. Furthermore, an additional risk arises when an insurance company operates overseas and some cash flows are made in different currencies (see Eisenberg and Palmowski 2020). Thus, we construct a surplus1 process whose cash flow can be summarized by the following equation:

$$U_t=u+B_t+N_t+D_t+L_t,$$

(1)

where $u>0$ is the initial capital and $B_0=W_0=D_0=L_0=0$. The process $B_t$ is a continuous predictable process of finite variation, characterizing a stable flow of income payments, including premiums and other sources; $N_t$ is a continuous local martingale representing a random perturbation. $D_t$ and $L_t$ are right-continuous and left-continuous jump processes, respectively. The process $L_t$ may model dividend payments or capital injections (where they were modeled by left-continuous increasing processes as suggested by Gajek and Kucinski 2017). The process $D_t$ includes a sum of negative jumps representing accumulated claims. In addition, $D_t$ may also consist of jumps formed by any non-anticipated shocks, such as those related to currency risk. All these processes are optional and adapted to the filtration ${\mathcal{F}}_t$.

Jump processes $D_t$ and $L_t$ are not restricted by their sign and can have positive or negative jumps in general. To consider an example of this model, assume that $U_t$ is a surplus process of some company at time t, and

$$U_t=u+ct+\sigma W_t-\sum _{k=1}^{N_t^r}{Y}_k+\sum _{i=1}^{N_t^g}{Z}_i$$

(2)

where c and $\sigma$ are some constant parameters; W is a Wiener process; and $N^r$ and $N^g$ are a Poisson process and a left-continuous modification of a Poisson process with intensities ${\lambda }^r$ and ${\lambda }^g$, respectively. $Z_i$ and $Y_k$ denote the left and right jump sizes, respectively, with some specified distributions. In this case, $ct$ is included in the income process B, $N_t=\sigma W_t$, and $L_t={\sum }_{i=1}^{N_t^g}{Z}_i$ and ${\sum }_{k=1}^{N_t^r}{Y}_k$ are a part of the process D.

2.2. Gaussian Risk Model

Another model that we construct is based on an optional Gaussian process. As we continue our examination of semimartingale risk models, we consider a general surplus process for an insurance company whose cash flow can be modeled by the following equation:

$$U\left(t\right)=u+B_t+M_t+Z_t$$

(3)

where $u>0$, $B={\left(B_t\right)}_{t\ge 0}$ is a predictable, non-decreasing optional process representing premium payments, and $Z={\left(Z_t\right)}_{t\ge 0}$ is a compound Poisson process with intensity $\lambda$ for aggregated claims, with a distribution function $P\left(x\right)$ for the jumps and a density $p\left(x\right)=P^{\prime }\left(x\right),x>0$. The process $M={\left(M_t\right)}_{t\ge 0}$ is an optional Gaussian martingale with stationary increments, independent of Z, and models the random environment with a density function $f\left(x\right)$. All these processes start at zero, i.e., $B_0=Z_0=M_0=0$.

Before proceeding further, we revisit the definition and characteristics of Gaussian martingales, as presented in Chapter 4 of Lipster and Shiryaev (1986). A stochastic process $X={\left(X_t\right)}_{t\ge 0}$ is called Gaussian if all of its finite-dimensional distributions are Gaussian. Furthermore, the following theorem highlights the properties of Gaussian martingales.

Theorem 1. 

Let X be a Gaussian local martingale such that $X_0=0$. Then, the following properties hold:

I.

X is a square-integrable martingale, and $\mathbf{E}{X}_t\equiv 0$.

II.

X has independent increments, and $\mathbf{E}{X}_t^2={⟨X⟩}_t$

III.

X can be decomposed as $X_t=X_t^c+X_t^d$, where $X^c$ denotes the continuous component and $X^d$ represents the discrete component. Together, $(X^c,X^d)$ form a Gaussian system.

IV.

The jump times of the trajectories of X are deterministic.

2.3. Counting Risk Model

The last formulated risk model describes the cash flow of insurance company by

$$U\left(t\right)=u+B_t+M_t-X_t$$

(4)

where $u>0$, $M={\left(M_t\right)}_{t\ge 0}$ is an optional Gaussian martingale with stationary increments, independent of X and B, modeling the random environment with a density function $f^M\left(x\right)$ and initial condition $M_0=0$. The premium process $B={\left(B_t\right)}_{t\ge 0}$ is a non-decreasing optional counting process, and $X={\left(X_t\right)}_{t\ge 0}$ is a counting process representing aggregated claims with jumps at random times.

Assume $N^X\left(t\right)$ represents the total number of claims up to time t with $N^X\left(0\right)=0$, and $N^X\left(t\right)={\sum }_{n\ge 1}{I}_{\{{\tau }_n<t\}},$ for $t\ge 0$, where claims occur at random stopping times ${\left({\tau }_n\right)}_{n\ge 1}$ such that

$$\left\{\begin{array}{cc}{\tau }_1>0,\hfill & P-a.s,\hfill \\ {\tau }_n<{\tau }_{n+1},\hfill & {\tau }_n<+\infty ,\hfill \\ {\tau }_n={\tau }_{n+1},\hfill & {\tau }_n=\infty .\hfill \end{array}\right.$$

(5)

Assume ${\left(x_n\right)}_{n\ge 1}$ is an identically independent distributed sequence with values in $\mathbb{N}$, representing the claim amounts with the following distribution function and expected value:

$$f_n^X=\mathbf{P}(\omega :x_i=n),{\mu }^X=\mathbb{E}\left[x_i\right].$$

Additionally, assume that $x_n$ and ${\tau }_n$ are independent for all $n\ge 1$. The risk process $X\left(t\right)$ is given by $X\left(t\right)={\sum }_{n=1}^{N^X\left(t\right)}{x}_n.$ To calculate the expectation of the risk process $X\left(t\right)$, we use Campbell’s formula for point processes, which states that the intensity measure of the point process $N^X\left(t\right)$ is defined as $\mathbb{E}\left[N^X\left(t\right)\right]={\lambda }^X\left(t\right),$ where ${\lambda }^X$ is measurable. Thus, the expectation of the risk process $X\left(t\right)$ is

$$\mathbb{E}\left[X\left(t\right)\right]={\mu }^X{\lambda }^X\left(t\right)$$

Similarly, we define the premium counting process. Assume $N^B\left(t\right)$ represents the total number of premiums up to time t with $N^B\left(0\right)=0$ and $N^B\left(t\right)={\sum }_{n\ge 1}{I}_{\{{\sigma }_n\le t\}},$ for $t\ge 0$, where premiums occur at random stopping times ${\left({\sigma }_n\right)}_{n\ge 1}$ with the same properties as mentioned in (5). Let ${\left(b_n\right)}_{n\ge 1}$ be an identically independent distributed sequence with values in $\mathbb{N}$, representing premium amounts, with the following distribution function and expected value:

$$f_n^B=\mathbf{P}(\omega :b_i=n),{\mu }^B=\mathbb{E}\left[b_i\right].$$

Assume that $b_n$ and ${\sigma }_n$ for $n\ge 1$ are independent for all $n\ge 1$. The premium process $B\left(t\right)$ is given by $B\left(t\right)={\sum }_{n=1}^{N^B\left(t\right)}{b}_n.$ Assuming ${\lambda }^B$ is the intensity measure of the point process $N^B\left(t\right)$, the expectation of the premium process $B\left(t\right)$ is $\mathbb{E}\left[B\left(t\right)\right]={\mu }^B{\lambda }^B\left(t\right)$. Finally, we can rewrite the counting risk model as follows:

$$U\left(t\right)=u+\sum _{n=1}^{N^B\left(t\right)}{b}_n+M_t-\sum _{n=1}^{N^X\left(t\right)}{x}_n.$$

3. Probability of Ruin

In our risk analysis, we explore the probability of the time of ruin, denoted by $\mathbf{P}(\tau <\infty )$, based on the optional risk model introduced by Equation (1), where

$$\tau =inf\left\{t>0:U_t<0\right\}$$

To facilitate this investigation, we formulate a theorem under the following assumptions:

I.

Negative and positive jumps are required to be bounded, which is satisfied by

$${\int }_{]0,t]}{\int }_{{\mathbb{R}}_0}xd{\mu }^r\in {\mathcal{A}}_{loc}\mathrm{and}{\int }_{[0,t[}{\int }_{{\mathbb{R}}_0}xd{\mu }^g\in {\mathcal{A}}_{loc},$$

(6)

where ${\mu }^r(\omega ,dt,dx)$ and ${\mu }^g(\omega ,dt,dx)$ are random measures describing jumps of the processes $D_t$ and $L_t$, respectively, on $(\mathcal{B}\left({\mathbb{R}}_{+}\right),\mathcal{B}\left({\mathbb{R}}_0\right))$, with ${\mathbb{R}}_0=\mathbb{R}\setminus \left\{0\right\}$. These measures are defined as

$$\begin{array}{c}{\mu }^r(dt,dx)=\sum _{s>0}{I}_{\{\Delta D_s\ne 0\}}{\delta }_{(s,\Delta D_s)}(dt,dx),\\ {\mu }^g(dt,dx)=\sum _{s\ge 0}{I}_{\{{\Delta }^{+}{L}_s\ne 0\}}{\delta }_{(s,{\Delta }^{+}{L}_s)}(dt,dx),\end{array}$$

where $I_{\omega }$ is the indicator function of a set $\omega$, and ${\delta }_{(s,y)}(dt,dx)$ denotes the Dirac measure.

II.

The negative and positive jumps must be integrable with respect to their compensated measures. This holds if $z_0>0$ exists such that

$${\int }_{]0,t]}{\int }_{-\infty }^{-1}{e}^{-zx}d{\nu }^r<\infty ,{\int }_{[0,t[}{\int }_{-\infty }^{-1}{e}^{-zx}d{\nu }^g<\infty$$

(7)

holds almost surely for all $t>0$ and $0<z\le z_0$. Here, ${\nu }^r,{\nu }^g$ are compensators of ${\mu }^r,{\mu }^g$, respectively.

Furthermore, we formulate the optional cumulant process for the surplus process U from Equation (1). For all $t>0$ and $z\in (0,z_0]$, the cumulant process $G_t\left(z\right)$ is given by:

$$\begin{array}{cc}\hfill G_t\left(z\right)& =-z{B}_t+{\displaystyle \frac{z^2}{2}}{⟨N⟩}_t+{\int }_{]0,t]}{\int }_{{\mathbb{R}}_0}(e^{-zx}-1)d{\nu }^r\hfill \\ & +{\int }_{[0,t[}{\int }_{{\mathbb{R}}_0}(e^{-zx}-1)d{\nu }^g\hfill \end{array}$$

The stochastic exponential of the cumulant process, denoted by $\mathcal{E}\left(G\right)$ under the conditions such as $\Delta G>-1$ and ${\Delta }^{+}G>-1$, can be represented as

$$\begin{array}{c}\hfill {\mathcal{E}}_t\left(G\right)=exp(G_t+\sum _{0<s\le t}(log(1+\Delta G_s)-\Delta G_s)\\ \hfill +\sum _{0\le s<t}(log(1+{\Delta }^{+}{G}_s)-{\Delta }^{+}{G}_s))\end{array}$$

With these assumptions and definitions in place, we present the following theorem:

Theorem 2. 

Given the optional risk model (1), suppose assumptions (6) and (8) hold.

1.

If $\Delta G>-1$ and ${\Delta }^{+}G>-1$, then

$$\mathbf{P}(\tau \le t)\le {\displaystyle \frac{e^{-zu}}{\mathbf{E}\left({\mathcal{E}}_{\tau }^{-1}\left[G\left(z\right)\right]\right|\tau \le t)}}$$

for all $z\in (0,z_0]$ and $t>0$.

2.

If $G_{\tau }^{\prime }\left(0\right)<0$, $\Delta G=0$, and ${\Delta }^{+}G=0$, then there is a unique $z^{*}\in (0,z_0]$ for which the right-hand side of

$$\mathbf{P}(\tau \le t)\le {\displaystyle \frac{e^{-z^{*}u}}{\mathbf{E}(exp[-G_{\tau }\left(z^{*}\right)]|\tau \le t)}}$$

attains its minimum.

3.

If $z_t>0$ exists for which $\mathbf{E}\left({\mathcal{E}}_{\tau }^{-1}\left[G\left(z_t\right)\right]\right|\tau \le t)=1$, then

$$\mathbf{P}(\tau \le t)\le e^{-u{z}_t}.$$

Moreover, if $z_t$ exists for all $t>0$ and if $\widehat{z}={lim}_{t\to \infty }{z}_t$ exists, then

$$\mathbf{P}(\tau <\infty )\le e^{-u\widehat{z}}.$$

Proof. 

Given U, our main goal is to evaluate the ruin probability $\mathbf{P}(\tau <\infty )$, where $\tau =inf\left\{t>0:U_t<0\right\}$. Assume that there is $z_0>0$ such that

$${\int }_{]0,t]}{\int }_{-\infty }^{-1}{e}^{-zx}d{\nu }^r<\infty ,{\int }_{[0,t[}{\int }_{-\infty }^{-1}{e}^{-zx}d{\nu }^g<\infty$$

(8)

almost surely for all $t>0$ and $0<z\le z_0$. Define the optional cumulant process for the surplus process U with the same domain for t and z:

$$\begin{array}{c}\hfill G_t\left(z\right)=-z{B}_t+{\displaystyle \frac{z^2}{2}}{⟨N⟩}_t+{\int }_{]0,t]}{\int }_{{\mathbb{R}}_0}(e^{-zx}-1)d{\nu }^r+{\int }_{[0,t[}{\int }_{{\mathbb{R}}_0}(e^{-zx}-1)d{\nu }^g.\end{array}$$

Define a process

$$M_t\left(z\right)=exp[-z(U_t-u)]{\mathcal{E}}_t^{-1}\left(G\left(z\right)\right).$$

(9)

It follows from Theorem A1 in Appendix B that the process $M_t\left(z\right)$ is an optional local martingale for every $z\in (0,z_0]$, if $\Delta G>-1$ and ${\Delta }^{+}G>-1$. We use the optional local martingale M in a manner similar to Sorensen (1996), which is standard in risk theory. We know that a non-negative optional local martingale is an optional supermartingale (see Lemma A4 in Appendix A). Thus, since $M_0\left(z\right)=1$, it follows that

$$1\ge \mathbf{E}\left(M_{\tau \wedge t}\left(z\right)\right)=\mathbf{E}(M_{\tau }\left(z\right)I_{\tau \le t}+M_t\left(z\right)I_{\tau >t})\ge \mathbf{E}\left(M_{\tau }\left(z\right)I_{\tau \le t}\right)=\mathbf{E}\left(M_{\tau }\left(z\right)\right|\tau \le t)\mathbf{P}(\tau \le t)$$

for every z in $(0,z_0]$ and $t>0$. Hence,

$$\mathbf{P}(\tau \le t)\le {\displaystyle \frac{1}{\mathbf{E}\left(M_{\tau }\right|\tau \le t)}}={\displaystyle \frac{1}{\mathbf{E}\left(e^{-z(U_{\tau }-u)}{\mathcal{E}}_{\tau }^{-1}\left(G\left(z\right)\right)\right|\tau \le t)}}.$$

Since $U_{\tau }\le 0$ on $\{\tau \le t\}$, we obtain

$$\mathbf{P}(\tau \le t)\le {\displaystyle \frac{e^{-zu}}{\mathbf{E}\left({\mathcal{E}}_{\tau }^{-1}\left[G\left(z\right)\right]\right|\tau \le t)}}$$

(10)

for all $z\in (0,z_0]$. At this point, if we want to obtain a better estimate of the upper bound of the ruin probability in (10), we must impose additional assumptions. Firstly, suppose that

$$-G_{\tau }^{\prime }\left(0\right)=B_{\tau }+{\int }_{]0,\tau ]}{\int }_{{\mathbb{R}}_0}xd{\nu }_s^r+{\int }_{[0,\tau [}{\int }_{{\mathbb{R}}_0}xd{\nu }_{s+}^g>0.$$

(11)

Note that differentiation with respect to z under the integral sign with respect to ${\nu }^j,j=r,g,$ in (11) is possible because zero is an interior point in the range of z-values for which the integral exists. If we assume that $\Delta G=0$ and ${\Delta }^{+}G=0$ (see Remark 4), then

$${\mathcal{E}}_t\left(G\left(z\right)\right)=exp\left(G_t\left(z\right)\right).$$

(12)

Using (12) and Jensen’s inequality, we obtain from (10) that

$$\begin{array}{cc}\hfill \mathbf{P}(\tau \le t)& \le e^{-zu}{\left[\mathbf{E}\left({\mathcal{E}}_{\tau }^{-1}\left[G\left(z\right)\right]|\tau \le t\right)\right]}^{-1}\hfill \\ \hfill & \le e^{-zu}\mathbf{E}\left({\mathcal{E}}_{\tau }\left[G\left(z\right)\right]\right|\tau \le t)\hfill \\ & =\mathbf{E}(exp[-zu+G_{\tau }\left(z\right)]|\tau \le t),\hfill \end{array}$$

(13)

for all $z\in (0,z_0]$ and $t>0$. The function $G_{\tau }\left(z\right)$ is a strictly convex function of z with $G_{\tau }\left(0\right)=0$. Further, due to assumption (11), the function $\mathbf{E}(exp[-zu+G_{\tau }\left(z\right)]|\tau \le t)$ is a strictly convex function of z decreasing from 1 at $z=0$. Furthermore, under assumption (8) with $z_0=\infty$, the function $\mathbf{E}(exp[-zu+G_{\tau }\left(z\right)]|\tau \le t)$ increases to $+\infty$ as $z\to \infty$. Thus, the function in the right-hand side of (13) has a unique minimum $z^{*}\in (0,z_0]$; therefore, we have

$$\mathbf{P}(\tau \le t)\le {\displaystyle \frac{e^{-z^{*}u}}{\mathbf{E}(exp[-G_{\tau }\left(z^{*}\right)]|\tau \le t)}}.$$

(14)

We can find a more explicit estimate of the ruin probability if we can choose the z-value $z_t>0$, for which the denominator of (10) equals one. Note that this z-value is not necessarily unique, and it often depends on t. In this case, we obtain

$$\mathbf{P}(\tau \le t)\le e^{-u{z}_t}.$$

Further, if $z_t$ exists for all $t>0$ and if $\widehat{z}={lim}_{t\to \infty }{z}_t$ exists, then

$$\mathbf{P}(\tau <\infty )\le e^{-u\widehat{z}}.$$

□

Remark 1. 

The inequality (3) is analogous to the classical Cramér–Lundberg bound (see Grandell 1991), obtained in a very general optional setting.

Remark 2. 

If the process D represents the accumulated claims (with all jumps being downward) while the process $L=0$, then the condition (11) simplifies to

$$-G_t^{\prime }\left(0\right)=B_t+{\int }_{]0,t]}{\int }_{-\infty }^{0}xd{\nu }^r>0$$

for all $t>0$, which is known as the net-profit condition in risk theory. This condition implies that the insurance company adopts a prudent premium policy, ensuring that premiums follow the claims’ intensity.

Remark 3. 

Note that in literature (see, for example, Asmussen and Albrecher 2010, p. 338), such a surplus process R is referred to as a process with light-tailed negative jumps.

Remark 4. 

By Lemma A5 in Appendix B, the condition

$$\Delta G=\int (e^{-zx}-1){\nu }^r(\left\{t\right\},dx)=0$$

is satisfied if $\Delta U_T=0$ almost surely on the set $\{T<\infty \}$ for every predictable stopping time T (i.e., the quasi-left-continuity of U; see Lipster and Shiryaev 1986).

On the other hand, the condition

$${\Delta }^{+}G=\int (e^{-zx}-1){\nu }^g(\left\{t\right\},dx)=0$$

is satisfied if ${\Delta }^{+}{U}_T=0$ almost surely on the set $\{T<\infty \}$ for every stopping time T (i.e., the quasi-right-continuity of U).

From the perspective of risk theory, this means that negative jumps (such as claims) or positive jumps (such as capital injections) in the surplus process cannot be predicted beforehand, which aligns with practical observations.

4. Expected Discounted Penalty

Another aspect of our risk theory analysis involves exploring the expected discounted value of the penalty at ruin, which is based on the Gaussian risk model introduced in Equation (3). Ruin can occur in two ways: either through oscillation, where $U\left(\tau \right)=0$, or through a jump, where $U\left(\tau \right)<0$.

Considering a penalty scheme suggested by Gerber and Landry (1998), which uses a constant ${\omega }_0$ and a function $\omega (-y)$ for $y>0$, the penalty is equal to ${\omega }_0$ if ruin occurs by oscillation and to $\omega \left(U\right(\tau \left)\right)$ if ruin occurs through a jump. Therefore, we are interested in the following expected value at risk:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \Phi \left(u\right)={\omega }_0\mathbf{E}\left(e^{-\delta \tau }{I}_{(\tau <\infty ,U(\tau )=0)}|U\left(0\right)=u\right)+\\ \hfill \mathbf{E}\left(e^{-\delta \tau }\omega \left(U\left(\tau \right)\right)I_{(\tau <\infty ,U(\tau )<0)}|U\left(0\right)=u\right)\end{array}\end{array}$$

(15)

Our approach to computing the expected discounted penalty at ruin begins with an examination of an optional version of Lundberg’s equation. To determine the expected discounted penalty or characteristic equation of a risk model, we first need to define Lundberg’s equation.

Definition 1. 

Given $\delta >0$, we are in search of a value ξ such that the process $\left\{e^{-\delta t+\xi U\left(t\right)},t\ge 0\right\}$, where $U\left(t\right)$ is a risk model with $U\left(0\right)=u$, which is an optional martingale.

With this equation properly defined, we proceed to derive a renewal equation for the model. This leads to the formulation of the following theorem for calculating the expected discounted penalty at ruin. The operator ∗ represents the convolution operator, defined as:

$$\Phi \ast g\left(u\right)={\int }_0^u\Phi \left(x\right)g(u-x)dx$$

Theorem 3. 

Given the Gaussian risk model (3), suppose that the following Lundberg equation has a unique positive solution $\xi =\rho$:

$${\displaystyle \frac{1}{2}}{\xi }^2⟨M_1⟩-\delta +\xi B_1+\lambda \left[{\int }_0^{\infty }{e}^{-\xi x}p\left(x\right)dx-1\right]=0$$

Then,

I.

When ruin occurs by oscillation, the penalty function $\Phi \left(u\right)$ satisfies the following renewal equation:

$$\tilde{M}{\Phi }^{″}\left(u\right)+\tilde{B}{\Phi }^{\prime }\left(u\right)+\lambda {\int }_0^u\Phi (u-y)p\left(y\right)dy-(\lambda +\delta )\Phi \left(u\right)=0$$

where $\tilde{M}={\displaystyle \frac{1}{2}}{\displaystyle \frac{{⟨M⟩}_t}{t}}$ and $\tilde{B}={\displaystyle \frac{B\left(t\right)}{t}}$. The solution is given by:

$$\Phi \left(u\right)={\omega }_0{e}^{-\beta v}+\Phi \ast g\left(u\right)$$

(16)

where $\beta =\alpha -\rho$, $\alpha ={\displaystyle \frac{\tilde{B}}{\tilde{M}}}+2\rho$, and

$$g\left(z\right)={\displaystyle \frac{\lambda }{\tilde{M}}}{e}^{\rho z}{\int }_0^z{e}^{-\alpha (z-s)}{\int }_s^{\infty }{e}^{-\rho y}p\left(y\right)dyds$$

II.

When ruin is caused by both jump and oscillation2, the penalty function $\Phi \left(u\right)$ is given by:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \Phi \left(u\right)& ={\int }_0^u\Phi (u-y)g\left(y\right)dy+{\omega }_0{e}^{-\beta u}\hfill \\ \hfill & +{\int }_u^{\infty }\omega (u-y)g\left(y\right)dy-e^{-\beta u}{\int }_0^{\infty }\omega (-y)g\left(y\right)dy\hfill \end{array}\end{array}$$

Proof. 

First, we need to find the solution to Lundberg’s equation as defined in Definition 1. To find $\xi$, define $V\left(t\right)=e^{-\delta t+\xi U\left(t\right)}$. It is evident that $V\left(t\right)$ is a martingale if $\mathbf{E}\left[V\right(t\left)\right]=V\left(0\right)$ for any $t\ge 0$. Given that ${\left(U\left(t\right)\right)}_{t\ge 0}$ is an optional semimartingale with stationary and independent increments, this is equivalent to the condition that:

$$e^{\xi u}{e}^{-\delta }\mathbf{E}\left[e^{\xi B_1+\xi M_1+\xi Z_1}\right]=e^{\xi u}$$

Since M is independent of Z and B is a deterministic process, we can simplify this equality as follows:

$$e^{-\delta +\xi B_1}\mathbf{E}\left[e^{\xi M_1}\right]\mathbf{E}\left[e^{\xi Z_1}\right]=1$$

Due to Theorem 1, we can state that $V\left(t\right)$ is a martingale if:

$${\displaystyle \frac{1}{2}}{\xi }^2⟨M_1⟩-\delta +\xi B_1+\lambda \left[{\int }_0^{\infty }{e}^{-\xi x}p\left(x\right)dx-1\right]=0$$

(17)

This equation has a unique positive solution $\xi =\rho$, which plays a crucial role in the following. Next, consider the case where there is a penalty of ${\omega }_0$ if ruin occurs by oscillation, and no penalty if ruin is caused by a jump, that is, $\omega (-y)\equiv 0$. Thus, we show that the function:

$$\Phi \left(u\right)={\omega }_0\mathbf{E}\left(e^{-\delta \tau }{I}_{(\tau <\infty ,U(\tau )=0)}|U\left(0\right)=u\right)$$

(18)

satisfies the following differential equation:

$$\tilde{M}{\Phi }^{″}\left(u\right)+\tilde{B}{\Phi }^{\prime }\left(u\right)+\lambda {\int }_0^u\Phi (u-y)p\left(y\right)dy-(\lambda +\delta )\Phi \left(u\right)=0$$

(19)

where $\tilde{M}={\displaystyle \frac{1}{2}}{\displaystyle \frac{{⟨M⟩}_t}{t}}$ and $\tilde{B}={\displaystyle \frac{B\left(t\right)}{t}}$. During the short time interval from 0 to $dt$, the discount factor is $1-\delta dt$. The process ${\left(Z\left(t\right)\right)}_{t\ge 0}$ has no jumps with probability $1-\lambda dt$ or has exactly one jump with probability $\lambda dt$. By conditioning on this, taking into account the size of the jump and the values of the processes M and B at time $dt$ (including the magnitude of the jump and the value of $M\left(dt\right)$), we can rewrite (18) as follows:

$$\Phi \left(u\right)=(1-\lambda dt)(1-\delta dt)\mathbf{E}(\Phi (u+B\left(dt\right)+M\left(dt\right)))+\lambda dt{\int }_0^u\Phi (u-y)p\left(y\right)dy$$

(20)

Proceed to calculate $\mathbf{E}(\Phi (u+B\left(dt\right)+M\left(dt\right)\left)\right)$. Based on the definition, M is a Gaussian martingale and thus follows a Gaussian distribution with a probability density function given by

$$f\left(x\right)={\displaystyle \frac{1}{{⟨M⟩}_t\sqrt{2\pi }}}exp\{-{\displaystyle \frac{1}{2}}{\displaystyle \frac{x^2}{\sqrt{{⟨M⟩}_t}}}\}$$

Therefore,

$$\begin{array}{c}\hfill \mathbf{E}(\Phi (u+B\left(dt\right)+M\left(dt\right)))=\Phi \left(u\right)+B\left(dt\right){\Phi }^{\prime }\left(u\right)+{\displaystyle \frac{1}{2}}{⟨M⟩}_t{\Phi }^{″}\left(u\right)\end{array}$$

By substituting this into Equation (20) and simplifying, we obtain:

$$\begin{array}{c}\hfill B\left(dt\right){\Phi }^{\prime }\left(u\right)+{\displaystyle \frac{1}{2}}{⟨M⟩}_t{\Phi }^{″}\left(u\right)-(\lambda +\delta )dt\Phi \left(u\right)+\lambda dt{\int }_0^u\Phi (u-y)p\left(y\right)dy=0\end{array}$$

(21)

Dividing this equation by $dt$, we arrive at the differential Equation (19). The next step is to convert this equation into a renewal equation. Integrate this equation twice, following the approach used in Gerber and Landry (1998). To proceed, multiply the differential Equation (19) by $e^{-\rho u}$ to obtain:

$$\tilde{M}{e}^{-\rho u}{\Phi }^{″}\left(u\right)+\tilde{B}{e}^{-\rho u}{\Phi }^{\prime }\left(u\right)+\lambda e^{-\rho u}{\int }_0^u\Phi (u-y)p\left(y\right)dy-(\lambda +\delta )e^{-\rho u}\Phi \left(u\right)=0$$

(22)

We can rewrite this equation in terms of the function:

$${\Phi }_{\rho }\left(u\right)=e^{-\rho u}\Phi \left(u\right)$$

(23)

with the following properties:

• ${\Phi }_{\rho }\left(0\right)={\omega }_0$
• ${\Phi }_{\rho }^{\prime }\left(u\right)=-\rho {\Phi }_{\rho }\left(u\right)+e^{-\rho u}{\Phi }^{\prime }\left(u\right)$
• ${\Phi }_{\rho }^{″}\left(u\right)={\rho }^2{\Phi }_{\rho }\left(u\right)-2\rho e^{-\rho u}{\Phi }^{\prime }\left(u\right)+e^{-\rho u}{\Phi }^{″}\left(u\right)$
• ${\Phi }_{\rho }\left(u\right)\to 0$ as $u\to \infty$. Also, ${\Phi }_{\rho }^{\prime }\left(u\right)\to 0$ as $u\to \infty$

Thus, we can rewrite (22) as follows:

$$\begin{array}{cc}\hfill & \tilde{M}{\Phi }_{\rho }^{″}\left(u\right)+(\tilde{B}+2\rho \tilde{M}){\Phi }_{\rho }^{\prime }\left(u\right)+\lambda {\int }_0^u{\Phi }_{\rho }(u-y)e^{-\rho y}p\left(y\right)dy+\hfill \\ \hfill & (\tilde{M}{\rho }^2+\tilde{B}\rho -\lambda -\delta ){\Phi }_{\rho }\left(u\right)=0\hfill \end{array}$$

Given that $\rho$ satisfies Lundberg’s Equation (17), we can reformulate this equation as follows:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & \tilde{M}{\Phi }_{\rho }^{″}\left(u\right)+(\tilde{B}+2\rho \tilde{M}){\Phi }_{\rho }^{\prime }\left(u\right)+\lambda {\int }_0^u{\Phi }_{\rho }(u-y)e^{-\rho y}p\left(y\right)dy\hfill \\ \hfill & -\lambda {\Phi }_{\rho }\left(u\right){\int }_0^{\infty }{e}^{-\rho y}p\left(y\right)dy=0\hfill \end{array}\end{array}$$

(24)

To proceed, integrate Equation (24) from $u=0$ to $u=z$

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & \tilde{M}({\Phi }_{\rho }^{\prime }\left(z\right)-{\Phi }_{\rho }^{\prime }\left(0\right))+(\tilde{B}+2\rho \tilde{M})({\Phi }_{\rho }\left(z\right)-{\omega }_0)\hfill \\ \hfill & +\lambda {\int }_0^z{\int }_0^u{\Phi }_{\rho }(u-y)e^{-\rho y}p\left(y\right)dydu-\lambda {\int }_0^z{\Phi }_{\rho }\left(u\right){\int }_0^{\infty }{e}^{-\rho y}p\left(y\right)dydu=0\hfill \end{array}\end{array}$$

(25)

To evaluate the double integrals, initiate a change of variables in the first double integral by substituting u with $x=u-y$. Additionally, in the second integral, note that the inner integral is independent of u. Thus,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & \lambda {\int }_0^z{\Phi }_{\rho }\left(x\right){\int }_0^{z-x}{e}^{-\rho y}p\left(y\right)dydx-\lambda {\int }_0^z{\Phi }_{\rho }\left(x\right)dx{\int }_0^{\infty }{e}^{-\rho y}p\left(y\right)dy\hfill \\ \hfill & =-\lambda {\int }_0^z{\Phi }_{\rho }\left(x\right){\int }_{z-x}^{\infty }{e}^{-\rho y}p\left(y\right)dydx\hfill \end{array}\end{array}$$

(26)

By substituting the double integrals from Equation (25) into (26), we obtain

$$\tilde{M}({\Phi }_{\rho }^{\prime }\left(z\right)-{\Phi }_{\rho }^{\prime }\left(0\right))+(\tilde{B}+2\rho \tilde{M})({\Phi }_{\rho }\left(z\right)-{\omega }_0)-\lambda {\int }_0^z{\Phi }_{\rho }\left(x\right){\int }_{z-x}^{\infty }{e}^{-\rho y}p\left(y\right)dydx=0$$

(27)

Taking the limit as z approaches infinity, we obtain the following equality:

$$\tilde{M}{\Phi }_{\rho }^{\prime }\left(0\right)+(\tilde{B}+2\rho \tilde{M}){\omega }_0=0$$

(28)

This equality helps simplify Equation (27) as follows:

$$\tilde{M}{\Phi }_{\rho }^{\prime }\left(z\right)+(\tilde{B}+2\rho \tilde{M}){\Phi }_{\rho }\left(z\right)=\lambda {\int }_0^z{\Phi }_{\rho }\left(x\right){\int }_{z-x}^{\infty }{e}^{-\rho y}p\left(y\right)dydx$$

(29)

It is important to note that this is a first-order differential equation that can be solved using integrating factors. To facilitate this process, we need to ensure that the coefficient of the derivative is equal to one. Therefore, we divide Equation (29) by $\tilde{M}$ and calculate the integrating factor using the notation $\alpha ={\displaystyle \frac{\tilde{B}}{\tilde{M}}}+2\rho$:

$$\mu \left(z\right)=exp\left({\int }_0^z\alpha dz\right)=e^{\alpha z}$$

By multiplying the version of Equation (29) (divided by $\tilde{M}$) by the integrating factor $\mu \left(z\right)$, and integrating from 0 to v, we obtain the following simplified version:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\Phi }_{\rho }\left(v\right)={\omega }_0{e}^{-\alpha v}+{\displaystyle \frac{\lambda }{\tilde{M}}}{\int }_0^v{e}^{-\alpha (v-z)}{\int }_0^z{\Phi }_{\rho }\left(x\right){\int }_{z-x}^{\infty }{e}^{-\rho y}p\left(y\right)dydxdz\end{array}\end{array}$$

(30)

By changing the order of integration, and replacing z with new variable $s=z-x$, we can simplify further and obtain:

$${\Phi }_{\rho }\left(v\right)={\omega }_0{e}^{-\alpha v}+{\displaystyle \frac{\lambda }{\tilde{M}}}{\int }_0^v{\Phi }_{\rho }\left(x\right){\int }_0^{v-x}{e}^{-\alpha (v-s-x)}{\int }_s^{\infty }{e}^{-\rho y}p\left(y\right)dydsdx$$

Since ${\Phi }_{\rho }\left(v\right)=e^{-\rho v}\Phi \left(v\right)$, we have:

$$\Phi \left(v\right)={\omega }_0{e}^{-v(\alpha -\rho )}+{\displaystyle \frac{\lambda }{\tilde{M}}}{\int }_0^v\Phi \left(x\right)e^{\rho (v-x)}{\int }_0^{v-x}{e}^{-\alpha (v-s-x)}{\int }_s^{\infty }{e}^{-\rho y}p\left(y\right)dydsdx$$

(31)

By denoting $\beta =\alpha -\rho$, this result can be written more succinctly:

$$\Phi \left(v\right)={\omega }_0{e}^{-\beta v}+\Phi \ast g\left(v\right)$$

(32)

where

$$g\left(z\right)={\displaystyle \frac{\lambda }{\tilde{M}}}{e}^{\rho z}{\int }_0^z{e}^{-\alpha (z-s)}{\int }_s^{\infty }{e}^{-\rho y}p\left(y\right)dyds$$

(33)

It is worth noting that Equation (32) is a renewal equation for the function $\Phi \left(v\right)$, and it can be solved using renewal theory techniques. Additionally, we can express the function g as a convolution of two functions:

$$h\left(s\right)=e^{-\beta s},\gamma \left(s\right)={\displaystyle \frac{\lambda }{\tilde{M}}}{e}^{\rho s}{\int }_s^{\infty }{e}^{-\rho y}p\left(y\right)dy$$

(34)

Thus, we can rewrite (32) as follows:

$$\Phi \left(v\right)={\omega }_0{e}^{-\beta v}+\Phi \ast h\ast \gamma \left(v\right)$$

(35)

This represents, as previously mentioned, the expected discounted penalty. According to the terminology of Feller (1968), Equation (32) qualifies as a defective renewal equation if ${\int }_0^{\infty }g\left(y\right)dy<1$. To prove this, consider the following:

$${\int }_0^{\infty }h\left(s\right)ds={\displaystyle \frac{\tilde{M}}{\tilde{B}+\rho \tilde{M}}}$$

(36)

By definition $\rho$, we have

$${\int }_0^{\infty }\gamma \left(s\right)ds={\displaystyle \frac{\tilde{M}{\rho }^2+\tilde{B}\rho -\delta }{\tilde{M}\rho }}$$

(37)

Hence,

$${\int }_0^{\infty }g\left(y\right)dy=\left({\int }_0^{\infty }h\left(s\right)ds\right)\left({\int }_0^{\infty }\gamma \left(s\right)ds\right)=1-{\displaystyle \frac{\delta }{\tilde{B}\rho +\tilde{M}{\rho }^2}}<1$$

(38)

Keep in mind that Equation (32) is specific to the scenario where a penalty is incurred only if ruin happens through oscillation. Consider the more general case where a penalty of ${\omega }_0$ is imposed if ruin occurs through oscillation, and $\omega (-y)$ is applied if ruin results from a jump. Here, y represents the deficit in the event of ruin, which can be calculated as the difference between the initial capital u and the capital at the time of the jump, denoted $U\left({\tau }_j\right)$ if the jump occurs at ${\tau }_j$.

Returning to the expected discounted penalty as defined in (15), we can follow the same approach as in the previous case, along with a probabilistic interpretation:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \Phi \left(u\right)& ={\int }_0^u\Phi (u-y)g\left(y\right)dy+{\omega }_0{e}^{-\beta u}\hfill \\ \hfill & +{\int }_u^{\infty }\omega (u-y)g\left(y\right)dy-e^{-\beta u}{\int }_0^{\infty }\omega (-y)g\left(y\right)dy\hfill \end{array}\end{array}$$

(39)

This decomposition can be achieved by conditioning as follows: The first term reflects the case where the deficit y is caused by the first jump being positive. The second term represents the scenario where ruin occurs before the first jump. The third and fourth terms account for the situation in which ruin occurs because of a jump in the compound Poisson process. The third term also includes an additional contribution for the case where ruin occurs through oscillation before the first jump, and the fourth term serves as the corresponding compensation. For a more comprehensive explanation of the probabilistic approach and further details, see Gerber and Landry (1998). □

Remark 5. 

It is important to highlight that $e^{-\beta v}$ represents the expected discounted value of a contingent claim, which is a payment of 1 due at ruin.

5. Characteristics Equation and Particular Models and Examples

An important aspect of theoretical analysis involves examining the characteristic equation for the probability of survival over a finite interval, denoted by $\phi (t,u)$, and an infinite interval, denoted by $\phi \left(t\right)$, as defined below.

$$\begin{array}{c}\hfill \phi (t,u)=\mathbf{P}\left(\right\{\omega :U\left(s\right)>0,\forall s\le t\left\}\right),U\left(0\right)=u\\ \hfill \phi \left(t\right)=\mathbf{P}\left(\right\{\omega :U\left(t\right)>0,\forall t\ge 0\left\}\right),U\left(0\right)=u\end{array}$$

Our aim is to conduct this analysis to examine the probability of ruin as a measure of risk exposure in cases where the premium process exhibits a complex structure based on the counting risk model introduced in Equation (4). First, assume that $\mathbb{E}\left[B\right(t\left)\right]>\mathbb{E}\left[X\right(t\left)\right]$, meaning that the pure income is positive. From the calculation provided in Section 2.3, we have

$${\mu }^B{\lambda }^B\left(t\right)>{\mu }^X{\lambda }^X\left(t\right),$$

which is the condition for positive pure income. Based on the Cramer–Lundberg inequality:

$$\phi \left(t\right)\le e^{-Rt}$$

where R is the positive solution to Lundberg’s equation defined in Definition 1. By solving this equation, we determine that the constant R satisfies the following characteristic equation:

$${\displaystyle \frac{1}{2}}{R}^2{⟨M⟩}_t+{\lambda }^B\left(t\right)\mathbb{E}\left[e^{-R{b}_i}\right]+{\lambda }^X\left(t\right)\mathbb{E}\left[e^{R{x}_i}\right]={\lambda }^B\left(t\right)+{\lambda }^X\left(t\right).$$

Another aspect of risk analysis, as previously discussed, is calculating the probability of ruin—or, more precisely, the probability of the time of ruin. We determined this probability for the optional risk model in Theorem 2. We aim to calculate the probability of the time of ruin for the counting risk model given in (4). To accomplish this, we first need to prove that the surplus process described by the counting risk model is an optional semimartingale process. To remind the reader, this surplus process is given by:

$$U\left(t\right)=u+\sum _{n=1}^{N^B\left(t\right)}{b}_n+M_t-\sum _{n=1}^{N^X\left(t\right)}{x}_n,$$

where u is ${\mathcal{F}}_0$-measurable. Based on the properties of the counting risk model, we know that $B\left(t\right)$ is a predictable, right-continuous process; $X\left(t\right)$ is a predictable, left-continuous process; and $M\left(t\right)$ is an optional Gaussian martingale. Consequently, this surplus process—having both right and left limits but lacking continuity on either side—qualifies as an optional semimartingale process if it has finite variation.

For this process to exhibit finite variation, as defined in (A1) in Appendix A, it is sufficient that the following inequality holds for $t\ge 0$ in the claim and premium processes:

$$\sum _{0\le s\le t}|X_{s^{+}}-X_s|+{\int }_{0^{+}}^t\left|d{B}^r\right|<\infty$$

(40)

This condition is satisfied since the changes in claims and premiums must be finite to ensure a well-defined surplus process. Having established that the surplus process in the counting risk model is an optional semimartingale, given assumption (40), we can now apply Theorem 2 to calculate the probability of ruin.

Let us apply our results to various risk models. Specifically, we examine examples that illustrate how our findings can be utilized in different contexts. These examples demonstrate the practical implications of our theoretical framework and showcase the versatility of the optional semimartingale methods we have developed. By analyzing these models, we aim to highlight the effectiveness of our approach in assessing risk exposure and calculating ruin probabilities across different scenarios.

Example 1. 

We consider a specific type of the optional risk model studied in Section 2. Specifically, we assume that

$$U_t=u+B_t+{\int }_0^t{\sigma }_{s}d{W}_s+\sum _{i=1}^{N_t^g}{Z}_i-\sum _{k=1}^{N_t^r}{Y}_k$$

where B is a continuous process of finite variation, W is a standard Wiener process, and σ is a predictable process. The processes $N_s^g$ and $N_s^r$ are left- and right-continuous counting processes with intensities ${\lambda }_s^g$ and ${\lambda }_s^r$, respectively, such that $N_t^j-{\int }_0^t{\lambda }_s^{j}ds$ (for $j=r,g$) are optional local martingales. The positive random variables $Y_k$ and $Z_i$ are assumed to be mutually independent. The distribution of the claim $Y_k$ depends on the time of the k-th jump but is otherwise independent of the $N^r$ process. Thus, the $Y_k$ values can depend on the $N^r$ process solely through the time-dependence of the distributions of the $Y_k$ variables. The same applies to the random variables $Z_i$ and the process $N^g$. A case of time dependence occurs when the claims are affected by factors such as inflation or interest rates (see, e.g., Cai and Dickson 2003). Under these assumptions, we have

$${\nu }^r(dt,dx)={\lambda }_t^r(1-F_t^r(-dx))dt$$

and

$${\nu }^g(dt,dx)={\lambda }_t^g{F}_t^g\left(dx\right)dt$$

where $F_t^r$ and $F_t^g$ are the respective cumulative distributions of $Y_k$ and $Z_i$ at time t. Hence,

$$G_t\left(z\right)=-z{B}_t+{\displaystyle \frac{z^2}{2}}{\int }_{[0,t]}{\sigma }_s^{2}ds+{\int }_{]0,t]}[{\phi }_s^r(-z)-1]{\lambda }_s^{r}ds+{\int }_{[0,t[}[{\phi }_s^g\left(z\right)-1]{\lambda }_s^{g}ds$$

where ${\phi }_s^j\left(z\right)=\int e^{-zx}d{F}_s^j\left(x\right)$ denotes the Laplace transform of $F_s^j,j=r,g$. For this model, from condition (11) we obtain

$$B_t+{\int }_0^t{\mu }_s^g{\lambda }_s^{g}ds>{\int }_0^t{\mu }_s^r{\lambda }_s^{r}ds,$$

for all $t>0$ where ${\mu }_s^g$ and ${\mu }_s^r$ denote the mean of $Z_i$ and $Y_k$ at time s, respectively. In the rest of this example, we discuss simple situations where the ruin probability can easily be evaluated. We suppose that for each $t>0$, there is a distribution function ${\tilde{F}}^j,j=r,g$ such that $F_s^r\left(x\right)\ge {\tilde{F}}_t^r\left(x\right)$ and $F_s^g\left(x\right)\le {\tilde{F}}_t^g\left(x\right)$ for all $x>0$ and all $s\le t,j=r,g$. Under these conditions, ${\mu }_s^j\le {\tilde{\mu }}_t^j,$ for $s\le t$ and

$$\begin{array}{cc}\hfill {\int }_0^s[{\phi }_u^r(-z)-1]{\lambda }_u^{r}du& \le [{\tilde{\phi }}_t^r(-z)-1]{\mathsf{\Lambda }}_t^r,\hfill \\ \hfill {\int }_0^s[{\phi }_u^g\left(z\right)-1]{\lambda }_u^{g}du& \le [{\tilde{\phi }}_t^g\left(z\right)-1]{\mathsf{\Lambda }}_t^g\hfill \end{array}$$

(41)

where ${\tilde{\mu }}_t^j$ denotes the mean value of ${\tilde{F}}_t^j$, ${\tilde{\phi }}_t^j\left(z\right)=\int e^{-zx}d{\tilde{F}}_t^j\left(x\right)$, and ${\mathsf{\Lambda }}_s^j={\int }_0^t{\lambda }_s^{j}ds$ is the integrated intensity of $N^j,j=r,g$. Assume that a company adopts a policy such that for some constant $c>1$

$$B_t\ge c\left({\tilde{\mu }}_t^r{\int }_0^t{\lambda }_s^{r}ds-{\tilde{\mu }}_t^g{\int }_0^t{\lambda }_s^{g}ds\right).$$

(42)

for $s\le t$. If, in a scenario, ${\sigma }_s^2$ is bounded by a constant ${\zeta }_t^2$ for $s\le t$, (41) and (42) imply that

$$\begin{array}{cc}\hfill G_s\left(z\right)& \le [-zc{\tilde{\mu }}_t^r+{\tilde{\phi }}_t^r(-z)-1]{\mathsf{\Lambda }}_s^r+[-zc{\tilde{\mu }}_t^g+{\tilde{\phi }}_s^g\left(z\right)-1]{\mathsf{\Lambda }}_{s+}^g+{\displaystyle \frac{1}{2}}{z}^2{\zeta }_t^{2}s\hfill \\ \hfill & =(g_t^r\left(z\right),g_t^g\left(z\right))\circ {\mathsf{\Lambda }}_s+{\displaystyle \frac{1}{2}}{z}^2{\zeta }_t^{2}s\hfill \end{array}$$

for all $s\le t$ where

$$g_t^r\left(z\right)=-zc{\tilde{\mu }}_t^r+{\tilde{\phi }}_t^r(-z)-1,g_t^g\left(z\right)=-zc{\tilde{\mu }}_t^g+{\tilde{\phi }}_s^g\left(z\right)-1$$

and ${\mathsf{\Lambda }}_s={\mathsf{\Lambda }}_s^r+{\mathsf{\Lambda }}_s^g$ (refer to Appendix A for the definition of ∘ in (A2).). Under these conditions, $g_t^j\left(z\right)$ is convex, $g_t^j\left(0\right)=0$, and ${\left(g_t^j\right)}^{\prime }\left(0\right)<0$, so there is a range $[0,z_t]$ of z-values for which $g_t^j\left(z\right)\le 0,j=r,g.$ For $z\in [0,z_0]$, it follows from (10) that

$$\mathbf{P}(\tau \le t)\le {\displaystyle \frac{e^{-zu+\frac{1}{2}{z}^2{\zeta }_t^{2}t}}{\mathbf{E}(exp[-(g_t^r\left(z\right),g_t^g\left(z\right))\circ {\mathsf{\Lambda }}_{\tau }]|\tau \le t)}}$$

(43)

The Laplace transform of Λ is rarely known, but when the Laplace transform of Λ is known, it is sometimes possible to proceed in a way analogous to the derivation of the upper bound (46) demonstrated in Example 2. Quite generally we can use the fact that $-zu+{\displaystyle \frac{1}{2}}{z}^2{\zeta }_t^{2}t$ has a minimum at $z=u/\left(t{\zeta }_t^2\right)$, which then implies

$$\mathbf{P}(\tau \le t)\le e^{-{\displaystyle \frac{1}{2}}{u}^2/\left({\zeta }_t^{2}t\right)}$$

provided $u/\left(t{\zeta }_t^2\right)\le z_t$. Finally, we have the following result:

$$\mathbf{P}(\tau \le t)\le exp\left(-z_{t}u+{\displaystyle \frac{1}{2}}{z}_t^2{\zeta }_t^{2}t\right).$$

Example 2. 

Consider the specific case of the classical compound Poisson risk model, which incorporates additional random positive left-continuous jumps of size $Z_i$ and is perturbed by a Wiener process W such that

$$U_t=u+ct+\sigma W_t+\sum _{i=1}^{N_t^g}{Z}_i-\sum _{k=1}^{N_t^r}{Y}_k$$

(44)

where c is the premium rate; $N^r$ is a Poisson process; and $N^g$ is a left-continuous modification of a Poisson process with intensities ${\lambda }^r$ and ${\lambda }^g$. The random variables $Z_i$ and $Y_k$ are positive, independent, and identically distributed, with distribution functions $F^g$ and $F^r$, respectively. We assume that W, $N^r$, $N^g$, $\left\{Z_i\right\}$, and $\left\{Y_k\right\}$ are all mutually independent. In this specific scenario, we have $B_t=ct$, ${⟨N⟩}_t={\sigma }^{2}t$, ${\nu }^r(\omega ;dt,dx)={\lambda }^r(1-F^r(-dx))dt$, and ${\nu }^g(\omega ;dt,dx)={\lambda }^g{F}^g\left(dx\right)dt$, so

$$G_t\left(z\right)=\left(-zc+{\displaystyle \frac{1}{2}}{\sigma }^2{z}^2+{\lambda }^r\left[{\phi }_{F^r}(-z)-1\right]+{\lambda }^g\left[{\phi }_{F^g}\left(z\right)-1\right]\right)t,$$

where ${\phi }_{F^j}\left(z\right)=\int e^{-zx}d{F}^j\left(x\right)$ represents the Laplace transform of $F^j$ for $j=r,g$. Since the process $U_t$ in this scenario has independent increments, $M_t\left(z\right)$ in (9) is an optional martingale for every z within the domain of ${\phi }_{F^r}$ and ${\phi }_{F^g}$. We observe that $z_t=\widehat{z}$ is the positive solution to $g\left(z\right)=0$. Therefore, for all $u\ge 0$

$$\mathbf{P}(\tau <\infty )\le e^{-u\widehat{z}}.$$

We can sometimes achieve a more precise upper bound for the finite time of ruin. For $z\in [\widehat{z},z_0]$, we have $g\left(z\right)\ge 0$; thus, by (10), we have

$$\mathbf{P}(\tau \le t)\le exp[-zu+g(z\left)t\right]$$

(45)

The right-hand side of (45) attains its minimum at $z^{*}$, which is given as the solution of $g^{\prime }\left(z^{*}\right)=u/t$, provided a solution exists in $[0,z_0]$. Otherwise, the minimum is attained at $z^{*}=z_0$, in which case $g^{\prime }\left(z^{*}\right)<u/t$. If $t\le u/g^{\prime }\left(\widehat{z}\right)$, the convexity of g implies that $z^{*}\ge \widehat{z}$; therefore,

$$\mathbf{P}(\tau \le t)\le exp[-z^{*}u+g\left(z^{*}\right)t]$$

(46)

Since $g\left(z^{*}\right)<(z^{*}-\widehat{z})u/t$ for $z^{*}>\widehat{z}$ (using again the convexity of g and the fact that $g\left(\widehat{z}\right)=0$), the right hand side of (46) is strictly smaller than $exp(-u\widehat{z})$ when $t<u/g^{\prime }\left(\widehat{z}\right)$.

Example 3. 

Consider another specific case of the model in (44) with the following cumulative distribution functions of $Z_i$ and $Y_i$:

$$F^g\left(x\right)=1-e^{-bx},b>0,$$

$$F^r\left(x\right)=1-e^{-ax},a>0.$$

Additionally, we assume that the net profit condition is satisfied and $c>{\displaystyle \frac{{\lambda }^r}{a}}-{\displaystyle \frac{{\lambda }^g}{b}}$. For $z\in (0,a]$, we obtain

$$G_t\left(z\right)=g\left(z\right)t=\left(-cz+{\displaystyle \frac{1}{2}}{\sigma }^2{z}^2+{\lambda }^r\left[{\displaystyle \frac{a}{a-z}}-1\right]+{\lambda }^g\left[{\displaystyle \frac{b}{b+z}}-1\right]\right)t,$$

Rewriting $g\left(z\right)={\displaystyle \frac{zh\left(z\right)}{2(a-z)(b+z)}},$ where

$$\begin{array}{cc}\hfill h\left(z\right)& =-{\sigma }^2{z}^3+({\sigma }^2(a-b)+2c)z^2\hfill \\ & +({\sigma }^{2}ab-2c(a-b)+2({\lambda }^r+{\lambda }^g))z\hfill \\ & +2({\lambda }^{r}b-{\lambda }^{g}a-cba)\hfill \end{array}$$

(47)

Thus, if the equation $h\left(z\right)=0$ has a solution $\widehat{z}\in (0,a]$, then by Theorem 2, it follows that

$$P(t<\infty )\le e^{-\widehat{z}u}.$$

This result generalizes the following special cases:

1.

Cadlag case: if $c>0,\sigma >0$ and there are no positive jumps, i.e., $Z_i=0$. The net profit condition is $c>{\displaystyle \frac{{\lambda }^r}{a}}$, and (47) becomes

$$h\left(z\right)=-{\sigma }^2{z}^2+({\sigma }^{2}a+2c)z-2(ca-{\lambda }^r).$$

Thus, the quadratic equation $h\left(z\right)=0$ has exactly two real roots for $z\in (0,\infty )$, given by:

$${\widehat{z}}_{\pm }={\displaystyle \frac{{\sigma }^{2}a+2c\pm \sqrt{\Delta }}{2{\sigma }^2}},$$

where $\Delta :={({\sigma }^{2}a-2c)}^2+8{\sigma }^2{\lambda }^r.$ Since ${({\sigma }^{2}a-2c)}^2\le \Delta \le {({\sigma }^{2}a+2c)}^2,$ we have ${\widehat{z}}_{-}\ge 0$ and ${\widehat{z}}_{+}\ge a.$ If ${\widehat{z}}_{-}\le a$; then, by Theorem 2, we obtain for all $u\ge 0$

$$\mathbf{P}(\tau <\infty )\le e^{-u{\widehat{z}}_{-}}.$$

2.

Pure jumps case (see Melnikov 2011, in Chapter 8, Section 3): If $c=0,\sigma =0$, then the net profit condition is ${\displaystyle \frac{{\lambda }^g}{b}}>{\displaystyle \frac{{\lambda }^r}{a}}$. And (47) becomes

$$h\left(z\right)=2({\lambda }^r+{\lambda }^g))z+2({\lambda }^{r}b-{\lambda }^{g}a).$$

Thus, the equation $h\left(z\right)=0$ has a unique real root in the interval $(0,a)$, given by

$$\widehat{z}={\displaystyle \frac{{\lambda }^{g}a-{\lambda }^{r}b}{{\lambda }^g+{\lambda }^r}}.$$

Therefore, by Theorem 2, for all $u\ge 0$

$$\mathbf{P}(\tau <\infty )\le e^{-u\widehat{z}}.$$

6. Conclusions

In this paper, we introduced an extended framework for risk modeling using optional semimartingales, allowing for a broader class of stochastic processes that do not require the usual conditions of completeness and right-continuity of the filtration. By formulating three distinct risk models—the Optional Risk Model, the Gaussian Risk Model, and the Counting Risk Model—we provided a flexible approach to capturing various financial risks, and stochastic premium and claim cash flows.

Furthermore, we derived a probabilistic analysis of ruin, incorporating jump processes and Gaussian martingales to account for both continuous and discrete risk fluctuations. Our findings demonstrate that the probability of ruin can be effectively bounded using the optional cumulant process and its stochastic exponential. This extends classical ruin probability results to a more general setting, encompassing complex financial environments where standard assumptions may not hold.

Additionally, we investigated the expected discounted penalty at ruin within the Gaussian Risk Model framework. We explored scenarios where ruin occurs either through oscillation or via a jump, introducing a penalty function that accounts for these distinct cases. Using an optional version of Lundberg’s equation, we derived renewal equations and formulated explicit solutions for the expected discounted penalty. Our analysis revealed that the penalty function satisfies a differential equation influenced by the Gaussian nature of the risk model and the underlying jump process.

Future research can explore optimal capital allocation strategies and hedging techniques, such as derivatives, reinsurance, and catastrophe bonds, to mitigate financial risks. Additionally, future work could examine dynamic hedging strategies that adapt to market fluctuations. By leveraging the optional semimartingale approach, our work contributes to a deeper understanding of risk assessment and management in insurance and financial mathematics.