In this study, we explore the pricing of vulnerable European options within a framework of a double exponential jump-diffusion model, incorporating stochastic volatility and stochastic jump intensity. The dynamics of volatility and jump intensity are characterized using mean-reverting processes, with long-run mean modulated by a continuous-time Markov process. This model can capture jump clustering and information transfer lag effectively. We obtain a closed-form solution for the joint characteristic functions using the iterative expectation formula. Then we employ the Fast Fourier Transform (FFT) approach for estimating option prices and validate the accuracy of this method through Monte Carlo simulations. Our numerical analysis delves into the impact of Markov regime-switching, stochastic volatility and stochastic jump intensity on the pricing of vulnerable European call options.

We propose a method-of-moments (MoM) approach to identify the drift and diffusion parameters of arithmetic and geometric Brownian motion from independent observations of the first hitting times for a fixed boundary. Unlike techniques that assume an inverse Gaussian or any specific distribution for the hitting times, our estimator matches only the first two sample moments to theoretical expressions derived from the backward Kolmogorov equation. This yields closed-form formulas that require no numerical optimization or distribution fitting, a distinct advantage for high-throughput microfluidic experiments where only threshold-crossing events are recorded, often for thousands of cells. We establish identifiability, prove consistency and asymptotic normality, and provide finite-sample bias corrections. Numerical simulations demonstrate accurate inference of the effective production rate and noise strength in a synthetic genetic reporter system with sample sizes as small as 50 cells. The proposed moment-based MoM estimator is computationally instantaneous and exhibits robustness against mild model misspecification, making it a practical tool for event-based parameter inference in single-cell biology.

Systemic risk measures are crucial for the stability of financial markets. However, traditional frameworks, such as those relying on fixed exponent spaces, rest on an assumption of globally consistent moment conditions, which fails to capture the time-varying and state-dependent nature of financial market volatility. In this paper, we propose a new framework for systemic risk measurement on the variable-exponent Bochner–Lebesgue space Lp(·), where the exponent p(·) is a random variable rather than a deterministic constant parameter. The variable exponent p(·) can be regarded as an endogenous risk-sensitivity adjuster. This design imbues the systemic risk measure with inherent adaptability, enabling it to move beyond applying a fixed scale across all market states and instead allowing its criterion to adjust dynamically in response to the evolving nature of risk amid shifts in market conditions. By constructing suitable deterministic auxiliary functions and single-firm risk measures, we decompose the quantification of systemic risk in Lp(·) into two sequential steps, ultimately deriving its dual representations. Several examples are provided to illustrate the theoretical results.

Given that most states in real-world systems are inaccessible, it is critical to study the inverse problem of an irreversibly stationary Markov chain regarding how a generator matrix can be identified using minimal observations. The hitting-time distribution of an irreversibly stationary Markov chain is first generalized from a reversible case. The hitting-time distribution is then decoded via the taboo rate, and the results show remarkably that under mild conditions, the generator matrix of a reversible Markov chain or a specific case of irreversibly stationary ones can be identified by utilizing observations from all leaves and two adjacent states in each cycle. Several algorithms are proposed for calculating the generator matrix accurately, and numerical examples are presented to confirm their validity and efficiency. An application to neurophysiology is provided to demonstrate the applicability of such statistics to real-world data. This means that partially observable data can be used to identify the generator matrix of a stationary Markov chain.

In this paper, we study the optimal investment and proportional reinsurance problem for an insurer with short-selling and borrowing constraints under the expected value premium principle. The claim process follows a Brownian risk model with a drift. The insurer’s surplus is allowed to invest in one risk-free asset and one risky asset. By using the dynamic programming approach and solving the corresponding boundary-value problems, the optimization objective of maximizing the probability of drawup before drowdown is considered initially. The optimal strategy and the corresponding value function are derived through solving the Hamilton-Jacobi-Bellman (HJB) equation. Moreover, numerical examples are performed to illustrate the effects of model parameters on the optimal strategy. In addition, we verify the optimality of the strategies obtained from the dynamic programming principle by Euler method.

Starting from the global financial crisis to the more recent disruptions brought about by geopolitical tensions and public health crises, the volatility of risk in financial markets has increased significantly. This underscores the necessity for comprehensive risk measures capable of capturing the complexity and heightened fluctuations in market volatility. This need is crucial not only for new financial assets but also for the traditional financial market in the face of a rapidly changing financial environment and global landscape. In this paper, we consider the risk measures on a special space L^p(·), where the variable exponent p(·) is no longer a given real number as in the conventional risk measure space L^p, but rather a random variable reflecting potential fluctuations in volatility within financial markets. Through further development of axioms related to this class of risk measures, we also establish dual representations for them.

The authors consider a robust optimal reinsurance and investment problem in a risk model with two dependent classes of insurance business for an Ambiguity-Averse insurer (AAI). The insurer aims to minimize the goal-reaching probability that the value of the wealth process reaches a low barrier before a high goal. Using the stochastic control approach based on the Hamilton-Jacobi-Bellman (HJB) equation, the authors derive the robust optimal reinsurance and investment strategies, as well as the corresponding value function. The authors conclude that the robust optimal investment-reinsurance strategy coincides with the one without model ambiguity, but the value function differs. As a consequence, ignoring model uncertainty leads to significant value function loss for the AAI. Besides, it is worth noting that if the insurer has only one business, the sum of the degenerated value function and the one of (Luo, et al., 2019) is equal to 1 both for ambiguity and ambiguity-neutral. Finally, numerical examples are given to illustrate our results.

This paper considers the non-zero-sum stochastic differential game problem between two ambiguity-averse insurers (AAIs) with common shock. Each AAI’s surplus process consists of a proportional reinsurance protection and an investment in a money account, a stock and a credit default swap (CDS) with the objective of maximizing the expected utility of her relative terminal surplus with respect to that of her competitors. We consider default contagion risk of CDSs through a Markovian model with interacting default intensities. It is worthwhile to consider the uncertainty of the model on both the insurer herself and her competitors. In our model, we describe the surplus processes of two insurers by two jump-diffusion models with a common shock. Under jump-diffusion models, the robust Nash equilibrium strategies and the value functions for the all-default, one-default and all-alive case are derived under a worst-case scenario, respectively. Finally, through some numerical examples, we found some interesting results about the effects of some model parameters on the robust Nash equilibrium strategies, such as, the common shocks and the individual claims have the opposite effect on reinsurance investment.

This paper considers a robust optimal investment and reinsurance problem with constraints for an Ambiguity-Averse Insurer (AAI). The criterion is to minimize the goal-reaching probability, namely, the probability that the value of the wealth process reaches a low barrier before a high goal. The robust optimal investment-reinsurance strategy and closed-form expression of the associated value function are derived explicitly by applying stochastic dynamic programming and solving the corresponding Hamiliton-Jacobi-Bellman (HJB) equation. It is extremely interesting that the sum of our value function and the value function of Luo et al. [23] is equal to 1 in two cases of ambiguity and ambiguity-neutral. Finally, numerical examples are given to illustrate the influence of typical parameters on our results.

Long-term memory behavior is one of the most important phenomena that has appeared in the time series analysis. Different from most definitions of second-order properties, an excess entropy approach is developed for stationary time series to classify long-term and short-term memory. A stationary sequence with finite block entropy is long-term memory if its excess entropy is infinite. The simulation results are graphically demonstrated after some theoretical results are simply presented by various stochastic sequences. Such an approach has advantages over the traditional ways that the excess entropy of stationary sequence with finite block entropy is invariant under instantaneous one-to-one transformation, and that it only requires very weak moment conditions rather than second-order moment conditions and thus can be applied to distinguish the LTM behavior of stationary sequences with unbounded second moment (e.g., heavy tail distribution). Finally, several applications on real data are exhibited.

This paper studies the optimal portfolio selection for defined contribution (DC) pension fund with mispricing. We adopt the general hyperbolic absolute risk averse (HARA) utility to describe the risk performance of the pension fund managers. The financial market comprises a risk-free asset, a pair of mispriced stocks, and the market index. Using the dynamic programming approach, we construct the Hamilton-Jacobi-Bellman (HJB) equation and obtain the explicit expressions for optimal portfolio choices with two methods. Finally, numerical analysis is presented to illustrate the sensitivity of the optimal portfolios to parameters of the financial market and contribution process. 200 words.

In this article, the dual risk model with two-sided jumps and two different randomized observations is considered. The dividend observation and ruin observation are supervised by two departments respectively. While in practice, the financial position of a company is usually monitored frequently, dividend decisions are only made periodically along with the publication of its books. So there are two situations. First, dividend observation and ruin observation are independent of each other, under this circumstance, we researched the integral-differential equation of the expected discounted dividend function until ruin. Second, dividend decision time is a multiple of ruin observation time. We deduced the expected discounted dividend function until ruin. Moreover, numerical analyses are provided to illustrate our results.

In this paper, we consider a mixed dividend strategy in a dual risk model. The mixed dividend strategy is the combination of a threshold dividend and a Parisian implementation delays dividend under periodic observation. Given a series of discrete observation points, when the surplus level is larger than the predetermined bonus barrier at observation point, the Parisian implementation delays dividend is immediately carried out, and the threshold dividend is performed continuously during the delayed period. We study the Gerber-Shiu expected discounted penalty function and the expected discounted dividend payments before ruin in such a dual risk model. Numerical illustrations are given to study the influence of relevant parameters on the ruin-related quantities and the selection of the optimal dividend barrier for a given initial surplus level.

In this work, we study the optimal investment and premium control problem with the short-selling constraint under the mean-variance criterion. The claim process is assumed to follow the non-homogeneous compound Poisson process. The insurer invests the surplus in one risk-free asset and one risky asset described by the Heston model. Under these, we consider an optimization objective that maximizes the return (the expectation of terminal wealth) and minimizes the risk (the variance of terminal wealth). By constructing the extended Hamilton–Jacobi–Bellman (HJB) system with the dynamic programming method, the time-consistent strategies and the corresponding value function are obtained. Furthermore, we provide numerical examples to illustrate the effects of the model parameters on the optimal policies.

In this paper, we study a discrete interaction risk model with delayed claims and randomized dividends payable at a non-negative threshold level. The recursive formula and the defective renewal equation for the Gerber-Shiu discounted penalty function are derived. Furthermore, the explicit expression for the discount-free Gerber-Shiu function is obtained. As an application, the joint distributions of the surplus immediately prior to ruin and the deficit at ruin and numerical illustration from a specific example are presented.

Abstract This article studies the optimal mean-variance reinsurance-investment selection for insurers with mispricing. Assuming that insurers wish to purchase proportional/excess-of-loss reinsurance and exchange among a risk-free asset, a pair of mispriced stocks, and the market index to maximize their return and minimize the risk. Using the approach developed by Björk, Khapko, and Murgoci (Finance and Stochastics 2017; 21 (2):331–60), we derive the equilibrium strategies and the corresponding equilibrium value functions under two cases through solving the extended Hamilton–Jacobi–Bellman system. Moreover, numerical analyses are provided to illustrate our results.

In this paper, we prove several results involving a general draw-down time from the running maximum for refracted spectrally negative Lévy processes. Using an approximation method, which is excursion theory at its heart, we find expressions for the Laplace transforms for the two-sided exit problems which are related to the draw-down time and an expression for the associated potential measure. The results are expressed in terms of scale functions.

This work proposes the concept of uncorrelation for fuzzy random variables, which is weaker than independence. For the sequence of uncorrelated fuzzy random variables, weak and strong laws of large numbers are studied under the uniform Hausdorff metric d H ∞ . The results generalize the law of large numbers for independent fuzzy random variables.

In this paper, we analyze a robust optimal investment-reinsurance problem involving a defaultable security for an ambiguity-averse insurer(AAI), who worries about uncertainty in model parameters. The insurer can trade in a risk-free asset, a stock and a defaultable corporate bond. The price process of the stock is described by a constant elasticity of variance(CEV) model. In particular, the reinsurance premium is calculated according to the generalized mean-variance premium principle. Using the dynamic programing approach, we study the pre-default case and the post-default case respectively, and then derive the optimal strategies and the corresponding value functions under the worst-case scenario. Moreover, the verification theorem is given under an inequality condition. Finally, we give some numerical examples to illustrate our main results.

In this paper, we adopt a Poisson approach to find Laplace transforms of joint occupation times over n disjoint intervals for pre-exit diffusion processes. Then we generalize previous result for the 2-dimensional case and the 3-dimensional case.