This paper studies an insurer's optimal investment portfolio under the mean-variance criterion. The financial market consists of a riskless bond and a risky asset, and the latter's volatility is random. We are extending the Cox–Ingersoll–Ross (CIR) model to the case with jumps, where it is modeled by a jump-diffusion stochastic differential equation (SDE). We use a Lévy SDE to describe the risk process we have, in which we extend the classic Cramér-Lundberg model to the Lévy process, and additionally introduce the stochastic volatility into this model. We assume that the insurer in question is a mean-variance optimizer. In other words, the decision that this insurer faces is to simultaneously maximize and minimize the mean and variance of his/her terminal wealth by selecting an optimal portfolio. We have uncovered closed-form solutions to the mean-variance problem with respect to the efficient strategy and efficient frontier by solving for expected utility maximization of a quadratic function through the martingale method. Finally, we give a numerical example that analyzes the economic behavior of the efficient frontier.

This work considered strong convergence of the Euler-Maruyama (EM) method for a stochastic volatility jump-diffusion model (SVJD model, for short). In this model, the underlying asset price follows a jump-diffusion geometric Brownian motion with stochastic volatility, and the volatility process obeys a mean-reverting square root process with Poisson jumps. As preliminary results, the existence and uniqueness of nonnegative solutions for the SVJD model was shown by means of Tanaka's formula and the comparison theorem. Also, some moment properties of the solution to the SVJD model were given. In view of unavailability of an explicit solution for the SVJD model, we used the EM method to approximate the exact solution and proved strong convergence of the EM approximation in the $ L^2 $ sense. In addition, the EM approximation for the SVJD model was applied to approximately compute expected payoffs of a European option and a barrier option. Finally, simulations were presented to verify the theoretical analysis.

The truncated Euler–Maruyama (EM) method for stochastic differential equations with Poisson jumps (SDEwPJs) has been proposed by Deng et al. in 2019. Although the finite-time $L^r$-convergence theory has been established, the strong convergence theory remains absent. In this paper, the strong convergence refers to the use of an $L^2$ measure and places the supremum over time inside the expectation operation. Our version can be used to justify the method within Monte Carlo simulations that compute the expected payoff of financial products. Noting that the conditions imposed are too strict, this paper presents an existence and uniqueness theorem for SDEwPJs under general conditions and proves the convergence of the truncated EM method for these equations. Finally, two examples are considered to illustrate the application of the truncated EM method in option price calculation.

This paper considers the issue of optimal investment and consumption strategies for an investor with stochastic economic factor in a defaultable market. In our model, the price process is composed of a money market account and a default-free risky asset, assuming they rely on a stochastic economic factor described by a diffusion process. A defaultable perpetual bond is depicted by the reduced-form model, and both the default risk premium and the default intensity of it rely on the stochastic economic factor. Our goal is to maximize the infinite horizon expected discounted power utility of the consumption. Applying the dynamic programming principle, we derive the Hamilton–Jacobi–Bellman (HJB) equations and analyze them using the so-called sub-super solution method to prove the existence and uniqueness of their classical solutions. Next, we use a verification theorem to derive the explicit formula for optimal investment and consumption strategies. Finally, we provide a sensitivity analysis.