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This paper studies optimal reinsurance and investment strategies that maximize expected utility of the terminal wealth for an insurer in a stochastic market. The insurer's preference is represented by a two-piece utility function which can be regarded as a generalization of traditional concave utility functions. We employ martingale approach and convex optimization method to transform the dynamic maximization problem into an equivalent static optimization problem. By solving the optimization problem, we derive explicit expressions of the optimal reinsurance and investment strategy and the optimal wealth process.

*et al.*[2]. This paper considers American step options, including perpetual case and finite expiration time case. In perpetual case, we find that the optimal exercise time is the first crossing time of the optimal level. The closed price formula for perpetual step option could be derived through Feynman-Kac formula. As for the latter, we present a system of variational inequalities satisfied by the option price. Using the explicit finite difference method we could get the numerical option price.

The issue aims to look at leading-edge research on the interface between derivatives, insurance, securities and quantitative finance. As financial mathematics and statistics are two essential components in these four areas, the issue, as we hope, will give the readers a survey of the important tools of mathematics and statistics being used in the modern financial institutions.

In this special issue, 7 papers are included. The papers cover mathematical finance topics, such as option pricing, interest models and stochastic volatility; topics in risk management, such as Value at Risk, liquidity risk management; and actuarial science topics, such as ruin theory.

The papers in the issue were selected with a view towards readers coming from finance, actuarial science, mathematics or statistics. Hopefully this is a first step to provide a platform for people who are interested in the interplay among theory and practice of these disciplines.

This paper deals with the optimal liability and dividend strategies for an insurance company in Markov regime-switching models. The objective is to maximize the total expected discounted utility of dividend payment in the infinite time horizon in the logarithm and power utility cases, respectively. The switching process, which is interpreted by a hidden Markov chain, is not completely observable. By using the technique of the Wonham filter, the partially observed system is converted to a completely observed one and the necessary information is recovered. The upper-lower solution method is used to show the existence of classical solution of the associated second-order nonlinear Hamilton-Jacobi-Bellman equation in the two-regime case. The explicit solution of the value function is derived and the corresponding optimal dividend policies and liability ratios are obtained. In the multi-regime case, a general setting of the Wonham filter is presented, and the value function is proved to be a viscosity solution of the associated system of Hamilton-Jacobi-Bellman equations.

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