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In this paper, we consider an optimal reinsurance problem. The surplus model of the insurance company is approximated by a diffusion model with the drift coefficient μ. The insurance company employs reinsurance to reduce the risk. π is the proportion of each claim paid by the company and the remainder proportion of the claim is paid by the reinsurer. λ(1 − π) is the rate at which the premiums are diverted to the reinsurer, thus it holds λ ≥ μ > 0 in general. We discuss two cases: (i) non-cheap reinsurance (when λ > μ), (ii) cheap reinsurance (when λ = μ). The objective of the insurance company is to make an optimal decision on reinsurance to reach a goal (a given surplus level) in minimal expected time. We disclose that for some case it is not suitable to search optimal decisions by minimizing the expected time to reach a goal. In order to deal with this case, we study two other methodologies (ruin probability minimization and expected discount factor maximization) for the optimal strategy selection problem in reinsurance decision. [ABSTRACT FROM AUTHOR]

We consider a multi-stock market model. The processes of stock prices are governed by stochastic differential equations with stock return rates and volatilities driven by a finite-state Markov process. Each volatility is also disturbed by a Brownian motion; more exactly, it follows a Markov-driven Ornstein–Uhlenbeck process. Investors can observe the stock prices only. Both the underlying Brownian motion and the Markov process are unobservable. We study a discretized version, which is a discrete-time hidden Markov process. The objective is to control trading at each time step to maximize an expected utility function of terminal wealth. Exploiting dynamic programming techniques, we derive an approximate optimal trading strategy that results in an expected utility function close to the optimal value function. Necessary filtering and forecasting techniques are developed to compute the near-optimal trading strategy. [ABSTRACT FROM AUTHOR]