Optimal risk control with investment decisions

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Zhang, Yu
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Modern businesses face different kinds of risk that may affect their management operations and cause significant financial losses. It is thus very crucial to identify, assess and control risks to reduce their potential impact. One important objective for insurance businesses is implementing strategies to control the risk of ruin, which is naturally measured by the ruin probability. In this study, we consider optimal risk control problems with investment decisions and aim to assess the impact of investment on minimizing the ruin probability. Specifically, we apply stochastic control in insurance to determine optimal investment strategies. We first consider the problem of controlling ruin probability by investment decisions under a discrete-time risk process. An insurance company may invest its reserve into a riskless asset and a risky asset. Our goal is concentrated on finding the optimal investment strategy to minimize the ruin probability, in the case that the claim size distribution has an unknown mean parameter. Applying the Bayesian approach and the dynamic programming method, we find the minimal ruin probability function and the corresponding optimal investment decisions. We also investigate some structural properties of the optimal strategy. We investigate the problem of minimizing the ruin probability with joint decisions of investment and excess-of-loss reinsurance for a continuous-time risk model. The reserve of an insurance company is modeled by a diffusion process and may be invested in a financial market which follows the Black-Scholes model consisting of a risky asset and a riskless asset. However, a constraint is imposed on investment decisions and the ratio between the amount invested in the risky asset and the total reserve should lie below a given bound. Meanwhile, the insurance company may purchase an excess-of-loss reinsurance to reduce risk. We characterize and derive jointly optimal decisions of investment and reinsurance to minimize ruin probability. We solve the corresponding Hamilton-Jacobi-Bellman equation and provide an explicit form for the minimal ruin probability function. In addition, we present several numerical examples to illustrate our results, which indicate a positive impact of investment on controlling the risk of ruin.
Investment, Ruin probability, Stochastic dynamic programming, Reinsurance, Risk control