Optimal reinsurance policies for an insurer with a bivariate reserve risk process in a dynamic setting

Lihua Bai, Jun Cai and Ming Zhou ()

Insurance: Mathematics and Economics, 2013, vol. 53, issue 3, 664-670

Abstract: Assume that an insurer has two dependent lines of business. The reserves of the two lines of business are modeled by a two-dimensional compound Poisson risk process or a common shock model. To protect from large losses and to reduce the ruin probability of the insurer, the insurer applies a reinsurance policy to each line of business, thus the two policies form a two-dimensional reinsurance policy. In this paper, we investigate the two-dimensional reinsurance policy in a dynamic setting. By using the martingale central limit theorem, we first derive a two-dimensional diffusion approximation to the two-dimensional compound Poisson reserve risk process. We then formulate the total reserve of the insurer by a controlled diffusion process and reduce the problem of optimal reinsurance strategies to a dynamic control problem for the controlled diffusion process. Under this setting, we show that a two-dimensional excess-of-loss reinsurance policy is an optimal form that minimizes the ruin probability of the controlled diffusion process. By solving a HJB equation with two dependent controls, we derive the explicit expressions of the optimal two-dimensional retention levels of the optimal two-dimensional excess-of-loss reinsurance policy and the minimized ruin probability. The results show that optimal dynamic two-dimensional retention levels are constant and the optimal retention levels are related by a deterministic function. We also illustrate the results by numerical examples.

Keywords: Two-dimensional compound Poisson process; Common shock model; Two-dimensional Brownian motion; Martingale central limit theorem; Two-dimensional diffusion approximation; HJB equation; Ruin probability; Excess-of-loss reinsurance (search for similar items in EconPapers)
Date: 2013
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Citations: View citations in EconPapers (20)

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Persistent link: https://EconPapers.repec.org/RePEc:eee:insuma:v:53:y:2013:i:3:p:664-670

DOI: 10.1016/j.insmatheco.2013.09.008

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