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Optimal proportional reinsurance policies for diffusion models with transaction costs. (English) Zbl 1093.91518

Summary: This paper extends the results of B. Højgaard and M. I. Taksar [Scand. Actuar. J. 1998, No. 2, 166–180 (1998; Zbl 1075.91559)] to the case of positive transactions costs. The setting here and in Højgaard and Taksar’s paper is the following: When applying a proportional reinsurance policy \(\pi\) the reserve of the insurance company \(\{R^\pi_t\}\) is governed by a SDE \[ dR^\pi_t=(\mu-(1-a_\pi(t))\lambda) dt+a_\pi(t)\sigma\, dW_t, \] where \(\{W_t\}\) is a standard Brownian motion, \(\mu\), \(\sigma>0\) are constants and \(\lambda\geq\mu\). The stochastic process \(\{a_\pi(t)\}\) satisfying \(0\leq a_\pi(t)\leq 1\) is the control process, where \(1-a_\pi(t)\) denotes the fraction of all incoming claims that is reinsured at time \(t\). The aim of this paper is to find a policy that maximizes the return function \(V_\pi(x)=\mathbf E\int^{\tau_\pi}_0e^{-ct}R^\pi_t\,dt\), where \(c>0\), \(\tau_\pi\) is the time of ruin and \(x\) refers to the initial reserve. In [loc. cit.] a closed form solution is found in case of \(\lambda=\mu\) by means of stochastic control theory. In this paper we generalize this method to the more general case where we find that if \(\lambda\leq 2\mu\), the optimal policy is not to reinsure, and if \(\mu<\lambda<2\mu\), the optimal fraction of reinsurance as a function of the current reserve monotonically increases from \(2(\lambda-\mu)/\lambda\) to 1 on \((0,x_1)\) for some constant \(x_1\) determined by exogenous parameters.

MSC:

91B30 Risk theory, insurance (MSC2010)
62P05 Applications of statistics to actuarial sciences and financial mathematics
60H30 Applications of stochastic analysis (to PDEs, etc.)
60J65 Brownian motion

Citations:

Zbl 1075.91559
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References:

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