The Optimal Control of a Jump Mutual Insurance Process

We shall define a mutual insurance firm as a firm whose stockholders are the bearers of the insurance contracts issued by the firm. The firm's insurance is then viewed as a collective process of say N persons seeking to protect themselves against claims that may occur to any one of them. For example, large employers protecting their employees by pooling risks and deducting for protection given amounts from salaries may be a case in point. In this latter case, the employer may match withdrawals from employees salaries and provide in the process a fringe benefit and increase employees loyalty to the firm. Alternatively, agricultural collectives have in some cases established mutual insurance firms whose purposes are to protect them, at a cost, from the uncertainty implicit in their production processes and the fluctuations of agricultural markets. Since these firms do not work for profit, contingent payments, or fund reimbursement in case of excess cash holdings are typical control policies which help cover extraordinary claims and at the same time are assumed the best investment policies. To further protect themselves against extraordinary claims, mutual insurers can turn to reinsurance firms, “selling†for example the excess claims of, say, a given amount R (e.g., see Tapiero and Zuckerman (1982)). The purpose of this paper is to consider such a mutual insurance firm facing a jump stochastic claims process, as is often assumed in the insurance literature (e.g., Feller (1971), Borch (1974)). For example, Poisson and Compound Poisson processes are typical jump processes treated in this paper, although other processes could be considered as well (see Srinivasan (1973) and Srinivasan and Mehata (1976), Bensoussan and Tapiero (1982)). First we define the mutual insurer problem as a jump process stochastic control problem which we solve analytically under the assumption of a gamma density claim sizes distribution. A solution is obtained by applying arguments from stochastic dynamic programming and by solving the resultant functional equation by application of Laplace transforms (e.g., Colombo and Lavoine (1972), Miller (1956), Tapiero, Chapter V). Subsequently, the effects of reinsurance are introduced and preliminary results obtained. As in Tapiero and Zuckerman (1982), we assume that the mutual insurance firm is a direct underwriter and that the reinsurer is a leader in a Stackleberg game (Stackleberg (1952), Simman and Cruz (1973), Luce and Raiffa (1957)).