CONTROLLED DIFFUSION-MODELS FOR OPTIMAL DIVIDEND PAY-OUT

The reserve r(t) of an insurance company at time t is assumed to be go verned by the stochastic differential equation dr(t) = (mu - a(t)) dt + sigma dw(t), where w is standard Brownian motion, mu, sigma > 0 cons tants and a(t) the rate of dividend payment at time t (0 acts as absor bing barrier for r(t)). The function a(t) is subject to dynamic alloca tion and the objective is to find the one which maximizes EJ(x)(a(.)), where J(x) = integral(0)(infinity) e(-ct) a(t) dt is the total (disco unted) pay-out of dividend and x refers to r(0) = x. Two situations ar e considered: (a) The dividend rate is restricted so that the function a(t) varies in [0, a(0)] for some a(0) < infinity. It is shown that i f a(0) is smaller than some critical value, the optimal strategy is to always pay the maximal dividend rate no. Otherwise, the optimal polic y prescribes to pay nothing when the reserve is below some critical le vel m, and to pay maximal dividend rate a(0) when the reserve is above m. (b) The dividend rate is unrestriced so that a(t) is allowed to va ry in all of [0, infinity). Then the optimal strategy is of singular c ontrol type in the sense that it prescribes to pay out whatever amount exceeds some critical level pn, but not pay out dividend when the res erve is below m.