Controlled diffusion models for optimal dividend pay-out

The reserve r(t) of an insurance company at time t is assumed to be governed by the stochastic differential equation dr(t) = (mu - a(t)) dt + sigma dw(t), where w is standard Brownian motion, mu, sigma > 0 constants and a(t) the rate of dividend payment at time t (0 acts as absorbing barrier for r(t)). The function a(t) is subject to dynamic allocation 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 (discounted) pay-out of dividend and x refers to r(0) = x. Two situations are 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 if a(0) is smaller than some critical value, the optimal strategy is to always pay the maximal dividend rate no. Otherwise, the optimal policy prescribes to pay nothing when the reserve is below some critical level 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 vary in all of [0, infinity). Then the optimal strategy is of singular control type in the sense that it prescribes to pay out whatever amount exceeds some critical level pn, but not pay out dividend when the reserve is below m.