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Optimal dividend payouts for diffusions with solvency constraints. (English) Zbl 1038.60081
This paper is concerned with the classical problem of optimal dividend payouts for a company. The author considers a company where surplus follows a diffusion process and whose objective is to maximize expected discounted dividend payouts to the shareholders, more exactly, to find a payout-scheme that maximizes the expected present value of all payouts until ruin occurs. The following restrictions are imposed: no dividends are allowed to be paid out unless surplus is at least ${b}_{0}$, and the surplus immediately after payment cannot be below ${b}_{0}$. Also, there is a level ${b}^{*}$ so that whenever surplus goes above ${b}^{*}$, the excess is paid out as dividends. If ${b}_{0}>{b}^{*}$, it is shown that an optimal restricted policy is to use a barrier strategy at ${b}_{0}$. This barrier is such that the probability of negative surplus within time $T$ does not exceed prescribed $\epsilon >0$. It is discussed how this ${b}_{0}$ can be calculated and a complete treatment is given when the surplus follows a Brownian motion with drift.

##### MSC:
 60J70 Applications of Brownian motions and diffusion theory 49K05 Free problems in one independent variable (optimality conditions) 91B30 Risk theory, insurance