## Optimal choice of dividend barriers for a risk process with stochastic return on investments.(English)Zbl 0894.90048

Summary: We consider a risk process with stochastic return on investments and we are interested in expected present value of all dividends paid until ruin occurs when the company uses a simple barrier strategy, i.e. when it pays dividends whenever its surplus reaches a level $$b$$. It is shown that given the barrier $$b$$, this expected value can be found by solving a boundary value problem for an integro-differential equation. The solution is then found in two special cases; when return on investments is constant and the surplus generating process is compound Poisson with exponentially distributed claims, and also when both return on investments as well as the surplus generating process are Brownian motions with drift. Also in this latter case, we are able to find the optimal barrier $$b^*$$, i.e. the barrier that gives the highest expected present value of dividends. Parallel with this, we treat the problem of finding the Laplace transform of the distribution of the time to ruin when a barrier strategy is employed, noting that the probability of eventual ruin is 1 in this case. The paper ends with a short discussion of the same problems when a time dependent barrier is employed.

### MSC:

 91B30 Risk theory, insurance (MSC2010) 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 45J05 Integro-ordinary differential equations
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### References:

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