Abstract
We consider optimal dividend payment under the constraint that the with-dividend ruin probability does not exceed a given value α. This is done in most simple discrete De Finetti models. We characterize the value function V(s,α) for initial surplus s of this problem, characterize the corresponding optimal dividend strategies, and present an algorithm for its computation. In an earlier solution to this problem, a Hamilton-Jacobi-Bellman equation for V(s,α) can be found which leads to its representation as the limit of a monotone iteration scheme. However, this scheme is too complex for numerical computations. Here, we introduce the class of two-barrier dividend strategies with the following property: when dividends are paid above a barrier B, i.e., a dividend of size 1 is paid when reaching B+1 from B, then we repeat this dividend payment until reaching a limit L for some 0≤L≤B. For these strategies we obtain explicit formulas for ruin probabilities and present values of dividend payments, as well as simplifications of the above iteration scheme. The results of numerical experiments show that the values V(s,α) obtained in earlier work can be improved, they are suboptimal.
Highlights
We consider the computation of optimal dividend payment under the constraint that the ruin probability with possible dividend payment does not exceed a given value α.1 This is done in most simple De Finetti models with risk process tS ( t ) = s + ∑ Xi, i =1 where X1, X2, ... are independent with P{ Xi = 1} = p, P{ Xi = −1} = 1 − p, 1/2 < p < 1, q = 1 − p, and initial surplus s ≥ 0, which are discrete in time and space, are stationary, have independent increments, and are skip free
In Hipp (2003) it is shown that (1) has exactly one solution V (s, α) which is the present value of an optimal dividend payment function δ(t, s, α), t ≥ 0, for initial surplus s and allowed ruin probability α
De Finetti models is rather simple, and at the same time these strategies are flexible enough to yield large dividend values. Their computation can be done for a single initial value s0 and allowed ruin probability α
Summary
We consider the computation of optimal dividend payment under the constraint that the ruin probability with possible dividend payment does not exceed a given value α.1 This is done in most simple De Finetti models with risk process t. In Hipp (2003) it is shown that (1) has exactly one solution V (s, α) which is the present value of an optimal dividend payment function δ(t, s, α), t ≥ 0, for initial surplus s and allowed ruin probability α. It is attained according to Lemma 2.e in Hipp (2003) This Bellman equation is rather complex: the numerical computations need maximization over a continuous variable 0 ≤ β 1 ≤ 1, and the running ruin probability α(t) is defined in the optimization step. Consider the dividend strategy δ2 paying k at s + 1 instead, and an additional 1 at B + 1 This dividend strategy has the same ruin probability as δ1 , but the present value of dividends of δ2 is larger than the one of δ1 since the payment of size 1 is paid earlier.
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