Optimizing expected utility of dividend payments for a Cramer-Lundberg risk proces

(1.1)

where Y 1 , Y 2 , . . . are i.i.d positive random variables with absolutely continuous d.f. F Y representing the claims, N = (N t ) t≥0 is an independent Poisson process with intensity λ > 0 modeling the times at which the claims occur, x ≥ 0 denotes the initial surplus and µ is a premium intensity. We define above Poisson process N and the sequence {Y i , i ∈ N} on a common probability space (Ω, F , P).

For the classical dividend problem, apart of the reserve process (1.1), we consider the dividend payments. Let C = (C t ) t≥0 be an adapted and nondecreasing process representing all accumulated dividend payments up to time t. Then the regulated process X = (X t ) t≥0 is given by: (1.2)

We observe the regulated process X t until the time of ruin:

Obviously the time of the ruin of an insurance company depend on dividend strategy and after ruin occurs no dividends are paid. As usual we assume that net profit condition µ > λE(Y 1 ) for Cramér-Lundberg risk process is fulfilled.

For the classical dividend problem we define the value of the dividends as:

where β > 0 is a discount factor, E x means expectation with respect to P x (•) = P (•|X 0 = x) and the value function as:

for C being the set of all admissible accumulated dividend strategies (C t ) t≥0 .

In the mathematical finance and actuarial literature, there is a good deal of work being done on dividend barrier models and the problem of finding an optimal policy of paying dividends, see Schmidli [11] for an overview. In this paper we assume additionally that (C t ) t≥0 is absolutely continuous with respect to the Lebesgue measure; see e.g. Hubalek and Schachermayer [6]. Then the process C admits a density process denoted by c = (c t ) t≥0 modeling the intensity of the dividend payments in continuous time. That is, for t ≥ 0:

Then we can consider the discounted cumulative utility of dividend payments:

where U is some fixed utility function, and the value function equals:

for C being the set of all admissible strategies (c t ) t≥0 . We assume that dividend density process (c t ) t≥0 is admissible if it is a nonnegative, adapted and càdlàg process and there is no dividend after ruin occurs: c t = 0 for all t ≥ τ . As noted in [11,Rem. 2.1] we choose càdlàg strategies instead of more often used left-continuous ones. We study this optimization problem under the constraint that only dividend strategies with dividend rate bounded by a fixed constant are admissible, i.e. 0 ≤ c t ≤ c 0 < ∞ for all t ≥ 0 and some fixed number c 0 > 0. Finally, we assume that the ruin cannot be caused by the dividend payment.

Under this set-up we prove that value function is differentiable and it solves the Hamilton-Jacobi-Bellman (HJB) equation.

Recall that the regulated process is given by:

and the value function are defined as in (1.3).

From now on we will assume that U : R ≥0 → R ≥0 is continuous, nonnegative, strictly increasing, strictly concave and U(0) = 0.

Lemma 2.1. The optimal value function v(x) is nonnegative and bounded by U (c 0 ) β and converges to U (c 0 ) β a.s. as x → ∞. Furthermore, v(x) is an increasing, Lipschitz continuous and therefore absolutely continuous.

Proof. Using the same strategy for two initial capitals shows that v

β . Consider the strategy c * = (c * t ) which pay dividends at constant rate c 0 for all t ≥ 0 and associated with strategy c * and initial capital x ruin time τ c *

x . Then:

We will show now that the ruin time τ c * x converges to infinity almost surely as x → ∞. Choose y ≥ x and one realization ω ∈ Ω of Nt i=1 Y i . We will denote it by adding superscript ω to the respective counterparts. Since

x has either finite or infinite limit. Without lost of generality we can consider the regulated risk process X stopped at the ruin time. If we assume then that there exists ω such that the limit τ c * of the τ c *

x is finite then we get a contradiction though. Indeed, in this case we can always find x large enough to get

This contradiction completes the proof that the ruin time τ c * x converges to infinity a.s. as x → ∞.

By bounded convergence the quantity E x e -βτ c * x appearing in (2.1) converges to zero as x → ∞. Therefore v(x) converges to U (c 0 ) β as x → ∞. Let us now prove that v(x) is absolutely continuous. Let y > x ≥ 0. We denote by c = (c t ) a strategy for the initial capital y. We take now the strategy c = (c t ) that starts at initial reserve x, pays no dividends if X c t < y and follows strategy c after reaching level y. This strategy c is of course admissible. In the event of no claims, the reserve process X c t reaches y at time t 0 = y-x µ . Since the p

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