Optimal control with absolutely continuous strategies for spectrally negative Lévy processes.

*(English)*Zbl 1253.93001Let \(X=(X_t)_{t\geq 0}\) be a spectrally negative Lévy process modeling the surplus wealth of an insurance company. Suppose that the dividends are paid out to shareholders according to the strategy \(\pi=(L_t^{\pi})_{t\geq 0}\) – an adapted, left-continuous, non-negative, non-decreasing process. The quantity \(L_t^{\pi}\) represents the cumulative dividends paid out up to time \(t\geq 0\). De Finetti’s dividend problem amounts to solving the following control problem: Let \(U_t^{\pi}=X_t-L_z^{\pi}\) and let \(\sigma^{\pi}=\inf\{t>0:\, U_t^{\pi} <0\}\) be the ruin time provided the dividends are paid out according to the strategy \(\pi\). Let further
\[
v_{\pi}(x)=\mathbb{E}_x\left[\int_{[0,\sigma^{\pi}}e^{-qt}\, dL_t^{\pi}\right]
\]
be the expected net present value of the strategy \(\pi\) in some admissible set of strategies \(\Pi\) with discount rate \(q>0\) and the initial capital \(x\geq 0\). De Finetti’s dividend problem is to find the optimal value function \(v_*(x):=\sup_{\pi \in \Pi} v_{\pi}(x)\) and, if possible, the optimal strategy.

In case \(\Pi\) consists of all strategies, this problem has been successfully solved several years ago, the optimal value function being the result of the so-called reflection strategy. A possible drawback to this solution is the fact that, following the reflection strategy, the company will be ruined in finite time with probability one. This is one reason while in the current paper the authors study the following modification of de Finetti’s problem: The admissible set of strategies \(\Pi\) admits only absolutely continuous strategies \(L_t^{\pi}=\int_0^t \ell^{\pi}(s)\, ds \), where the rate \(\ell\) satisfies \(0\leq \ell^{\pi}(t)\leq \delta\), \(\delta\) being a ceiling rate (plus an additional condition in case of bounded variation \(X\)). The main result of the paper is that if the Lévy measure of \(X\) has a completely monotone density, the optimal solution of the above modification of de Finetti’s problem is the so-called refraction strategy – if \(U^{\pi}\) is above a certain level \(b\geq 0\), then dividends are paid out at the ceiling rate \(\delta\), while otherwise no dividends are paid out. The critical level \(b\) is also identified. The crucial role in the proofs is played by the scale function.

In case \(\Pi\) consists of all strategies, this problem has been successfully solved several years ago, the optimal value function being the result of the so-called reflection strategy. A possible drawback to this solution is the fact that, following the reflection strategy, the company will be ruined in finite time with probability one. This is one reason while in the current paper the authors study the following modification of de Finetti’s problem: The admissible set of strategies \(\Pi\) admits only absolutely continuous strategies \(L_t^{\pi}=\int_0^t \ell^{\pi}(s)\, ds \), where the rate \(\ell\) satisfies \(0\leq \ell^{\pi}(t)\leq \delta\), \(\delta\) being a ceiling rate (plus an additional condition in case of bounded variation \(X\)). The main result of the paper is that if the Lévy measure of \(X\) has a completely monotone density, the optimal solution of the above modification of de Finetti’s problem is the so-called refraction strategy – if \(U^{\pi}\) is above a certain level \(b\geq 0\), then dividends are paid out at the ceiling rate \(\delta\), while otherwise no dividends are paid out. The critical level \(b\) is also identified. The crucial role in the proofs is played by the scale function.

Reviewer: Zoran Vondraček (Zagreb)