Optimal investment and dividend policy in an insurance company: a varied bound for dividend rates (Q2321145)

Consider a classical risk process, where the surplus is invested in a Black-Scholes type risky asset. At most the surplus can be invested and no short positions are allowed. A dividend can be paid but the dividend rate is bounded. That is, the surplus is under the investment/dividend strategy \(\pi\) \[ X_t^\pi = x + \int_0^t (p + \gamma_s r X_s)\;d s + \int_0^t \gamma_s \sigma X_s \;d W_s - \sum_{k=1}^{N_t} Y_k - L_t\;, \] where \(x\) is the initial capital, \(N\) is a Poisson process, \(\{Y_k\}\) are iid positive claim sizes, \(W\) is a Brownian motion, \(r,\sigma > 0\), \(\gamma_t \in [0,1]\) is the fraction of the surplus invested in the risky asset, and the accumulated dividend payment is \(L_t = \int_0^t \ell_s \;d s\). The dividend rate is bounded by \(0 \le \ell_t \le g(X_t^\pi)\), for some increasing Lipschitz-continuous and linearly bounded function \(g(x)\). The value of a dividend strategy is \(V^\pi(x) = \mathbb{E}[\int_0^{\tau^\pi} \ell_t e^{-\delta t}\;d t]\) where \(\delta > r\) is a preference parameter and \(\tau^\pi = \inf\{t > 0: X_t^\pi < 0\}\) is the time of ruin. The goal is to maximise the value, \(V(x) = \sup_\pi V^\pi(x)\), where the supremum is taken over all adapted strategies. It is proved that \(V(x)\) is the smallest viscosity solution to the corresponding Hamilton-Jacobi-Bellman equation. It turns out that there is an optimal strategy. The optimal dividend strategy is of band type. That is, \(\ell_t^* = 0\) on some band \(\mathcal{A}\), \(\ell_t^* = g(X_t^*)\) on some other band \(\mathcal{B}\). The article follows very closely the article by \textit{P. Azcue} and \textit{N. Muler} [Ann. Appl. Probab. 20, No. 4, 1253--1302 (2010; Zbl 1196.91033)]. In the latter paper, there is no restriction on the dividend payment. However, one should prove in addition that the process under the ``optimal strategy'' exists. The problem is the term \(\int_0^t g(X_s^*)\mathcal{I}_{\mathcal B}(X_s^*)\;d s\), because the Brownian motion in the investment moves infinitely often from \(\mathcal A\) to \(\mathcal B\) and vice versa if the surplus is that the boundary. The problem does not appear in the article by Azcue and Muler because there the process is reflected at such a boundary.