Suppose the wealth of an insurance company is modeled by a Brownian motion \(R_t, t\geq 0\), with drift, \(C_t,\;t\geq 0\) denotes the accumulated dividend process, and \(X_t=R_t - C_t\). Let \(\tau = \inf \{t: X_t\leq 0\}\) be the time of ruin under the dividend policy \(C\). The authors consider the following optimization problem: \[ \max _C E[U(\int_0^{\infty} \exp(-\beta t)dC_t)],\;\text{where}\;U(t)=(1-\exp(-\gamma x))/\gamma, \] i.e., they want to maximize the expected exponential utility of discount dividend payments. Here \(U(t)\) denotes the utility function with the risk aversion parameter \(\gamma\), and \(\beta\) is a discount factor.
It is shown that if a certain integral equation has a solution then the optimal strategy is a barrier strategy. The barrier function is a solution of this integral equation and turns out to be time-dependent. In addition, the authors study the optimization problem from a different point of view, namely by using a certain series ansatz for the boundary and value functions. They obtain recursions for the coefficients of the series, and numerical experiments indicate that such series produce meaningful results.