Minimal expected time in drawdown through investment for an insurance diffusion model

Consider an insurance company whose surplus is modelled by an arithmetic Brownian motion of not necessarily positive drift. Additionally, the insurer has the possibility to invest in a stock modelled by a geometric Brownian motion independent of the surplus. Our key variable is the (absolute) drawdown Δ of the surplus X, defined as the distance to its running maximum X̄. Large, long-lasting drawdowns are unfavourable for the insurance company. We consider the stochastic optimisation problem of minimising the expected time that the drawdown is larger than a positive critical value (weighted by a discounting factor) under investment. A fixed-point argument is used to show that the value function is the unique solution to the Hamilton-Jacobi-Bellman equation related to the problem. It turns out that the optimal investment strategy is given by a piecewise monotone and continuously differentiable function of the current drawdown. Several numerical examples illustrate our findings.