Minimal Expected Time in Drawdown through Investment for an Insurance Diffusion Model

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