A Note on Applications of Stochastic Ordering to Control Problems in Insurance and Finance

Abstract

We consider a controlled diffusion process $(X_t)_{t\ge 0}$ where the controller is allowed to choose the drift $\mu_t$ and the volatility $\sigma_t$ from a set \K(x) \subset \R\times (0,\infty) when $X_t=x$. By choosing the largest $\frac{\mu}{\sigma^2}$ at every point in time an extremal process is constructed which is under suitable time changes stochastically larger than any other admissible process. This observation immediately leads to a very simple solution of problems where ruin or hitting probabilities have to be minimized. Under further conditions this extremal process also minimizes "drawdown" probabilities.Comment: To appear in Stochastics. Keywords: Time changed continuous Martingale, Stochastic Ordering, Ruin Proble