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

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 $\text{K}(x)\subset \mathbb{R}\times (0,\mathrm{\infty })$ 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.