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.

To appear in Stochastics. Keywords: Time changed continuous Martingale, Stochastic Ordering, Ruin Problem

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