We consider a controlled diffusion process $(X_t)_{t\ge 0}$ where thecontroller 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 thelargest $\frac{\mu}{\sigma^2}$ at every point in time an extremal process isconstructed which is under suitable time changes stochastically larger than anyother admissible process. This observation immediately leads to a very simplesolution of problems where ruin or hitting probabilities have to be minimized.Under further conditions this extremal process also minimizes "drawdown"probabilities.
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