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

TL;DR

This paper explores how stochastic ordering can be used to control diffusion processes in insurance and finance, providing simple solutions to minimize ruin and drawdown probabilities through extremal process construction.

Contribution

It introduces a method to construct extremal controlled diffusion processes that are stochastically larger, simplifying the solution of control problems involving ruin and drawdown probabilities.

Findings

Extremal process maximizes the ratio μ/σ² at each point.

Extremal process minimizes ruin and hitting probabilities.

Under certain conditions, it also minimizes drawdown probabilities.

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.