The Bellman equation for control of the running max of a diffusion and applications to look-back options

Abstract

We will prove that the value function for the optimal control of the running max of a diffusion is the viscosity solution of an oblique derivative problem. The result is derived using L^p approximations. Then we apply the technique to derive the totally risk averse price of an option, called a look back option, on the running minimum of a stock.

Keywords:

• Optimal stochastic control
• running maximum
• viscosity solutions
• look-back options
• total risk aversion

AMS::

• 49B60
• 90A09