Boundary crossing optimal stopping and optimal bayesian sequential declsion procedures

Abstract

Bayesian sequential decision problems can be reduced to an optimal stopping problem for a stochastic process of the form A+B, where A is a submartingale and B a supermartingale, and by the Doob Meyer decomposition, A = C+N and B = V+M, where N and M are local martingales, C is an increasing and V a decreasing predictable process. If N and M are uniformly integrable, then we have an equivalent optimal stopping problem for the process C+V. Often, C and V do not only have continuous but also "differentiable" paths. The stopping time T of first horizontal tangent is suggested as an approximation of an optimal stopping time. Often, T leads to a problem of curved boundary crossing for a basic observable process, where the boundary may be non random or random.

Keywords:

• boundary crossing
• optimal stopping
• Bayesian sequential decision procedures
• Doob Meyer decomposition
• Ito formula
• supermartingales
• local martingales
• stochastic processes with independent increments