Maximal Exponential Inequalities for Certain Diffusion Processes

Abstract

In this paper, we prove maximal exponential inequalities for a class of diffusion processes under certain conditions. The present results extend the well-known maximal inequalities of the power-type proved in the case of a Brownian motion, and in the geometric Brownian motion case. The method of proof is essentially based on explicit forms of optimal stopping problems related to those treated by Shepp and Shiryaev [Ann. Probab. Appl., 3 (1993), pp. 631--640], but considered here with a multiplicative reward functional. The key tool in the proof of one of our main results is a comparison principle for solutions of a system of first-order nonlinear differential equations. This is one of the novel features in this paper.