MPS-RR 2000-38
September 2000

On Criteria for the Uniform Integrability of Brownian Stochastic Exponentials

by:

A.S. Cherny, A.N. Shiryaev

Abstract

This paper deals with various sufficient (as well as necessary and sufficient) conditions for the uniform integrability of the exponential martingales of the form Z_t=exp Bigl {B_{t wedge tau}-frac12 ,t wedge tau Bigr }, t ge 0, where B is a Brownian motion and tau is a stopping time. We give an overview of the known results and present some new criteria (Theorems 3.2, 4.1). As an auxiliary lemma, we prove the following statement that is interesting in itself: for any function phi: R_{+} to R, the upper limit limsup_{t ua infty}(B_t- phi(t)) either equals + infty a.s. or equals -infty a.s. This provides a simple criterion for distinguishing lower and upper functions of a Brownian motion.

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This paper has now been published in Optimal Control and Partial Differential Equations. IOS Press, 2001. In honour of Alain Bensoussan's 60th birthday, pp. 80-92.