MPS-RR 2000-38

September 2000

# On Criteria for the Uniform Integrability of Brownian Stochastic Exponentials

by:

### A.S. Cherny, A.N. Shiryaev

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.*