Uniform Integrability Of The Stochastic Exponential Martingales

In this paper，we apply property of Brownian motion, continuous martingale,Girsanov ’s theorem and the low function of a Brownnian motion to gain sufcientcondition for stochastic exponentialε(M)=Zt=exp{Mt1/2〔M〕t},to be uniformly integrable martingales. The case of M=Wtis extended to the casethat M is a local martingale. And it is important that two results of Cherny andShiryaev’ are unifed, which improves Kazamaki conditions and Novikov conditions.Finally, we give some examples to illustrate the generality of the results in this paper.As an auxiliary lemma, we prove that for any function φ: R+â†'R, Supremumlim sup(Wtâ†'∞t) φ(t) is either equal to+∞or equal to∞. This result can easilydistinguish low and upper function of a Brownian motion.