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Sufficient Conditions Of The Uniform Integrability Of Exponential Martingales

Dmitry Kramkov, A. Shiryaev
Published 1998 · Mathematics

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The process Z = (Z t ,^ ,P) t>o is a positive (exponential) martingale and by Doob's convergence theorem ([1], Chapter 1, §le), with P-probability one there exists a limit Z^ = lim^oo Zt\ in addition, Z^ = 0 (P-a.s.) and therefore EZoo 0. It is of interest in many issues of the stochastic calculus to describe the Markov times r = r(u) such that EZr = 1, which is equivalent to the condition that the family of random variables {Z iAr ,£ > 0} be uniformly integrable with respect to P. Obviously, EZ r = 1 for each bounded Markov time r , that is, for a time such that T(UJ) < N for some constant N and all tu G ft. Is was shown in [2] that if there exists e > 0 that Eexp((l + e)r) < oo, then EZ r = 1. This condition was weakened in [3]; namely, it was shown there that if Eexp((l/2 + e)r) < oo for some e > 0, then EZr = 1. Novikov [4] showed that one can set e = 0 in the above condition, that is,
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