The Finiteness Of Moments Of Stochastic Exponentials

The theories concerning martingales are applied to the researches on the finance, stochastic differential equations, stochastic control theory and filtering theory, accelerating the development of correlativ subjects. Stochastic differential equations developed to a fruitful branch of stochastic analysis, so as the related fields. In some specific matters, for instant it is necessary to know whether the moments of a stochastic exponential are finite and under which conditions, when considering the finiteness of the solutions of some kinds of stochastic differential equations. Therefore, the problem of finding effective criteria for a stochastic exponential to be a martingale or the finiteness of its moments has aroused significant interest in the literature.In this thesis, by taking use of the methods in Yan Jiaan’s papers in which some sufficient conditions of the finiteness of a stochastic exponential were given, the objects, named the increasing process connected to a discontinuous square integrable martingale or local martingale, are replaced with the predictable dual projections of them in the sufficient conditions, and then the conditions are extended. Secondly, considering the contact between the moments and the coefficients of the local martingale or square integrable martingale in the sufficient conditions, this paper separately shows that for some moments, there is a coefficient such that things work. And then the conclusions above are compared to that of the case in which for some coefficients, there is a moment such that things work. Deserve to be mentioned, we give a corollary concerning the finiteness of the condition expection of a stochastic exponential to, which is applied to show that the moments of the stochastic exponential of a stochastic integral with respect to a Brownian motion are all finite, provided the integrand is a Brownian functional of linear growth, however the proof of it in the reference being improved.