The Minimal Entropy Martingale Measures For Multi-dimensional Jump Processes

The study of transportation inequalities is a very important topic in probability and statistics.Talagrand's inequalities are closely related with measure concentration phenomenon,log Sobolev's inequality and Poincare inequality.This paper mainly studies the transportation inequalities of a backward stochastic differential equation with mean reflection and a nolinear stochastic heat equation driven by Gaussian noise,which is white in time and has a fractional Brownian motion in the space.This paper is divided into four chapters.We introduces the basic knowledge and the current research status around the world of transportation inequality in the first chapter.We introduces the research background of backward stochastic differential equations with mean reflection and the definition of deterministic flat solutions at the beginning of the second chapter,then we give the coefficient assumptions and prove the existence and uniquness of a solution to the equation in the following sections.In the last two sections of the second chapter,we prove that the distribution of the deterministic flat solution to the equation satisfies transportation inequalities with respect to the uniform norm and the L2 norm and give some applications of transportation inequalities to the concentration inequalities.In the first section of the third chapter,we mainly introduces the research progress of the stochastic heat equation and the definition of stochastic integral and its properties.In the second section,We introduces the function space in which the solution exists and the coefficient hypothesis of stochastic heat equation.In the third and fourth sections,we prove that the distribution of the solution to the equation with respect to the weighted L2 norm satisfies transportation inequality.In the last section,we study the applications of the transportation inequality to the three specific concentration inequalities.Finally,in the fourth chapter,we summarize the paper and propose some improvements that could be attained in our following research.