On the theory of backward stochastic differential equations and their applications

In this paper, we discuss conditions of existence of solutions of backward stochastic differential equations with respect to general filtrations. A solution of a linear backward stochastic differential equation is found using classical theory of differential equations. We also study a special class of backward stochastic differential equations. Using solution properties of this type of equations, we give a new direct proof of Doob-Meyer theorem on a decomposition of a supermartingale from class $DL$ into a difference of a martingale and an increasing predictive process. We also prove a new theorem on transposition of an integral of a stochastic process and a conditional mathematical expectation.