The Generalized Sample Solutions Of Stochastic Differential Equations

Stochastic differential equations have attracted the interest of most the pure scientist and applied scientists since K .Ito? first published "On stochastic differential equations" in 1961. As a tool of modern mathematics, stochastic differential equation, has made remarkable achievements in many areas , and played a great role in social development.The classical theory of stochastic differential equations is perturbed by Brownian motion. However, there are great limitations in application due to the special nature of Brownian motion. Therefore many researchers have attempted to promote the theory of the classical stochastic differential equations, and have achieved a lot.In this paper, firstly, we discussed a kind of stochastic differential equations which is defined on the Complete probability space(Ω, F ,P, ), perturbed by a continuous Square integrable martingale, and it's drift coefficient and fluctuation coefficient is function which meets local linear growth conditions and local Lipchitz conditions, and [ M ,M ]( t )is absolutely continuous about t. we used the random time transformation to change the stochastic differential equations, and then testified the existence and uniqueness of it's Solution using the Picard iteration.Getting the solutions of Stochastic differential equation has been the important issues of practical application, and to get the solutions of stochastic differential equations also is a difficult problem, In this paper we proposed a new numerical method about a special form stochastic differential equations on the basis of predecessors' research. we discussed the random sample of generalized solutions of above stochastic differential equations, we transform the problem this stochastic differential equation into ordinary differential equation's problem, and proofed the rationality of this transformation and their solutions are the same. And this paper give the method of getting the random sample of generalized solutions of above stochastic differential equations.