Existence and Uniqueness Solution for a Semimartingale Stochastic Integral Equation

This paper studied existence and uniqueness of a solution for a semimartingale stochastic integral equation by using Existence and Uniqueness Theorem on the martingale process. Using the concept of convergence Cauchy sequence  to a cadlag process , where  , we can find a convergence Cauchy sequence  to  a cadlag process  on the space  of martingales, where  is a square-integrable cadlag martingale on a probability space , as           = .  And some important assumptions are is a map from the space  into the space  of  -matrices.  satisfies a spatial Lipschitz condition uniformly in the other variables: for each  there exists a finite constant  such that this holds for  and all :   .  ii. Given any adapted -valued cadlag process  on  , the function  is a predictable process, and there exist stopping times  such that  is bounded for each .