Existence and uniqueness theorem of associated solutions of the stochastic differential system with measures

The Cauchy problem for stochastic differential system with measures is considered in the paper. Finite-difference with averaging system of stochastic differential equations with correspondence to Cauchy problem is investigated. Stochastic integral equation system with the solution as a limit of the finite-difference with averaging system of stochastic differential equations is studied and also the space of the solutions is described. Moreover, the Cauchy problem associated solutions of the stochastic differential system with measures are defined. Existence and uniqueness theorem of the associated solutions is proved. Besides that, Cauchy problem for higher-order linear stochastic differential equation is considered. Associated solutions of the Cauchy problem are investigated. Also the theorem of the representation of the associated solutions of the higher-order linear stochastic differential equation with help of associated fundamental matrices of the correspondent homogeneous equation system is proved.