On the existence and uniqueness of solutions to stochastic differential equations of mixed Brownian and Poissonian sheet type

Let Xz=X0+∫Rzf(ξ) dMξ+∫Rzu(ξ) dAξ+∫Rzα(ξ)Λ(dξ)+∫Rzβ(ξ)N(dξ) be a two-parameter semimartingale, where M is a continuous martingale, Λ is the character of the Poisson point measure Y, N=Y−Λ, we prove that f(Xz) is expressible as such a sum once again via the partial differentiation formula, where f is a twice continuously differentiable function. Then, we prove a new theorem on the existence and uniqueness of solutions to the mixed Brownian and Poissonian sheet type stochastic differential equations with non-Lipschitz coefficients by applying the partial differentiation formula.