Estimates on moments of the solutions to stochastic differential equations with respect to martingales in the plane

Abstract

Let M = { M z , z ϵ R 2 +} be a two-parameter strong martingale, A be a two-parameter increasing process on R 2 + = [0, + ∞) × [0, + ∞). Consider the following stochastic differential equations in the plane: X z = X 0 + ∞ R z a(ξ,X) dM ξ + ∞ R z b(ξ,X) dA ξ for z ϵ R 2 +. Under some assumptions on the coefficients a, b and the integrators M, A, we prove the existence and uniqueness of solutions for the equations, and obtain some estimates on moments of solution.