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

• Published in: Stochastic processes and their applications Vol. 62; no. 2; pp. 263 - 276
• Main Authors: Liang, Zong-xia, Zheng, Ming-li
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.07.1996; Elsevier
• Series: Stochastic Processes and their Applications
• Subjects: 60H15; 60H15 60H20 Two-parameter stochastic differential equation Two-parameter strong martingale Two-parameter Ito's formula Gronwall's inequality; 60H20; Gronwall's inequality; Two-parameter Ito's formula; Two-parameter stochastic differential equation; Two-parameter strong martingale; 60H20; Two-parameter Ito's formula; Two-parameter stochastic differential equation; 60H15; Gronwall's inequality; Two-parameter strong martingale
• ISSN: 0304-4149; 1879-209X
• DOI: 10.1016/0304-4149(96)00057-9

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.