Non-Lipschitz SDE Driven By Multi-parameter Brownian Motions

In this paper, we mainly study the following three aspects: the first is the extension of Bihari's inequality in multi-parameter case, the second is the existence and uniqueness of the solution to stochastic differential equations with non-Lipschitz coefficients driven by multi-parameter Brownian motions. Then we also study two discretizing schemes for this type of equation, and obtain their L~2-convergence speeds.Firstly, we introduct several results on stochastic integrals in the plane, which play a basal role in proving the main results of the following parts. Then we consider the following stochastic differential equations(SDE) driven by W_zsubjected to the boundary condition , where W is an m dimension Brownian motions with d parameters.σ(x) : R~n→R~n×R~m and b{x) : R~n→R~n are continuous function.In this paper, we shall prove the existence and uniqueness for Eq.(1) under some non-Lipschitz conditions. Most of the well known results for non-Lipschitz SDEs with one or two parameters are based on the Ito formula. The strongly Ito formula will become non applicable in d≥3 as a result of the complexity of the form. To avoid using the Ito formula, we shall extend Bihari's inequality to the multi-parameter case.With this extension in hand, the existence and uniqueness for Eq.(1) are simple by successive approximation. And we shall prove thatTheorem 1 Letρ: R~+→R~+ be a continuous non-decreasing and concave function satisfyingρ(0) = 0 andAssume thatThen there is a unique continuous F~z-adapted solution denoted by {X~z, z∈T} to Eq.(l)On the other hand, in order to simulate the numerical calculation,we usually need to discretize SDE(1). Here we shall discuss two discretizing manners. One is similar to the Euler scheme in one parameter case, Another way is that we discretize the multi-parameter Brownian motion, and obtain their L~2-convergence speeds.