Path-by-path uniqueness of multidimensional SDE's on the plane with nondecreasing coefficients

Abstract

In this paper with study a uniqueness result (in the sense of [13]) for multidimensional stochastic differential equations driven by the Brownian sheet. We assume that the drift coefficient is unbounded, verifies a spacial linear growth condition and is componentwise nondeacreasing. We first show the result when the drift is bounded measurable and nondecreasing. Our proofs rely on a local time-space representation of Brownian sheet and a type of law of the iterated logarithm for the Brownian sheet. The result in the unbounded case then follows by using the Gronwall's lemma on the plane. As a by product, we obtain the existence of a unique strong solution of multidimensional SDEs driven by Brownian sheet when the drift is non-decreasing and satisfies a spacial linear growth condition.