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

TL;DR

This paper establishes path-by-path uniqueness for multidimensional SDEs driven by Brownian sheets with nondecreasing, unbounded drifts satisfying linear growth, using local time-space representation and a Gronwall lemma.

Contribution

It proves path-by-path uniqueness for multidimensional SDEs with nondecreasing, unbounded drifts driven by Brownian sheets, extending previous results to more general conditions.

Findings

Proves path-by-path uniqueness under specified conditions.

Establishes existence of a unique strong solution.

Utilizes local time-space representation and Gronwall lemma.

Abstract

In this paper we study path-by-path uniqueness for multidimensional stochastic differential equations driven by the Brownian sheet. We assume that the drift coefficient is unbounded, verifies a spatial linear growth condition and is componentwise nondeacreasing. Our approach consists of showing the result for bounded and componentwise nondecreasing drift using both a local time-space representation and a law of iterated logarithm for Brownian sheets. The desired result follows using a Gronwall type lemma on the plane. As a by product, we obtain the existence of a unique strong solution of multidimensional SDEs driven by the Brownian sheet when the drift is non-decreasing and satisfies a spatial linear growth condition.