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arXiv · 2112.00393

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

Antoine-Marie Bogso·Moustapha Dieye·Olivier Menoukeu-Pamen
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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.

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Antoine-Marie Bogso, Moustapha Dieye, Olivier Menoukeu-Pamen. 2022-09-23. Path-by-path uniqueness of multidimensional SDE's on the plane with nondecreasing coefficients. https://doi.org/10.1214/22-ejp844

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