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.
Funding Statement
The project on which this publication is based has been carried out with funding provided by the Alexander von Humboldt Foundation, under the programme financed by the German Federal Ministry of Education and Research entitled German Research Chair No 01DG15010.
Acknowledgments
The authors wish to thank an anonymous referee and the editor for their valuable comments and suggestions.
Citation
Antoine-Marie Bogso. Moustapha Dieye. Olivier Menoukeu Pamen. "Path-by-path uniqueness of multidimensional SDE’s on the plane with nondecreasing coefficients." Electron. J. Probab. 27 1 - 26, 2022. https://doi.org/10.1214/22-EJP844
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