Path-by-path uniqueness of multidimensional SDE’s on the plane with nondecreasing coefficients
Copyright policy )Path-by-path uniqueness of multidimensional SDE’s on the plane with nondecreasing coefficients
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.
24 pages
60H05, 60H15, stochastic wave equations, Stochastic partial differential equations (aspects of stochastic analysis), Brownian sheet, Probability (math.PR), FOS: Mathematics, Regularization by noise, path-by-path uniqueness, SDEs on the plane, Mathematics - Probability, Stochastic ordinary differential equations (aspects of stochastic analysis)
60H05, 60H15, stochastic wave equations, Stochastic partial differential equations (aspects of stochastic analysis), Brownian sheet, Probability (math.PR), FOS: Mathematics, Regularization by noise, path-by-path uniqueness, SDEs on the plane, Mathematics - Probability, Stochastic ordinary differential equations (aspects of stochastic analysis)
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