Exponential Bounds for the Density of the Law of the Solution of an SDE with Locally Lipschitz Coefficients

Cristina Anton ()
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Cristina Anton: Department of Mathematics and Statistics, Grant MacEwan University, 5-103C, 10700-104 Ave., Edmonton, AB T5J 4S2, Canada

Mathematics, 2025, vol. 13, issue 5, 1-20

Abstract: Under the uniform Hörmander hypothesis, we study the smoothness and exponential bounds of the density of the law of the solution of a stochastic differential equation (SDE) with locally Lipschitz drift that satisfies a monotonicity condition. We extend the approach used for SDEs with globally Lipschitz coefficients and obtain estimates for the Malliavin covariance matrix and its inverse. Based on these estimates and using the Malliavin differentiability of any order of the solution of the SDE, we prove exponential bounds of the solution’s density law. These results can be used to study the convergence of implicit numerical schemes for SDEs.

Keywords: Malliavin covariance matrix; Hörmander’s condition; exponential bounds for density; monotone growth; stochastic differential equation (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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