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Local Lipschitz Continuity in the Initial Value and Strong Completeness for Nonlinear Stochastic Differential Equations
About this Title
Sonja Cox, Martin Hutzenthaler and Arnulf Jentzen
Publication: Memoirs of the American Mathematical Society
Publication Year:
2024; Volume 296, Number 1481
ISBNs: 978-1-4704-6701-2 (print); 978-1-4704-7818-6 (online)
DOI: https://doi.org/10.1090/memo/1481
Published electronically: May 1, 2024
Keywords: Non-linear stochastic ordinary differential equations,
non-linear stochastic partial differential equations,
regularity with respect to initial value,
strong completeness,
stochastic Van der Pol equation,
stochastic Duffing-Van der Pol equation,
stochastic Burgers equations,
Cahn-Hilliard-Cook equation,
non-linear stochastic wave equation
Table of Contents
Chapters
- 1. Introduction
- 2. Strong stability analysis for solutions of SDEs
- 3. Strong completeness of SDEs
- 4. Examples of SODEs
- 5. Examples of SPDEs
Abstract
Recently, Hairer et al. (2015) showed that there exist stochastic differential equations (SDEs) with infinitely often differentiable and globally bounded coefficient functions whose solutions fail to be locally Lipschitz continuous in the strong $L^p$-sense with respect to the initial value for every $p\in (0,\infty ]$. In this article we provide conditions on the coefficient functions of the SDE and on $p \in (0,\infty ]$ that are sufficient for local Lipschitz continuity in the strong $L^p$-sense with respect to the initial value and we establish explicit estimates for the local Lipschitz continuity constants. In particular, we prove local Lipschitz continuity in the initial value for several nonlinear stochastic ordinary and stochastic partial differential equations in the literature such as the stochastic van der Pol oscillator, Brownian dynamics, the Cox-Ingersoll-Ross processes and the Cahn-Hilliard-Cook equation. As an application of our estimates, we obtain strong completeness for several nonlinear SDEs.- Aurélien Alfonsi, On the discretization schemes for the CIR (and Bessel squared) processes, Monte Carlo Methods Appl. 11 (2005), no. 4, 355–384. MR 2186814, DOI 10.1163/156939605777438569
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