Local Lipschitz continuity in the initial value and strong completeness for nonlinear stochastic differential equations

Cox, Sonja; Hutzenthaler, Martin; Jentzen, Arnulf

doi:10.48550/arxiv.1309.5595

Cited by 33 publications

(82 citation statements)

References 32 publications

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“…The next result, Lemma 2.6, is a minor generalization of Corollary 2.4 in [4] to arbitrary nonnegative starting times.…”

Section: Lemma 25 (A Lyapunov Estimate) Assume Setting 24 and Letmentioning

confidence: 70%

“…For the convenience of the reader, we recall the following well-known Lyapunov estimate consequence of e.g., Theorem 2.4 in [11], Lemma 2.2 in [4] or the proof of Lemma 2.2 in Gyöngy & Krylov [9]).…”

Section: Exponential Moment Estimatesmentioning

confidence: 99%

“…Strong completeness does not hold in general, in fact, there exist counterexample SDEs with smooth and globally bounded coefficients; see [20]. Under suitable general assumptions on the coefficient functions, however, strong completeness of the SDE (1) holds; see [4] and also, e.g., [24,1,8,7,6,22,21,19]…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On moments and strong local Hölder regularity of solutions of stochastic differential equations and of their spatial derivative processes

Hudde¹,

Hutzenthaler²,

Mazzonetto³

2019

Preprint

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Spatial differentiability of solutions of stochastic differential equations (SDEs) is required for the Itô-Alekseev-Gröbner formula and other applications. In the literature, this differentiability is only derived if the coefficient functions of the SDE have bounded derivatives and this property is rarely satisfied in applications. In this article we establish existence of continuously differentiable solutions of SDEs whose coefficients satisfy a suitable local monotonicity property and further conditions. These conditions are satisfied by many SDEs from applications.

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“…The next result, Lemma 2.6, is a minor generalization of Corollary 2.4 in [4] to arbitrary nonnegative starting times.…”

Section: Lemma 25 (A Lyapunov Estimate) Assume Setting 24 and Letmentioning

confidence: 70%

Section: Exponential Moment Estimatesmentioning

confidence: 99%

See 1 more Smart Citation

On moments and strong local Hölder regularity of solutions of stochastic differential equations and of their spatial derivative processes

Hudde¹,

Hutzenthaler²,

Mazzonetto³

2019

Preprint

Self Cite

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“…More importantly, already in the case of standard ODEs there are examples of finite time blowup for suitable choices of b and continuous Y , see [17,Section 3.3]; the monotonicity assumption can be replaced by more refined criteria, see [43] and the recent work [9], which however don't seem to transfer easily to the distribution-dependent setting.…”

Section: 1mentioning

confidence: 99%

Distribution dependent SDEs driven by additive continuous noise

Galeati,

Harang,

Mayorcas

2021

Preprint

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We study distribution dependent stochastic differential equation driven by a continuous process, without any specification on its law, following the approach initiated in [16]. We provide several criteria for existence and uniqueness of solutions which go beyond the classical globally Lipschitz setting. In particular we show well-posedness of the equation, as well as almost sure convergence of the associated particle system, for drifts satisfying either Osgood-continuity, monotonicity, local Lipschitz or Sobolev differentiability type assumptions. Contents 1. Introduction 1 1.1. Notations, conventions and well-known facts 4 2. Well-Posedness Under Lipschitz Assumptions 5 3. Refined criteria for existence and uniqueness 9 3.1. Existence 11 3.2. Uniqueness 13 4. DDSDEs with convolutional structure 21 5. Mean Field Convergence 28 5.1. Abstract criteria 28 5.2. Applications to particular drifts 30 Appendix A. Approximation in metric spaces 33 Appendix B. Analytic lemmas and inequalities involving maximal functions 34 References 37

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“…This, the fact that the function (0, ∞) ∋ y → y 1 p ∈ R is increasing and continuous, and assumption (10) imply that…”

Section: A Nonlinear Gronwall-bellman-opial Inequalitymentioning

confidence: 99%

A stochastic Gronwall inequality and applications to moments, strong completeness, strong local Lipschitz continuity, and perturbations

Hudde,

Hutzenthaler,

Mazzonetto

2019

Preprint

Self Cite

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There are numerous applications of the classical (deterministic) Gronwall inequality. Recently, Michael Scheutzow discovered a stochastic Gronwall inequality which provides upper bounds for p-th moments, p ∈ (0, 1), of the supremum of nonnegative scalar continuous processes which satisfy a linear integral inequality. In this article we complement this with upper bounds for p-th moments, p ∈ [2, ∞), of the supremum of general Itô processes which satisfy a suitable one-sided affine-linear growth condition. As example applications, we improve known results on strong local Lipschitz continuity in the starting point of solutions of stochastic differential equations (SDEs), on (exponential) moment estimates for SDEs, on strong completeness of SDEs, and on perturbation estimates for SDEs.

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Local Lipschitz continuity in the initial value and strong completeness for nonlinear stochastic differential equations

Cited by 33 publications

References 32 publications

On moments and strong local Hölder regularity of solutions of stochastic differential equations and of their spatial derivative processes

On moments and strong local Hölder regularity of solutions of stochastic differential equations and of their spatial derivative processes

Distribution dependent SDEs driven by additive continuous noise

A stochastic Gronwall inequality and applications to moments, strong completeness, strong local Lipschitz continuity, and perturbations

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