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

Abstract

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.

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

“…For convenience of the reader the following proposition, Proposition 2.2, formulates Proposition 4.5 in [11] which is a version of the Kolmogorov-Chentsov continuity theorem.…”

“…For convenience of the reader the following proposition, Proposition 2.3, formulates a part of Corollary 2.5 in [11]. ), T, δ ∈ (0, ∞) let (Ω, F , P) be a probability space with a filtration (F t ) t∈[0,T ] which satisfies that {B ∈ F : P(B) = 0} ⊆ F 0 , and let (W t ) t∈[0,T ] be a cylindrical Id U -Wiener process, let X, a…”

“…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]).…”

“…Without loss of regularity we additionally assume throughout this proof that s 1 ≤ s 2 (otherwise exchange the roles of (s 1 , t 1 , x 1 ) and of (s 2 , t 2 , x 2 ). Lemma 3.8 in [11] (applied with…”

Counterexamples to local Lipschitz and local Hölder continuity with respect to the initial values for additive noise driven SDEs with smooth drift coefficient functions with at most polynomially growing derivatives

“…The regularity of stochastic differential equations (SDEs) with respect to their initial values naturally arises as an important problem in stochastic analysis (cf., e.g., Chen & Li [1], Cox et al [2], Fang et al [3], Hairer et al [4], Hairer & Mattingly [5], Hudde et al [7], Krylov [12], Li [13], Li & Scheutzow [14], Liu & Röckner [15], and Scheutzow & Schulze [16]). At the same time this problem has strong links to the analysis of numerical approximations for SDEs (cf., e.g., Hudde et al [6], Hutzenthaler & Jentzen [8], Hutzenthaler et al [9], and Zhang [17]).…”