Strong completeness for a class of stochastic differential equations with irregular coefficients

Abstract

We prove the strong completeness for a class of non-degenerate SDEs, whose coefficients are not necessarily uniformly elliptic nor locally Lipschitz continuous nor bounded. Moreover, for each $t$, the solution flow $F_t$ is weakly differentiable and for each $p>0$ there is a positive number $T(p)$ such that for all $t<T(p)$, the solution flow $F_t(\cdot )$ belongs to the Sobolev space $W_{\loc}^{1,p}$. The main tool for this is the approximation of the associated derivative flow equations. As an application a differential formula is also obtained.

References in corpus (7)

• Formulae for the derivatives of heat semigroups
• Loss of regularity for Kolmogorov equations
• Sobolev differentiable stochastic flows for SDEs with singular coefficients: Applications to the transport equation
• Local Lipschitz continuity in the initial value and strong completeness for nonlinear stochastic differential equations
• Global flows for stochastic differential equations without global Lipschitz conditions
• Stochastic Flows on Non-compact Manifolds
• An approximation scheme for SDEs with non-smooth coefficients

Cited by in corpus (4)

• Solution theory of fractional SDEs in complete subcritical regimes
• A stochastic Gronwall inequality and applications to moments, strong completeness, strong local Lipschitz continuity, and perturbations
• Strong solutions of stochastic differential equations with coefficients in mixed-norm spaces
• 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