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Regularity of stochastic differential equations on the Wiener space by coupling

Stefan Geiss, Xilin Zhou

TL;DR

The paper develops a robust coupling framework on Wiener space to study regularity properties of stochastic differential equations with random, path-dependent coefficients. By combining transference, Besov spaces via real interpolation, and two coupling variants (uniform and cut-off), it establishes Malliavin-Sobolev and Besov regularity results for Lipschitz SDEs and reveals limitations under Hölder diffusion. It also links these forward- SDE regularities to backward SDEs, using Zvonkin and Lamperti transforms to enhance initial-value dependence analysis and to derive Lp-variation bounds. A central finding is that Lipschitz systems enjoy strong regularity across Malliavin/Besov scales, while Hölder coefficients can break D_{1,2}-membership, with precise rates given for Hölder dispersion in dimension one. These results have implications for Malliavin differentiability, real interpolation of stochastic processes, and the numerical treatment of BSDEs in fully path-dependent settings.

Abstract

Using the coupling method introduced in \cite{Geiss:Ylinen:21}, we investigate regularity properties of stochastic differential equations, where we consider the Lipschitz case in $\R^d$ and allow for Hölder continuity of the diffusion coefficient of scalar valued stochastic differential equations. Two cases of the coupling method are of special interest: The uniform coupling to treat the Malliavin Sobolev space $\D_{1,2}$ and real interpolation spaces, and secondly a cut-off coupling to treat the $L_p$-variation of backward stochastic differential equations where the forward process is the investigated stochastic differential equation.

Regularity of stochastic differential equations on the Wiener space by coupling

TL;DR

The paper develops a robust coupling framework on Wiener space to study regularity properties of stochastic differential equations with random, path-dependent coefficients. By combining transference, Besov spaces via real interpolation, and two coupling variants (uniform and cut-off), it establishes Malliavin-Sobolev and Besov regularity results for Lipschitz SDEs and reveals limitations under Hölder diffusion. It also links these forward- SDE regularities to backward SDEs, using Zvonkin and Lamperti transforms to enhance initial-value dependence analysis and to derive Lp-variation bounds. A central finding is that Lipschitz systems enjoy strong regularity across Malliavin/Besov scales, while Hölder coefficients can break D_{1,2}-membership, with precise rates given for Hölder dispersion in dimension one. These results have implications for Malliavin differentiability, real interpolation of stochastic processes, and the numerical treatment of BSDEs in fully path-dependent settings.

Abstract

Using the coupling method introduced in \cite{Geiss:Ylinen:21}, we investigate regularity properties of stochastic differential equations, where we consider the Lipschitz case in and allow for Hölder continuity of the diffusion coefficient of scalar valued stochastic differential equations. Two cases of the coupling method are of special interest: The uniform coupling to treat the Malliavin Sobolev space and real interpolation spaces, and secondly a cut-off coupling to treat the -variation of backward stochastic differential equations where the forward process is the investigated stochastic differential equation.

Paper Structure

This paper contains 26 sections, 38 theorems, 236 equations.

Table of Contents

  1. Introduction
  2. General setting and preliminaries
  3. General notation
  4. Stochastic basis
  5. The Malliavin Sobolev space $\mathbb D_{1,2}$
  6. The $\sigma$-algebra of predictable events
  7. Fefferman's inequality
  8. Other inequalities
  9. Transference of the Wiener space
  10. Couplings of the Wiener space and Besov spaces
  11. Couplings of the Wiener space
  12. Besov spaces $\mathbb B_p^\Phi$
  13. The spaces $\mathbb B_p^{\Phi_\alpha}$
  14. Characterization of real interpolation spaces
  15. Characterization of $\mathbb D_{1,2}$
  16. ...and 11 more sections

Key Result

Lemma 2.1

Let $t\in [0,T)$, $L:[t,T]\times \Omega\to \mathbb R$ with $L_t\equiv 0$, and $\widetilde{L}: [0,T]\times \Omega\to \mathbb R$ with $\widetilde{L}_s:= L_{s\vee t}$. Then the following assertions are equivalent:

Theorems & Definitions (82)

  • Lemma 2.1
  • Lemma 2.2
  • Proposition 2.3
  • proof
  • Definition 2.4: Geiss:Ylinen:21
  • Lemma 2.5: Fefferman's inequality, Geiss:Ylinen:21
  • Lemma 2.6: Lenglart's inequality
  • Lemma 2.7
  • proof
  • Lemma 3.1
  • ...and 72 more