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A norm equivalence result for stochastic differential equations with locally Lipschitz coefficients

Kyo Yamazaki

TL;DR

The paper develops a two-sided norm equivalence (weighted $L^p$ bounds) for stochastic differential equations with time-dependent locally Lipschitz coefficients and explicit growth controls, yielding constants $c=e^{-5\|\widetilde{K}\|_{L^1([0,T])}}$ and $C=e^{5\|\widetilde{K}\|_{L^1([0,T])}}$. A central tool is the inverse stochastic-flow analysis, for which the inverse-flow SDE is established via a backward Itô integral under reduced regularity, with mollification stability and a stochastic Liouville-type Jacobian representation. The paper also proves weak differentiability of solutions with respect to initial data in the globally Lipschitz setting, deriving a linear SDE for the Jacobian and determinant bounds. The main theorem is proven by a careful approximation scheme, change-of-variables on weakly differentiable maps, and a limiting argument to obtain the weighted norm bounds, which are then specialized to polynomial and exponential weights. As an application, the norm equivalence yields weighted integrability estimates for solutions to associated parabolic PDEs, linking probabilistic representations (BSDE/FBSDE) with analytic Sobolev-type estimates and providing a priori bounds and nonnegativity results.

Abstract

We establish two-sided weighted integrability estimates, often referred to as a norm equivalence result, for stochastic differential equations (SDEs) with locally Lipschitz coefficients. As a key ingredient in our approach, we also derive an SDE satisfied by the inverse stochastic flow under reduced regularity assumptions in the globally Lipschitz setting.

A norm equivalence result for stochastic differential equations with locally Lipschitz coefficients

TL;DR

The paper develops a two-sided norm equivalence (weighted bounds) for stochastic differential equations with time-dependent locally Lipschitz coefficients and explicit growth controls, yielding constants and . A central tool is the inverse stochastic-flow analysis, for which the inverse-flow SDE is established via a backward Itô integral under reduced regularity, with mollification stability and a stochastic Liouville-type Jacobian representation. The paper also proves weak differentiability of solutions with respect to initial data in the globally Lipschitz setting, deriving a linear SDE for the Jacobian and determinant bounds. The main theorem is proven by a careful approximation scheme, change-of-variables on weakly differentiable maps, and a limiting argument to obtain the weighted norm bounds, which are then specialized to polynomial and exponential weights. As an application, the norm equivalence yields weighted integrability estimates for solutions to associated parabolic PDEs, linking probabilistic representations (BSDE/FBSDE) with analytic Sobolev-type estimates and providing a priori bounds and nonnegativity results.

Abstract

We establish two-sided weighted integrability estimates, often referred to as a norm equivalence result, for stochastic differential equations (SDEs) with locally Lipschitz coefficients. As a key ingredient in our approach, we also derive an SDE satisfied by the inverse stochastic flow under reduced regularity assumptions in the globally Lipschitz setting.
Paper Structure (16 sections, 27 theorems, 65 equations)

This paper contains 16 sections, 27 theorems, 65 equations.

Table of Contents

  1. Main results
  2. Inverse flow
  3. Approximation by mollifiers
  4. Backward Itô integral
  5. Inverse flow
  6. Weak Differentiability of Solutions to SDEs
  7. Proof of the main theorem
  8. Application
  9. A priori estimates and stability for SDEs
  10. Uniform convergence in probability of random fields
  11. Weak derivatives
  12. Joint measurability of weak derivatives
  13. Chain rule for weak derivatives
  14. Proof of Lemma \ref{['th:estim_of_approximation_of_quadratic_varitaions_for_g(r,X)']}
  15. Proof of Lemma \ref{['th:differentiability_of_sol_of_SDE']}
  16. ...and 1 more sections

Key Result

Theorem 1.1

Assume item:asummption_of_main_theorem_2--item:asummption_of_main_theorem_4. Let $\rho:\mathbb{R}^d\to(0,\infty)$ be a $C^2$ function, and suppose that there exists a nonnegative function $\widetilde{K}\in L^1([0,T])$ such that for every $x\in\mathbb{R}^d$ and $r\in[0,T]$, and such that where $\sigma_{(k)}$ denotes the $k$-th column of $\sigma$. Then, for every Borel measurable function $\varphi

Theorems & Definitions (39)

  • Theorem 1.1: Main theorem
  • Corollary 1.2
  • Remark 1.3
  • Remark 1.4
  • Lemma 2.1
  • Lemma 2.2
  • Definition 2.3
  • Remark 2.4
  • Lemma 2.6
  • Lemma 2.7
  • ...and 29 more