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$L^{α-1}$ distance between two one-dimensional stochastic differential equations with drift terms driven by a symmetric $α$-stable process

Takuya Nakagawa

Abstract

This paper establishes a quantitative stability theory for one-dimensional stochastic differential equations (SDEs) with non-zero drift, driven by a symmetric $α$-stable process for $α\in(1,2)$. Our work generalizes the classical pathwise comparison method, pioneered by Komatsu for uniqueness problems, to address the stability of SDEs featuring both non-zero drift and, crucially, time-dependent coefficients. We provide the first explicit convergence rates for this broad class of SDEs. The main result is a Hölder-type estimate for the $L^{α-1}(Ω)$ distance between two solution paths, which quantifies the stability with respect to the initial values and coefficients. A key innovation of our approach is the measurement of the distance between coefficients. Instead of using a standard supremum norm, which would impose restrictive conditions, we introduce a weighted integral norm constructed from the transition probability density of the baseline solution. This technique, which generalizes the framework of Nakagawa \cite{Nakagawa}, is essential for handling time-dependent perturbations and effectively localizes the error analysis. The proof is based on a refined analysis of a mollified auxiliary function, for which we establish a new, sharper derivative estimate to control the drift terms. Finally, we apply these stability results to derive corresponding convergence rates in probability, providing an upper bound for the tail probability of the uniform distance between solution paths.

$L^{α-1}$ distance between two one-dimensional stochastic differential equations with drift terms driven by a symmetric $α$-stable process

Abstract

This paper establishes a quantitative stability theory for one-dimensional stochastic differential equations (SDEs) with non-zero drift, driven by a symmetric -stable process for . Our work generalizes the classical pathwise comparison method, pioneered by Komatsu for uniqueness problems, to address the stability of SDEs featuring both non-zero drift and, crucially, time-dependent coefficients. We provide the first explicit convergence rates for this broad class of SDEs. The main result is a Hölder-type estimate for the distance between two solution paths, which quantifies the stability with respect to the initial values and coefficients. A key innovation of our approach is the measurement of the distance between coefficients. Instead of using a standard supremum norm, which would impose restrictive conditions, we introduce a weighted integral norm constructed from the transition probability density of the baseline solution. This technique, which generalizes the framework of Nakagawa \cite{Nakagawa}, is essential for handling time-dependent perturbations and effectively localizes the error analysis. The proof is based on a refined analysis of a mollified auxiliary function, for which we establish a new, sharper derivative estimate to control the drift terms. Finally, we apply these stability results to derive corresponding convergence rates in probability, providing an upper bound for the tail probability of the uniform distance between solution paths.

Paper Structure

This paper contains 16 sections, 10 theorems, 76 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. General notation
  4. The symmetric $\alpha$-stable process
  5. The transition density function and its estimates for SDE \ref{['SDE']}
  6. Distance between solutions in the $L^{\alpha-1}(\Omega)$-norm
  7. Komatsu's method and properties of the mollifying Function
  8. Proof of Theorem \ref{['mainthm']}
  9. Convergence in probability of the solution paths
  10. Proof strategy for Theorem \ref{['mainthm2']}
  11. Appendices
  12. The time-homogeneous case
  13. An alternative framework: The supremum norm case
  14. Proof of the martingale property for $M^{\delta,\varepsilon}$
  15. Regarding the well-definedness of a function $L_\alpha u_{\delta,\varepsilon}$
  16. ...and 1 more sections

Key Result

Theorem 3.1

Let $Z$ be a one-dimensional symmetric $\alpha$-stable process with $\alpha \in (1,2)$. Consider two processes, $X$ and $\tilde{X}$, satisfying the SDEs for $t\in[0,T]$ with initial values $x_0, \tilde{x}_0\in\mathbb{R}$. Suppose the coefficients satisfy the following conditions for some constants $K, k, K'>0$ and $\tilde{\eta} \in [1/\alpha,1]$: Then, there exists a positive constant $C = C\lef

Theorems & Definitions (18)

  • Theorem 3.1
  • Remark 1
  • Lemma 3.2
  • Lemma 3.3
  • proof
  • Theorem 4.1
  • Lemma 4.2
  • Corollary 5.1
  • proof
  • Theorem 5.2
  • ...and 8 more