[Paper Review] Pathwise uniqueness for singular SDEs driven by stable processes

Enrico Priola|Osaka City University (Osaka City University)|May 23, 2010

Stochastic processes and financial applications15 references18 citations

TL;DR

This paper establishes pathwise uniqueness for multidimensional stochastic differential equations (SDEs) driven by non-degenerate symmetric $α$-stable Lévy processes with bounded, Hölder continuous drifts satisfying $\beta > 1 - \alpha/2$ and $\alpha \in [1,2)$. The proof relies on novel global Schauder estimates for associated Kolmogorov-type integro-differential operators, ensuring the existence of strong solutions and establishing differentiability and homeomorphism properties of the solution flow with respect to initial conditions.

ABSTRACT

We prove pathwise uniqueness for stochastic differential equations driven by non-degenerate symmetric $α$-stable Lévy processes with values in $\R^d$ having a bounded and $β$-Hölder continuous drift term. We assume $β> 1 - \fracα{2} $ and $α\in [ 1, 2)$. The proof requires analytic regularity results for associated integro-differential operators of Kolmogorov type. We also study differentiability of solutions with respect to initial conditions and the homeomorphism property.

Motivation & Objective

• To establish pathwise uniqueness for SDEs driven by non-degenerate symmetric $\alpha$-stable Lévy processes when the drift is bounded and $\beta$-Hölder continuous.
• To extend the classical pathwise uniqueness framework beyond the Itô-Stratonovich setting to Lévy-driven SDEs with non-Gaussian stable noise.
• To develop analytic regularity results for integro-differential operators of Kolmogorov type associated with $\alpha$-stable processes.
• To establish differentiability and homeomorphism properties of the solution flow with respect to initial conditions under minimal regularity on the drift.

Proposed method

• Derives global Schauder estimates for the resolvent equation $\lambda u - \mathcal{L}u - b \cdot Du = g$ on $\mathbb{R}^d$, where $\mathcal{L}$ is the generator of the $\alpha$-stable process and $g \in C_b^\beta(\mathbb{R}^d)$.
• Applies an Itô-Tanaka-type argument to reduce pathwise uniqueness to the regularity of solutions to the resolvent equation.
• Uses interpolation and Hölder continuity estimates to control the difference of increments of the drift term in the resolvent equation.
• Establishes that $\|Du_{\lambda}\|_0 < 1/3$ for sufficiently large $\lambda$, enabling contraction arguments in the proof of pathwise uniqueness.
• Applies results from [14] on stochastic flows to deduce that the solution map $x \mapsto X_t^x$ is a $C^1$-diffeomorphism almost surely.
• Relies on the non-degeneracy of the $\alpha$-stable process and the condition $\alpha + \beta > 1$ to ensure integrability of the Lévy measure in the regularity estimates.

Experimental results

Research questions

• RQ1 Under what conditions on the drift $b$ and the index $\alpha$ of the symmetric $\alpha$-stable process does pathwise uniqueness hold for SDEs of the form $X_t = x + \int_0^t b(X_s)ds + L_t$?
• RQ2 Can Schauder-type estimates be established for Kolmogorov-type integro-differential operators driven by $\alpha$-stable Lévy processes with $\alpha \in [1,2)$?
• RQ3 Does the solution flow $x \mapsto X_t^x$ remain a $C^1$-diffeomorphism almost surely when $b$ is only $\beta$-Hölder continuous and $\beta > 1 - \alpha/2$?
• RQ4 What is the regularity of the derivative $D_x X_t^x$ with respect to initial conditions, and does it satisfy a stochastic integral representation?

Key findings

• Pathwise uniqueness holds for SDEs driven by non-degenerate symmetric $\alpha$-stable processes with $\alpha \in [1,2)$ and bounded $\beta$-Hölder continuous drifts satisfying $\beta > 1 - \alpha/2$.
• The solution map $x \mapsto X_t^x$ is a homeomorphism from $\mathbb{R}^d$ onto itself almost surely for all $t \geq 0$.
• For any $t \geq 0$ and $p \geq 1$, the $p$-th moment of the supremum of the difference of solutions satisfies $\mathbb{E}[\sup_{0\leq s\leq t}|X_s^x - X_s^y|^p] \leq C(t,p)|x-y|^p$, with $C(t,p)$ depending on $\alpha$, $\beta$, and the process $L$.
• The solution map $x \mapsto X_t^x$ is almost surely $C^1$-differentiable, and the derivative satisfies a stochastic integral equation involving the drift derivative and the Lévy noise.
• The proof relies on global Schauder estimates for the resolvent equation $\lambda u - \mathcal{L}u - b \cdot Du = g$, which are established under the condition $\alpha + \beta > 1$.
• The result implies the existence of a stochastic flow $\xi_{s,t}(x)$ satisfying the Markov property and the flow composition rule $\xi_{s,t} = \xi_{u,t} \circ \xi_{s,u}$ almost surely.