[Paper Review] Parametrix method and the weak solution to an SDE driven by an $\alpha$-stable noise

Victoria Knopova, Alexei Kulik|arXiv (Cornell University)|Dec 30, 2014

Stochastic processes and financial applications7 citations

TL;DR

This paper develops a novel parametrix method to construct the transition density and establish two-sided estimates for the weak solution of a stochastic differential equation (SDE) driven by $α$-stable Lévy noise, where the generator includes both a non-degenerate $α$-stable jump component and a drift term. The key contribution is handling the case $0 < \alpha \leq 1$ with non-zero drift without requiring the drift to be dominated by the jump component.

ABSTRACT

Let $L:= a(x) (-\Delta)^{-\alpha/2}+ (b(x), abla)$, where $\alpha\in (0,2)$, and $a:\mathbb{R}^d o \mathbb{R}$, $b: \mathbb{R}^d o \mathbb{R}^d$ are H\older continuous. We show that the $C_\infty(\mathbb{R}^d)$-closure of $(L, C_\infty^2(\mathbb{R}^d))$ is the generator of a Feller Markov process $X$, which possesses a transition probability density $p_t(x,y)$. Complete description of this process is given both in terms of a martingale problem and as a weak solution to an SDE driven by an $\alpha$-stable noise. To construct the transition probability density and to obtain the two-sided estimates for it, we develop a new version of the parametrix method, which allows one to handle the case where $0<\alpha\leq 1$ and $b eq 0$; that is, in our approach the gradient part of the generator is not required to be dominated by the jump part.

Motivation & Objective

• To construct a Feller Markov process with generator $L = a(x)(-Δ)^{-\alpha/2} + (b(x), \nabla)$ for $\alpha \in (0,2)$.
• To establish the existence of a transition probability density $p_t(x,y)$ for the associated process.
• To provide a complete characterization of the process via both a martingale problem and as a weak solution to an SDE with $α$-stable noise.
• To develop a parametrix method that works without requiring the drift term to be dominated by the jump part, especially in the challenging case $0 < \alpha \leq 1$.

Proposed method

• A new version of the parametrix method is developed to handle non-local operators with Hölder continuous coefficients $a(x)$ and $b(x)$.
• The method constructs a fundamental solution by iteratively correcting the transition density of a stable process using the drift and diffusion components.
• The parametrix series is shown to converge under Hölder continuity assumptions on $a$ and $b$, even when $\alpha \leq 1$.
• The convergence is established via estimates on the iterated kernels, relying on the scaling properties of $α$-stable processes.
• The method allows for the derivation of two-sided bounds on the transition density $p_t(x,y)$ without requiring the drift to be lower-order compared to the jump component.
• The process is characterized as the weak solution of an SDE driven by an $α$-stable Lévy process, and its Feller property is verified via the closure of the generator in $C_\infty(\mathbb{R}^d)$.

Experimental results

Research questions

• RQ1 Can a parametrix method be extended to construct transition densities for SDEs with $α$-stable noise when $0 < \alpha \leq 1$ and the drift is non-zero?
• RQ2 Is it possible to derive two-sided estimates for the transition density without assuming the drift is dominated by the jump component?
• RQ3 How can the generator $L = a(x)(-Δ)^{-\alpha/2} + (b(x), \nabla)$ be realized as the generator of a Feller process?
• RQ4 What conditions on $a(x)$ and $b(x)$ ensure the existence of a weak solution to the SDE driven by $α$-stable noise?
• RQ5 Can the process be characterized equivalently via a martingale problem and a weak SDE formulation?

Key findings

• The $C_\infty(\mathbb{R}^d)$-closure of the operator $(L, C_\infty^2(\mathbb{R}^d))$ generates a Feller Markov process.
• The process admits a transition probability density $p_t(x,y)$ that is jointly continuous in $(t,x,y)$ and satisfies two-sided estimates.
• The parametrix method successfully constructs the density and estimates even when $0 < \alpha \leq 1$, overcoming the challenge of non-dominant drift.
• The method does not require the drift term to be dominated by the jump component, extending applicability to previously intractable regimes.
• The process is characterized as the weak solution of an SDE driven by an $α$-stable Lévy process.
• The process satisfies the martingale problem associated with the generator $L$, confirming its probabilistic consistency.