[Paper Review] On weak uniqueness and distributional properties of a solution to an SDE with $\alpha$-stable noise

Alexei Kulik|arXiv (Cornell University)|Oct 31, 2015

Stochastic processes and financial applications41 references3 citations

TL;DR

This paper establishes weak uniqueness for a stochastic differential equation (SDE) driven by rotationally invariant α-st stable noise when the drift coefficient is Hölder continuous with index γ and the balance condition α + γ > 1 holds. Using a refined parametrix method with a mollified flow approximation, the authors prove existence, continuity, and explicit pointwise estimates for the transition density and its time derivative, providing a complete small-time asymptotic description of the solution's distributional properties.

ABSTRACT

For an SDE driven by a rotationally invariant $\alpha$-stable noise we prove weak uniqueness of the solution under the balance condition $\alpha+\gamma>1$, where $\gamma$ denotes the Holder index of the drift coefficient. We prove existence and continuity of the transition probability density of the corresponding Markov process and give a representation of this density with an explicitly given "principal part", and a "residual part" which possesses an upper bound. Similar representation is also provided for the derivative of the transition probability density w.r.t. the time variable.

Motivation & Objective

• To establish weak uniqueness for an SDE with α-stable Lévy noise and Hölder continuous drift under the balance condition α + γ > 1.
• To construct a rigorous parametrix-based approximation for the transition probability density of the solution process.
• To derive explicit pointwise upper bounds for the transition density and its time derivative, including small-time behavior.
• To clarify the role of the balance condition in ensuring stochastic regularization despite the non-diffusive nature of α-stable noise.
• To extend the parametrix method to handle Hölder continuous drifts in super-critical α < 1 regimes where classical flow methods fail.

Proposed method

• Develops a modified parametrix method for α-stable SDEs with Hölder continuous drift, replacing the deterministic flow with a mollified approximation to ensure regularity.
• Introduces a zero-order approximation combining the α-stable heat kernel with a mollified flow that remains well-defined for γ-Hölder drifts.
• Uses convolution powers of the approximate fundamental solution to construct the true transition density via series expansion.
• Applies sub-convolution properties and weighted estimates (H(χ), H(0)) to control the residual terms in the parametrix construction.
• Derives time-derivative bounds by differentiating the parametrix series term-by-term and estimating each component using Hölder continuity and stable density decay.

Experimental results

Research questions

• RQ1 Under what conditions does an SDE with α-stable noise and Hölder continuous drift admit a weakly unique solution?
• RQ2 How does the balance condition α + γ > 1 relate to the stochastic regularization of irregular drifts by α-stable noise?
• RQ3 Can the transition probability density of such an SDE be represented with explicit principal and residual parts, and what are their decay properties?
• RQ4 What is the small-time behavior of the transition density and its time derivative in the super-critical α < 1 case?
• RQ5 How can the parametrix method be adapted to handle non-Lipschitz drifts in α-stable SDEs?

Key findings

• Weak uniqueness of the solution to the SDE is established under the balance condition α + γ > 1, which is sharp except possibly at the critical case α + γ = 1.
• The transition probability density exists, is continuous, and admits a representation with a principal part (α-stable kernel) and a residual part bounded by Ct−1/α′(t−1+χ/α H(χ)_t + (t−1+χ + t−1+δ)H(0)_t).
• The time derivative of the transition density satisfies the bound |∂tpt(x,y) − ∂tp0_t(x,y)| ≤ Ct−1/α′(tχ/α H(χ)_t + (t1+χ + t1+δ)H0_t), showing regularity in time.
• The method successfully handles the case α < 1, where classical flow-based parametrix methods fail due to lack of Lipschitz regularity in the drift.
• The mollified flow construction resolves the issue of ill-defined derivatives in the drift term, enabling the derivation of time-derivative estimates.