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Regularity properties of densities of SDEs using the Fourier analytic approach

Simon Ellinger

TL;DR

The paper addresses regularity properties of local densities for one-dimensional SDEs using a direct Fourier-analytic approach. It combines an Euler-type approximation, localization via stopping times, and a Lamperti-type transformation to reduce to unit diffusion, obtaining explicit bounds on the Fourier transform of localized densities. These bounds yield the existence of local densities and, under locally Hölder coefficients, Hölder continuity in space and continuity in time for the local densities, even when coefficients are only locally regular. Compared with Malliavin-calculus or Besov/difference-operator methods, this work presents a simpler, elementary route that extends to time-parameter regularity and to coefficients with piecewise Hölder structure, broadening applicability to practical SDEs with nonuniform regularity.

Abstract

We show regularity properties of local densities of solutions of stochastic differential equations (SDEs) with the Fourier analytic approach. With this simple method, statements that were previously derived with approaches using Malliavin calculus or difference operators can be recovered and extended to include regularity properties with respect to the time variable. For example, we derive the Hölder continuity and joint continuity of local densities in the case of drift coefficients that are locally piecewise Hölder continuous. To this end, we derive fairly general bounds for the Fourier transform of the local density of a solution of the SDE when the drift is locally bounded and the diffusion is locally sufficiently regular.

Regularity properties of densities of SDEs using the Fourier analytic approach

TL;DR

The paper addresses regularity properties of local densities for one-dimensional SDEs using a direct Fourier-analytic approach. It combines an Euler-type approximation, localization via stopping times, and a Lamperti-type transformation to reduce to unit diffusion, obtaining explicit bounds on the Fourier transform of localized densities. These bounds yield the existence of local densities and, under locally Hölder coefficients, Hölder continuity in space and continuity in time for the local densities, even when coefficients are only locally regular. Compared with Malliavin-calculus or Besov/difference-operator methods, this work presents a simpler, elementary route that extends to time-parameter regularity and to coefficients with piecewise Hölder structure, broadening applicability to practical SDEs with nonuniform regularity.

Abstract

We show regularity properties of local densities of solutions of stochastic differential equations (SDEs) with the Fourier analytic approach. With this simple method, statements that were previously derived with approaches using Malliavin calculus or difference operators can be recovered and extended to include regularity properties with respect to the time variable. For example, we derive the Hölder continuity and joint continuity of local densities in the case of drift coefficients that are locally piecewise Hölder continuous. To this end, we derive fairly general bounds for the Fourier transform of the local density of a solution of the SDE when the drift is locally bounded and the diffusion is locally sufficiently regular.
Paper Structure (3 sections, 5 theorems, 64 equations)

This paper contains 3 sections, 5 theorems, 64 equations.

Table of Contents

  1. Notation
  2. Proof of Theorem \ref{['thm:localDensity']}
  3. Proof of Corollary \ref{['cor:HoelderReg']}

Key Result

Theorem 1

Let $\mu, \sigma\colon {\mathbb R} \rightarrow {\mathbb R}$ be measurable functions and assume that there exist $\xi \in {\mathbb R}$, $\delta \in (0,\infty)$ such that $\mu\vert_{[\xi - \delta, \xi+\delta]}$ is bounded, $\inf_{x \in B_\delta(\xi)} |\sigma(x)| \ge l_\sigma > 0$ for $l_\sigma \in (0, Let $X$ be a strong solution of the SDE sde0, $\delta_0 \in (0, \delta)$ and $\varphi\colon {\mathb

Theorems & Definitions (11)

  • Theorem 1
  • Remark 1
  • Corollary 1
  • Corollary 2
  • Remark 2
  • Proposition 1
  • proof
  • proof : Proof of Theorem \ref{['thm:localDensity']}
  • Lemma 1
  • proof
  • ...and 1 more