Regularity properties of densities of SDEs using the Fourier analytic approach

2025/04/30 by Simon Ellinger, Ellinger, Simon
Decision Sciences · Economics, Econometrics and Finance · Engineering · #60H10 #FOS: Mathematics #Probability (math.PR) #Risk and Portfolio Optimization #Stability and Control of Uncertain Systems #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.2504.21516

openalex publication_date 2025/04/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• Sharp lower error bounds for strong approximation of SDEs with discontinuous drift coefficient by coupling of noise
• On the performance of the Euler-Maruyama scheme for SDEs with discontinuous drift coefficient
• A simple method for the existence of a density for stochastic evolutions\n with rough coefficients
• H "older continuous densities of solutions of SDEs with measurable and\n path dependent drift coefficients
• Hölder regularity of the densities for the Navier--Stokes equations with noise
• A numerical method for SDEs with discontinuous drift
• Malliavin and flow regularity of SDEs. Application to the study of densities and the stochastic transport equation
• Time regularity of the densities for the Navier--Stokes equations with noise
• Smooth density for the Solution of Scalar SDEs with Locally Lipschitz Coefficients under Hörmander Condition
• Local Hölder continuity property of the Densities of Solutions of SDEs with Singular Coefficients
• Existence of densities for the 3D Navier-Stokes equations driven by Gaussian noise
• Smoothness and asymptotic estimates of densities for SDEs with locally smooth coefficients and applications to square root-type diffusions
• Integration by parts formula and applications to equations with jumps
• Absolute continuity for some one-dimensional processes

Cited by

• The local coupling of noise technique and its application to lower error bounds for strong approximation of SDEs with irregular coefficients

Related

• Local Hölder continuity property of the Densities of Solutions of SDEs with Singular Coefficients
• H "older continuous densities of solutions of SDEs with measurable and\n path dependent drift coefficients
• Smooth densities of the laws of perturbed diffusion processes
• On moments and strong local Hölder regularity of solutions of stochastic differential equations and of their spatial derivative processes
• Malliavin and flow regularity of SDEs. Application to the study of densities and the stochastic transport equation