[Paper Review] Probability density function of SDEs with unbounded and path--dependent drift coefficient

Dai Taguchi, Akihiro Tanaka|arXiv (Cornell University)|Nov 17, 2018

Stochastic processes and financial applications58 references3 citations

TL;DR

This paper establishes the existence, explicit representation, Gaussian two-sided bounds, and Hölder continuity of the probability density function (pdf) for solutions of d-dimensional SDEs with unbounded and path-dependent drift coefficients. By combining the Maruyama-Girsanov transformation with local Novikov conditions and parametrix methods, the authors derive sharp bounds and apply them to prove convergence rates for Euler-Maruyama schemes and construct unbiased simulation schemes.

ABSTRACT

In this paper, we first prove that the existence of a solution of SDEs under the assumptions that the drift coefficient is of linear growth and path--dependent, and diffusion coefficient is bounded, uniformly elliptic and Hölder continuous. We apply Gaussian upper bound for a probability density function of a solution of SDE without drift coefficient and local Novikov condition, in order to use Maruyama--Girsanov transformation. The aim of this paper is to prove the existence with explicit representations (under linear/super--linear growth condition), Gaussian two--sided bound and Hölder continuity (under sub--linear growth condition) of a probability density function of a solution of SDEs with path--dependent drift coefficient. As an application of explicit representation, we provide the rate of convergence for an Euler--Maruyama (type) approximation, and an unbiased simulation scheme.

Motivation & Objective

• To establish the existence and regularity of the probability density function (pdf) for solutions of SDEs with unbounded and path-dependent drift coefficients.
• To derive explicit representations of the pdf under linear and super-linear growth conditions on the drift.
• To prove Gaussian two-sided bounds and Hölder continuity of the pdf under sub-linear growth conditions.
• To apply the theoretical results to analyze convergence rates of Euler-Maruyama-type approximation schemes.
• To develop an unbiased simulation scheme for the solution of such SDEs using the derived pdf properties.

Proposed method

• Utilizes the Maruyama-Girsanov theorem to transform the SDE with path-dependent drift into a diffusion process under a new measure.
• Applies a local Novikov condition to ensure the validity of the Girsanov transformation under unbounded drift.
• Employs the parametrix method to construct a fundamental solution and derive bounds for the pdf.
• Uses Gaussian upper bounds on the transition density of the diffusion without drift as a reference measure.
• Applies Fourier analysis and characteristic function estimates to control the $L^2$-norm of the characteristic function, ensuring integrability and thus existence of the pdf.
• Combines moment estimates and Hölder continuity of the diffusion coefficient to derive regularity properties of the pdf.

Experimental results

Research questions

• RQ1 Does a solution exist for SDEs with unbounded and path-dependent drift coefficients under linear or super-linear growth conditions?
• RQ2 Can the probability density function of the solution be explicitly represented in such cases?
• RQ3 What are the sharp Gaussian two-sided bounds for the pdf under sub-linear growth of the drift?
• RQ4 Is the pdf Hölder continuous under sub-linear drift growth?
• RQ5 What is the rate of convergence for Euler-Maruyama-type approximations of the solution, and can an unbiased simulation scheme be constructed?

Key findings

• The solution to the SDE with unbounded and path-dependent drift exists under linear growth and uniform ellipticity of the diffusion coefficient.
• An explicit representation of the pdf is derived under linear and super-linear growth conditions on the drift.
• Gaussian two-sided bounds are established for the pdf under sub-linear growth of the drift coefficient.
• The pdf is shown to be Hölder continuous under sub-linear drift growth, ensuring smoothness properties.
• The rate of convergence for the Euler-Maruyama-type approximation scheme is quantified via moment and characteristic function estimates.
• An unbiased simulation scheme is constructed based on the derived pdf properties and characteristic function bounds.