Probability density function of SDEs with unbounded and path-dependent drift coefficient

Dai Taguchi, Akihiro Tanaka

Research output: Contribution to journalArticlepeer-review

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.

Original languageEnglish
Pages (from-to)5243-5289
Number of pages47
JournalStochastic Processes and their Applications
Volume130
Issue number9
DOIs
Publication statusPublished - Sep 2020

Keywords

  • Euler–Maruyama scheme
  • Gaussian two-sided bound
  • Maruyama–Girsanov theorem
  • Parametrix method
  • Probability density function
  • Unbiased simulation

ASJC Scopus subject areas

  • Statistics and Probability
  • Modelling and Simulation
  • Applied Mathematics

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