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

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  • Taguchi, Dai
  • Tanaka, Akihiro

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.

Suggested Citation

  • Taguchi, Dai & Tanaka, Akihiro, 2020. "Probability density function of SDEs with unbounded and path-dependent drift coefficient," Stochastic Processes and their Applications, Elsevier, vol. 130(9), pages 5243-5289.
    • Handle: RePEc:eee:spapps:v:130:y:2020:i:9:p:5243-5289
    DOI: 10.1016/j.spa.2020.03.006
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    References listed on IDEAS

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    1. Kusuoka, Seiichiro, 2017. "Continuity and Gaussian two-sided bounds of the density functions of the solutions to path-dependent stochastic differential equations via perturbation," Stochastic Processes and their Applications, Elsevier, vol. 127(2), pages 359-384.
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