TY - JOUR

T1 - Probability density function of SDEs with unbounded and path-dependent drift coefficient

AU - Taguchi, Dai

AU - Tanaka, Akihiro

N1 - Funding Information:
The authors would like to thank an anonymous referee for his/her careful readings and comments. The first author was supported by JSPS KAKENHI Grant Number 17H06833 and 19K14552 . The second author was supported by Sumitomo Mitsui Banking Corporation .

PY - 2020/9

Y1 - 2020/9

N2 - 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.

AB - 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.

KW - Euler–Maruyama scheme

KW - Gaussian two-sided bound

KW - Maruyama–Girsanov theorem

KW - Parametrix method

KW - Probability density function

KW - Unbiased simulation

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U2 - 10.1016/j.spa.2020.03.006

DO - 10.1016/j.spa.2020.03.006

M3 - Article

AN - SCOPUS:85083733102

VL - 130

SP - 5243

EP - 5289

JO - Stochastic Processes and their Applications

JF - Stochastic Processes and their Applications

SN - 0304-4149

IS - 9

ER -