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 .
Publisher Copyright:
© 2020 Elsevier B.V.
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
SN - 0304-4149
VL - 130
SP - 5243
EP - 5289
JO - Stochastic Processes and their Applications
JF - Stochastic Processes and their Applications
IS - 9
ER -