Probability density function of SDEs with unbounded and path-dependent drift coefficient |
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| Institution: | 1. School of Mathematics and Statistics, Jiangsu Normal University, 221116, Xuzhou, China;2. Fakultät für Mathematik, Universität Bielefeld, D-33501, Bielefeld, Germany;3. Academy of Mathematics and Systems Science, Chinese Academy of Sciences (CAS), 100190, Beijing, China;1. Department of Mathematics, Ehime University, Matsuyama, 7908577, Japan;2. Kyushu University, Fukuoka, 8190395, Japan;3. Kanazawa University, Kanazawa 9201192, Japan;1. Center for Applied Mathematics, Tianjin University, Tianjin 300072, China;2. Department of Mathematics, City University of HongKong, HongKong, 83 Tat Chee Avenue, China |
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| 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. |
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| Keywords: | Probability density function Maruyama–Girsanov theorem Gaussian two-sided bound Parametrix method Euler–Maruyama scheme Unbiased simulation |
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