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Probability density function of SDEs with unbounded and path-dependent drift coefficient. (English) Zbl 07242827
Summary: 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.
MSC:
65C30 Numerical solutions to stochastic differential and integral equations
62G07 Density estimation
35K08 Heat kernel
60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
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