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 Holder 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 Holder 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.

中文翻译：

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具有无界和路径相关漂移系数的 SDE 概率密度函数

在本文中，我们首先证明了在漂移系数为线性增长和路径依赖的假设下，SDEs 解的存在性，并且扩散系数是有界的、均匀椭圆和Holder 连续的。我们将高斯上限应用于没有漂移系数和局部诺维科夫条件的 SDE 解的概率密度函数，以便使用 Maruyama--Girsanov 变换。本文的目的是证明一个概率密度函数的显式表示（在线性/超线性增长条件下）、高斯双边界和Holder连续性（在亚线性增长条件下）的存在性。具有路径相关漂移系数的 SDE 解。作为显式表示的应用，