We prove the existence and uniqueness of the fundamental solution for Kolmogorov operators associated to some stochastic processes, that arise in the Black & Scholes setting for the pricing problem relevant to path dependent options. We improve previous results in that we provide a closed form expression for the solution of the Cauchy problem under weak regularity assumptions on the coefficients of the differential operator. Our method is based on a limiting procedure, whose convergence relies on some barrier arguments and uniform a priori estimates recently discovered.

中文翻译：

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与亚式期权相关的偏微分方程的基本解的存在性

我们证明了与一些随机过程相关的 Kolmogorov 算子的基本解的存在性和唯一性，这些随机过程出现在 Black & Scholes 设置中，用于与路径相关选项相关的定价问题。我们改进了先前的结果，因为我们在微分算子系数的弱正则假设下为柯西问题的解决方案提供了一个封闭形式的表达式。我们的方法基于一个限制程序，其收敛依赖于最近发现的一些障碍参数和统一的先验估计。