Existence of a Fundamental Solution of Partial Differential Equations associated to Asian Options

TL;DR

This paper establishes the existence and uniqueness of fundamental solutions for Kolmogorov operators related to Asian options, providing a closed-form solution under weak regularity assumptions, advancing mathematical finance modeling.

Contribution

It introduces a new method using a limiting procedure with barrier arguments to derive closed-form solutions for path-dependent option pricing models.

Findings

Proved existence and uniqueness of fundamental solutions.

Derived a closed-form expression for the solution.

Extended results to operators with weak regularity assumptions.

Abstract

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.