A Yosida's parametrix approach to Varadhan's estimates for a degenerate diffusion under the weak H\"ormander condition

TL;DR

This paper extends Yosida's parametrix method to derive Varadhan-type asymptotic estimates for the transition density of a degenerate diffusion under weak H"ormander conditions, with applications to financial derivatives and numerical approximations.

Contribution

It adapts Yosida's parametrix approach to a degenerate diffusion under weak H"ormander conditions, providing new asymptotic estimates and an explicit expansion of the cost function.

Findings

Derived a partial proof of the Varadhan formula for degenerate diffusions.

Provided an asymptotic expansion of the cost function in elementary terms.

Presented numerical evidence supporting the key inequality for the proof.

Abstract

We adapt and extend Yosida's parametrix method, originally introduced for the construction of the fundamental solution to a parabolic operator on a Riemannian manifold, to derive Varadhan-type asymptotic estimates for the transition density of a degenerate diffusion under the weak H\"ormander condition. This diffusion process, widely studied by Yor in a series of papers, finds direct application in the study of a class of path-dependent financial derivatives known as Asian options. We obtain the Varadhan formula \begin{equation} \frac{-2 \log p(t,x;T,y) } { \Psi(t,x;T,y) } \to 1, \qquad \text{as } \quad T-t \to 0^+, \end{equation} where $p$ denotes the transition density and $\Psi$ denotes the optimal cost function of a deterministic control problem associated to the diffusion. We provide a partial proof of this formula, and present numerical evidence to support the validity of an intermediate inequality that is required to complete the proof. We also derive an asymptotic expansion of the cost function $\Psi$, expressed in terms of elementary functions, which is useful in order to design efficient approximation formulas for the transition density.