The G\"{a}rtner-Ellis theorem, homogenization, and affine processes

TL;DR

This paper extends the Gärtner-Ellis theorem to include first-order large deviation estimates, introduces a method to construct functions with similar Laplace expansions using heat kernel techniques, and applies these ideas to affine processes including the Heston model.

Contribution

It provides a first-order extension of large deviation estimates and explicit homogenization expansions for affine processes, including the Heston model.

Findings

First-order large deviation estimates for the Gärtner-Ellis theorem.

Explicit homogenization expansion for affine processes.

Coefficients computed for the Heston model.

Abstract

We obtain a first order extension of the large deviation estimates in the G\"{a}rtner-Ellis theorem. In addition, for a given family of measures, we find a special family of functions having a similar Laplace principle expansion up to order one to that of the original family of measures. The construction of the special family of functions mentioned above is based on heat kernel expansions. Some of the ideas employed in the paper come from the theory of affine stochastic processes. For instance, we provide an explicit expansion with respect to the homogenization parameter of the rescaled cumulant generating function in the case of a generic continuous affine process. We also compute the coefficients in the homogenization expansion for the Heston model that is one of the most popular stock price models with stochastic volatility.