Probabilistic solution of the American options

TL;DR

This paper proves the existence and uniqueness of probabilistic solutions for American options using advanced stochastic calculus, extending Itô's formula to handle hypoelliptic diffusions in financial modeling.

Contribution

It introduces an extended Itô formula applicable to hypoelliptic diffusions, enabling the analysis of American options under broader conditions than previously possible.

Findings

Proved existence and uniqueness of solutions for American options

Extended Itô formula for hypoelliptic diffusions

Applicable to general diffusion processes in finance

Abstract

The existence and uniqueness of probabilistic solutions of variational inequalities for the general American options are proved under the hypothesis of hypoellipticity of the infinitesimal generator of the underlying diffusion process which represents the risky assets of the stock market with which the option is created. The main tool is an extension of the It\^o formula which is valid for the tempered distributions on $\R^d$ and for nondegenerate It\^o processes in the sense of the Malliavin calculus.