On backward stochastic differential equations approach to valuation of American options

TL;DR

This paper explores the valuation of American options on dividend-paying stocks using backward stochastic differential equations and semilinear PDEs, providing dual mathematical representations of the option value.

Contribution

It introduces a dual approach to American option valuation through backward stochastic differential equations and semilinear PDEs, linking stochastic and analytical methods.

Findings

Value function represented via nonlinear backward stochastic differential equations

Value function also characterized as a weak solution to semilinear PDEs

Provides a mathematical framework connecting stochastic and PDE approaches

Abstract

We consider the problem of valuation of American (call and put) options written on a dividend paying stock governed by the geometric Brownian motion. We show that the value function has two different but related representations: by means of a solution of some nonlinear backward stochastic differential equation and weak solution to some semilinear partial differential equation.