Stochastic representation of solutions to degenerate elliptic and parabolic boundary value and obstacle problems with Dirichlet boundary conditions

TL;DR

This paper establishes the existence and uniqueness of stochastic representations for solutions to degenerate elliptic and parabolic boundary value and obstacle problems, focusing on the Heston stochastic volatility model used in finance.

Contribution

It provides a rigorous stochastic framework for degenerate PDEs associated with the Heston model, including boundary and obstacle problems, with applications to American options valuation.

Findings

Proved existence and uniqueness of solutions

Applied to the Heston stochastic volatility process

Connected solutions to American option pricing

Abstract

We prove existence and uniqueness of stochastic representations for solutions to elliptic and parabolic boundary value and obstacle problems associated with a degenerate Markov diffusion process. In particular, our article focuses on the Heston stochastic volatility process, which is widely used as an asset price model in mathematical finance and a paradigm for a degenerate diffusion process where the degeneracy in the diffusion coefficient is proportional to the square root of the distance to the boundary of the half-plane. The generator of this process with killing, called the elliptic Heston operator, is a second-order, degenerate, elliptic partial differential operator whose coefficients have linear growth in the spatial variables and where the degeneracy in the operator symbol is proportional to the distance to the boundary of the half-plane. In mathematical finance, solutions to terminal/boundary value or obstacle problems for the parabolic Heston operator correspond to value functions for American-style options on the underlying asset.