Feynman-Kac Formulas for Solutions to Degenerate Elliptic and Parabolic Boundary-Value and Obstacle Problems with Dirichlet Boundary Conditions

Abstract

We prove Feynman-Kac formulas for solutions to elliptic and parabolic boundary value and obstacle problems associated with a general Markov diffusion process. Our diffusion model covers several popular stochastic volatility models, such as the Heston model, the CEV model and the SABR model, which are widely used as asset pricing models in mathematical finance. The generator of this Markov process with killing is a second-order, degenerate, elliptic partial differential operator, where the degeneracy in the operator symbol is proportional to the $2\alpha$-power of the distance to the boundary of the half-plane, with $\alpha\in(0,1]$. Our stochastic representation formulas provide the unique solutions to the elliptic boundary value and obstacle problems, when we seek solutions which are suitably smooth up to the boundary portion $\Gamma_{0}$ contained in the boundary of the upper half-plane. In the case when the full Dirichlet condition is given, our stochastic representation formulas provide the unique solutions which are not guaranteed to be any more than continuous up to the boundary portion $\Gamma_{0}$.