  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 $\Gamma_{0}$.
