  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.
