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We prove existence and uniqueness of stochastic representations for solutionsto elliptic and parabolic boundary value and obstacle problems associated witha degenerate Markov diffusion process. In particular, our article focuses onthe Heston stochastic volatility process, which is widely used as an assetprice model in mathematical finance and a paradigm for a degenerate diffusionprocess where the degeneracy in the diffusion coefficient is proportional tothe square root of the distance to the boundary of the half-plane. Thegenerator of this process with killing, called the elliptic Heston operator, isa second-order, degenerate, elliptic partial differential operator whosecoefficients have linear growth in the spatial variables and where thedegeneracy in the operator symbol is proportional to the distance to theboundary of the half-plane. In mathematical finance, solutions toterminal/boundary value or obstacle problems for the parabolic Heston operatorcorrespond to value functions for American-style options on the underlyingasset.

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