  The Heston stochastic volatility process, which is widely used as an asset
price model in mathematical finance, is 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. With the aid of weighted Sobolev spaces, we prove
supremum bounds, a Harnack inequality, and H\"older continuity near the
boundary for solutions to variational equations defined by the elliptic Heston
operator, as well as H\"older continuity up to the boundary for solutions to
variational inequalities defined by the elliptic Heston operator. In
mathematical finance, solutions to obstacle problems for the elliptic Heston
operator correspond to value functions for perpetual American-style options on
the underlying asset.
