  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
existence, uniqueness, and global regularity of solutions to stationary
variational inequalities and obstacle problems for the elliptic Heston operator
on unbounded subdomains of the half-plane. 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.
