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The Heston stochastic volatility process, which is widely used as an assetprice model in mathematical finance, is 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. With the aid of weighted Sobolev spaces, we proveexistence, uniqueness, and global regularity of solutions to stationaryvariational inequalities and obstacle problems for the elliptic Heston operatoron unbounded subdomains of the half-plane. In mathematical finance, solutionsto obstacle problems for the elliptic Heston operator correspond to valuefunctions for perpetual American-style options on the underlying asset.

Existence, uniqueness, and global regularity for degenerate elliptic obstacle problems in mathematical finance