C1,1 regularity for degenerate elliptic obstacle problems in mathematical finance
Panagiota Daskalopoulos and Paul Feehan
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Submission date: 05. Jun. 2012
MSC-Numbers: 35J70, 35J86, 49J40, 35R45
Keywords and phrases: American-style option, degenerate elliptic differential operator, degenerate diffusion process, free boundary problem, Heston stochastic volatility process, mathematical finance, Obstacle Problem, variational inequality
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The Heston stochastic volatility process is 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, where the degeneracy in the operator symbol is proportional to the distance to the boundary of the half-plane. In mathematical finance, solutions to obstacle problem for the elliptic Heston operator correspond to value functions for perpetual American-style options on the underlying asset. With the aid of weighted Sobolev spaces and weighted Hölder spaces, we establish the optimal C1,1 regularity (up to the boundary of the half-plane) for solutions to obstacle problems for the elliptic Heston operator when the obstacle functions are sufficiently smooth.