128 research outputs found
Schauder a priori estimates and regularity of solutions to boundary-degenerate elliptic linear second-order partial differential equations
We establish Schauder a priori estimates and regularity for solutions to a
class of boundary-degenerate elliptic linear second-order partial differential
equations. Furthermore, given a smooth source function, we prove regularity of
solutions up to the portion of the boundary where the operator is degenerate.
Degenerate-elliptic operators of the kind described in our article appear in a
diverse range of applications, including as generators of affine diffusion
processes employed in stochastic volatility models in mathematical finance,
generators of diffusion processes arising in mathematical biology, and the
study of porous media.Comment: 58 pages, 1 figure. To appear in the Journal of Differential
Equations. Incorporates final galley proof corrections corresponding to
published versio
Stochastic representation of solutions to degenerate elliptic and parabolic boundary value and obstacle problems with Dirichlet boundary conditions
We prove existence and uniqueness of stochastic representations for solutions
to elliptic and parabolic boundary value and obstacle problems associated with
a degenerate Markov diffusion process. In particular, our article focuses on
the Heston stochastic volatility process, which is widely used as an asset
price model in mathematical finance and 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. In mathematical finance, solutions to
terminal/boundary value or obstacle problems for the parabolic Heston operator
correspond to value functions for American-style options on the underlying
asset.Comment: 47 pages; to appear in Transactions of the American Mathematical
Societ
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