We solve four intertwined problems, motivated by mathematical finance, concerning degenerate-parabolic partial differential operators and degenerate diffusion processes. First, we consider a parabolic partial differential equation on a half-space whose coefficients are suitably Holder continuous and allowed to grow linearly in the spatial variable and which becomes degenerate along the boundary of the half-space. We establish existence and uniqueness of solutions in weighted Holder spaces which incorporate both the degeneracy at the boundary and the unboundedness of the coefficients. Second, we ] show that the martingale problem associated with a degenerate elliptic differential operator with unbounded, locally Holder continuous coefficients on a half-space is well-posed in the sense of Stroock and Varadhan . Third, we prove existence, uniqueness, and the strong Markov property for weak solutions to a stochastic differential equation with degenerate diffusion and unbounded coefficients with suitable H\"older continuity properties. Fourth, for an Ito process with degenerate diffusion and unbounded but appropriately regular coefficients, we prove existence of a strong Markov process, unique in the sense of probability law, whose one-dimensional marginal probability distributions match those of the given Ito process .
Motivated by applications to probability and mathematical finance, we consider a parabolic partial differential equation on a half-space whose coefficients are suitably Holder continuous and allowed to grow linearly in the spatial variable and which become degenerate along the boundary of the half-space. We establish existence and uniqueness of solutions in weighted Holder spaces which incorporate both the degeneracy at the boundary and the unboundedness of the coefficients. In our companion article arXiv:1211.4636], we apply the main result of this article to show that the martingale problem associated with a degenerate-elliptic partial differential operator is well-posed in the sense of Stroock and Varadhan .
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We solve four intertwined problems, motivated by
Motivated by applications to probability and
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concerning degenerate-parabolic partial differential operators and degenerate diffusion processes. First,
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Second, we
In our companion article
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arXiv:1211.4636
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, we apply the main result of this article to
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degenerate elliptic differential operator with unbounded, locally Holder continuous coefficients on a half-space
degenerate-elliptic partial differential operator
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. Third, we prove existence, uniqueness, and the strong Markov property for weak solutions to a stochastic differential equation with degenerate diffusion and unbounded coefficients with suitable H\"older continuity properties. Fourth, for an Ito process with degenerate diffusion and unbounded but appropriately regular coefficients, we prove existence of a strong Markov process, unique in the sense of probability law, whose one-dimensional marginal probability distributions match those of the given Ito process