Degenerate-parabolic partial differential equations with unbounded coefficients, martingale problems, and a mimicking theorem for Ito processes

Details Degenerate-parabolic partial differential equations with unbounded coefficients, martingale problems, and a mimicking theorem for Ito processes Degenerate-parabolic partial differential equations with unbounded coefficients, martingale problems, and a mimicking theorem for Ito processes

2011-12-20

arxiv.org/abs/1112.4824v1

Mathematics

Probability

61 pages

Scientific paper

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.

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