Degenerate stochastic differential equations with Hölder continuous coefficients and super-Markov chains

Abstract:

We consider the operator \[ \sum _{i,j=1}^d \sqrt {x_ix_j}\gamma _{ij}(x) \frac {\partial ^2}{\partial x_i \partial x_j}+\sum _{i=1}^d b_i(x) \frac {\partial }{\partial x_i}\] acting on functions in $C_b^2(\mathbb {R}^d_+)$. We prove uniqueness of the martingale problem for this degenerate operator under suitable nonnegativity and regularity conditions on $\gamma _{ij}$ and $b_i$. In contrast to previous work, the $b_i$ need only be nonnegative on the boundary rather than strictly positive, at the expense of the $\gamma _{ij}$ and $b_i$ being Hölder continuous. Applications to super-Markov chains are given. The proof follows Stroock and Varadhan’s perturbation argument, but the underlying function space is now a weighted Hölder space and each component of the constant coefficient process being perturbed is the square of a Bessel process.