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Transactions of the American Mathematical Society

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

Authors: Richard F. Bass and Edwin A. Perkins
Journal: Trans. Amer. Math. Soc. 355 (2003), 373-405
MSC (2000): Primary 60H10
Published electronically: September 6, 2002
MathSciNet review: 1928092
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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.

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Additional Information

Richard F. Bass
Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269

Edwin A. Perkins
Affiliation: Department of Mathematics, University of British Columbia, Vancouver, B.C., Canada V6T 1Z2

Keywords: Stochastic differential equations, margingale problem, elliptic operators, degenerate operators, diffusions, Bessel processes, superprocesses, Hölder norms
Received by editor(s): February 1, 2002
Received by editor(s) in revised form: June 6, 2002
Published electronically: September 6, 2002
Additional Notes: The first author’s research was supported in part by NSF grant DMS9988496
The second author’s research was supported in part by an NSERC grant
Article copyright: © Copyright 2002 American Mathematical Society