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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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

\begin{displaymath}\sum_{i,j=1}^d \sqrt{x_ix_j}\gamma_{ij}(x) \frac{\partial^2}... ...\partial x_j}+\sum_{i=1}^d b_i(x) \frac{\partial}{\partial x_i}\end{displaymath}

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
Email: bass@math.uconn.edu

Edwin A. Perkins
Affiliation: Department of Mathematics, University of British Columbia, Vancouver, B.C., Canada V6T 1Z2
Email: perkins@math.ubc.ca

DOI: http://dx.doi.org/10.1090/S0002-9947-02-03120-3
PII: S 0002-9947(02)03120-3
Keywords: Stochastic differential equations, margingale problem, elliptic operators, degenerate operators, diffusions, Bessel processes, superprocesses, H\"older 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