Abstract
We study the martingale problem associated with the operator
where d 0 ≤ d. We show that the martingale problem is well-posed when the function a is continuous and strictly positive definite on \({\mathbb{R}^{d_0}}\) and the matrix B takes a particular lower-diagonal, block form. We then localize this result to show that the martingale problem remains well-posed when B is replaced by a sufficiently smooth vector field whose Jacobian matrix satisfies a nondegeneracy condition.
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Brunick, G. Uniqueness in law for a class of degenerate diffusions with continuous covariance. Probab. Theory Relat. Fields 155, 265–302 (2013). https://doi.org/10.1007/s00440-011-0398-8
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DOI: https://doi.org/10.1007/s00440-011-0398-8
Keywords
- Martingale problem
- Stochastic differential equations
- Degenerate parabolic operators
- Homogeneous groups
Mathematics Subject Classification (2000)
- 60H10
- 35K65