Uniqueness in law for a class of degenerate diffusions with continuous covariance

• 成果类型: Article
• 署名作者: Brunick, Gerard
• 署名单位: University of California System; University of California Santa Barbara
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-011-0398-8
• 发表日期: 2013
• 页码: 265-302
• 关键词: STOCHASTIC DIFFERENTIAL-EQUATIONS; weak uniqueness; EXISTENCE

摘要

We study the martingale problem associated with the operator Lu(s, x) = partial derivative(s)u(s, x) + 1/2 Sigma(d0)(i,j=1) a(ij)(s, x)partial derivative(ij)u(s, x) + Sigma(d)(i,j=1) B(ij)x(j)partial derivative(i)u(s, x), where d(0) < d. We show that the martingale problem is well-posed when the function a is continuous and strictly positive definite on R-d0 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.