Uniqueness in Law for a Class of Degenerate Diffusions with Continuous Covariance

2011/05/09 by Gerard Brunick, Brunick, Gerard
Mathematics · #FOS: Mathematics #Probability (math.PR) #math.PR

paper · pdf · doi:10.48550/arxiv.1105.1821

arxiv created 2011/05/09 · arxiv updated 2011/05/11

Abstract

We study the martingale problem associated with the operator L u = ∂_s u + 1/2 ∑_i,j=1^d₀ a^ij ∂_ij u + ∑_i,j=1^d B^ij x^j ∂_i u, where d₀ ≤ d. We show that the martingale problem is well-posed when the function a is continuous and strictly positive-definite on \bb 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.

Related

• Uniqueness in law for singular degenerate SDEs with respect to a (sub-)invariant measure
• Martingale problems for some degenerate Kolmogorov equations
• Parametrix techniques and martingale problem for some degenerate Kolmogorov's equations
• Uniqueness in Cauchy problems for diffusive real-valued strict local\n martingales
• Well-posedness of Multidimensional Diffusion Processes with Weakly\n Differentiable Coefficients