SUPERCRITICAL SDES DRIVEN BY MULTIPLICATIVE STABLE-LIKE LEVY PROCESSES

In this paper, we study the following time-dependent stochastic differential equation (SDE) in R-d: dX(t) = sigma(t, X-t-)dZ(t) + b(t, X-t)dt, X-0 = x is an element of R-d, where Z is a d-dimensional non-degenerate a-stable-like process with alpha is an element of (0,2), and uniform in t >= 0, x bar right arrow sigma(t,x) : R-d -> R-d circle times R(d )is beta-order Holder continuous and uniformly elliptic with beta is an element of ((1 - alpha)(+), 1), and x bar right arrow b(t, x) is beta-order Holder continuous. The Levy measure of the Levy process Z can be anisotropic or singular with respect to the Lebesgue measure on R-d and its support can be a proper subset of R-d. We show in this paper that for every starting point x is an element of R-d, the above SDE has a unique weak solution. We further show that the above SDE has a unique strong solution if x bar right arrow sigma(t, x) is Lipschitz continuous and x bar right arrow b(t, x) is beta-order Holder continuous with beta is an element of (1 - alpha/2, 1). When sigma(t,x) = I-dxd, the d x d identity matrix, and Z is an arbitrary non-degenerate a-stable process with 0 < alpha < 1, our strong well-posedness result in particular gives an affirmative answer to the open problem in a paper by Priola.