Supercritical sdes driven by multiplicative stable-like Lévy processes

In this paper, we study the following time-dependent stochastic differential equation (SDE) in Rd: dXt = σ(t, Xt-)dZt + b(t, Xt)dt, X0 = x ∈ Rd, where Z is a d-dimensional non-degenerate α-stable-like process with α ∈ (0, 2), and uniform in t ≥ 0, x → σ(t, x): Rd → Rd ⊗ Rd is β-order Hölder continuous and uniformly elliptic with β ∈ ((1 - α)+, 1), and x → b(t, x) is β-order Hölder continuous. The Lévy measure of the Lévy process Z can be anisotropic or singular with respect to the Lebesgue measure on Rd and its support can be a proper subset of Rd. We show in this paper that for every starting point x ∈ Rd, the above SDE has a unique weak solution. We further show that the above SDE has a unique strong solution if x _→ σ(t, x) is Lipschitz continuous and x → b(t, x) is β-order Hölder continuous with β ∈ (1-α/2, 1). When σ(t, x) = Id×d, the d × d identity matrix, and Z is an arbitrary nondegenerate α-stable process with 0 < α < 1, our strong well-posedness result in particular gives an affirmative answer to the open problem in a paper by Priola.