2024/04/21 by Mytnik, Leonid, Weinberger, Johanna
#60G52 #60H10 #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.2404.13729
We consider the one-dimensional stochastic differential equation Xt = x0 + Lt + ∫0t μ(Xs)ds, t ≥ 0, where μ is a finite measure of Kato class Kη with η∈ (0,α-1] and (Lt)t ≥ 0 is a symmetric α-stable process with α∈ (1,2). We derive weak and strong well posedness for this equation when η≤α-1 and η< α-1, respectively, and show that the condition η≤ α-1 is sharp for weak existence. We furthermore reformulate the equation in terms of the local time of the solution (Xt)t ≥ 0 and prove its well posedness. To this end, we also derive a Tanaka-type formula for a symmetric, α-stable processes with α∈ (1,2) that is perturbed by an adapted, right-continuous process of finite variation.