On the pathwise uniqueness of solutions of stochastic differential equations driven by symmetric stable Lévy processes

On the Pathwise Uniqueness of Solutions of One-Dimensional Stochastic Differential Equations with Jumps

“…We can apply our results to stochastic differential equation driven by symmetric stable processes studied by [4], taken by [5].…”

Approximation and Stability of Solutions of SDEs Driven by a Symmetric α Stable Process with Non-Lipschitz Coefficients

“…In multidimensional case, Tsuchiya [17] considered rather recently the pathwise uniqueness of solutions of SDEs driven by a symmetric α process. Belfadli-Ouknine condition which was found very recently can be seen as the counterpart of Nakao-Le Gall condition in the Brownian motion case ( [3], [13], [14]).…”

“…By stopping X and X n , when one of them first leaves a compact set, we can assume |X| ∨ |X n | ≤ M for every t ≥ 0. Letting λ ↓ 0 in the last inequality in the proof of Lemma 2.2 in [3]. We get the following inequality: [a] sup 0≤t≤T |X n (t) − X(t)| → 0 in probability (n → ∞), [b] the family of random variables sup 0≤t≤T |X n (t) − X(t)| β , n = 1, 2, · · · is uniformly integrable.…”

“…Remark 4. Belfadli and Ouknine [3] show that under the (ii) of (C) the pathwise uniqueness holds for each solutions of ( 17) and (18).…”

“…In the theory of SDEs driven by a symmetric α stable process, some non-Lipschitz conditions for coefficients which guarantee the pathwise uniqueness for solutions are known ( [2], [3], [7], [12], [17], [19]). Komatsu condition ( [12], see also [2]) is an analogue of Yamada-Watanabe condition for one-dimensional Brownian SDEs.…”