On One-dimensional Stochastic Equations Driven by Symmetric Stable Processes

Ergodicity for Time Changed Symmetric Stable Processes

In this paper we study the ergodicity and the related semigroup property for a class of symmetric Markov jump processes associated with time changed symmetric α-stable processes. For this purpose, explicit and sharp criteria for Poincaré type inequalities (including Poincaré, super Poincaré and weak Poincaré inequalities) of the corresponding non-local Dirichlet forms are derived. Moreover, our...

Stochastic Differential Equations Driven by Stable Processes for Which Pathwise Uniqueness Fails

Let Zt be a one-dimensional symmetric stable process of order α with α ∈ (0, 2) and consider the stochastic differential equation dXt = φ(Xt−)dZt. For β < 1 α ∧ 1, we show there exists a function φ that is bounded above and below by positive constants and which is Hölder continuous of order β but for which pathwise uniqueness of the stochastic differential equation does not hold. This result is...

PATHWISE UNIQUENESS FOR SINGULAR SDEs DRIVEN BY STABLE PROCESSES

We prove pathwise uniqueness for stochastic differential equations driven by non-degenerate symmetric α-stable Lévy processes with values in Rd having a bounded and β-Hölder continuous drift term. We assume β > 1− α/2 and α ∈ [1, 2). The proof requires analytic regularity results for the associated integro-differential operators of Kolmogorov type. We also study differentiability of solutions w...

Existence and Measurability of the Solution of the Stochastic Differential Equations Driven by Fractional Brownian Motion

Equivalence of Uniqueness in Law and Joint Uniqueness in Law for SDEs Driven by Poisson Processes

We give an extension result of Watanabe’s characterization for 2-dimensional Poisson processes. By using this result, the equivalence of uniqueness in law and joint uniqueness in law is proved for one-dimensional stochastic differential equations driven by Poisson processes. After that, we give a simplified Engelbert theorem for the stochastic differential equations of this type.