Yamada-Watanabe Results for Stochastic Differential Equations with Jumps

Mátyás Barczy,Gyula Pap,Zenghu Li

doi:10.1155/2015/460472

Abstract

Recently, Kurtz (2007, 2014) obtained a general version of the Yamada-Watanabe and Engelbert theorems relating existence and uniqueness of weak and strong solutions of stochastic equations covering also the case of stochastic differential equations with jumps. Following the original method of Yamada and Watanabe (1971), we give alternative proofs for the following two statements: pathwise uniqueness implies uniqueness in the sense of probability law, and weak existence together with pathwise uniqueness implies strong existence for stochastic differential equations with jumps.

Highlights

In order to prove existence and pathwise uniqueness of a strong solution for stochastic differential equations, it is an important issue to clarify the connections between weak and strong solutions
We investigate stochastic differential equations with jumps
Let b : [0, ∞) × Rd → Rd, σ : [0, ∞) × Rd → Rd×r, f : [0, ∞) × Rd × U → Rd, and g : [0, ∞) × Rd × U → Rd be Borel measurable functions, where [0, ∞) × Rd × U is equipped with its Borel σ-algebra B([0, ∞) × Rd × U) = B([0, ∞)) ⊗ B(Rd) ⊗ B(U)

Summary

Introduction

In order to prove existence and pathwise uniqueness of a strong solution for stochastic differential equations, it is an important issue to clarify the connections between weak and strong solutions. Where (Wt)t⩾0 is an r-dimensional standard Brownian motion, N(ds, du) is a Poisson random measure on (0, ∞)×U with intensity measure dsm(du), Ñ(ds, du) := N(ds, du) − dsm(du), and (Xt)t⩾0 is a suitable process with values in Rd. Yamada and Watanabe [1] proved that weak existence and pathwise uniqueness imply uniqueness in the sense of probability law and strong existence for the SDE (1) with f = 0 and g = 0. Kurtz [5, 10] continued the direction of Engelbert [3] and Jacod [6] He studied general stochastic models which relate stochastic inputs with stochastic outputs and obtained a general version of the Yamada-Watanabe and Engelbert theorems relating existence and uniqueness of weak and strong solutions of stochastic models with the message that the original results are not limited to SDEs driven by Wiener processes. Comparing with the results of the present paper, note that we explicitly stated and proved in Theorem 1 that pathwise uniqueness for the SDE (1) implies uniqueness in the sense of probability law

Preliminaries

Notions of Weak and Strong Solutions

Proof of Theorem 1

Precise Formulation and Proof of Theorem 2