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.
Bahlali, K., Mezerdi, B., N’zi, M. and Ouknine, Y. (2007) Weak Solutions and a Yamada Watanabe Theorem for FBSDEs. Random Operators and Stochastic Equations, 15, 271-286. http://dx.doi.org/10.1515/rose.2007.016
Engelbert, H. (1991) On the Theorem of T. Yamada and S. Watanabe, Stochastics. An International Journal of Probability and Stochastic Processes, 36, 205-216. http://dx.doi.org/10.1080/17442509108833718
Cherny, A.S. (2002) On the Uniqueness in Law and the Pathwise Uniqueness for Stochastic Differential Equations. Theory of Probability and Its Applications, 46, 406-419. http://dx.doi.org/10.1137/S0040585X97979093