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

DOI: 10.4236/am.2016.78070, PP. 784-792

Keywords: Uniqueness in Law, Joint Uniqueness in Law, Poisson Process, Engelbert Theorem

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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.


[1]  Yamada, T. and Watanabe, S. (1971) On the Uniqueness of Solutions of Stochastic Differential Equations. Journal of Mathematics-Kyoto University, 11, 155-167.
[2]  Ondrejat, M. (2004) Uniqueness for Stochastic Evolution Equations in Banach Spaces. Dissertationes Mathematicae, 426, 1-63.
[3]  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.
[4]  Rockner, M.M., Schmuland, B. and Zhang, X. (2008) Yamada-Watanabe Theorem for Stochastic Evolution Equations in Infinite Dimensions. Condensed Matter Physics, 11, 247-259.
[5]  Kurtz, T.G. (2007) The Yamada-Watanabe-Engelbert Theorem for General Stochastic Equations and Inequalities. Electronic Journal of probability, 12, 951-965.
[6]  Tappe, S. (2013) The Yamada-Watanabe Theorem for Mild Solutions to Stochastic Partial Differential Equations. Electronic Communications in Probability, 18, 1-13.
[7]  Kurtz, T.G. (2013) Weak and Strong Solutions of General Stochastic Models. arXiv:1305.6747v1
[8]  Graczyk, P. and Malecki, J. (2013) Multidimensional Yamada-Watanabe Theorem and Its Applications to Particle Systems. Journal of Mathematical Physics, 54, 021503.
[9]  Barczy, M., Li, Z. and Pap, G. (2013) Yamada-Watanabe Results for Stochastic Differential Equations with Jumps. arXiv:1312.4485
[10]  Zhao, H. (2014) Yamada-Watanabe Theorem for stochastic Evolution Equation Driven by Poisson Random Measure. ISRN Probability and Statistics, 7 p.
[11]  Jean, J. (1980) Weak and Strong Solutions of Stochastic Differential Equations. Stochastics, 3, 171-191.
[12]  Engelbert, H. (1991) On the Theorem of T. Yamada and S. Watanabe, Stochastics. An International Journal of Probability and Stochastic Processes, 36, 205-216.
[13]  Brossard, J. (2003) Deux notions quivalentes dunicit en loi pour les quations diffrentielles stochastiques, Sminaire de Probabilits XXXVII. Springer Berlin Heidelberg, 246-250.
[14]  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.
[15]  Qiao, H.J. (2010) A Theorem Dual to Yamada-Watanabe Theorem for Stochastic Evolution Equations. Stochastics and Dynamics, 10, 367.
[16]  Watanabe, S. (1964) On Discontinuous Additive Functionals and Lvy Measures of a Markov Process. Japanese Journal of Mathematics, 34, 53-70.


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