The Yamada-Watanabe-Engelbert theorem for general stochastic equations and inequalities

Electronic Journal of Probability

Volumn 12, Issue , 2007, Pages 951-965

  Kurtz, Thomas G ^a  

^a UNIVERSITY OF WISCONSIN MADISON   (United States)

Author keywords

Multidimensionalindex; Pathwise uniqueness; Stochastic differential equations; Stochastic partial differential equations; Strong solution; Weak solution

Indexed keywords

EID: 34547668507     PISSN: None     EISSN: 10836489     Source Type: Journal    
DOI: 10.1214/EJP.v12-431     Document Type: Article

References (21)

1. ANTONELLI, F. AND MA, J. (2003). WEAK SOLUTIONS OF FORWARD-BACKWARD SDE'S. Stochastic Anal. Appl. 21, 3, 493-514.
2. BARLOW, M. T. (1982). ONE-DIMENSIONAL STOCHASTIC DIFFERENTIAL EQUATIONS WITH NO STRONG SOLUTION. J. London Math. Soc. (2) 26, 2, 335-347.
3. BARLOW, M. T. AND PERKINS, E. (1983). STRONG EXISTENCE, UNIQUENESS AND NONUNIQUENESS IN AN EQUATION INVOLVING LOCAL TIME. IN Seminar on probability, XVII. Lecture Notes in Math., Vol. 986. Springer, Berlin, 32-61.
4. BLACKWELL, D. AND DUBINS, L. E. (1983). AN EXTENSION OF SKOROHOD'S ALMOST SURE REPRESENTATION THEOREM. Proc. Amer. Math. Soc. 89, 4, 691-692.
5. BUCKDAHN, R., ENGELBERT, H.-J., AND RASCANU, A. (2004). ON WEAK SOLUTIONS OF BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS. Teor. Veroyatn. Primen. 49, 1, 70-108.
6. A. S. Cherny. On the uniqueness in law and the pathwise uniqueness for stochastic differential equations. Theory Probab. Appl., 46(3):406-419, 2003.
7. ENGELBERT, H. J. (1991). ON THE THEOREM OF T. YAMADA AND S. WATANABE. Stochastics Stochastics Rep. 36, 3-4, 205-216.
8. ENGELBERT, H. J. AND SCHMIDT, W. (1985). ON ONE-DIMENSIONAL STOCHASTIC DIFFERENTIAL EQUATIONS WITH GENERALIZED DRIFT. IN Stochastic differential systems (Marseille-Luminy, 1984). Lecture Notes in Control and Inform. Sci., Vol. 69. Springer, Berlin, 143-155.
9. ETHIER, S. N. AND KURTZ, T. G. (1986). Markov processes. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. John Wiley & Sons Inc., New York. Characterization and convergence.
10. GARCIA, N. L. AND KURTZ, T. G. (2006). SPATIAL BIRTH AND DEATH PROCESSES AS SOLUTIONS OF STOCHASTIC EQUATIONS. ALEA Lat. Am. J. Probab. Math. Stat. 1, 281-303 (electronic).
11. HOLLEY, R. A. AND STROOCK, D. W. (1976). A MARTINGALE APPROACH TO INFINITE SYSTEMS OF INTERACTING PROCESSES. Ann. Probability 4, 2, 195-228.
12. JACOD, J. (1980). WEAK AND STRONG SOLUTIONS OF STOCHASTIC DIFFERENTIAL EQUATIONS. Stochastics 3, 3, 171-191.
13. JACOD, J. AND MÉMIN, J. (1981). WEAK AND STRONG SOLUTIONS OF STOCHASTIC DIFFERENTIAL EQUATIONS: EXISTENCE AND STABILITY. IN Stochastic integrals (Proc. Sympos., Univ. Durham, Durham, 1980). Lecture Notes in Math., Vol. 851. Springer, Berlin, 169-212.
14. KARANDIKAR, R. L. (1995). ON PATHWISE STOCHASTIC INTEGRATION. Stochastic Process. Appl. 57, 1, 11-18.
15. KURTZ, T. G. (1980). REPRESENTATIONS OF MARKOV PROCESSES AS MULTIPARAMETER TIME CHANGES. Ann. Probab. 8, 4, 682-715.
16. LÉ GALL, J.-F. (1983). APPLICATIONS DU TEMPS LOCAL AUX EQUATIONS DIFFÉRENTIELLES STOCHASTIQUES UNIDIMENSIONNELLES. IN Seminar on probability, XVII. Lecture Notes in Math., Vol. 986. Springer, Berlin, 15-31.
17. Jin Ma, Jianfeng Zhang, and Ziyu Zheng. Weak solutions for forward-backward SDEs: A martingale problem approach. preprint, 30 pages.
18. NIÉ, Z. K. AND ZHANG, D. F. (1993). WEAK SOLUTIONS TO TWO-PARAMETER STOCHASTIC DIFFERENTIAL EQUATIONS AND UNIQUENESS OF THEIR DISTRIBUTION. J. Math. (Wuhan) 13, 2, 127-136.
19. WAGNÉR, D. H. (1977). SURVEY OF MEASURABLE SELECTION THEOREMS. SIAM J. Control Optimization 15, 5, 859-903.
20. YAMADA, T. AND WATANABÉ, S. (1971). ON THE UNIQUENESS OF SOLUTIONS OF STOCHASTIC DIFFERENTIAL EQUATIONS. J. Math. Kyoto Univ. 11, 155-167.
21. YÉH, J. (1987). UNIQUENESS OF STRONG SOLUTIONS TO STOCHASTIC DIFFERENTIAL EQUATIONS IN THE PLANE WITH DETERMINISTIC BOUNDARY PROCESS. Pacific J. Math. 128, 2, 391-400.