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 Electronic Journal of Probability > Vol. 12 (2007) > Paper 33 open journal systems 


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

Thomas G Kurtz, University of Wisconsin-Madison


Abstract
A general version of the Yamada-Watanabe and Engelbert results relating existence and uniqueness of strong and weak solutions for stochastic equations is given. The results apply to a wide variety of stochastic equations including classical stochastic differential equations, stochastic partial differential equations, and equations involving multiple time transformations.


Full text: PDF

Pages: 951-965

Published on: August 2, 2007
Bibliography
  1. Antonelli, Fabio; Ma, Jin. Weak solutions of forward-backward SDE's. Stochastic Anal. Appl. 21 (2003), no. 3, 493--514. MR1978231 (2004b:60140)
  2. Barlow, M. T. One-dimensional stochastic differential equations with no strong solution. J. London Math. Soc. (2) 26 (1982), no. 2, 335--347. MR0675177 (84d:60083)
  3. Barlow, M. T.; Perkins, E. Strong existence, uniqueness and nonuniqueness in an equation involving local time. Seminar on probability, XVII, 32--61, Lecture Notes in Math., 986, Springer, Berlin, 1983. MR0770394 (86j:60171)
  4. Blackwell, David; Dubins, Lester E. An extension of Skorohod's almost sure representation theorem. Proc. Amer. Math. Soc. 89 (1983), no. 4, 691--692. MR0718998 (86b:60005)
  5. Buckdahn, R.; Engelbert, H.-J.; Ru ac scanu, A. On weak solutions of backward stochastic differential equations. Teor. Veroyatn. Primen. 49 (2004), no. 1, 70--108; translation in Theory Probab. Appl. 49 (2005), no. 1, 16--50 MR2141331 (2006h:60096)
  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. On the theorem of T. Yamada and S. Watanabe. Stochastics Stochastics Rep. 36 (1991), no. 3-4, 205--216. MR1128494 (92k:60127)
  8. Engelbert, H. J.; Schmidt, W. On one-dimensional stochastic differential equations with generalized drift. Stochastic differential systems (Marseille-Luminy, 1984), 143--155, Lecture Notes in Control and Inform. Sci., 69, Springer, Berlin, 1985. MR0798317 (86m:60144)
  9. Ethier, Stewart N.; Kurtz, Thomas G. Markov processes. Characterization and convergence. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. John Wiley & Sons, Inc., New York, 1986. x+534 pp. ISBN: 0-471-08186-8 MR0838085 (88a:60130)
  10. Garcia, Nancy L.; Kurtz, Thomas G. Spatial birth and death processes as solutions of stochastic equations. ALEA Lat. Am. J. Probab. Math. Stat. 1 (2006), 281--303 (electronic). MR2249658
  11. Holley, R. A.; Stroock, D. W. A martingale approach to infinite systems of interacting processes. Ann. Probability 4 (1976), no. 2, 195--228. MR0397927 (53 #1782)
  12. Jacod, Jean. Weak and strong solutions of stochastic differential equations. Stochastics 3, no. 3, 171--191. (1980), MR0573202 (82a:60086)
  13. Jacod, Jean; Mémin, Jean. Weak and strong solutions of stochastic differential equations: existence and stability. Stochastic integrals (Proc. Sympos., Univ. Durham, Durham, 1980), pp. 169--212, Lecture Notes in Math., 851, Springer, Berlin-New York, 1981. MR0620991 (83h:60062)
  14. Karandikar, Rajeeva L. On pathwise stochastic integration. Stochastic Process. Appl. 57 (1995), no. 1, 11--18. MR1327950 (96c:60067)
  15. Kurtz, Thomas G. Representations of Markov processes as multiparameter time changes. Ann. Probab. 8 (1980), no. 4, 682--715. MR0577310 (82d:60130)
  16. Le Gall, J.-F. Applications du temps local aux équations différentielles stochastiques unidimensionnelles. (French) [Local time applications to one-dimensional stochastic differential equations] Seminar on probability, XVII, 15--31, Lecture Notes in Math., 986, Springer, Berlin, 1983. MR0770393 (86c:60088)
  17. Jin Ma, Jianfeng Zhang, and Ziyu Zheng. newblock Weak solutions for forward-backward {SDE}s: A martingale problem approach. newblock preprint, 30~pages.
  18. Nie, Zan Kan; Zhang, Dao Fa. Weak solutions to two-parameter stochastic differential equations and uniqueness of their distribution. (Chinese) J. Math. (Wuhan) 13 (1993), no. 2, 127--136. MR1257738 (95a:60085)
  19. Wagner, Daniel H. Survey of measurable selection theorems. SIAM J. Control Optimization 15 (1977), no. 5, 859--903. MR0486391 (58 #6137)
  20. Yamada, Toshio; Watanabe, Shinzo. On the uniqueness of solutions of stochastic differential equations. J. Math. Kyoto Univ. 11 1971 155--167. MR0278420 (43 #4150)
  21. Yeh, J. Uniqueness of strong solutions to stochastic differential equations in the plane with deterministic boundary process. Pacific J. Math. 128 (1987), no. 2, 391--400. MR0888527 (88i:60103)
















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Electronic Journal of Probability. ISSN: 1083-6489