Electronic Communications in Probability

Weak and strong solutions of general stochastic models

Thomas Kurtz

Full-text: Open access

Abstract

Typically, a stochastic model relates stochastic “inputs” and, perhaps, controls tostochastic “outputs”. A general version of the Yamada-Watanabe and Engelbert the-orems relating existence and uniqueness of weak and strong solutions of stochasticequations is given in this context. A notion of compatibility between inputs and out-puts is critical in relating the general result to its classical forebears. The usualformulation of stochastic differential equations driven by semimartingales does notrequire compatibility, so a notion of partial compatibility is introduced which doeshold. Since compatibility implies partial compatibility, classical strong uniquenessresults imply strong uniqueness for compatible solutions. Weak existence argumentstypically give existence of compatible solutions (not just partially compatible solu-tions), and as in the original Yamada-Watanabe theorem, existence of strong solutionsfollows.

Article information

Source
Electron. Commun. Probab., Volume 19 (2014), paper no. 58, 16 pp.

Dates
Accepted: 25 August 2014
First available in Project Euclid: 7 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465316760

Digital Object Identifier
doi:10.1214/ECP.v19-2833

Mathematical Reviews number (MathSciNet)
MR3254737

Zentralblatt MATH identifier
1301.60035

Subjects
Primary: 60G05: Foundations of stochastic processes

Keywords
weak solution strong solution stochastic models pointwise uniqueness pathwise uniqueness compatible solutions stochastic differential equations stochastic partial differential equations backward stochastic differential equations Meyer-Zheng condi

Rights
This work is licensed under a Creative Commons Attribution 3.0 License.

Citation

Kurtz, Thomas. Weak and strong solutions of general stochastic models. Electron. Commun. Probab. 19 (2014), paper no. 58, 16 pp. doi:10.1214/ECP.v19-2833. https://projecteuclid.org/euclid.ecp/1465316760


Export citation

References

  • Blackwell, David; Dubins, Lester E. An extension of Skorohod's almost sure representation theorem. Proc. Amer. Math. Soc. 89 (1983), no. 4, 691–692. http://dx.doi.org/10.2307/2044607
  • Buckdahn, R.; Engelbert, H.-J.; Răşcanu, 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 http://dx.doi.org/10.1137/S0040585X97980877
  • ChernyÄ­, A. S. On strong and weak uniqueness for stochastic differential equations. (Russian) Teor. Veroyatnost. i Primenen. 46 (2001), no. 3, 483–497; translation in Theory Probab. Appl. 46 (2003), no. 3, 406–419
  • Engelbert, H. J. On the theorem of T. Yamada and S. Watanabe. Stochastics Stochastics Rep. 36 (1991), no. 3-4, 205–216.
  • 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
  • Jacod, Jean. Weak and strong solutions of stochastic differential equations. Stochastics 3, no. 3, 171–191. (1980),
  • Jacod, Jean; Memin, Jean. Existence of weak solutions for stochastic differential equations with driving semimartingales. Stochastics 4 (1980/81), no. 4, 317–337. http://dx.doi.org/10.1080/17442508108833169
  • Jakubowski, Adam. A non-Skorohod topology on the Skorohod space. Electron. J. Probab. 2 (1997), no. 4, 21 pp. (electronic). http://dx.doi.org/10.1214/EJP.v2-18
  • Kurtz, Thomas G. Random time changes and convergence in distribution under the Meyer-Zheng conditions. Ann. Probab. 19 (1991), no. 3, 1010–1034. http://www.jstor.org/stable/2244471
  • Kurtz, Thomas G. The Yamada-Watanabe-Engelbert theorem for general stochastic equations and inequalities. Electron. J. Probab. 12 (2007), 951–965. http://dx.doi.org/10.1214/EJP.v12-431
  • Kurtz, Thomas G.; Protter, Philip. Weak limit theorems for stochastic integrals and stochastic differential equations. Ann. Probab. 19 (1991), no. 3, 1035–1070.
  • Lebedev, V. A. On the existence of weak solutions for stochastic differential equations with driving martingales and random measures. Stochastics 9 (1983), no. 1-2, 37–76. http://dx.doi.org/10.1080/17442508308833247
  • Meyer, P.-A.; Zheng, W. A. Tightness criteria for laws of semimartingales. Ann. Inst. H. Poincare Probab. Statist. 20 (1984), no. 4, 353–372. http://www.numdam.org/item?id=AIHPB_1984__20_4_353_0
  • Yamada, Toshio; Watanabe, Shinzo. On the uniqueness of solutions of stochastic differential equations. J. Math. Kyoto Univ. 11 1971 155–167.