Skip to main content

Advertisement

SpringerLink
Log in
Menu
Find a journal Publish with us
Search
Cart
  1. Home
  2. Probability Theory and Related Fields
  3. Article
Well-posedness and regularity of backward stochastic Volterra integral equations
Download PDF
Download PDF
  • Published: 08 August 2007

Well-posedness and regularity of backward stochastic Volterra integral equations

  • Jiongmin Yong1 

Probability Theory and Related Fields volume 142, pages 21–77 (2008)Cite this article

  • 498 Accesses

  • 58 Citations

  • Metrics details

Abstract

Backward stochastic Volterra integral equations (BSVIEs, for short) are studied. Notion of adapted M-solution is introduced. Well-posedness of BSVIEs is established and some regularity results are proved for the adapted M-solutions via Malliavin calculus. A Pontryagin type maximum principle is presented for optimal controls of stochastic Volterra integral equations.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Artzner Ph., Delbaen F., Jean-Marc E. and Heath D. (1999). Coherent measures of risk. Math. Finance 9: 203–228

    Article  MATH  MathSciNet  Google Scholar 

  2. Belbas S.A. (1999). Iterative schemes for optimal control of Volterra integral equations. Nonlinear Anal. 37: 57–79

    Article  MathSciNet  Google Scholar 

  3. Belbas S.A. and Schmidt W.H. (2005). Optimal control of Volterra equations with impulses. Appl. Math. Comput. 166: 696–723

    Article  MATH  MathSciNet  Google Scholar 

  4. Bismut, J.B.: Théorie Probabiliste du Contrôle des Diffusions. Mem. Am. Math. Soc. 176, Providence, Rhode Island (1973)

  5. Briand P., Delyon B., Hu Y., Pardoux E. and Stoica L. (2003). L p solutions of backward stochastic differential equations. Stoch. Proc. Appl. 108: 109–129

    Article  MATH  MathSciNet  Google Scholar 

  6. Burnap C. and Kazemi M. (1999). Optimal control of a system governed by nonlinear Volterra integral equations with delay. IMA J. Math. Control Inform. 16: 73–89

    Article  MATH  MathSciNet  Google Scholar 

  7. Carlson D.A. (1987). An elementary proof of the maximum principle for optimal control problems governed by a Volterra integral equation. J. Optim. Theory Appl. 54: 43–61

    Article  MathSciNet  Google Scholar 

  8. Cheridito P., Delbaen F. and Kupper M. (2004). Coherent and convex monetary risk measures for bounded càdlàg processes. Stoch. Proc. Appl. 112: 1–22

    Article  MATH  MathSciNet  Google Scholar 

  9. Duffie D. and Epstein L. (1992). Stochastic differential utility. Econometrica 60: 353–394

    Article  MATH  MathSciNet  Google Scholar 

  10. Duffie, D., Huang, C.F.: Stochastic production-exchange equilibria. Research paper No. 974, Graduate School of Business, Stanford University, Stanford (1986)

  11. Ekeland, I., Lazrak, A.: Non-commitment in continuous time. Preprint (2006)

  12. El Karoui N., Peng S. and Quenez M.C. (1997). Backward stochastic differential equations in finance. Math. Finance 7: 1–71

    Article  MATH  MathSciNet  Google Scholar 

  13. Kamien M.I. and Muller E. (1976). Optimal control with integral state equations. Rev. Econ. Stud. 43: 469–473

    Article  MATH  Google Scholar 

  14. Karatzas I. and Shreve S.E. (1988). Brownian Motion and Stochastic Calculus. Springer, Heidelberg

    MATH  Google Scholar 

  15. Kobylanski M. (2000). Backward stochastic differential equations and partial differential equations with quadratic growth. Ann. Probab. 28: 558–602

    Article  MATH  MathSciNet  Google Scholar 

  16. Laibson D. (1997). Golden eggs and hyperbolic discounting. Q. J. Econ. 42: 443–477

    Google Scholar 

  17. Lin J. (2002). Adapted solution of a backward stochastic nonlinear Volterra integral equation. Stoch. Anal. Appl. 20: 165–183

    Article  MATH  Google Scholar 

  18. Ma J. and Yong J. (1999). Forward-Backward Stochastic Differential Equations and Their Applications, Springer-Verlag, Berlin. Springer, Berlin

    Google Scholar 

  19. Medhin N.G. (1986). Optimal processes governed by integral equations. J. Math. Anal. Appl. 120: 1–12

    Article  MATH  MathSciNet  Google Scholar 

  20. Neustadt L.W. and Warga J. (1970). Comments on the paper “Optimal control of processes described by integral equations. I” by V.R. Vinokurov. SIAM J. Control 8: 572

    Article  MathSciNet  Google Scholar 

  21. Nualart D. (1995). The Malliavin Calculus and Related Topics. Springer, Heidelberg

    MATH  Google Scholar 

  22. Pardoux E. and Peng S. (1990). Adapted solutions of backward stochastic equations. Syst. Control Lett. 14: 55–61

    Article  MATH  MathSciNet  Google Scholar 

  23. Pardoux E. and Protter P. (1990). Stochastic Volterra equations with anticipating coefficients. Ann. Probab. 18: 1635–1655

    Article  MATH  MathSciNet  Google Scholar 

  24. Peng S. (1990). A general stochastic maximum principle for optimal control problems. SIAM J. Control Optim. 28: 966–979

    Article  MATH  MathSciNet  Google Scholar 

  25. Peng, S.: Nonlinear expectations, nonlinear evaluations and risk measures. Stochastic Methods in Finance, Lecture Notes in Math., vol. 1856, pp. 165–253. Springer, Heidelberg (2004)

  26. Pritchard A.J. and You Y. (1996). Causal feedback optimal control for Volterra integral equations. SIAM J. Control Optim. 34: 1874–1890

    Article  MATH  MathSciNet  Google Scholar 

  27. Protter P. (1985). Volterra equations driven by semimartingales. Ann. Probab. 13: 519–530

    Article  MATH  MathSciNet  Google Scholar 

  28. Schroder M. and Skiadas C. (1999). Optimal consumption and portfolio selection with stochastic utility. J. Econ. Theory 89: 68–126

    Article  MATH  MathSciNet  Google Scholar 

  29. Skiadas C. (2003). Robust control and recursive utility. Finance Stoch. 7: 475–489

    Article  MATH  MathSciNet  Google Scholar 

  30. Strotz R.H. (1956). Myopia and inconsistency in dynamic utility maximization. Rev. Econ. Stud. 23: 165–180

    Google Scholar 

  31. Vinokurov, V.R.: Optimal control of processes described by integral equations, I, II, III, Izv. Vysš. Učebn. Zaved. Matematika 7(62), 21–33; 8(63) 16–23; 9(64), 16–25; (in Russian) English transl. in SIAM J. Control 7 324–336, 337–345, 346–355 (1967)

  32. Wang, T.: A class of dynamic risk measures. Preprint, University of British Columbia, BC (2002)

  33. Yong J. (2006). Backward stochastic Volterra integral equations and some related problems. Stoch. Proc. Appl. 116: 779–795

    Article  MATH  MathSciNet  Google Scholar 

  34. Yong J. and Zhou X.Y. (1999). Stochastic Controls: Hamiltonian Systems and HJB Equations. Springer, New York

    MATH  Google Scholar 

  35. You Y. (2000). Quadratic integral games and causal synthesis. Trans. Am. Math. Soc. 352: 2737–2764

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Central Florida, Orlando, FL, 32816, USA

    Jiongmin Yong

Authors
  1. Jiongmin Yong
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jiongmin Yong.

Additional information

This work is supported in part by NSF Grant DMS-0604309.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Yong, J. Well-posedness and regularity of backward stochastic Volterra integral equations. Probab. Theory Relat. Fields 142, 21–77 (2008). https://doi.org/10.1007/s00440-007-0098-6

Download citation

  • Received: 28 May 2006

  • Revised: 08 July 2007

  • Published: 08 August 2007

  • Issue Date: September 2008

  • DOI: https://doi.org/10.1007/s00440-007-0098-6

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Keywords

  • Backward stochastic Volterra integral equation
  • Adapted M-solution
  • Malliavin calculus
  • Optimal control
  • Pontryagin maximum principle

Mathematics Subject Classification (2000)

  • 60H20
  • 60H07
  • 93E20
  • 49K22
Download PDF

Working on a manuscript?

Avoid the common mistakes

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support

167.114.118.210

Not affiliated

Springer Nature

© 2023 Springer Nature