Skip to main content
SpringerLink
Log in

Non-linear pricing theory and backward stochastic differential equations

  • Chapter
  • First Online:
Financial Mathematics

Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 1656))

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 54.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 74.95
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • Antonelli, F. (1992) “Backward-Forward Stochastic Differential Equations”, Annals of Applied Probability 3. 777–793 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  • Barles, G. (1994) “Solutions de viscocité des équations de Hamilton-Jacobi du premier ordre et Applications”, Mathématiques et Applications 17, Springer 1994.

    Google Scholar 

  • Barles, G. & Kobylansky, M. (1996): “Existence and uniqueness results for backward stochastic differential equationswhen the generator has a quadratic growth”. Université de Tours, preprint.

    Google Scholar 

  • Benes, V.E. (1971) “Existence of Optimal Stochastic Control Law” SIAM J. of Control, 9, 446–472.

    Article  MathSciNet  MATH  Google Scholar 

  • Bergman, Y. (1991) “Option Pricing with Divergent Borrowing and Lending Rates,” Working Paper, Dpt of Economics Brown University.

    Google Scholar 

  • Bensoussan, A.: “On the Theory of Option Pricing” Acta Applicandae Mathematicae 2, 139–158, 1984.

    MathSciNet  MATH  Google Scholar 

  • Bensoussan, A., & Lions, J. L. (1978): “Applications des Inéquations Varitionnelles en Contrôle Stochastique”, Dunod, Paris.

    MATH  Google Scholar 

  • Bismut, J.M. (1973) “Conjugate Convex Functions in Optimal Stochastic Control,” J. Math. Anal. Apl., 44, 384–404.

    Article  MathSciNet  MATH  Google Scholar 

  • Bismut, J.M. (1978) “Contrôle des systèmes linéaires quadratiques: applications de l’intégrale stochastique,” Sémin. Proba. XII., Lect. Notes in Math., 649, 180–264, Springer.

    Article  MathSciNet  MATH  Google Scholar 

  • Buckdahn, R. (1993) “Backward Stochastic Differential Equations driven by a Martingale,” Preprint.

    Google Scholar 

  • Cinlar, E., Jacod, J., Protter, P., & Sharpe, M.J. (1980) “Semimartingale and Markov Processes,” Z.f.W., 54, 161–219.

    Article  MathSciNet  MATH  Google Scholar 

  • Crandall, M., Ishii, H., & Lions P.L. (1992): “User’s guide to the viscosity solutions of second order partial differential equations,” Bull. A.M.S. 27, 1–67

    Article  MathSciNet  MATH  Google Scholar 

  • Cvitanic, J. (1996) “Optimal Trading under constraints” this volume.

    Google Scholar 

  • Cvitanic, J., & Karatzas, I. (1992) “Convex duality in Constrained Portfolio Optimization,” Annals of Applied Probability 2, pp. 767–818.

    Article  MathSciNet  MATH  Google Scholar 

  • Cvitanic, J., & Karatzas, I. (1993) “Hedging Contingent Claims with Constrained Portfolios,” Annals of Applied Probability 3, pp. 652–681.

    Article  MathSciNet  MATH  Google Scholar 

  • Cvitanic, J., & Ma, J. (1996) “Hedging Options for a Large Investor and Forward-Backward SDE’s,” To appear in Annals of Applied Probability.

    Google Scholar 

  • Delbaen, F., & Schachermayer, W. (1994) “A General Version of the Fundamental Theorem of Asset Pricing,” Math. Annal, 123.

    Google Scholar 

  • Dellacherie, C. (1977) “Sur l’existence de certains essinf et essup de familles de processus mesurables,” Sem. Proba. XII. Lectures. Notes in Math. 649. Springer Verlag

    Google Scholar 

  • Dellacherie, C. & Meyer, P.A. (1980) “Probabilités et Potentiel” Chap V to VIII, Théorie des martingales. Hermann

    Google Scholar 

  • Dellacherie, C. & Meyer, P.A. (1980) “Probabilités et Potentiel” Chap XII to XVI, Théorie du potentiel, Processus de Markov. Hermann

    Google Scholar 

  • Duffie, D. (1992) Dynamic Asset Pricing Theory Princeton University Press.

    Google Scholar 

  • Duffie, D., & Epstein, L. (1992) “Stochastic Differential Utility,” Econometrica, 60, n.2, 353–394.

    Article  MathSciNet  MATH  Google Scholar 

  • Duffie, D., Ma, J., & Yong, J. (1994) “Black’s Consol Rate Conjecture”, Working paper, Graduate School of Business, Stanford University.

    Google Scholar 

  • Duffie, D., & Skiadias, C. (1991) “Continuous-time Security Pricing: A Utility Gradient Approach”, Working paper, Graduate School of Business, Stanford University.

    Google Scholar 

  • Ekeland, I., & Turnbull, T. (1979) Infinite Dimensional Optimization and Convexity Chicago Lectures in Math.

    Google Scholar 

  • El Karoui, N. (1981) “Les aspects probabilistes du contrôle stochastique” Lectures Notes in Math. 816. Springer-Verlag.

    Google Scholar 

  • El Karoui, N. & Huang S.J. (1996) “A General Result of Existence and Uniqueness of Backward Stochastic Differential Equations” to appear Pitman Series ed: El Karoui, N and Mazliak, L.

    Google Scholar 

  • El Karoui, N., Kapoudjian, C., Pardoux, E., Peng, S. & Quenez, M.C. (1995) “Reflected solutions of backward SDE’s, and Related Obstacle Problems for PDE’s,” to appear Annals of Probability.

    Google Scholar 

  • El Karoui, N., Peng, S., & Quenez, M.C. (1994) “Optimization of Utility Functions” Working Paper, Paris VI University.

    Google Scholar 

  • El Karoui, N., Peng, S., & Quenez, M.C. (1997) “Backward Stochastic Differential Equations in Finance” Mathematical Finance, to appear in January 1997.

    Google Scholar 

  • El Karoui, N., & Quenez, M.C. (1991) “Programmation dynamique et évaluation des actifs contingents en marché incomplet,” C.R. Acad. Sci. Paris, t.313, pp 851–854.

    MATH  Google Scholar 

  • El Karoui, N., & Quenez, M.C. (1995) “Dynamic Programming and Pricing of Contingent Claims in Incomplete Market,” Siam J. of Control and Opti., 33, n.1.

    Google Scholar 

    Google Scholar 

  • Epstein, L., & Zin, S. (1989) “Substitution, Risk Aversion and the Temporal Behavior of Consumption and Asset Returns: A Theorical Framework,” Econometrica, 57, n.4, 837–969.

    Article  MathSciNet  MATH  Google Scholar 

  • Fleming, W.F., & Soner, M. (1993) Controlled Markov processes and Viscosity Solutions Springer-Verlag, New York, Heidelberg, Berlin.

    MATH  Google Scholar 

  • Föllmer, H., & Schweizer, M. (1990) “Hedging of Contingent Claims under Incomplete Information,” (1990), Applied stochastic analysis eds. M.H.A. Davis and R.J. Elliot, Gordon and Breach, London.

    Google Scholar 

  • Geoffard, P.Y. (1995) “Discounting and Optimizing, Utility maximization: a Lagrange Variational Formulation as a Minmax Problem,” Journal of Economic Theory.

    Google Scholar 

  • Hamadene, S., & Lepeltier, J.P. (1995) “Zero-Sum Stochastic Differential Games and Backward Equations”, Systems and Control Letters, 24, 259–263.

    Article  MathSciNet  MATH  Google Scholar 

  • Harrison, M., & Kreps, D. (1979) “Martingales and Arbitrage in Multiperiod Securities Markets”, Journal of Economic Theory, 20, 381–408.

    Article  MathSciNet  MATH  Google Scholar 

  • Harisson, M., & Pliska, S.P. (1981) “Martingales and Stochastic Integrals in the Theory of Continuous Trading,” Stochastic Processes and their Applications, 11, 215–260.

    Article  MathSciNet  MATH  Google Scholar 

  • Harisson, M., & Pliska, S.P. (1983) “A Stochastic Calculus Model of Continuous Trading: Complete Markets,” Stochastic Processes and their Applications, 15, 313–316.

    Article  MathSciNet  MATH  Google Scholar 

  • He, H., & Pearson, N.D. (1991): “Consumption and Portfolio Policies with Incomplete Markets and Short-Shale Constraints; the infinite dimensional case”, Journal of Economic Theory 54 259–304

    Article  MathSciNet  MATH  Google Scholar 

  • Hu & Peng, S (1995): “Solution of Forward-Backward Stochastic Differential Equations” Proba. Theory. Rel. Fields, 103 pp. 273–283.

    Article  MathSciNet  MATH  Google Scholar 

  • Jacka, S. (1993): “Local times, optimal stopping and semimartingales,” The Annals of Probability 21, 329–339.

    Article  MathSciNet  MATH  Google Scholar 

  • Jouini, E., & Kallal, H. (1992) “Arbitrage and Equilibrium in Securities Markets with Shortsale Constraints,” Working Paper, Univ. Paris I.

    Google Scholar 

  • Karatzas, I. (1988) “On the pricing of American Options” Appl. Math. Optimization, 17, pp. 37–60

    Article  MathSciNet  MATH  Google Scholar 

  • Karatzas, I. (1989) “Optimization problems in the theory of continuous trading”, SIAM. J. Control. Optimization, 27, pp. 1221–1259

    Article  MathSciNet  MATH  Google Scholar 

  • Karatzas, I., & Shreve, S.E. (1996) “Methods on Mathematical Finance”, Book to appear

    Google Scholar 

  • Karatzas, I., & Shreve, S.E. (1987) “Brownian Motion and Stochastic Calculus”, Springer Verlag, New York.

    MATH  Google Scholar 

  • Korn, R. (1992) “Option Pricing in a Model with a Higher Interest Rate for Borrowing than for Lending,” Working paper.

    Google Scholar 

  • Kramkov, D.O. (1996) “Optional decomposition of supermartingales and hedging contingent claims in incomplete security markets,” Probability Theory and Related Fields 105, 459–479 (1996).

    Article  MathSciNet  MATH  Google Scholar 

  • Krylov, N. (1980) Controlled Diffusion Processes, New York: Springer Verlag.

    Book  MATH  Google Scholar 

  • Lepeltier J-P. & San Martin J. (1996) “Backward stochastic differential equations with continuous generator” to appear in Statistics and Probability Letters

    Google Scholar 

  • Ma, J., Protter, P., & Yong, J. (1994) “Solving forward-backward stochastic differential equations explicity—a four step schem” Proba. Theory Relat. Fields 98, 339–359 (1994).

    Article  MathSciNet  MATH  Google Scholar 

  • Merton, R. (1971) “Optimum Consumption and Portfolio Rules in a Continuous Time Model”, Journal of Economic Theory. 3, pp. 373–413.

    Article  MathSciNet  MATH  Google Scholar 

  • Müller, S. (1987) “Arbitrage pricing of contingent claims” Lecture Notes in Economics and Mathematical Systems, 254 Springer Verlag.

    Google Scholar 

  • Musiela, M & Rutkowoski (1997) “Arbitrage Pricing of Derivative Securities. Theory and Aplplications” Book to appear Springer-verlag.

    Google Scholar 

  • Pardoux, E. (1996) “BSDE’s and semilinear PDE’s”, Working paper, Geilo Lectures 1996.

    Google Scholar 

  • Pardoux, E., & Peng, S. (1990) “Adapted Solution of a Backward Stochastic Differential Equation”, Systems and Control Letters, bf 14, 55–61.

    Article  MathSciNet  MATH  Google Scholar 

  • Pardoux, E., & Peng, S. (1992) “Backward Stochastic Differential equations and Quasilinear Parabolic Partial Differential Equations” Lecture Notes in CIS 176, 200–217, Springer.

    MathSciNet  MATH  Google Scholar 

  • Pardoux, E., & Peng, S. (1995) “Some backward SDEs with non Lipschitz coefficients”, Proc. Conf. Metz, to appear.

    Google Scholar 

  • Peng, S. (1991) “Probabilistic Interpretation for Systems of Quasilinear Parabolic Partial Differential Equations”, Stochastics, 37, 61–74.

    MathSciNet  MATH  Google Scholar 

  • Peng, S. (1992a) “A Generalized Dynamic Programming Principle and Hamilton-Jacobi-Bellman equation,” Stochastics, Vol. 38, 119–134.

    MathSciNet  MATH  Google Scholar 

  • Peng, S. (1992b) “A Nonlinear Feynman-Kac Formula and Applications,” Proceedings of Symposium of System Sciences and Control theory, Chen & Yong ed. 173–184, World Scientific, Singapore.

    Google Scholar 

  • Peng, S. (1993) “Backward Stochastic Differential Equation and It’s Application in Optimal Control,” Appl. Math. & Optim. 27:125–144.

    Article  MathSciNet  Google Scholar 

  • Quenez, M.C. (1993) “Méthodes de contrôle stochastique en Finance,” Thèse de doctorat de l’Université Pierre et Marie Curie.

    Google Scholar 

  • Revuz, D. & Yor, M. (1994) “Continuous Martingales and Brownian Motion”, Springer Verlag 1994.

    Google Scholar 

  • Schweizer, M. (1992) “Mean-Variance hedging for general claims,” The Annals of Applied Probability, 2, pp. 171–179.

    Article  MathSciNet  MATH  Google Scholar 

  • Svensson, L.E.O., & Werner, J. (1990) “Portfolio Choice and Asset Pricing with non-traded Assets,” Research Paper, 2005, pp.. Graduate School of Business, Stanford University.

    Google Scholar 

  • Uzawa, H. (1968) “Time preference, the consumption-function, and optimal asset holdings”, in Value, capital, and Growth: Papers in Honor of Sir John Hicks. J.N. Wolfe ed., Edinburgh University, Edinburgh.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratoire de Probabilités, CNRS-URA 224, Université de Paris VI, 4 Place Jussieu, 75232, Paris Cedex 05

    N. El Karoui

  2. Equipe de Mathématiques, Université de Marne la Vallée, 2 rue de la Butte Verte, 93.166, Noisy-Le-Grand

    M.C. Quenez

Authors
  1. N. El Karoui
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M.C. Quenez
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Wolfgang J. Runggaldier

Rights and permissions

Reprints and permissions

Copyright information

© 1997 Springer-Verlag

About this chapter

Cite this chapter

El Karoui, N., Quenez, M. (1997). Non-linear pricing theory and backward stochastic differential equations. In: Runggaldier, W.J. (eds) Financial Mathematics. Lecture Notes in Mathematics, vol 1656. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0092001

Download citation

  • DOI: https://doi.org/10.1007/BFb0092001

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-62642-8

  • Online ISBN: 978-3-540-68356-8

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation