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Comparison Theorems for Finite State Backward Stochastic Differential Equations

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Contemporary Quantitative Finance

Abstract

Most previous contributions on BSDEs, and the related theories of non linear expectation and dynamic risk measures, have been in the framework of continuous time diffusions or jump diffusions. Using solutions of BSDEs on spaces related to finite state, continuous time Markov Chains, we discuss a theory of nonlinear expectations in the spirit of Peng (math/0501415 (2005)). We prove basic properties of these expectations, and show their applications to dynamic risk measures on such spaces. In particular, we prove comparison theorems for scalar and vector valued solutions to BSDEs, and discuss arbitrage and risk measures in the scalar case.

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References

  1. Artzner, P., Delbaen, F., Eber, J.M., Heath, D.: Coherent measures of risk. Math. Finance 9(3), 203–228 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  2. Barles, G., Buckdahn, R., Pardoux, E.: Backward stochastic differential equations and integral-partial differential equations. Stoch. Stoch. Rep. 60(1), 57–83 (1997)

    MathSciNet  MATH  Google Scholar 

  3. Barrieu, P., El Karoui, N.: Optimal derivatives design under dynamic risk measures. In: Yin, G., Zhang, Q. (eds.) Mathematics of Finance. Contemporary Mathematics, vol. 351, pp. 13–26. AMS, Providence (2004)

    Chapter  Google Scholar 

  4. Becherer, D.: Bounded solutions to backward SDE’s with jumps for utility optimization and indifference hedging. Ann. Appl. Probab. 16(4), 2027–2054 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cohen, S.N., Elliott, R.J.: Solutions of backward stochastic differential equations on Markov chains. Commun. Stoch. Anal. 2(2), 251–262 (2008)

    MathSciNet  Google Scholar 

  6. Cohen, S.N., Elliott, R.J.: Comparisons for backward stochastic differential equations on Markov chains and related no-arbitrage conditions. Ann. Appl. Probab. 20(1), 267–311 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cohen, S.N., Elliott, R.J.: A general theory of infinite state backward stochastic difference equations. Stoch. Process. Appl. 120(4), 442–466 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. Coquet, F., Hu, Y., Memin, J., Peng, S.: Filtration consistent nonlinear expectations and related g-expectations. Probab. Theory Relat. Fields 123(1), 1–27 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  9. Delbaen, F., Peng, S., Rosazza Gianin, E.: Representation of the penalty term of dynamic concave utilities. Working Paper. Available at 0802.1121 (2008)

  10. Duffie, D., Epstein, L.G.: Asset pricing with stochastic differential utility. Rev. Financ. Stud. 5(3), 411–436 (1992)

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  12. El Otmani, M.: Backward stochastic differential equations associated with Lévy processes and partial integro-differential equations. Commun. Stoch. Anal. 2(2), 277–288 (2008)

    MathSciNet  Google Scholar 

  13. Elliott, R.J.: Stochastic Calculus and its Applications. Springer, Berlin (1982)

    Google Scholar 

  14. Elliott, R.J., Aggoun, L., Moore, J.: Hidden Markov Models: Estimation and Control. Springer, Berlin (1994)

    Google Scholar 

  15. Föllmer, H., Schied, A.: Stochastic Finance: An Introduction in Discrete Time. Studies in Mathematics, vol. 27. de Gruyter, Berlin (2002)

    Book  MATH  Google Scholar 

  16. Gantmacher, F.: Matrix Theory, vol. 2. Chelsea, New York (1960)

    Google Scholar 

  17. Gill, R.D., Johansen, S.: A survey of product-integration with a view toward application in survival analysis. Ann. Stat. 18(4), 1501–1555 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  18. Hu, Y., Ma, J., Peng, S., Yao, S.: Representation theorems for quadratic -consistent nonlinear expectations. Stoch. Process. Appl. 118, 1518–1551 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  19. Jacod, J., Shiryaev, A.N.: Limit Theorems for Stochastic Processes. Grundlehren der mathematischen Wissenschaften, vol. 288. Springer, Berlin (2003)

    MATH  Google Scholar 

  20. Lin, Q.Q.: Nonlinear Doob-Meyer decomposition with jumps. Acta Math. Sin. Engl. Ser. 19(1), 69–78 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  21. Peng, S.: A generalized dynamic programming principle and Hamilton-Jacobi-Bellman equation. Stoch. Stoch. Rep. 38, 119–134 (1992)

    MATH  Google Scholar 

  22. Peng, S.: Dynamically consistent nonlinear evaluations and expectations. Preprint No. 2004-1, Institute of Mathematics, Shandong University. Available at math/0501415 (2005)

  23. Rosazza Gianin, E.: Risk measures via g-expectations. Insur. Math. Econ. 39, 19–34 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  24. Royer, M.: Backward stochastic differential equations with jumps and related non-linear expectations. Stoch. Process. Appl. 116(10), 1358–1376 (2006)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. School of Mathematical Sciences, University of Adelaide, Adelaide, 5005, Australia

    Samuel N. Cohen & Robert J. Elliott

  2. Haskayne School of Business, University of Calgary, Calgary, T2N 1N4, Canada

    Robert J. Elliott

Authors
  1. Samuel N. Cohen
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  2. Robert J. Elliott
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Correspondence to Robert J. Elliott .

Editor information

Editors and Affiliations

  1. School of Finance and Economics Department of Mathematical Sciences, The University of Technology, Sydney, P.O. Box 123, Broadway 2007, NSW, Australia

    Carl Chiarella  & Alexander Novikov  & 

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Cohen, S.N., Elliott, R.J. (2010). Comparison Theorems for Finite State Backward Stochastic Differential Equations. In: Chiarella, C., Novikov, A. (eds) Contemporary Quantitative Finance. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03479-4_8

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