The Annals of Applied Probability

Comparisons for backward stochastic differential equations on Markov chains and related no-arbitrage conditions

Samuel N. Cohen and Robert J. Elliott

Full-text: Open access


Most previous contributions to BSDEs, and the related theories of nonlinear 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 develop a theory of nonlinear expectations in the spirit of [Dynamically consistent nonlinear evaluations and expectations (2005) Shandong Univ.]. 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.

Article information

Ann. Appl. Probab., Volume 20, Number 1 (2010), 267-311.

First available in Project Euclid: 8 January 2010

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H10: Stochastic ordinary differential equations [See also 34F05]
Secondary: 91B70: Stochastic models

Backward stochastic differential equation Markov chains nonlinear expectation dynamic risk measures comparison theorem


Cohen, Samuel N.; Elliott, Robert J. Comparisons for backward stochastic differential equations on Markov chains and related no-arbitrage conditions. Ann. Appl. Probab. 20 (2010), no. 1, 267--311. doi:10.1214/09-AAP619.

Export citation


  • [1] Artzner, P., Delbaen, F., Eber, J.-M. and Heath, D. (1999). Coherent measures of risk. Math. Finance 9 203–228.
  • [2] Barrieu, P. and El Karoui, N. (2004). Optimal derivatives design under dynamic risk measures. In Mathematics of Finance (G. Yin and Q. Zhang, eds.). Contemporary Mathematics 351 13–25. Amer. Math. Soc., Providence, RI.
  • [3] Bellman, R. (1970). Introduction to Matrix Analysis, 2nd ed. McGraw-Hill, New York.
  • [4] Cohen, S. N. and Elliott, R. J. (2008). Solutions of backward stochastic differential equations on Markov chains. Commun. Stoch. Anal. 2 251–262.
  • [5] Cohen, S. N. and Elliott, R. J. (2009). A general theory of finite state backward stochastic difference equations. Forthcoming. Available at
  • [6] Cohen, S. N., Elliott, R. J. and Pearce, C. E. M. (2009). A ring isomorphism and corresponding pseudoinverses. Forthcoming. Available at
  • [7] El Karoui, N., Peng, S. and Quenez, M. C. (1997). Backward stochastic differential equations in finance. Math. Finance 7 1–71.
  • [8] Elliott, R. J. (1982). Stochastic Calculus and Applications. Applications of Mathematics 18. Springer, New York.
  • [9] Elliott, R. J., Aggoun, L. and Moore, J. B. (1995). Hidden Markov Models: Estimation and Control. Applications of Mathematics 29. Springer, New York.
  • [10] Elliott, R. J. and Kopp, P. E. (2005). Mathematics of Financial Markets, 2nd ed. Springer, New York.
  • [11] Föllmer, H. and Schied, A. (2002). Stochastic Finance: An Introduction in Discrete Time. De Gruyter Studies in Mathematics 27. de Gruyter, Berlin.
  • [12] Gantmacher, F. (1960). Matrix Theory 2. Chelsea, New York.
  • [13] Gill, R. D. and Johansen, S. (1990). A survey of product-integration with a view toward application in survival analysis. Ann. Statist. 18 1501–1555.
  • [14] Jacod, J. and Shiryaev, A. N. (2003). Limit Theorems for Stochastic Processes, 2nd ed. Springer, Berlin.
  • [15] Peng, S. (2005). Dynamically consistent nonlinear evaluations and expectations. Preprint No. 2004-1. Institute of Mathematics, Shandong Univ. Available at
  • [16] Rosazza Gianin, E. (2006). Risk measures via g-expectations. Insurance Math. Econom. 39 19–34.