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A general comparison theorem for backward stochastic differential equations

Part of: Stochastic analysis Stochastic systems and control

Published online by Cambridge University Press:  01 July 2016

Samuel N. Cohen ,
Robert J. Elliott  and
Charles E. M. Pearce
Samuel N. Cohen*
Affiliation:
University of Adelaide
Robert J. Elliott*
Affiliation:
University of Adelaide and University of Calgary
Charles E. M. Pearce*
Affiliation:
University of Adelaide
*
∗ Postal address: School of Mathematical Sciences, University of Adelaide, Adelaide, 5005, Australia.
∗∗∗ Postal address: Haskayne School of Business, University of Calgary, Calgary, T2N 1N4, Canada. Email address: [email protected]
∗ Postal address: School of Mathematical Sciences, University of Adelaide, Adelaide, 5005, Australia.
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Abstract

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A useful result when dealing with backward stochastic differential equations is the comparison theorem of Peng (1992). When the equations are not based on Brownian motion, the comparison theorem no longer holds in general. In this paper we present a condition for a comparison theorem to hold for backward stochastic differential equations based on arbitrary martingales. This theorem applies to both vector and scalar situations. Applications to the theory of nonlinear expectations are also explored.

Keywords

BSDEcomparison theoremnonlinear expectationdynamic risk measure

MSC classification

Primary: 60H10: Stochastic ordinary differential equations
Secondary: 93E20: Optimal stochastic control
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 42 , Issue 3 , September 2010 , pp. 878 - 898
Copyright
Copyright © Applied Probability Trust 2010 

References

Artzner, P. et al. (2007). Coherent multiperiod risk adjusted values and Bellman's principle. Ann. Operat. Res. 152, 5–22.CrossRefGoogle Scholar
Barles, G., Buckdahn, R. and Pardoux, E. (1997). Backward stochastic differential equations and integral-partial differential equations. Stoch. Stoch. Reports 60, 57–83.CrossRefGoogle Scholar
Barrieu, P. and El Karoui, N. (2004). Optimal derivatives design under dynamic risk measures. In Mathematics of Finance (Contemp. Math. 351), eds Yin, G. and Zhang, Q., American Mathematical Society, Providence, RI, pp. 13–25.CrossRefGoogle Scholar
Cheredito, P. and Stadje, M. (2010). BSΔEs and BSDEs with non-Lipschitz drivers: comparison, convergence and robustness. Preprint. Available at http://arxiv.org/abs/1002.0175v1.Google Scholar
Cohen, S. N. and Elliott, R. J. (2008). Solutions of backward stochastic differential equations on Markov chains. Commun. Stoch. Anal. 2, 251–262.Google Scholar
Cohen, S. N. and Elliott, R. J. (2010). A general theory of finite state backward stochastic difference equations. Stoch. Process. Appl. 120, 442–466.CrossRefGoogle Scholar
Cohen, S. N. and Elliott, R. J. (2010). Backward stochastic difference equations and nearly-time-consistent nonlinear expectations. In preparation.Google Scholar
Cohen, S. N. and Elliott, R. J. (2010). Comparisons for backward stochastic differential equations on Markov chains and related no-arbitrage conditions. Ann. Appl. Prob. 20, 267–311.CrossRefGoogle Scholar
Cohen, S. N. and Elliott, R. J. (2010). Existence and comparisons for BSDEs in general spaces. In preparation.Google Scholar
Coquet, F., Hu, Y., Memin, J. and Peng, S. (2002). Filtration-consistent nonlinear expectations and related g-expectations. Prob. Theory Relat. Fields 123, 1–27.CrossRefGoogle Scholar
Davis, M. H. A. (1976). The representation of martingales of Jump processes. SIAM J. Control Optimization 14, 623–638.CrossRefGoogle Scholar
Delbaen, F. and Schachermayer, W. (2006). The Mathematics of Arbitrage. Springer, Berlin.Google Scholar
El Karoui, N. and Mazliak, L. (eds) (1997). Backward Stochastic Differential Equations. Pitman, London.Google Scholar
Elliott, R. J. (1982). Stochastic Calculus and Applications. Springer, New York.Google Scholar
Elliott, R. J. and Kopp, P. E. (2005). Mathematics of Financial Markets, 2nd edn. Springer, New York.Google Scholar
El Otmani, M. (2008). Backward stochastic differential equations associated with Lévy processes and partial integro-differential equations. Commun. Stoch. Anal. 2, 277–288.Google Scholar
Föllmer, H. and Schied, A. (2002). Stochastic Finance (de Gruyter Stud. Math 27). Walter de Gruyter, Berlin.CrossRefGoogle Scholar
Hu, Y. and Peng, S. (2006). On the comparison theorem for multidimensional BSDEs. C. R. Acad. Sci. Paris 343, 135–140.CrossRefGoogle Scholar
Jacod, J. and Shiryaev, A. N. (2003). Limit theorems for Stochastic Processes, 2nd edn. Springer, Berlin.CrossRefGoogle Scholar
Jia, G. (2009). Some uniqueness results for one-dimensional BSDEs with uniformly continuous coefficients. Stat. Prob. Lett. 79, 436–441.CrossRefGoogle Scholar
Jobert, A. and Rogers, L. C. G. (2008). Valuations and dynamic convex risk measures. Math. Finance 18, 1–22.CrossRefGoogle Scholar
Klöppel, S. and Schweizer, M. (2007). Dynamic indifference valuation via convex risk measures. Math. Finance 17, 599–627.CrossRefGoogle Scholar
Lin, Q. Q. (2003). Nonlinear Doob–Meyer decomposition with Jumps. Acta Math. Sinica 19, 69–78.CrossRefGoogle Scholar
Peng, S. G. (1992). A generalized dynamic programming principle and Hamilton–Jacobi–Bellman equation. Stoch. Stoch. Reports 38, 119–134.CrossRefGoogle Scholar
Peng, S.-G. (2005). Dynamically consistent nonlinear evaluations and expectations. Preprint. Available at http://arxiv.org/abs/math/0501415.Google Scholar
Peng, S. and Xu, X. (2009). BSDEs with random default time and their applications to default risk. Preprint. Available at http://arxiv.org/abs/0910.2091v1.Google Scholar
Rosazza Gianin, E. (2006). Risk measures via g-expectations. Insurance Math. Econom. 39, 19–34.CrossRefGoogle Scholar
Royer, M. (2006). Backward stochastic differential equations with Jumps and related non-linear expectations. Stoch. Process. Appl. 116, 1358–1376.CrossRefGoogle Scholar
Situ, R. (2005). Theory of Stochastic Differential Equations with Jumps and Applications. Springer, New York.Google Scholar
Zhou, Q. (2007). On comparison theorem and solutions of BSDEs for Lévy processes. Acta Math. Appl. Sinica 23, 513–522.CrossRefGoogle Scholar