Electronic Communications in Probability

Comparison and converse comparison theorems for backward stochastic differential equations with Markov chain noise

Zhe Yang, Dimbinirina Ramarimbahoaka, and Robert J. Elliott

Full-text: Open access

Abstract

Comparison and converse comparison theorems are important parts of the research on backward stochastic differential equations. In this paper, we obtain comparison results for one dimensional backward stochastic differential equations with Markov chain noise, adapting previous results under simplified hypotheses. We introduce a type of nonlinear expectation, the $f$-expectation, which is an interpretation of the solution to a BSDE, and use it to establish a converse comparison theorem for the same type of equations as those in the comparison results.

Article information

Source
Electron. Commun. Probab., Volume 21 (2016), paper no. 25, 10 pp.

Dates
Received: 1 August 2014
Accepted: 12 February 2016
First available in Project Euclid: 10 March 2016

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1457617916

Digital Object Identifier
doi:10.1214/16-ECP4102

Mathematical Reviews number (MathSciNet)
MR3485394

Zentralblatt MATH identifier
1338.60164

Subjects
Primary: 60H15: Stochastic partial differential equations [See also 35R60]

Keywords
BSDEs comparison theorem converse comparison Markov chain

Rights
Creative Commons Attribution 4.0 International License.

Citation

Yang, Zhe; Ramarimbahoaka, Dimbinirina; Elliott, Robert J. Comparison and converse comparison theorems for backward stochastic differential equations with Markov chain noise. Electron. Commun. Probab. 21 (2016), paper no. 25, 10 pp. doi:10.1214/16-ECP4102. https://projecteuclid.org/euclid.ecp/1457617916


Export citation

References

  • [1] G. Barles, R. Buckdahn and E. Pardoux, Backward stochastic differential equations and integral-partial differential equations, Stochastics and Stochastics Report, 60, 57–83 (1996).
  • [2] P. Briand, F. Coquet, J.Memin, S. Peng, A Converse Comparison Theorem for BSDEs and Related Properties of $g$-expectation, Electron. Comm. Probab., 5, 101–117 (2000).
  • [3] L. Campbell and D. Meyer, Generalized inverses of linear transformations, SIAM, (2008).
  • [4] S. N. Cohen and R. J. Elliott, Solutions of Backward Stochastic Differential Equations in Markov Chains., Communications on Stochastic Analysis, 2, 251–262 (2008).
  • [5] S. N. Cohen and R. J. Elliott, Comparison Theorems for Finite State Backward Stochastic Differential Equations, in Contemporary Quantitative Finance, Springer (2010).
  • [6] S. N. Cohen and R. J. Elliott, Comparisons for Backward Stochastic Differential Equations on Markov Chains and Related No-arbitrage Conditions, Annals of Applied Probability, 20(1), 267–311 (2010).
  • [7] F. Coquet, Y. Hu, J. Mémin, and S. Peng, A general Converse Comparison Theorem for Backward Stochastic Differential Equations. C.R. Acad. Sci. Paris, 1, 577–581 (2001).
  • [8] S. Crepey and A. Matoussi, Reflected and doubly reflected BSDEs with jumps: a priori estimates and comparison, The Annals of Applied Probability, 18(5), 2041–2069 (2008).
  • [9] J. Cvitanic and I. Karatzas, Backward stochastic differential equations with reflection and Dynkin games, Ann. Probab., 24(4), 2024–2056 (1996).
  • [10] X. De Scheemaekere, A converse comparison theorem for backward stochastic differential equations with jumps, Statistics and Probability Letters, 81, 298–301 (2011).
  • [11] N. El Karoui, C. Kapoudjian, E. Pardoux, S. Peng and M. C. Quenez, Reflected solutions of backwards SDE’s, and related obstacle problems for PDE’s, The Annals of Probability 25, 702–737 (1997).
  • [12] N. El Karoui, S. Peng and M. C. Quenez, Backward stochastic differential equations in finance, Mathematical Finance 7(1), 1–71 (1997).
  • [13] R. J. Elliott, Stochastic calculus and applications, Springer-Verlag, New York, Heidelberg, Berlin (1982).
  • [14] R. J. Elliott, L. Aggoun and J. B. Moore, Hidden markov models: estimation and control, Applications of Mathematics, Springer-Verlag, Berlin, Heidelberg, New York 29 (1994).
  • [15] Y. Hu and S. Peng, On the comparison theorem for multidimensional BSDEs, C. R. Acad. Sci. Paris, Ser. I, 343, 135–140 (2006).
  • [16] L. Jiang, Converse comparison theorems for backward stochastic differential equations, Statistics and Probaility Letters, 71, 173–183 (2005).
  • [17] J. P. Lepeltier and J. San Martin, Backward SDEs with two barriers and continuous coefficient: an existence result, J. Appl. Prob., 41, 162–175 (2000).
  • [18] J. C. Liu and J. G. Ren, Comparison theorem for solutions of backward stochastic differential equations with continuous coefficient, Statist. Probab. Lett., 56(1), 93–100 (2002).
  • [19] E. Pardoux and S. Peng, Adapted solution of a backward differential equation, Systems Controls Lett., 14, 61–74 (1990).
  • [20] S. Peng, Monotonic limit theorem of BSDE and nonlinear decomposition theorem of Doob-Meyer’s type, Probab. Theory Related Fields, 113(4), 473–499 (1999).
  • [21] S. Peng, Nonlinear expectations, nonlinear evaluations and risk measures, Springer-Verlag, Berlin, Heidelberg 2004, 165–253 (2004).
  • [22] S. Peng and M. Y. Xu, The smallest g-supermartingale and reflected BSDE with single and double $L^{2}$ obstacles, Probabilités et Statistiques, 41, 605–630 (2005).
  • [23] M. Royer, Backward stochastic differential equations with jumps and related non-linear expectations, Stochastic Process, Appl. 116, 1358–1376 (2006).
  • [24] R. Situ, Comparison theorem of solutions to BSDE with jumps, and viscosity solution to a generalized Hamilton-Jacobi-Bellman equation, Control of distributed parameter and stochastic systems (Hangzhou, (1998)), 275–282, Kluwer Acad. Publ., Boston, MA (1999).
  • [25] J. van der Hoek and R. J. Elliott, Asset pricing using finite state Markov chain stochastic discount functions, Stochastic Analysis and Applications, 30, 865–894 (2010).
  • [26] J. van der Hoek and R. J. Elliott, American option prices in a Markov chain model, Applied Stochastic Models in Business and Industry, 28, 35–39 (2012).
  • [27] T. S. Zhang, A comparison theorem for solutions of backward stochastic differential equations with two reflecting barriers and its applications, Probabilistic methods in fluids, 324–331, World Sci. Publ., River Edge, NJ (2003).