Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-27T22:05:56.329Z Has data issue: false hasContentIssue false
  • English
  • Français

Undiscounted Markov Chain BSDEs to Stopping Times

Part of: Markov processes Stochastic systems and control Circuits, networks

Published online by Cambridge University Press:  30 January 2018

Samuel N. Cohen
Samuel N. Cohen*
Affiliation:
University of Oxford
*
∗ Postal address: Mathematical Institute, University of Oxford, 24-29 St Giles', Oxford OX1 3LB, UK. Email address: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We consider backward stochastic differential equations in a setting where noise is generated by a countable state, continuous time Markov chain, and the terminal value is prescribed at a stopping time. We show that, given sufficient integrability of the stopping time and a growth bound on the terminal value and BSDE driver, these equations admit unique solutions satisfying the same growth bound (up to multiplication by a constant). This holds without assuming that the driver is monotone in y, that is, our results do not require that the terminal value be discounted at some uniform rate. We show that the conditions are satisfied for hitting times of states of the chain, and hence present some novel applications of the theory of these BSDEs.

Keywords

BSDEMarkov chainuniform ergodicityrisk averse controlnon-Ohmic circuit

MSC classification

Primary: 60J27: Continuous-time Markov processes on discrete state spaces
Secondary: 93E20: Optimal stochastic control 94C05: Analytic circuit theory
Type
Research Article
Information
Journal of Applied Probability , Volume 51 , Issue 1 , March 2014 , pp. 262 - 281
Copyright
© Applied Probability Trust 

References

Bismut, J.-M. (1973). Conjugate convex functions in optimal stochastic control. J. Math. Anal. Appl. 44, 384–404.Google Scholar
Briand, P. and Hu, Y. (1998). Stability of BSDEs with random terminal time and homogenization of semilinear elliptic PDEs. J. Funct. Anal. 155, 455–494.Google Scholar
Cohen, S. N. (2012). Representing filtration consistent nonlinear expectations as g-expectations in general probability spaces. Stoch. Process. Appl. 122, 1601–1626.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). Comparisons for backward stochastic differential equations on Markov chains and related no-arbitrage conditions. Ann. Appl. Prob. 20, 267–311.Google Scholar
Cohen, S. N. and Elliott, R. J. (2012). Existence, uniqueness and comparisons for BSDEs in general spaces. Ann. Prob. 40, 2264–2297.Google Scholar
Cohen, S. N. and Hu, Y. (2013). Ergodic BSDEs driven by Markov chains. SIAM J. Control Optimization 51, 4138–4168.Google Scholar
Cohen, S. N. and Szpruch, L. (2012). On Markovian solutions to Markov chain BSDEs. Numer. Algebra Control Optimization 2, 257–269.Google Scholar
Cohen, S. N., Elliott, R. J. and Pearce, C. E. M. (2010). A general comparison theorem for backward stochastic differential equations. Adv. Appl. Prob. 42, 878–898.Google Scholar
Confortola, F. and Fuhrman, M. (2013). Backward stochastic differential equations and optimal control of marked point processes. SIAM J. Control Optimization 51, 3592–3623.Google Scholar
Crecraft, D. I. and Gergely, S. (2002). Analog Electronics: Circuits, Systems and Signal Processing. Butterworth-Heinemann, Oxford.Google Scholar
Darling, R. W. R. and Pardoux, E. (1997). Backwards SDE with random terminal time and applications to semilinear elliptic PDE. Ann. Prob. 25, 1135–1159.Google Scholar
Debussche, A., Hu, Y. and Tessitore, G. (2011). Ergodic BSDEs under weak dissipative assumptions. Stoch. Process. Appl. 121, 407–426.Google Scholar
Elliott, R. J., Aggoun, L. and Moore, J. B. (1995). Hidden Markov Models. Estimation and Control. Springer, New York.Google Scholar
Föllmer, H. and Schied, A. (2002). Stochastic Finance. An Introduction in Discrete Time (De Gruyter Studies Math. 27). De Gruyter, Berlin.Google Scholar
Grimmett, G. (2010). Probability on Graphs. Cambridge University Press.Google Scholar
McShane, E. J. and Warfield, R. B. Jr. (1967). On Filippov's implicit functions lemma. Proc. Amer. Math. Soc. 18, 41–47.Google Scholar
Meyn, S. and Tweedie, R. L. (2009). Markov Chains and Stochastic Stability, 2nd edn. Cambridge University Press.Google Scholar
Pardoux, E. and Peng, S. G. (1990). Adapted solution of a backward stochastic differential equation. Systems Control Lett. 14, 55–61.Google Scholar
Rogers, L. C. G. and Williams, D. (2000). Diffusions, Markov Processes, and Martingales, Vol. 1, Foundations, 2nd edn. Cambridge University Press.Google Scholar
Royer, M. (2004). BSDEs with a random terminal time driven by a monotone generator and their links with PDEs. Stoch. Stoch. Reports 76, 281–307.Google Scholar
Royer, M. (2006). Backward stochastic differential equations with Jumps and related non-linear expectations. Stoch. Process. Appl. 116, 1358–1376.Google Scholar