American Institute of Mathematical Sciences

Advanced
2012, 2(2): 257-269. doi: 10.3934/naco.2012.2.257

On Markovian solutions to Markov Chain BSDEs

Samuel N. Cohen 1, and  Lukasz Szpruch 1,

1. 

Mathematical Institute, University of Oxford, 24-29 St Giles, OX1 3LB, Oxford, United Kingdom, United Kingdom

Received  November 2011 Revised  March 2012 Published  May 2012

We study (backward) stochastic differential equations with noise coming from a finite state Markov chain. We show that, for the solutions of these equations to be `Markovian', in the sense that they are deterministic functions of the state of the underlying chain, the integrand must be of a specific form. This allows us to connect these equations to coupled systems of ODEs, and hence to give fast numerical methods for the evaluation of Markov-Chain BSDEs.
Keywords: BSDE, coupled ODE, Markov chain., numerical solution.
Mathematics Subject Classification: Primary: 60J27; Secondary: 60H05, 34A1.
Citation: Samuel N. Cohen, Lukasz Szpruch. On Markovian solutions to Markov Chain BSDEs. Numerical Algebra, Control & Optimization, 2012, 2 (2) : 257-269. doi: 10.3934/naco.2012.2.257
References:
[1]

C. Bender and R. Denk, A forward scheme for backward SDEs, Stochastic Processes and their Applications, 117 (2007), 1793-1812. doi: 10.1016/j.spa.2007.03.005.  Google Scholar

[2]

B. Bouchard and N. Touzi, Discrete-time approximation and monte carlo simulation of backward stochastic differential equations, Stochastic Processes and their Applications, 111 (2004), 175-206. doi: 10.1016/j.spa.2004.01.001.  Google Scholar

[3]

P. Carr, H. Geman, D. B. Madan and M. Yor, From local volatility to local Lévy models, Quantitative Finance, 4 (2004), 581-588. doi: 10.1080/14697680400000039.  Google Scholar

[4]

S. N. Cohen and R. J. Elliott, Solutions of backward stochastic differential equations on Markov chains, Communications on Stochastic Analysis, 2 (2008), 251-262.  Google Scholar

[5]

S. N. Cohen and R. J. Elliott, Comparisons for backward stochastic differential equations on Markov chains and related no-arbitrage conditions, The Annals of Applied Probability, 20 (2010), 267-311. doi: 10.1214/09-AAP619.  Google Scholar

[6]

N. El Karoui, S. Peng and M. C. Quenez, Backward stochastic differential equations in finance, Mathematical Finance, 7 (1997), 1-71. doi: 10.1111/1467-9965.00022.  Google Scholar

[7]

F. A. Longstaff and E. S. Schwartz, Valuing american options by simulation: a simple least-squares approach, Review of Financial Studies, 14 (2001), 113-147. doi: 10.1093/rfs/14.1.113.  Google Scholar

[8]

D. B. Madan, M. Pistorius and W.Schoutens, The valuation of structured products using Markov chain models, University of Maryland Working Paper, 2010. Available from http://www.ssrn.com/abstract=1563500. Google Scholar

[9]

E. Pardoux and S. Peng, Adapted solution of a backward stochastic differential equation, Systems & Control Letters, 14 (1990), 55-61. doi: 10.1016/0167-6911(90)90082-6.  Google Scholar

[10]

S. Peng., A generalized dynamic programming principle and Hamilton-Jacobi-Bellman equation, Stochastics and Stochastics Reports, 38 (1992), 119-134.  Google Scholar

[11]

J. Yong and X. Y. Zhou, "Stochastic Controls, Hamiltonian Systems and HJB Equations," Springer, Berlin-Heidelberg-New York, 1999. Google Scholar

show all references

References:
[1]

C. Bender and R. Denk, A forward scheme for backward SDEs, Stochastic Processes and their Applications, 117 (2007), 1793-1812. doi: 10.1016/j.spa.2007.03.005.  Google Scholar

[2]

B. Bouchard and N. Touzi, Discrete-time approximation and monte carlo simulation of backward stochastic differential equations, Stochastic Processes and their Applications, 111 (2004), 175-206. doi: 10.1016/j.spa.2004.01.001.  Google Scholar

[3]

P. Carr, H. Geman, D. B. Madan and M. Yor, From local volatility to local Lévy models, Quantitative Finance, 4 (2004), 581-588. doi: 10.1080/14697680400000039.  Google Scholar

[4]

S. N. Cohen and R. J. Elliott, Solutions of backward stochastic differential equations on Markov chains, Communications on Stochastic Analysis, 2 (2008), 251-262.  Google Scholar

[5]

S. N. Cohen and R. J. Elliott, Comparisons for backward stochastic differential equations on Markov chains and related no-arbitrage conditions, The Annals of Applied Probability, 20 (2010), 267-311. doi: 10.1214/09-AAP619.  Google Scholar

[6]

N. El Karoui, S. Peng and M. C. Quenez, Backward stochastic differential equations in finance, Mathematical Finance, 7 (1997), 1-71. doi: 10.1111/1467-9965.00022.  Google Scholar

[7]

F. A. Longstaff and E. S. Schwartz, Valuing american options by simulation: a simple least-squares approach, Review of Financial Studies, 14 (2001), 113-147. doi: 10.1093/rfs/14.1.113.  Google Scholar

[8]

D. B. Madan, M. Pistorius and W.Schoutens, The valuation of structured products using Markov chain models, University of Maryland Working Paper, 2010. Available from http://www.ssrn.com/abstract=1563500. Google Scholar

[9]

E. Pardoux and S. Peng, Adapted solution of a backward stochastic differential equation, Systems & Control Letters, 14 (1990), 55-61. doi: 10.1016/0167-6911(90)90082-6.  Google Scholar

[10]

S. Peng., A generalized dynamic programming principle and Hamilton-Jacobi-Bellman equation, Stochastics and Stochastics Reports, 38 (1992), 119-134.  Google Scholar

[11]

J. Yong and X. Y. Zhou, "Stochastic Controls, Hamiltonian Systems and HJB Equations," Springer, Berlin-Heidelberg-New York, 1999. Google Scholar

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