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 and 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[10]

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

[11]

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

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[10]

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

[11]

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

[1]

Matthias Gerdts, Sven-Joachim Kimmerle. Numerical optimal control of a coupled ODE-PDE model of a truck with a fluid basin. Conference Publications, 2015, 2015 (special) : 515-524. doi: 10.3934/proc.2015.0515
[2]

Jiequn Han, Jihao Long. Convergence of the deep BSDE method for coupled FBSDEs. Probability, Uncertainty and Quantitative Risk, 2020, 5 (0) : 5-. doi: 10.1186/s41546-020-00047-w
[3]

Alexandre Caboussat, Allison Leonard. Numerical solution and fast-slow decomposition of a population of weakly coupled systems. Conference Publications, 2009, 2009 (Special) : 123-132. doi: 10.3934/proc.2009.2009.123
[4]

Ajay Jasra, Kody J. H. Law, Yaxian Xu. Markov chain simulation for multilevel Monte Carlo. Foundations of Data Science, 2021, 3 (1) : 27-47. doi: 10.3934/fods.2021004
[5]

Ronald E. Mickens. Positivity preserving discrete model for the coupled ODE's modeling glycolysis. Conference Publications, 2003, 2003 (Special) : 623-629. doi: 10.3934/proc.2003.2003.623
[6]

Vanessa Baumgärtner, Simone Göttlich, Stephan Knapp. Feedback stabilization for a coupled PDE-ODE production system. Mathematical Control and Related Fields, 2020, 10 (2) : 405-424. doi: 10.3934/mcrf.2020003
[7]

Alexandre N. Carvalho, Luciano R. N. Rocha, José A. Langa, Rafael Obaya. Structure of non-autonomous attractors for a class of diffusively coupled ODE. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022083
[8]

Dmytro Marushkevych, Alexandre Popier. Limit behaviour of the minimal solution of a BSDE with singular terminal condition in the non Markovian setting. Probability, Uncertainty and Quantitative Risk, 2020, 5 (0) : 1-. doi: 10.1186/s41546-020-0043-5
[9]

Paweł Pilarczyk. Topological-numerical approach to the existence of periodic trajectories in ODE's. Conference Publications, 2003, 2003 (Special) : 701-708. doi: 10.3934/proc.2003.2003.701
[10]

Jingzhi Tie, Qing Zhang. An optimal mean-reversion trading rule under a Markov chain model. Mathematical Control and Related Fields, 2016, 6 (3) : 467-488. doi: 10.3934/mcrf.2016012
[11]

Ralf Banisch, Carsten Hartmann. A sparse Markov chain approximation of LQ-type stochastic control problems. Mathematical Control and Related Fields, 2016, 6 (3) : 363-389. doi: 10.3934/mcrf.2016007
[12]

Kun Fan, Yang Shen, Tak Kuen Siu, Rongming Wang. On a Markov chain approximation method for option pricing with regime switching. Journal of Industrial and Management Optimization, 2016, 12 (2) : 529-541. doi: 10.3934/jimo.2016.12.529
[13]

Karl Kunisch, Sérgio S. Rodrigues. Oblique projection based stabilizing feedback for nonautonomous coupled parabolic-ode systems. Discrete and Continuous Dynamical Systems, 2019, 39 (11) : 6355-6389. doi: 10.3934/dcds.2019276
[14]

Maria Laura Delle Monache, Paola Goatin. A front tracking method for a strongly coupled PDE-ODE system with moving density constraints in traffic flow. Discrete and Continuous Dynamical Systems - S, 2014, 7 (3) : 435-447. doi: 10.3934/dcdss.2014.7.435
[15]

Gervy Marie Angeles, Gilbert Peralta. Energy method for exponential stability of coupled one-dimensional hyperbolic PDE-ODE systems. Evolution Equations and Control Theory, 2022, 11 (1) : 199-224. doi: 10.3934/eect.2020108
[16]

Sondes khabthani, Lassaad Elasmi, François Feuillebois. Perturbation solution of the coupled Stokes-Darcy problem. Discrete and Continuous Dynamical Systems - B, 2011, 15 (4) : 971-990. doi: 10.3934/dcdsb.2011.15.971
[17]

Tetsuya Ishiwata, Takeshi Ohtsuka. Numerical analysis of an ODE and a level set methods for evolving spirals by crystalline eikonal-curvature flow. Discrete and Continuous Dynamical Systems - S, 2021, 14 (3) : 893-907. doi: 10.3934/dcdss.2020390
[18]

Ling Liu, Jiashan Zheng. Global existence and boundedness of solution of a parabolic-parabolic-ODE chemotaxis-haptotaxis model with (generalized) logistic source. Discrete and Continuous Dynamical Systems - B, 2019, 24 (7) : 3357-3377. doi: 10.3934/dcdsb.2018324
[19]

Song Wang. Numerical solution of an obstacle problem with interval coefficients. Numerical Algebra, Control and Optimization, 2020, 10 (1) : 23-38. doi: 10.3934/naco.2019030
[20]

Roberto Garrappa, Eleonora Messina, Antonia Vecchio. Effect of perturbation in the numerical solution of fractional differential equations. Discrete and Continuous Dynamical Systems - B, 2018, 23 (7) : 2679-2694. doi: 10.3934/dcdsb.2017188

 Impact Factor: 

Tools

Metrics

  • PDF downloads (167)
  • HTML views (0)
  • Cited by (16)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy