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Nash equilibria for nonzero-sum ergodic stochastic differential games

Part of: Stochastic systems and control Miscellaneous topics in calculus of variations and optimal control Game theory

Published online by Cambridge University Press:  30 November 2017

Samuel N. Cohen  and
Victor Fedyashov
Samuel N. Cohen*
Affiliation:
University of Oxford
Victor Fedyashov*
Affiliation:
University of Oxford
*
* Postal address: Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK.
* Postal address: Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK.
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Abstract

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We consider nonzero-sum games where multiple players control the drift of a process, and their payoffs depend on its ergodic behaviour. We establish their connection with systems of ergodic backward stochastic differential equations, and prove the existence of a Nash equilibrium under generalised Isaac's conditions. We also study the case of interacting players of different type.

Keywords

Ergodic gameNash equilibriumBSDE

MSC classification

Primary: 91A60: Probabilistic games; gambling
Secondary: 49N70: Differential games 93E20: Optimal stochastic control
Type
Research Papers
Information
Journal of Applied Probability , Volume 54 , Issue 4 , December 2017 , pp. 977 - 994
Copyright
Copyright © Applied Probability Trust 2017 

References

[1] Buckdahn, R., Li, J. and Quincampoix, M. (2014). Value in mixed strategies for zero-sum stochastic differential games without Isaacs condition. Ann. Prob. 42, 1724–1768. Google Scholar
[2] Cohen, S. N. and Elliott, R. J. (2015). Stochastic Calculus and Applications, 2nd edn. Springer, Cham. Google Scholar
[3] Cohen, S. N. and Fedyashov, V. (2015). Ergodic BSDEs with jumps and time dependence. Preprint. Available at https://arxiv.org/abs/1406.4329. Google Scholar
[4] Debussche, A., Hu, Y. and Tessitore, G. (2011). Ergodic BSDEs under weak dissipative assumptions. Stoch. Process. Appl. 121, 407–426. Google Scholar
[5] El Karoui, N. and Hamadène, S. (2003). BSDEs and risk-sensitive control, zero-sum and nonzero-sum game problems of stochastic functional differential equations. Stoch. Process. Appl. 107, 145–169. Google Scholar
[6] Fuhrman, M., Hu, Y. and Tessitore, G. (2009). Ergodic BSDEs and optimal ergodic control in Banach spaces. SIAM J. Control Optimization 48, 1542–1566. Google Scholar
[7] Hamadène, S. and Lepeltier, J.-P. (1995). Backward equations, stochastic control and zero-sum stochastic differential games. Stoch. Stoch. Reports 54, 221–231. CrossRefGoogle Scholar
[8] Hamadène, S. and Mu, R. (2015). Existence of Nash equilibrium points for Markovian non-zero-sum stochastic differential games with unbounded coefficients. Stochastics 87, 85–111. CrossRefGoogle Scholar
[9] Hamadène, S., Lepeltier, J. P. and Peng, S. (1997). BSDEs with continuous coefficients and stochastic differential games. In Backward Stochastic Differential Equations (Pitman Res. Notes Math. 364), eds N. El Karoui and L. Mazliak, Longman, Harlow, pp. 115–128.Google Scholar
[10] Hu, Y. and Tang, S. (2014). Multi-dimensional backward stochastic differential equations of diagonally quadratic generators. Preprint. Available at https://arxiv.org/abs/1408.4579. Google Scholar
[11] Hu, Y., Madec, P.-Y. and Richou, A. (2015). A probabilistic approach to large time behavior of mild solutions of HJB equations in infinite dimension. SIAM J. Control Optimization 53, 378–398. Google Scholar
[12] Karatzas, I. and Li, Q. (2012). BSDE approach to non-zero-sum stochastic differential games of control and stopping. In Stochastic Processes, Finance and Control, World Scientific, Hackensack, NJ, pp. 105–153. Google Scholar
[13] Lepeltier, J.-P. and San Martin, J. (1997). Backward stochastic differential equations with continuous coefficient. Statist. Prob. Lett. 32, 425–430. Google Scholar
[14] Mannucci, P. (2004). Nonzero-sum stochastic differential games with discontinuous feedback. SIAM J. Control Optimization 43, 1222–1233. Google Scholar
[15] Parthasarathy, K. R. (1967). Probability Measures on Metric Spaces. Academic Press, New York. Google Scholar