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2015, 35(11): 5435-5445. doi: 10.3934/dcds.2015.35.5435

## Some linear-quadratic stochastic differential games for equations in Hilbert spaces with fractional Brownian motions

 1 Department of Mathematics, University of Kansas, Lawrence, KS 66045, United States

Received  December 2012 Revised  February 2014 Published  May 2015

A noncooperative, two person, zero sum, stochastic differential game is formulated and solved that is described by a linear stochastic equation in a Hilbert space with a fractional Brownian motion and a quadratic payoff functional for the two players. The stochastic equation can model stochastic partial differential equations not only with distributed strategies and noise but also with control strategies and noise restricted to the boundary of the domain. The optimal strategies for the two players are given explicitly. The verification method is a generalization of completion of squares and provides the optimal strategies directly without solving partial differential equations or backward stochastic differential equations. Some examples of games described by stochastic partial differential equations are given.
Citation: Tyrone E. Duncan. Some linear-quadratic stochastic differential games for equations in Hilbert spaces with fractional Brownian motions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5435-5445. doi: 10.3934/dcds.2015.35.5435
##### References:
 [1] E. Bayraktar and H. V. Poor, Stochastic differential games in a non-Markovian setting,, SIAM J. Control Optim., 43 (2005), 1737. doi: 10.1137/S0363012902417632. [2] R. Buckdahn and J. Li, Stochastic differential games and viscosity solutions of Hamilton-Jacobi-Bellman-Isaacs equations,, SIAM J. Control Optim., 47 (2008), 444. doi: 10.1137/060671954. [3] T. E. Duncan, Prediction for some processes related to a fractional Brownian motion,, Stat. Prob. Lett., 76 (2006), 128. doi: 10.1016/j.spl.2005.06.014. [4] T. E. Duncan, Linear-exponential-quadratic Gaussian control,, IEEE Trans. Autom. Control, 58 (2013), 2910. doi: 10.1109/TAC.2013.2257610. [5] T. E. Duncan, Linear-quadratic stochastic differential games with general noise processes,, in Models and Methods in Economics and Management Science: Essays in Honor of Charles S. Tapiero (eds. F. El Ouardighi and K. Kogan), (2014), 17. doi: 10.1007/978-3-319-00669-7_2. [6] T. E. Duncan, J. Jakubowski and B. Pasik-Duncan, Stochastic integration for fractional Brownian motion in a Hilbert space,, Stoch. Dyn., 6 (2006), 53. doi: 10.1142/S0219493706001645. [7] T. E. Duncan, B. Maslowski and B. Pasik-Duncan, Stochastic equations in Hilbert space with a multiplicative fractional Gaussian noise,, Stoc. Proc. Appl., 115 (2005), 1357. doi: 10.1016/j.spa.2005.03.011. [8] T. E. Duncan, B. Maslowski and B. Pasik-Duncan, Semilinear stochastic equations in Hilbert space with a fractional Brownian motion,, SIAM J. Math. Anal., 40 (2009), 2286. doi: 10.1137/08071764X. [9] T. E. Duncan, B. Maslowski and B. Pasik-Duncan, Linear-quadratic control for stochastic equations in a Hilbert space with fractional Brownian motions,, SIAM J. Control Optim., 50 (2012), 507. doi: 10.1137/110831416. [10] T. E. Duncan and B. Pasik-Duncan, Stochastic linear-quadratic control for systems with a fractional Brownian motion,, in Proc.49th IEEE Conference on Decision and Control, (2010), 6163. doi: 10.1109/CDC.2010.5718045. [11] T. E. Duncan and B. Pasik-Duncan, Linear-exponential-quadratic Gaussian control for stochastic equations in a Hilbert space,, Dyn. Systems Applic. (special issue), 21 (2012), 407. [12] T. E. Duncan and B. Pasik-Duncan, Linear quadratic fractional Gaussian control,, SIAM J. Control Optim., 51 (2013), 4504. doi: 10.1137/120877283. [13] T. E. Duncan, B. Pasik-Duncan and B. Maslowski, Fractional Brownian motion and stochastic equations in Hilbert spaces,, Stoch. Dyn., 2 (2002), 225. doi: 10.1142/S0219493702000340. [14] F. Flandoli, Direct solution of a Riccati equation arising in a stochastic control problem with control and observations on the boundary,, Appl. Math. Optim., 14 (1986), 107. doi: 10.1007/BF01442231. [15] W. H. Fleming and D. Hernandez-Hernandez, On the value of stochastic differential games,, Commun. Stoch. Anal., 5 (2011), 341. [16] W. H. Fleming and P. E. Souganidis, On the existence of value functions of two player, zero sum stochastic differential games,, Indiana Math. J., 38 (1989), 293. doi: 10.1512/iumj.1989.38.38015. [17] H. E. Hurst, Long-term storage capacity in reservoirs,, Trans. Amer. Soc. Civil Eng., 116 (1951), 400. [18] R. Isaacs, Differential Games,, J. Wiley, (1965). [19] D. H. Jacobson, Optimal stochastic linear systems with exponential performance criteria and their relation to deterministic differential games,, IEEE Trans. Autom. Control, AC-18 (1973), 124. [20] A. N. Kolmogorov, Wienersche spiralen und einige andere interessante kurven in Hilbertschen Raum,, C.R. (Doklady) Acad. USSS (N.S.), 26 (1940), 115. [21] I. Lasiecka and R. Triggiani, Feedback semigroups and cosine operators for boundary feedback parabolic and hyperbolic equations,, J. Differential Equations, 47 (1983), 246. doi: 10.1016/0022-0396(83)90036-0. [22] I. Lasiecka and R. Triggiani, The regulator problem for parabolic equations with Dirichlet boundary control I,, Appl. Math. Optim., 16 (1987), 147. doi: 10.1007/BF01442189. [23] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Springer-Verlag, (1983). doi: 10.1007/978-1-4612-5561-1. [24] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives,, Gordon and Breach, (1993). [25] C. Tudor and M. Tudor, A Wong-Zakai aproximation for double Stratonovich integrals with respect to the fractional Brownian motion,, Math. Rep. (Bucar.), 7 (2005), 253. [26] J. Yong and X. Y. Zhou, Stochastic Controls,, Springer-Verlag, (1999). doi: 10.1007/978-1-4612-1466-3.

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##### References:
 [1] E. Bayraktar and H. V. Poor, Stochastic differential games in a non-Markovian setting,, SIAM J. Control Optim., 43 (2005), 1737. doi: 10.1137/S0363012902417632. [2] R. Buckdahn and J. Li, Stochastic differential games and viscosity solutions of Hamilton-Jacobi-Bellman-Isaacs equations,, SIAM J. Control Optim., 47 (2008), 444. doi: 10.1137/060671954. [3] T. E. Duncan, Prediction for some processes related to a fractional Brownian motion,, Stat. Prob. Lett., 76 (2006), 128. doi: 10.1016/j.spl.2005.06.014. [4] T. E. Duncan, Linear-exponential-quadratic Gaussian control,, IEEE Trans. Autom. Control, 58 (2013), 2910. doi: 10.1109/TAC.2013.2257610. [5] T. E. Duncan, Linear-quadratic stochastic differential games with general noise processes,, in Models and Methods in Economics and Management Science: Essays in Honor of Charles S. Tapiero (eds. F. El Ouardighi and K. Kogan), (2014), 17. doi: 10.1007/978-3-319-00669-7_2. [6] T. E. Duncan, J. Jakubowski and B. Pasik-Duncan, Stochastic integration for fractional Brownian motion in a Hilbert space,, Stoch. Dyn., 6 (2006), 53. doi: 10.1142/S0219493706001645. [7] T. E. Duncan, B. Maslowski and B. Pasik-Duncan, Stochastic equations in Hilbert space with a multiplicative fractional Gaussian noise,, Stoc. Proc. Appl., 115 (2005), 1357. doi: 10.1016/j.spa.2005.03.011. [8] T. E. Duncan, B. Maslowski and B. Pasik-Duncan, Semilinear stochastic equations in Hilbert space with a fractional Brownian motion,, SIAM J. Math. Anal., 40 (2009), 2286. doi: 10.1137/08071764X. [9] T. E. Duncan, B. Maslowski and B. Pasik-Duncan, Linear-quadratic control for stochastic equations in a Hilbert space with fractional Brownian motions,, SIAM J. Control Optim., 50 (2012), 507. doi: 10.1137/110831416. [10] T. E. Duncan and B. Pasik-Duncan, Stochastic linear-quadratic control for systems with a fractional Brownian motion,, in Proc.49th IEEE Conference on Decision and Control, (2010), 6163. doi: 10.1109/CDC.2010.5718045. [11] T. E. Duncan and B. Pasik-Duncan, Linear-exponential-quadratic Gaussian control for stochastic equations in a Hilbert space,, Dyn. Systems Applic. (special issue), 21 (2012), 407. [12] T. E. Duncan and B. Pasik-Duncan, Linear quadratic fractional Gaussian control,, SIAM J. Control Optim., 51 (2013), 4504. doi: 10.1137/120877283. [13] T. E. Duncan, B. Pasik-Duncan and B. Maslowski, Fractional Brownian motion and stochastic equations in Hilbert spaces,, Stoch. Dyn., 2 (2002), 225. doi: 10.1142/S0219493702000340. [14] F. Flandoli, Direct solution of a Riccati equation arising in a stochastic control problem with control and observations on the boundary,, Appl. Math. Optim., 14 (1986), 107. doi: 10.1007/BF01442231. [15] W. H. Fleming and D. Hernandez-Hernandez, On the value of stochastic differential games,, Commun. Stoch. Anal., 5 (2011), 341. [16] W. H. Fleming and P. E. Souganidis, On the existence of value functions of two player, zero sum stochastic differential games,, Indiana Math. J., 38 (1989), 293. doi: 10.1512/iumj.1989.38.38015. [17] H. E. Hurst, Long-term storage capacity in reservoirs,, Trans. Amer. Soc. Civil Eng., 116 (1951), 400. [18] R. Isaacs, Differential Games,, J. Wiley, (1965). [19] D. H. Jacobson, Optimal stochastic linear systems with exponential performance criteria and their relation to deterministic differential games,, IEEE Trans. Autom. Control, AC-18 (1973), 124. [20] A. N. Kolmogorov, Wienersche spiralen und einige andere interessante kurven in Hilbertschen Raum,, C.R. (Doklady) Acad. USSS (N.S.), 26 (1940), 115. [21] I. Lasiecka and R. Triggiani, Feedback semigroups and cosine operators for boundary feedback parabolic and hyperbolic equations,, J. Differential Equations, 47 (1983), 246. doi: 10.1016/0022-0396(83)90036-0. [22] I. Lasiecka and R. Triggiani, The regulator problem for parabolic equations with Dirichlet boundary control I,, Appl. Math. Optim., 16 (1987), 147. doi: 10.1007/BF01442189. [23] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Springer-Verlag, (1983). doi: 10.1007/978-1-4612-5561-1. [24] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives,, Gordon and Breach, (1993). [25] C. Tudor and M. Tudor, A Wong-Zakai aproximation for double Stratonovich integrals with respect to the fractional Brownian motion,, Math. Rep. (Bucar.), 7 (2005), 253. [26] J. Yong and X. Y. Zhou, Stochastic Controls,, Springer-Verlag, (1999). doi: 10.1007/978-1-4612-1466-3.
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