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November  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

Tyrone E. Duncan 1,

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.
Keywords: fractional Brownian motions, stochastic partial differential equations., Stochastic differential games.
Mathematics Subject Classification: 49N70, 91A25, 60G22, 60H1.
Citation: Tyrone E. Duncan. Some linear-quadratic stochastic differential games for equations in Hilbert spaces with fractional Brownian motions. Discrete & Continuous Dynamical Systems, 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), 1737-1756. doi: 10.1137/S0363012902417632.  Google Scholar

[2]

R. Buckdahn and J. Li, Stochastic differential games and viscosity solutions of Hamilton-Jacobi-Bellman-Isaacs equations, SIAM J. Control Optim., 47 (2008), 444-475. doi: 10.1137/060671954.  Google Scholar

[3]

T. E. Duncan, Prediction for some processes related to a fractional Brownian motion, Stat. Prob. Lett., 76 (2006), 128-134. doi: 10.1016/j.spl.2005.06.014.  Google Scholar

[4]

T. E. Duncan, Linear-exponential-quadratic Gaussian control, IEEE Trans. Autom. Control, 58 (2013), 2910-2911. doi: 10.1109/TAC.2013.2257610.  Google Scholar

[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), Operations Research and Management Series, 198, Springer Intern. Publishing, Switzerland, 2014, 17-25. doi: 10.1007/978-3-319-00669-7_2.  Google Scholar

[6]

T. E. Duncan, J. Jakubowski and B. Pasik-Duncan, Stochastic integration for fractional Brownian motion in a Hilbert space, Stoch. Dyn., 6 (2006), 53-75. doi: 10.1142/S0219493706001645.  Google Scholar

[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), 1357-1383. doi: 10.1016/j.spa.2005.03.011.  Google Scholar

[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), 2286-2315. doi: 10.1137/08071764X.  Google Scholar

[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), 507-531. doi: 10.1137/110831416.  Google Scholar

[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, Atlanta, 2010, 6163-6168. doi: 10.1109/CDC.2010.5718045.  Google Scholar

[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), 407-416.  Google Scholar

[12]

T. E. Duncan and B. Pasik-Duncan, Linear quadratic fractional Gaussian control, SIAM J. Control Optim., 51 (2013), 4504-4519. doi: 10.1137/120877283.  Google Scholar

[13]

T. E. Duncan, B. Pasik-Duncan and B. Maslowski, Fractional Brownian motion and stochastic equations in Hilbert spaces, Stoch. Dyn., 2 (2002), 225-250. doi: 10.1142/S0219493702000340.  Google Scholar

[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), 107-129. doi: 10.1007/BF01442231.  Google Scholar

[15]

W. H. Fleming and D. Hernandez-Hernandez, On the value of stochastic differential games, Commun. Stoch. Anal., 5 (2011), 341-351.  Google Scholar

[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), 293-314. doi: 10.1512/iumj.1989.38.38015.  Google Scholar

[17]

H. E. Hurst, Long-term storage capacity in reservoirs, Trans. Amer. Soc. Civil Eng., 116 (1951), 400-410. Google Scholar

[18]

R. Isaacs, Differential Games, J. Wiley, New York, 1965.  Google Scholar

[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), 124-131.  Google Scholar

[20]

A. N. Kolmogorov, Wienersche spiralen und einige andere interessante kurven in Hilbertschen Raum, C.R. (Doklady) Acad. USSS (N.S.), 26 (1940), 115-118.  Google Scholar

[21]

I. Lasiecka and R. Triggiani, Feedback semigroups and cosine operators for boundary feedback parabolic and hyperbolic equations, J. Differential Equations, 47 (1983), 246-272. doi: 10.1016/0022-0396(83)90036-0.  Google Scholar

[22]

I. Lasiecka and R. Triggiani, The regulator problem for parabolic equations with Dirichlet boundary control I, Appl. Math. Optim., 16 (1987), 147-168. doi: 10.1007/BF01442189.  Google Scholar

[23]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[24]

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach, Yverdon, 1993.  Google Scholar

[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), 253-263.  Google Scholar

[26]

J. Yong and X. Y. Zhou, Stochastic Controls, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4612-1466-3.  Google Scholar

show all references

References:
[1]

E. Bayraktar and H. V. Poor, Stochastic differential games in a non-Markovian setting, SIAM J. Control Optim., 43 (2005), 1737-1756. doi: 10.1137/S0363012902417632.  Google Scholar

[2]

R. Buckdahn and J. Li, Stochastic differential games and viscosity solutions of Hamilton-Jacobi-Bellman-Isaacs equations, SIAM J. Control Optim., 47 (2008), 444-475. doi: 10.1137/060671954.  Google Scholar

[3]

T. E. Duncan, Prediction for some processes related to a fractional Brownian motion, Stat. Prob. Lett., 76 (2006), 128-134. doi: 10.1016/j.spl.2005.06.014.  Google Scholar

[4]

T. E. Duncan, Linear-exponential-quadratic Gaussian control, IEEE Trans. Autom. Control, 58 (2013), 2910-2911. doi: 10.1109/TAC.2013.2257610.  Google Scholar

[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), Operations Research and Management Series, 198, Springer Intern. Publishing, Switzerland, 2014, 17-25. doi: 10.1007/978-3-319-00669-7_2.  Google Scholar

[6]

T. E. Duncan, J. Jakubowski and B. Pasik-Duncan, Stochastic integration for fractional Brownian motion in a Hilbert space, Stoch. Dyn., 6 (2006), 53-75. doi: 10.1142/S0219493706001645.  Google Scholar

[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), 1357-1383. doi: 10.1016/j.spa.2005.03.011.  Google Scholar

[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), 2286-2315. doi: 10.1137/08071764X.  Google Scholar

[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), 507-531. doi: 10.1137/110831416.  Google Scholar

[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, Atlanta, 2010, 6163-6168. doi: 10.1109/CDC.2010.5718045.  Google Scholar

[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), 407-416.  Google Scholar

[12]

T. E. Duncan and B. Pasik-Duncan, Linear quadratic fractional Gaussian control, SIAM J. Control Optim., 51 (2013), 4504-4519. doi: 10.1137/120877283.  Google Scholar

[13]

T. E. Duncan, B. Pasik-Duncan and B. Maslowski, Fractional Brownian motion and stochastic equations in Hilbert spaces, Stoch. Dyn., 2 (2002), 225-250. doi: 10.1142/S0219493702000340.  Google Scholar

[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), 107-129. doi: 10.1007/BF01442231.  Google Scholar

[15]

W. H. Fleming and D. Hernandez-Hernandez, On the value of stochastic differential games, Commun. Stoch. Anal., 5 (2011), 341-351.  Google Scholar

[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), 293-314. doi: 10.1512/iumj.1989.38.38015.  Google Scholar

[17]

H. E. Hurst, Long-term storage capacity in reservoirs, Trans. Amer. Soc. Civil Eng., 116 (1951), 400-410. Google Scholar

[18]

R. Isaacs, Differential Games, J. Wiley, New York, 1965.  Google Scholar

[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), 124-131.  Google Scholar

[20]

A. N. Kolmogorov, Wienersche spiralen und einige andere interessante kurven in Hilbertschen Raum, C.R. (Doklady) Acad. USSS (N.S.), 26 (1940), 115-118.  Google Scholar

[21]

I. Lasiecka and R. Triggiani, Feedback semigroups and cosine operators for boundary feedback parabolic and hyperbolic equations, J. Differential Equations, 47 (1983), 246-272. doi: 10.1016/0022-0396(83)90036-0.  Google Scholar

[22]

I. Lasiecka and R. Triggiani, The regulator problem for parabolic equations with Dirichlet boundary control I, Appl. Math. Optim., 16 (1987), 147-168. doi: 10.1007/BF01442189.  Google Scholar

[23]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[24]

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach, Yverdon, 1993.  Google Scholar

[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), 253-263.  Google Scholar

[26]

J. Yong and X. Y. Zhou, Stochastic Controls, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4612-1466-3.  Google Scholar

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