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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 |
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. |
[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. |
[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. |
[4] |
T. E. Duncan, Linear-exponential-quadratic Gaussian control, IEEE Trans. Autom. Control, 58 (2013), 2910-2911.
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), Operations Research and Management Series, 198, Springer Intern. Publishing, Switzerland, 2014, 17-25.
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), 53-75.
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), 1357-1383.
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), 2286-2315.
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), 507-531.
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, Atlanta, 2010, 6163-6168.
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), 407-416. |
[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. |
[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. |
[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. |
[15] |
W. H. Fleming and D. Hernandez-Hernandez, On the value of stochastic differential games, Commun. Stoch. Anal., 5 (2011), 341-351. |
[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. |
[17] |
H. E. Hurst, Long-term storage capacity in reservoirs, Trans. Amer. Soc. Civil Eng., 116 (1951), 400-410. |
[18] | |
[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. |
[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. |
[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. |
[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. |
[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. |
[24] |
S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach, Yverdon, 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), 253-263. |
[26] |
J. Yong and X. Y. Zhou, Stochastic Controls, Springer-Verlag, New York, 1999.
doi: 10.1007/978-1-4612-1466-3. |
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. |
[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. |
[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. |
[4] |
T. E. Duncan, Linear-exponential-quadratic Gaussian control, IEEE Trans. Autom. Control, 58 (2013), 2910-2911.
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), Operations Research and Management Series, 198, Springer Intern. Publishing, Switzerland, 2014, 17-25.
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), 53-75.
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), 1357-1383.
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), 2286-2315.
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), 507-531.
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, Atlanta, 2010, 6163-6168.
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), 407-416. |
[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. |
[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. |
[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. |
[15] |
W. H. Fleming and D. Hernandez-Hernandez, On the value of stochastic differential games, Commun. Stoch. Anal., 5 (2011), 341-351. |
[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. |
[17] |
H. E. Hurst, Long-term storage capacity in reservoirs, Trans. Amer. Soc. Civil Eng., 116 (1951), 400-410. |
[18] | |
[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. |
[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. |
[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. |
[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. |
[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. |
[24] |
S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach, Yverdon, 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), 253-263. |
[26] |
J. Yong and X. Y. Zhou, Stochastic Controls, Springer-Verlag, New York, 1999.
doi: 10.1007/978-1-4612-1466-3. |
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