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Feedback optimal control for stochastic Volterra equations with completely monotone kernels
1. | Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci, 32 - 20133 Milano |
2. | Dipartimento di statistica e metodi quantitativi, Universitá degli studi di Milano Bicocca, Piazza dell'Ateneo Nuovo, 1 - 20126, Milano |
  The first main result of the paper is the proof of existence and uniqueness of a mild solution for the corresponding Hamilton-Jacobi-Bellman (HJB) equation. The main technical point consists in the differentiability of the BSDE associated with the reformulated equation with respect to its initial datum $x$.
References:
[1] |
O. P. Agrawal, Solution for a fractional diffusion-wave equation defined in a bounded domain, Nonlinear Dynamics, 29 (2002), 145-155.
doi: 10.1023/A:1016539022492. |
[2] |
S. A. Belbas, A new method for optimal control of Volterra integral equations, Appl. Math. Comput., 189 (2007), 1902-1915.
doi: 10.1016/j.amc.2006.12.077. |
[3] |
S. Bonaccorsi, Some Applications in Malliavin Calculus, Ph.D thesis, Dept. Mathematics, Univ. Trento., 1998. |
[4] |
S. Bonaccorsi and G. Desch, Volterra equations in Banach spaces with completely monotone kernels, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 557-594.
doi: 10.1007/s00030-012-0167-0. |
[5] |
S. Bonaccorsi and E. Mastrogiacomo, An analytic approach to stochastic Volterra equations with completely monotone kernels, J. Evol. Equ., 9 (2009), 315-339.
doi: 10.1007/s00028-009-0010-1. |
[6] |
S. Bonaccorsi, F. Confortola and E. Mastrogiacomo, Optimal control for stochastic Volterra equations with completely monotone kernels, SIAM J. Control Optim., 50 (2012), 748-789.
doi: 10.1137/100782875. |
[7] |
P. Briand and F. Confortola, SDEs with stochastic Lipschitz condition and quadratic PDEs in Hilbert spaces, Stochastic Process. Appl., 118 (2008), 818-838.
doi: 10.1016/j.spa.2007.06.006. |
[8] |
F. Confortola and E. Mastrogiacomo, Optimal control for stochastic heat equation with memory, Evolution Equations and Control Theory, 3 (2014), 35-58.
doi: 10.3934/eect.2014.3.35. |
[9] |
G. Da Prato and J. Zabczyk, Evolution Equations with White-Noise Boundary Conditions, Cambridge Univ. Press, Cambridge, 1993.
doi: 10.1080/17442509308833817. |
[10] |
G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge Univ. Press, Cambridge, 1992.
doi: 10.1017/CBO9780511666223. |
[11] |
A. Debussche, M. Fuhrman and G. Tessitore, Optimal control of a stochastic heat equation with boundary-noise and boundary-control, ESAIM Control Optim. Calc. Var., 13 (2007), 178-205 (electronic).
doi: 10.1051/cocv:2007001. |
[12] |
W. Desch and R. K. Miller, Exponential stabilization of Volterra integral equations with singular kernels, J. Integral Equations Appl., 1 (1988), 397-433.
doi: 10.1216/JIE-1988-1-3-397. |
[13] |
K. J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations, Graduate Texts in Math., {194}, Springer-Verlag, Berlin, 2000. |
[14] |
W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions, Springer-Verlag, New York, 1993. |
[15] |
M. Fuhrman and G. Tessitore, Nonlinear Kolmogorov equations in infinite dimensional spaces: The backward stochastic differential equations approach and applications to optimal control, Ann. Probab., 30 (2002), 1397-1465.
doi: 10.1214/aop/1029867132. |
[16] |
G. Gripenberg, S. O. Londen and O. Staffans, Volterra Integral and Functional Equations, Cambridge University Press, 1990.
doi: 10.1017/CBO9780511662805. |
[17] |
A. Grorud and E. Pardoux, Intégrales Hilbertiennes anticipantes par rapport à un processus de Wienner cylindrique et calcul stochastique associé, Appl. Math. Optim., 25 (1992), 31-49.
doi: 10.1007/BF01184155. |
[18] |
K. W. Homan, An Analytic Semigroup Approach to Convolution Volterra Equations, Ph.D. Thesis, Delft University Press, 2003. |
[19] |
G. E. Ladas and V. Lakshmikantham, Differential Equations in Abstract Spaces, Mathematics in Science and Engineering, 85, Academic Press, New York, 1972, |
[20] |
A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, 2013 reprint of the 1995 original, Modern Birkhäuser Classics, Basel, 1995. |
[21] |
F. Masiero, A stochastic optimal control problem for the heat equation on the halfline with dirichlet boundary-noise and boundary-control, Appl. Math. Optim., 62 (2010), 253-294.
doi: 10.1007/s00245-010-9103-z. |
[22] |
R. K. Miller, Linear Volterra integrodifferential equations as semigroups, Funkcial. Ekvac., 17 (1974), 39-55. |
[23] |
D. Nualart, Malliavin Calculus and Related Topics, Probability and its Applications (New York), Springer-Verlag, Berlin, 2006. |
[24] |
B. Øksendal and T. Zhang, The stochastic Volterra equation, in Barcelona Seminar on Stochastic Analysis (St. Feliu de Guíxols, 1991), Progr. Probab., 32, Birkhäuser, Basel, 1993, 168-202. |
[25] |
B. Øksendal and T. Zhang, The general linear stochastic Volterra equations with anticipating coefficients, in Stochastic Analysis and Applications (Powys, 1995), World Sci. Publ., River Edge, NJ, 1996, 343-366. |
[26] |
E. Pardoux, Nonlinear filtering, prediction and smoothing equations,, Stochastics, 6 (): 193.
doi: 10.1080/17442508208833204. |
[27] |
J. Prüss, Evolutionary Integral Equations and Applications, Monographs in Mathematics, 87, Birkhäuser Verlag, Basel, 1993.
doi: 10.1007/978-3-0348-8570-6. |
[28] |
W. R. Schneider and W. Wyss, Fractional diffusion and wave equations, Journal of Mathematical Physics, 30 (1989), 134-144.
doi: 10.1063/1.528578. |
show all references
References:
[1] |
O. P. Agrawal, Solution for a fractional diffusion-wave equation defined in a bounded domain, Nonlinear Dynamics, 29 (2002), 145-155.
doi: 10.1023/A:1016539022492. |
[2] |
S. A. Belbas, A new method for optimal control of Volterra integral equations, Appl. Math. Comput., 189 (2007), 1902-1915.
doi: 10.1016/j.amc.2006.12.077. |
[3] |
S. Bonaccorsi, Some Applications in Malliavin Calculus, Ph.D thesis, Dept. Mathematics, Univ. Trento., 1998. |
[4] |
S. Bonaccorsi and G. Desch, Volterra equations in Banach spaces with completely monotone kernels, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 557-594.
doi: 10.1007/s00030-012-0167-0. |
[5] |
S. Bonaccorsi and E. Mastrogiacomo, An analytic approach to stochastic Volterra equations with completely monotone kernels, J. Evol. Equ., 9 (2009), 315-339.
doi: 10.1007/s00028-009-0010-1. |
[6] |
S. Bonaccorsi, F. Confortola and E. Mastrogiacomo, Optimal control for stochastic Volterra equations with completely monotone kernels, SIAM J. Control Optim., 50 (2012), 748-789.
doi: 10.1137/100782875. |
[7] |
P. Briand and F. Confortola, SDEs with stochastic Lipschitz condition and quadratic PDEs in Hilbert spaces, Stochastic Process. Appl., 118 (2008), 818-838.
doi: 10.1016/j.spa.2007.06.006. |
[8] |
F. Confortola and E. Mastrogiacomo, Optimal control for stochastic heat equation with memory, Evolution Equations and Control Theory, 3 (2014), 35-58.
doi: 10.3934/eect.2014.3.35. |
[9] |
G. Da Prato and J. Zabczyk, Evolution Equations with White-Noise Boundary Conditions, Cambridge Univ. Press, Cambridge, 1993.
doi: 10.1080/17442509308833817. |
[10] |
G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge Univ. Press, Cambridge, 1992.
doi: 10.1017/CBO9780511666223. |
[11] |
A. Debussche, M. Fuhrman and G. Tessitore, Optimal control of a stochastic heat equation with boundary-noise and boundary-control, ESAIM Control Optim. Calc. Var., 13 (2007), 178-205 (electronic).
doi: 10.1051/cocv:2007001. |
[12] |
W. Desch and R. K. Miller, Exponential stabilization of Volterra integral equations with singular kernels, J. Integral Equations Appl., 1 (1988), 397-433.
doi: 10.1216/JIE-1988-1-3-397. |
[13] |
K. J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations, Graduate Texts in Math., {194}, Springer-Verlag, Berlin, 2000. |
[14] |
W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions, Springer-Verlag, New York, 1993. |
[15] |
M. Fuhrman and G. Tessitore, Nonlinear Kolmogorov equations in infinite dimensional spaces: The backward stochastic differential equations approach and applications to optimal control, Ann. Probab., 30 (2002), 1397-1465.
doi: 10.1214/aop/1029867132. |
[16] |
G. Gripenberg, S. O. Londen and O. Staffans, Volterra Integral and Functional Equations, Cambridge University Press, 1990.
doi: 10.1017/CBO9780511662805. |
[17] |
A. Grorud and E. Pardoux, Intégrales Hilbertiennes anticipantes par rapport à un processus de Wienner cylindrique et calcul stochastique associé, Appl. Math. Optim., 25 (1992), 31-49.
doi: 10.1007/BF01184155. |
[18] |
K. W. Homan, An Analytic Semigroup Approach to Convolution Volterra Equations, Ph.D. Thesis, Delft University Press, 2003. |
[19] |
G. E. Ladas and V. Lakshmikantham, Differential Equations in Abstract Spaces, Mathematics in Science and Engineering, 85, Academic Press, New York, 1972, |
[20] |
A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, 2013 reprint of the 1995 original, Modern Birkhäuser Classics, Basel, 1995. |
[21] |
F. Masiero, A stochastic optimal control problem for the heat equation on the halfline with dirichlet boundary-noise and boundary-control, Appl. Math. Optim., 62 (2010), 253-294.
doi: 10.1007/s00245-010-9103-z. |
[22] |
R. K. Miller, Linear Volterra integrodifferential equations as semigroups, Funkcial. Ekvac., 17 (1974), 39-55. |
[23] |
D. Nualart, Malliavin Calculus and Related Topics, Probability and its Applications (New York), Springer-Verlag, Berlin, 2006. |
[24] |
B. Øksendal and T. Zhang, The stochastic Volterra equation, in Barcelona Seminar on Stochastic Analysis (St. Feliu de Guíxols, 1991), Progr. Probab., 32, Birkhäuser, Basel, 1993, 168-202. |
[25] |
B. Øksendal and T. Zhang, The general linear stochastic Volterra equations with anticipating coefficients, in Stochastic Analysis and Applications (Powys, 1995), World Sci. Publ., River Edge, NJ, 1996, 343-366. |
[26] |
E. Pardoux, Nonlinear filtering, prediction and smoothing equations,, Stochastics, 6 (): 193.
doi: 10.1080/17442508208833204. |
[27] |
J. Prüss, Evolutionary Integral Equations and Applications, Monographs in Mathematics, 87, Birkhäuser Verlag, Basel, 1993.
doi: 10.1007/978-3-0348-8570-6. |
[28] |
W. R. Schneider and W. Wyss, Fractional diffusion and wave equations, Journal of Mathematical Physics, 30 (1989), 134-144.
doi: 10.1063/1.528578. |
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