We introduce the notion of mild supersolution for an obstacle problem in an infinite dimensional Hilbert space. The minimal supersolution of this problem is given in terms of a reflected BSDEs in an infinite dimensional Markovian framework. The results are applied to an optimal control and stopping problem.
Accepté le :
DOI : 10.1051/cocv/2016059
Keywords: Reflected backward stochastic differential equations, obstacle problem, optimal stopping in infinite dimension
Fuhrman, Marco 1 ; Masiero, Federica 2 ; Tessitore, Gianmario 2
@article{COCV_2017__23_4_1419_0,
author = {Fuhrman, Marco and Masiero, Federica and Tessitore, Gianmario},
title = {Reflected {BSDEs,} optimal control and stopping for infinite-dimensional systems},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {1419--1445},
year = {2017},
publisher = {EDP Sciences},
volume = {23},
number = {4},
doi = {10.1051/cocv/2016059},
mrnumber = {3716927},
zbl = {1375.60106},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2016059/}
}
TY - JOUR AU - Fuhrman, Marco AU - Masiero, Federica AU - Tessitore, Gianmario TI - Reflected BSDEs, optimal control and stopping for infinite-dimensional systems JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2017 SP - 1419 EP - 1445 VL - 23 IS - 4 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2016059/ DO - 10.1051/cocv/2016059 LA - en ID - COCV_2017__23_4_1419_0 ER -
%0 Journal Article %A Fuhrman, Marco %A Masiero, Federica %A Tessitore, Gianmario %T Reflected BSDEs, optimal control and stopping for infinite-dimensional systems %J ESAIM: Control, Optimisation and Calculus of Variations %D 2017 %P 1419-1445 %V 23 %N 4 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/cocv/2016059/ %R 10.1051/cocv/2016059 %G en %F COCV_2017__23_4_1419_0
Fuhrman, Marco; Masiero, Federica; Tessitore, Gianmario. Reflected BSDEs, optimal control and stopping for infinite-dimensional systems. ESAIM: Control, Optimisation and Calculus of Variations, Tome 23 (2017) no. 4, pp. 1419-1445. doi: 10.1051/cocv/2016059
J.P. Aubin and H. Frankowska, Set-valued analysis, in Vol. 2 of Systems and Control: Foundations and Applications. Birkhäuser Boston Inc., Boston, MA (1990). | MR | Zbl
A. Bensoussan, Stochastic control by functional analysis methods. Studies in Mathematics and its Applications, Vol. 11. North-Holland Publishing Co., Amsterdam, New York (1982). | MR | Zbl
G. Da Prato and J. Zabczyk,Stochastic equations in infinite dimensions. Encyclopedia of Mathematics and its Applications, Vol. 44. Cambridge University Press (1992). | MR | Zbl
G. Da Prato and J. Zabczyk, Second order partial differential equations in Hilbert spaces. In Vol. 293 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge (2002). | MR | Zbl
C. Dellacherie and P.A. Meyer. Probability and Potential B: Theory of Martingales. North-Holland Amsterdam (1982). | MR | Zbl
, , , and , Reflected solutions of backward SDE’s, and related obstacle problems for PDE’s. Ann. Probab. 25 (1997) 702–737. | MR | Zbl | DOI
, and , Backward stochastic differential equations in finance. Math. Finance 7 (1997) 1–71. | MR | Zbl | DOI
I. Karatzas, S.E. Shreve, Steven, Brownian motion and stochastic calculus. Second edition. In Vol. 113 of Graduate Texts in Mathematics. Springer-Verlag, New York. | MR | Zbl
and , Viscosity solutions of an infinite-dimensional Black-Scholes-Barenblatt equation. Appl. Math. Optim. 47 (2003) 253–278. | MR | Zbl | DOI
and , Nonlinear Kolmogorov equations in infinite dimensional spaces: the backward stochastic differential equations approach and applications to optimal control. Ann. Probab. 30 (2002) 1397–1465. | MR | Zbl | DOI
and , The Bismut-Elworthy formula for backward SDEs and applications to nonlinear Kolmogorov equations and control in infinite dimensional spaces. Stoch. Stoch. Rep. 74 (2002) 429–464. | MR | Zbl | DOI
and , Infinite horizon backward stochastic differential equations and elliptic equations in Hilbert spaces. Ann. Probab. 32 (2004) 607–660. | MR | Zbl | DOI
and , Generalized directional gradients, backward stochastic differential equations and mild solutions of semilinear parabolic equations. Appl. Math. Optim. 51 (2005) 279–332. | MR | Zbl | DOI
and , BSDE on an infinite horizon and elliptic PDEs in infinite dimension. NoDEA Nonlinear Differ. Equ. Appl. 14 (2007) 825–846. | MR | Zbl | DOI
, Semilinear Kolmogorov equations and applications to stochastic optimal control. Appl. Math. Optim. 51 (2005) 201–250. | MR | Zbl | DOI
, Infinite horizon stochastic optimal control problems with degenerate noise and elliptic equations in Hilbert spaces. Appl. Math. Optim. 55 (2007) 285–326. | MR | Zbl | DOI
and , Adapted solution of a backward stochastic differential equation. Syst. Control Lett. 14 (1990) 55–61. | MR | Zbl | DOI
E. Pardoux and S. Peng, Backward stochastic differential equations and quasilinear parabolic partial differential equations, in: Stochastic partial differential equations and their applications, edited by B.L. Rozowskii and R.B. Sowers. In Vol. 176 of Lect. Notes Control Inf. Sci. Springer 176 (1992). | MR | Zbl
and , The generalized covariation process and It formula. Stochastic Processes Appl. 59 (1995) 81–104. | MR | Zbl | DOI
J. Yong and X.Y. Zhou, Stochastic controls, Hamiltonian systems and HJB equations, Applications of Mathematics. Springer, New York (1999). | MR | Zbl
Cité par Sources :
