In this paper, we investigate infinite horizon jump-diffusion forward-backward stochastic differential equations under some monotonicity conditions. We establish an existence and uniqueness theorem, two stability results and a comparison theorem for solutions to such kind of equations. Then the theoretical results are applied to study a kind of infinite horizon backward stochastic linear-quadratic optimal control problems, and then differential game problems. The unique optimal controls for the control problems and the unique Nash equilibrium points for the game problems are obtained in closed forms.
Accepté le :
DOI : 10.1051/cocv/2016055
Mots clés : Forward-backward stochastic differential equation, monotonicity condition, stochastic optimal control, nonzero-sum stochastic differential game, linear-quadratic problem
@article{COCV_2017__23_4_1331_0,
author = {Yu, Zhiyong},
title = {Infinite horizon jump-diffusion forward-backward stochastic differential equations and their application to backward linear-quadratic problems},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {1331--1359},
publisher = {EDP-Sciences},
volume = {23},
number = {4},
year = {2017},
doi = {10.1051/cocv/2016055},
mrnumber = {3716923},
zbl = {1375.60104},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2016055/}
}
TY - JOUR AU - Yu, Zhiyong TI - Infinite horizon jump-diffusion forward-backward stochastic differential equations and their application to backward linear-quadratic problems JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2017 SP - 1331 EP - 1359 VL - 23 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2016055/ DO - 10.1051/cocv/2016055 LA - en ID - COCV_2017__23_4_1331_0 ER -
%0 Journal Article %A Yu, Zhiyong %T Infinite horizon jump-diffusion forward-backward stochastic differential equations and their application to backward linear-quadratic problems %J ESAIM: Control, Optimisation and Calculus of Variations %D 2017 %P 1331-1359 %V 23 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2016055/ %R 10.1051/cocv/2016055 %G en %F COCV_2017__23_4_1331_0
Yu, Zhiyong. Infinite horizon jump-diffusion forward-backward stochastic differential equations and their application to backward linear-quadratic problems. ESAIM: Control, Optimisation and Calculus of Variations, Tome 23 (2017) no. 4, pp. 1331-1359. doi : 10.1051/cocv/2016055. http://www.numdam.org/articles/10.1051/cocv/2016055/
, Backward-forward stochastic differential equations. Ann. Appl. Probab. 3 (1993) 777–793. | DOI | MR | Zbl
R. Cont and P. Tankov, Financial modelling with jump processes. Financial Mathematics Series. Chapman & Hall/CRC (2004). | MR | Zbl
and , Hedging options for a large investor and forward-backward SDE’s. Ann. Appl. Probab. 6 (1996) 370–398. | DOI | MR | Zbl
, Backward-forward SDE’s and stochastic differential games. Stochastic Process. Appl. 77 (1998) 1–15. | DOI | MR | Zbl
and , Solution of forward-backward stochastic differential equations. Probab. Theory Related Fields 103 (1995) 273–283. | DOI | MR | Zbl
J. Ma and J. Yong, Forward-backward stochastic differential equations and their applications. In vol. 1702 of Lect. Notes Math. Springer-Verlag, Berlin (1999). | MR | Zbl
, and , Solving forward-backward stochastic differential equations explicitly-a four step scheme. Probab. Theory Related Fields 98 (1994) 339–359. | DOI | MR | Zbl
, , and , On wellposedness of forward-backward SDEs-a unified approach. Ann. Appl. Probab. 25 (2015) 2168–2214. | MR | Zbl
B. Øksendal and A. Sulem, Applied stochastic control of jump diffusions, Second edition. Universitext. Springer, Berlin (2007). | MR
and , Forward-backward stochastic differential equations and quasilinear parabolic PDEs. Probab. Theory Related Fields 114 (1999) 123–150. | DOI | MR | Zbl
and , Infinite horizon forward-backward stochastic differential equations. Stochastic Process. Appl. 85 (2000) 75–92. | DOI | MR | Zbl
and , Fully coupled forward-backward stochastic differential equations and applications to optimal control. SIAM J. Control Optim. 37 (1999) 825–843. | DOI | MR | Zbl
, and , Maximum principle for mean-field jump-diffusion stochastic delay differential equations and its application to finance. Automatica J. IFAC 50 (2014) 1565–1579. | DOI | MR | Zbl
, Fully coupled FBSDE with Brownian motion and Poisson process in stopping time duration. J. Aust. Math. Soc. 74 (2003) 249–266. | DOI | MR | Zbl
and , Probabilistic interpretation for a system of quasilinear parabolic partial differential equation combined with algebra equations. Stochastic Process. Appl. 124 (2014) 3921–3947. | DOI | MR | Zbl
, On solutions of a class of infinite horizon FBSDEs. Statist. Probab. Lett. 78 (2008) 2412–2419. | DOI | MR | Zbl
, Forward-backward SDEs with random terminal time and applications to pricing special European-type options for a large investor. Bull. Sci. Math. 135 (2011) 883–895. | DOI | MR | Zbl
, Finding adapted solutions of forward-backward stochastic differential equations: method of continuation. Probab. Theory Related Fields 107 (1997) 537–572. | DOI | MR | Zbl
, Forward-backward stochastic differential equations with mixed initial-terminal conditions. Trans. Amer. Math. Soc. 362 (2010) 1047–1096. | DOI | MR | Zbl
J. Yong and X. Zhou, Stochastic controls. Hamiltonian systems and HJB equations. Vol. 43 of Applications of Mathematics. Springer-Verlag, New York (1999). | MR | Zbl
, Linear-quadratic optimal control and nonzero-sum differential game of forward-backward stochastic system. Asian J. Control 14 (2012) 173–185. | DOI | MR | Zbl
E. Zeidler, Nonlinear functional analysis and its applications, II/B, Nonlinear monotone operators, Translated from the German by the author and Leo F. Boron. Springer-Verlag, New York (1990). | MR | Zbl
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