Infinite horizon backward stochastic Volterra integral equations (BSVIEs for short) are investigated. We prove the existence and uniqueness of the adapted M-solution in a weighted L2-space. Furthermore, we extend some important known results for finite horizon BSVIEs to the infinite horizon setting. We provide a variation of constant formula for a class of infinite horizon linear BSVIEs and prove a duality principle between a linear (forward) stochastic Volterra integral equation (SVIE for short) and an infinite horizon linear BSVIE in a weighted L2-space. As an application, we investigate infinite horizon stochastic control problems for SVIEs with discounted cost functional. We establish both necessary and sufficient conditions for optimality by means of Pontryagin’s maximum principle, where the adjoint equation is described as an infinite horizon BSVIE. These results are applied to discounted control problems for fractional stochastic differential equations and stochastic integro-differential equations.
Accepté le :
Première publication :
Publié le :
Keywords: Infinite horizon backward stochastic Volterra integral equation, stochastic Volterra integral equation, duality principle, discounted stochastic control
@article{COCV_2021__27_1_A103_0,
author = {Hamaguchi, Yushi},
title = {Infinite horizon backward stochastic {Volterra} integral equations and discounted control problems},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2021},
publisher = {EDP-Sciences},
volume = {27},
doi = {10.1051/cocv/2021098},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2021098/}
}
TY - JOUR AU - Hamaguchi, Yushi TI - Infinite horizon backward stochastic Volterra integral equations and discounted control problems JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2021 VL - 27 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2021098/ DO - 10.1051/cocv/2021098 LA - en ID - COCV_2021__27_1_A103_0 ER -
%0 Journal Article %A Hamaguchi, Yushi %T Infinite horizon backward stochastic Volterra integral equations and discounted control problems %J ESAIM: Control, Optimisation and Calculus of Variations %D 2021 %V 27 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/cocv/2021098/ %R 10.1051/cocv/2021098 %G en %F COCV_2021__27_1_A103_0
Hamaguchi, Yushi. Infinite horizon backward stochastic Volterra integral equations and discounted control problems. ESAIM: Control, Optimisation and Calculus of Variations, Tome 27 (2021), article no. 101. doi: 10.1051/cocv/2021098
[1] , and , Linear–Quadratic control for a class of stochastic Volterra equations: solvability and approximation. Preprint (2020). | arXiv
[2] , Dynamic risk measure for BSVIE with jumps and semimartingale issues. Stoch. Anal. Appl. 37 (2019) 361–376.
[3] and , Mallivain calculus and optimal control of stochastic Volterra equations. J. Optim. Theory Appl. 167 (2015) 1070–1094.
[4] , pth mean integrability and almost sure asymptotic stability of solutions of Itô–Volterra equations. J. Integral Equ. Appl. 15 (2003) 321–341.
[5] and , The term structure of Sharpe ratios and arbitrage-free asset pricing in continuous time. Probab. Uncertain. Quantit. Risk 6 (2021) 23–52.
[6] and , Volterra equations with Itô integrals—I. J. Integral Equ. 2 (1980) 187–245.
[7] and , A linear quadratic optimal control problem for stochastic Volterra integral equations. Control theory and related topics – in memory of professor Xunjing Li, Fudan university, China. (2007) 44–66.
[8] , Asymptotically periodic solutions to Volterra integral equations in epidemic models. J. Math. Anal. Appl. 110 (1985) 483–494.
[9] , and , Existence results for neutral integro-differential equations with unbounded delay. J. Integr. Equ. Appl. 23 (2011) 289–330.
[10] and , Infinite horizon BSDEs in infinite dimensions with continuous driver and applications. J. Evol. Equ. 6 (2006) 459–484.
[11] and Infinite horizon backward stochastic differential equations and elliptic equations in Hilbert spaces. Ann. Probab. 32 (2004) 607–660.
[12] , , and , Volatility is rough. Quant. Finance 18 (2018) 933–949.
[13] , Extended backward stochastic Volterra integral equations and their applications to time-inconsistent stochastic recursive control problems. Math. Control Relat. Fields 11 (2021) 197–242.
[14] , On the maximum principle for optimal control problems of stochastic Volterra integral equations with delay. Preprint: (2021). | arXiv
[15] and , Me, myself and I: a general theory of non-Markovian time-inconsistent stochastic control for sophisticated agents. Preprint: (2021). | arXiv
[16] , Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000).
[17] and , Linear Volterra backward stochastic integral equations. Stochastic Process. Appl. 129 (2019) 626–633.
[18] , and , Theory and Applications of Fractional Differential Equations. North-Holland, Amsterdam (2006).
[19] and , Differentiability of BSVIEs and dynamic capital allocations. Int. J. Theor. Appl. Finance 20 (2017) 1–26.
[20] , Adapted solution of a backward stochastic nonlinear Volterra integral equation. Stoch. Anal. Appl. 20 (2002) 165–183.
[21] and , Controlled singular Volterra integral equations and Pontryagin maximum principle. SIAM J. Control Optim. 58 (2020) 136–164.
[22] and , Forward-Backward Stochastic Differential Equations and Their Applications. Lecture Notes in Math. 1702, Springer-Verlag, New York (1999).
[23] and , Mean square stability of stochastic Volterra integro-differential equations. Syst. Control Lett. 55 (2006) 459–465.
[24] and , Sufficient stochastic maximum principle for discounted control problem. Appl. Math. Optim. 70 (2014) 225–252.
[25] and , Necessary stochastic maximum principle for dissipative systems on infinite time horizon. ESAIM: COCV 23 (2017) 337–371.
[26] and , Functional limit theorems for non-Markovian epidemic models. Preprint: (2021). | arXiv
[27] , BSDEs weak convergence and homogenizations of semilinear PDEs, in Nonlinear Analysis Differential Equations and Control, edited by , (1999) 503–509.
[28] and , Infinite horizon forward-backward stochastic differential equations. Stochastic Process. Appl. 85 (2000) 75–92.
[29] , and , Approximate controllability of nonlinear deterministic and stochastic systems with unbounded delay. Taiwan J. Math. 14 (2010) 1777–1797.
[30] , and , Mean-field backward stochastic Volterra integral equations. Discrete Contin. Dyn. Syst. 18 (2013) 1929–1967.
[31] , and , Optimal control problems of forward-backward stochastic Volterra integral equations. Math. Control Relat. Fields 5 (2015) 613–649.
[32] , and , Recursive utility processes, dynamic risk measures and quadratic backward stochastic Volterra integral equations. Appl. Math. Optim. 84 (2021) 145–190.
[33] and , Time-inconsistent stochastic optimal control problems and backward stochastic Volterra integral equations. ESAIM: COCV 27 (2021).
[34] , and , Path dependent Feynman–Kac formula for forward backward stochastic Volterra integral equations. To appear Ann. Inst. Henri Poincaré Probab. Stat. Preprint (2021). | arXiv
[35] , Linear quadratic control problems of stochastic Volterra integral equations. ESAIM: COCV 24 (2018) 1849–1879.
[36] , Necessary conditions of Pontraygin’s type for general controlled stochastic Volterra integral equations. ESAIM: COCV 26 (2020) 16.
[37] and , Comparison theorems for some backward stochastic Volterra integral equations. Stochastic Process. Appl. 125 (2015) 1756–1798.
[38] and , Optimal control problems of forward-backward stochastic Volterra integral equations with closed control regions. SIAM J. Control Optim. 55 (2017) 2574–2602.
[39] , and , Asymptotic behavior of stochastic lattice systems with a Caputo fractional time derivative. Nonlinear Anal. 135 (2016) 205–222.
[40] and , Maximum principle for stochastic recursive optimal control problems involving impulse controls. Abstr. Appl. Anal. 32 (2012 1–16.
[41] , On solutions of a class of infinite horizon FBSDE’s. Stat. Probab. Lett. 78 (2008) 2412–2419.
[42] , Backward stochastic Volterra integral equations and some related problems. Stoch. Anal. Appl. 116 (2006) 779–795.
[43] , Continuous-time dynamic risk measures by backward stochastic Volterra integral equations. Appl. Anal. 86 (2007) 1429–1442.
[44] , Well-posedness and regularity of backward stochastic Volterra integral equations. Probab. Theory Related Fields 142 (2008) 2–77.
[45] , Backward Stochastic Differential Equations; From Linear to Fully Nonlinear Theory. Springer (2017).
Cité par Sources :
The author was supported by JSPS KAKENHI Grant Number JP21J00460.
