An optimal control problem is considered for a stochastic differential equation with the cost functional determined by a backward stochastic Volterra integral equation (BSVIE, for short). This kind of cost functional can cover the general discounting (including exponential and non-exponential) situations with a recursive feature. It is known that such a problem is time-inconsistent in general. Therefore, instead of finding a global optimal control, we look for a time-consistent locally near optimal equilibrium strategy. With the idea of multi-person differential games, a family of approximate equilibrium strategies is constructed associated with partitions of the time intervals. By sending the mesh size of the time interval partition to zero, an equilibrium Hamilton–Jacobi–Bellman (HJB, for short) equation is derived, through which the equilibrium value function and an equilibrium strategy are obtained. Under certain conditions, a verification theorem is proved and the well-posedness of the equilibrium HJB is established. As a sort of Feynman–Kac formula for the equilibrium HJB equation, a new class of BSVIEs (containing the diagonal value Z(r, r) of Z(⋅ , ⋅)) is naturally introduced and the well-posedness of such kind of equations is briefly presented.
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Keywords: Time-inconsistent optimal control problem, backward stochastic Volterra integral equation, stochastic differential games, equilibrium strategy, equilibrium Hamilton–Jacobi–Bellman equation
@article{COCV_2021__27_1_A24_0,
author = {Wang, Hanxiao and Yong, Jiongmin},
editor = {Buttazzo, G. and Casas, E. and de Teresa, L. and Glowinski, R. and Leugering, G. and Tr\'elat, E. and Zhang, X.},
title = {Time-inconsistent stochastic optimal control problems and backward stochastic volterra integral equations},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2021},
publisher = {EDP-Sciences},
volume = {27},
doi = {10.1051/cocv/2021027},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2021027/}
}
TY - JOUR AU - Wang, Hanxiao AU - Yong, Jiongmin ED - Buttazzo, G. ED - Casas, E. ED - de Teresa, L. ED - Glowinski, R. ED - Leugering, G. ED - Trélat, E. ED - Zhang, X. TI - Time-inconsistent stochastic optimal control problems and backward stochastic volterra integral equations JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2021 VL - 27 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2021027/ DO - 10.1051/cocv/2021027 LA - en ID - COCV_2021__27_1_A24_0 ER -
%0 Journal Article %A Wang, Hanxiao %A Yong, Jiongmin %E Buttazzo, G. %E Casas, E. %E de Teresa, L. %E Glowinski, R. %E Leugering, G. %E Trélat, E. %E Zhang, X. %T Time-inconsistent stochastic optimal control problems and backward stochastic volterra integral equations %J ESAIM: Control, Optimisation and Calculus of Variations %D 2021 %V 27 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/cocv/2021027/ %R 10.1051/cocv/2021027 %G en %F COCV_2021__27_1_A24_0
Wang, Hanxiao; Yong, Jiongmin. Time-inconsistent stochastic optimal control problems and backward stochastic volterra integral equations. ESAIM: Control, Optimisation and Calculus of Variations, Tome 27 (2021), article no. 22. doi: 10.1051/cocv/2021027
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