This article is addressed to giving a solution to a unsolved problem, i.e., to establish the necessary optimality conditions of Pontraygin’s type for controlled stochastic Volterra integral equations (SVIEs) when the control region is non-convex and the control variable enters into the diffusion. This problem has been open since [J. Yong, Stochastic Process Appl. 116 (2006) 779–795] obtained the analogue result for the case of convex control region. The key is to introduce a pair of suitable second-order adjoint processes (SOAPs). It is found that the usual way of using only one SOAP in the maximum condition for the classical setting of controlled stochastic differential equations does not work here.
Mots-clés : Stochastic Volterra integral equations, maximum principles, second-order adjoint processes, non-convex control region
@article{COCV_2020__26_1_A16_0,
author = {Wang, Tianxiao},
title = {Necessary conditions of {Pontraygin{\textquoteright}s} type for general controlled stochastic {Volterra} integral equations},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
publisher = {EDP-Sciences},
volume = {26},
year = {2020},
doi = {10.1051/cocv/2020001},
mrnumber = {4064477},
zbl = {1441.93349},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2020001/}
}
TY - JOUR AU - Wang, Tianxiao TI - Necessary conditions of Pontraygin’s type for general controlled stochastic Volterra integral equations JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2020 VL - 26 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2020001/ DO - 10.1051/cocv/2020001 LA - en ID - COCV_2020__26_1_A16_0 ER -
%0 Journal Article %A Wang, Tianxiao %T Necessary conditions of Pontraygin’s type for general controlled stochastic Volterra integral equations %J ESAIM: Control, Optimisation and Calculus of Variations %D 2020 %V 26 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2020001/ %R 10.1051/cocv/2020001 %G en %F COCV_2020__26_1_A16_0
Wang, Tianxiao. Necessary conditions of Pontraygin’s type for general controlled stochastic Volterra integral equations. ESAIM: Control, Optimisation and Calculus of Variations, Tome 26 (2020), article no. 16. doi : 10.1051/cocv/2020001. http://www.numdam.org/articles/10.1051/cocv/2020001/
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This work is supported in part by NSF of China (Grant 11401404, 11971332, 11931011) and the Fundamental Research Funds for the central Universities (YJ201605).
