This paper is concerned with linear quadratic control problems of stochastic differential equations (SDEs, in short) and stochastic Volterra integral equations (SVIEs, in short). Notice that for stochastic systems, the control weight in the cost functional is allowed to be indefinite. This feature is demonstrated here only by open-loop optimal controls but not limited to closed-loop optimal controls in the literature. As to linear quadratic problem of SDEs, some examples are given to point out the issues left by existing papers, and new characterizations of optimal controls are obtained in different manners. For the study of SVIEs with deterministic coefficients, a class of stochastic Fredholm−Volterra integral equations is introduced to replace conventional forward-backward SVIEs. Eventually, instead of using convex variation, we use spike variation to obtain some additional optimality conditions of linear quadratic problems for SVIEs.

Accepted:

DOI: 10.1051/cocv/2017002

Keywords: Stochastic Volterra integral equations, stochastic Fredholm−Volterra integral equations, stochastic linear quadratic problems, spike variation

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@article{COCV_2018__24_4_1849_0, author = {Wang, Tianxiao}, title = {Linear quadratic control problems of stochastic {Volterra} integral equations}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {1849--1879}, publisher = {EDP-Sciences}, volume = {24}, number = {4}, year = {2018}, doi = {10.1051/cocv/2017002}, zbl = {1415.93296}, mrnumber = {3922428}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv/2017002/} }

TY - JOUR AU - Wang, Tianxiao TI - Linear quadratic control problems of stochastic Volterra integral equations JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2018 SP - 1849 EP - 1879 VL - 24 IS - 4 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2017002/ DO - 10.1051/cocv/2017002 LA - en ID - COCV_2018__24_4_1849_0 ER -

%0 Journal Article %A Wang, Tianxiao %T Linear quadratic control problems of stochastic Volterra integral equations %J ESAIM: Control, Optimisation and Calculus of Variations %D 2018 %P 1849-1879 %V 24 %N 4 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2017002/ %R 10.1051/cocv/2017002 %G en %F COCV_2018__24_4_1849_0

Wang, Tianxiao. Linear quadratic control problems of stochastic Volterra integral equations. ESAIM: Control, Optimisation and Calculus of Variations, Volume 24 (2018) no. 4, pp. 1849-1879. doi : 10.1051/cocv/2017002. http://archive.numdam.org/articles/10.1051/cocv/2017002/

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