In this paper, we investigate two families of fully coupled linear Forward-Backward Stochastic Differential Equations (FBSDEs) and its applications to optimal Linear Quadratic (LQ) problems. Within these families, one could get same well-posedness of FBSDEs with totally different coefficients. A family of FBSDEs is proved to be equivalent with respect to the Unified Approach. Thus one could get well-posedness of whole family once a member exists a unique solution. Another equivalent family of FBSDEs are investigated by introducing a linear transformation method. Owing to the coupling structure between forward and backward equations, it leads to a highly interdependence in solutions. We are able to decouple FBSDEs into partial coupling, by virtue of linear transformation, without losing the existence and uniqueness to solutions. Moreover, owing to non-degeneracy of transformation matrix, the solution to original FBSDEs is totally determined by solutions of FBSDEs after transformation. In addition, an example of optimal LQ problem is presented to illustrate.
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DOI : 10.1051/cocv/2022073
Keywords: Forward-backward stochastic differential equations, well-posedness, equivalent family, linear transformation, linear quadratic problem
@article{COCV_2022__28_1_A82_0,
author = {Liu, Ruyi and Wu, Zhen and Zhang, Detao},
title = {Two {Equivalent} {Families} of {Linear} {Fully} {Coupled} {Forward} {Backward} {Stochastic} {Differential} {Equations}},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2022},
publisher = {EDP-Sciences},
volume = {28},
doi = {10.1051/cocv/2022073},
mrnumber = {4525176},
zbl = {1511.60088},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2022073/}
}
TY - JOUR AU - Liu, Ruyi AU - Wu, Zhen AU - Zhang, Detao TI - Two Equivalent Families of Linear Fully Coupled Forward Backward Stochastic Differential Equations JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2022 VL - 28 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2022073/ DO - 10.1051/cocv/2022073 LA - en ID - COCV_2022__28_1_A82_0 ER -
%0 Journal Article %A Liu, Ruyi %A Wu, Zhen %A Zhang, Detao %T Two Equivalent Families of Linear Fully Coupled Forward Backward Stochastic Differential Equations %J ESAIM: Control, Optimisation and Calculus of Variations %D 2022 %V 28 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/cocv/2022073/ %R 10.1051/cocv/2022073 %G en %F COCV_2022__28_1_A82_0
Liu, Ruyi; Wu, Zhen; Zhang, Detao. Two Equivalent Families of Linear Fully Coupled Forward Backward Stochastic Differential Equations. ESAIM: Control, Optimisation and Calculus of Variations, Tome 28 (2022), article no. 82. doi: 10.1051/cocv/2022073
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