Volume 26, 2020
|Number of page(s)||36|
|Published online||21 October 2020|
Stochastic maximum principle, dynamic programming principle, and their relationship for fully coupled forward-backward stochastic controlled systems*
Zhongtai Securities Institute for Financial Studies, Shandong University,
Shandong 250100, P.R. China.
2 School of Management and Zhongtai Securities Institute for Financial Studies, Shandong University, Jinan 250100, P.R. China.
** Corresponding author: firstname.lastname@example.org
Accepted: 22 July 2020
Within the framework of viscosity solution, we study the relationship between the maximum principle (MP) from M. Hu, S. Ji and X. Xue [SIAM J. Control Optim. 56 (2018) 4309–4335] and the dynamic programming principle (DPP) from M. Hu, S. Ji and X. Xue [SIAM J. Control Optim. 57 (2019) 3911–3938] for a fully coupled forward–backward stochastic controlled system (FBSCS) with a nonconvex control domain. For a fully coupled FBSCS, both the corresponding MP and the corresponding Hamilton–Jacobi–Bellman (HJB) equation combine an algebra equation respectively. With the help of a new decoupling technique, we obtain the desirable estimates for the fully coupled forward–backward variational equations and establish the relationship. Furthermore, for the smooth case, we discover the connection between the derivatives of the solution to the algebra equation and some terms in the first-order and second-order adjoint equations. Finally, we study the local case under the monotonicity conditions as from J. Li and Q. Wei [SIAM J. Control Optim. 52 (2014) 1622–1662] and Z. Wu [Syst. Sci. Math. Sci. 11 (1998) 249–259], and obtain the relationship between the MP from Z. Wu [Syst. Sci. Math. Sci. 11 (1998) 249–259] and the DPP from J. Li and Q. Wei [SIAM J. Control Optim. 52 (2014) 1622–1662].
Mathematics Subject Classification: 93E20 / 60H10 / 35K15
Key words: Fully coupled forward–backward stochastic differential equations / global stochastic maximum principle / dynamic programming principle / viscosity solution / monotonicity condition
The first author's research supported by NSF (No. 11671231) and Young Scholars Program of Shandong University (No. 2016WLJH10). The second author's research supported by the National Key R&D Program of China (No. 2018YFA0703900) and NSF (Nos. 11971263 and 11871458). The third author's research supported by “The Fundamental Research Funds of Shandong University”, NSF (No. 61907022) and Natural Science Foundation of Shandong Province(ZR2019BF015).
© EDP Sciences, SMAI 2020
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