Mean-Field Type FBSDEs under Domination-Monotonicity Conditions and Application to LQ Problems

Tian, Ran; Yu, Zhiyong

doi:10.1137/21m144219x

Cited by 5 publications

(1 citation statement)

References 26 publications

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“…Furthermore, these domination-monotonicity conditions precisely correspond to four types of LQ problems (see Table 1 in [52] and Remark 4.9 in [43]). Due to the better applicability of such conditions, this framework has been applied to numerous notable literature including [25,41,42,43]. One piece of literature that needs to be singled out here is Wei and Yu [43] where they further modified the domination-monotonicity conditions to a new infinite horizon version.…”

Section: Introductionmentioning

confidence: 99%

Infinite Horizon Forward-Backward Stochastic Evolution Equations in Hilbert Space and Application to LQ Optimal Control Problems

Xu,

Tang,

Meng

2023

Preprint

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This paper focuses on the study of infinite horizon forward-backward stochastic evolution equations (FBSEEs) in Hilbert spaces. By imposing a restriction on the unbounded operators A and B, that is, both operators are maximal dissipative, the Itô inequality can be utilized to address the problem that Itô's formula is not directly applicable. Based on this, a pair of priori estimates for the solution to (forward) stochastic evolution equations (SEEs) and backward stochastic evolution equations (BSEEs), respectively are established. In the study of infinite horizon FBSEEs, for wider applicability, we apply a set of domination-monotonicity conditions that are more relaxed compared to general conditions. Within this framework, we employ the method of continuation to establish the unique solvability result and provide a pair of solution estimates. Notably, to address the challenges posed by the infinite-dimensional setting, we employ an approximation technique to obtain the dual relationship between x and y. Actually, these results find application in various linear- quadratic (LQ) problems, where the stochastic Hamiltonian systems precisely correspond to the FBSEEs satisfying the aforementioned domination-monotonicity conditions under certain assumptions. Consequently, by solving the corresponding stochastic Hamiltonian systems, we can obtain explicit expressions for the unique optimal controls. Finally, due to the features of the stochastic evolution systems, two specific examples are given to further demonstrate the application of our results.

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