Stochastic maximum principle for optimal control problems involving delayed systems
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. LETTER .
January 2021, Vol. 64 119206:1–119206:3 https://doi.org/10.1007/s11432-019-2826-3
Stochastic maximum principle for optimal control problems involving delayed systems Feng ZHANG School of Mathematics and Quantitative Economics, Shandong University of Finance and Economics, Jinan 250014, China Received 17 November 2019/Accepted 6 February 2020/Published online 24 November 2020
Citation Zhang F. Stochastic maximum principle for optimal control problems involving delayed systems. Sci China Inf Sci, 2021, 64(1): 119206, https://doi.org/10.1007/s11432-019-2826-3
Dear editor, The main objective of this study is to investigate one type of stochastic optimal control problem for a delayed system using the maximum principle method. The existing research can be categorized into two categories. In the first category, the adjoint equation comprises two backward stochastic differential equations (BSDEs) and one backward ordinary differential equation (BODE), which is assumed to have a zero solution; see e.g., [1–3]. However, there is still scope for improvement in this research direction. Here, a special setting is considered for the system parameters; see e.g., (3.9) and (3.12) in [2]. The BODE is assumed to result in a zero solution when the necessary maximum principle is strictly applied; see e.g., Theorem 5.1 in [3]. In the second category, a type of anticipated/time-advanced BSDE is introduced as the adjoint equation; see e.g., [1, 4–7]. In this study, (i) the averaged and point-wise time delays of the state and control processes are observed to be involved with the system equation and the cost function, ensuring that the problem that is being studied is in accordance with the general framework. (ii) Further, the necessary and sufficient maximum principles are clearly established, proving that the sufficient maximum principle conforms to some slightly relaxed conditions. (iii) Next, two types of adjoint equations are introduced, among which the first type contains an anticipated BSDE (see (3)) and a BSDE (see (4)), which can be expressed using one Hamiltonian H. The other type of adjoint equation comprises one time-advanced BSDE (see (5)) and one BSDE (see (6)), both of which can be expressed using another Hamiltonian H. BODE is not needed as part of the adjoint equation. Furthermore, to the best of the author’s knowledge, the two aforementioned types of adjoint equations are proved to be equivalent for the very first time. Thus, a unified adjoint equation can be obtained (see (2)). Formulation. Here, let (Ω, F , P) be a complete probability space, where the expectation is denoted by E[·]. Let W (t), t > 0 be a one-dimensional standard Brownian motion, the augmented filtration of which is F = {Ft , t > 0}. Let Et [·] = E[·|Ft ] for each value of t. Fur-
ther, assume that T is a finite time horizon, λ1 , λ2 are constants and δ1 , δ2 , δ3 are positive constants, such that max{δ1 , δ2 , δ3 } < T . Let L2F (0, T ; Rn ) denote the class of Rn -valued progressively measurable processes that are square integrable an
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