Stochastic global maximum principle for optimization with recursive utilities

The necessary conditions for an optimal control of a stochastic control problem with recursive utilities is investigated. The first order condition is the the well-known Pontryagin type maximum principle. When the optimal control satisfying such first-order necessary condition is singular in some sense, certain type of the second-order necessary condition will come in naturally. The aim of this paper is to explore such kind of conditions for our optimal control problem.

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“…For future purposes, we recall the standard estmates of BSDEs (see [10] and the refereneces therein). Lemma 1.…”

Section: Basic Assumptionsmentioning

confidence: 99%

Second-Order Necessary Conditions for Optimal Control with Recursive Utilities

Dong

Meng

2019

J Optim Theory Appl

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“…Proof. The proof can be directly obtained by combining Proposition 2.1 in [15] and Lemma 2 in [7]. Now we state our main result, which provides the necessary condition for a control to be near-optimal with order ε 1 2 .…”

Section: Resultsmentioning

confidence: 81%

“…Note the adjoint equation (11) is a forward-backward stochastic differential equation whose solution consists of an 7-tuple process (k(·), p(·), q 1 (·), q 2 (·), r(·), R 1 (·), R 2 (·)). Under Assumptions 2.1 and 2.2, by Proposition 2.1 in [15] and Lemma 2 in [7] , it is easily to see that the adjoint equation (11) admits a unique solution (k(·), p(·), q 1 (·), q 2 (·), r(·), R 1 (·), R 2 (·)), also called the adjoint process corresponding the admissible pair (u(·); x(·), y(·), z 1 (·), z 2 (·), ρ(·)). Particularly, we write (k u (·), p u (·), q u 1 (·), q u 2 (·), r u (·), R u 1 (·), R u 2 (·)) for the adjoint processes associated with any admissible pair (u(·); x u (·), y u (·), z u 1 (·), z u 2 (·), ρ u (·)), whenever we want to emphasize the dependence of (k(·), p(·), q 1 (·), q 2 (·), r(·), R 1 (·), R 2 (·)).…”

Section: Resultsmentioning

confidence: 97%

Partial Information Near-Optimal Control of Forward-Backward Stochastic Differential System with Observation Noise

Meng

Shi

Tang

2018

2018 37th Chinese Control Conference (CCC)

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This paper first makes an attempt to investigate the partial information near optimal control of systems governed by forward-backward stochastic differential equations with observation noise under the assumption of a convex control domain. By Ekeland's variational principle and some basic estimates for state processes and adjoint processes, we establish the necessary conditions for any ε-near optimal control in a local form with an error order of exact ε 1 2 . Moreover, under additional convexity conditions on Hamiltonian function, we prove that an ε-maximum condition in terms of the Hamiltonian in the integral form is sufficient for near-optimality. 1 3 in [22] to "exactly" ε 1 3 . Meanwhile, there have been growing interests on stochastic optimal control problems under partial information, partly due to the applications in mathematical finance. For the partial information optimal control problem, the objective is to find an optimal control for which the controller has less information than the complete information filtration. In particular, sometimes an economic model in which there are information gaps among economic agents can be formulated as a partial information optimal control problem (see Øksendal [16], Kohlmann and Xiong [10]). Baghery and Øksendal [1] established a maximum principle

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“…There are two main obstacles met when f depending on z nonlinearly (see Yong [26]): What is the second-order variational equation and how to obtain the corresponding second-order adjoint equation? Hence, this open problem was not solved completely until 2017 by Hu [18]. Hu overcome the above two difficulties by building a new second-order Taylor expansion of the variation of Y .…”

Section: Introductionmentioning

confidence: 99%

“…Based on the previous works [18], [15], [16], [4], [5], a natural question is: Is it possible to develop the necessary condition for optimality of the general mean-field forward-backward control systems with delay? We confirmed the question.…”

Section: Introductionmentioning

confidence: 99%

A global maximum principle for optimal control of general mean-field forward-backward stochastic systems with jumps

Hao

Meng

2020

ESAIM: COCV

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The purpose of this paper is to explore the necessary conditions for optimality of meanfield forward-backward delay control systems. A new estimate is proved, which is a powerful tool to deal with the optimal control problems of mean-field type with delay. Different from the classical situation, in our case the first-order adjoint system is an anticipated mean-field backward stochastic differential equation, and the second-order adjoint system is a system of matrix-valued process, not mean-field type. With the help of two adjoint systems, the second-order expansion of the variation of the state Y is proved, and therewith the Peng's stochastic maximum principle. As an illustrative example, we apply our result to the meanfield game in Finance. Although we just investigate the case of one pointwise delay for convenience, but our method is adequate for analysing the case of pointwise delay.

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Stochastic global maximum principle for optimization with recursive utilities

Cited by 73 publications

References 18 publications

Second-Order Necessary Conditions for Optimal Control with Recursive Utilities

Second-Order Necessary Conditions for Optimal Control with Recursive Utilities

Partial Information Near-Optimal Control of Forward-Backward Stochastic Differential System with Observation Noise

A global maximum principle for optimal control of general mean-field forward-backward stochastic systems with jumps

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