A global stochastic maximum principle for fully coupled forward-backward stochastic systems

Hu, Mingshang; Ji, Shaolin; Xue, Xiaole

doi:10.48550/arxiv.1803.02109

Cited by 3 publications

(15 citation statements)

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“…By Assumption 2.1, b i , σ i , g i and φ 1 are bounded for i = 1, 2, 3. Then (3.5) holds by Theorem 2.2 in [7].…”

Section: The Dynamic Programming Principlementioning

confidence: 84%

“…It is easy to check that (5.2) satisfies the assumptions of Theorem 2.2 in [7]. Consequently, it has a unique…”

Section: The Smooth Casementioning

confidence: 88%

“…Since |ψ(s, 0, 0, 0, u)| ≤ L for s ∈ [t, T ] and u ∈ U where ψ = b, σ, g, by Theorem 2.2 in [7], we obtain…”

Section: The Dynamic Programming Principlementioning

confidence: 94%

“…is the Lipschitz constant of value function W with respect to x, C 2 and C 4 are defined in Lemma 5.1 in [7].…”

Section: The Value Function and The Hjb Equationmentioning

confidence: 99%

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The existence and uniqueness of viscosity solution to a kind of Hamilton-Jacobi-Bellman equations

Hu,

Ji,

Xue

2018

Preprint

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In this paper, we study the existence and uniqueness of viscosity solutions to generalized Hamilton-Jacobi-Bellman (HJB) equations combined with algebra equations. This generalized HJB equation is related to a stochastic optimal control problem for which the state equation is described by a fully coupled forward-backward stochastic differential equation (FBSDE). By extending Peng's backward semigroup approach to this problem, we obtain the dynamic programming principle (DPP) and show that the value function is a viscosity solution to this generalized HJB equation. As for the proof of the uniqueness of viscosity solution, the analysis method in Barles, Buckdahn and Pardoux [2] usually does not work for this fully coupled case. With the help of the uniqueness of the solution to FBSDEs, we propose a novel probabilistic approach to study the uniqueness of the solution to this generalized HJB equation. We obtain that the value function is the minimum viscosity solution to this generalized HJB equation. Especially, when the coefficients are independent of the control variable or the solution is smooth, the value function is the unique viscosity solution.

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“…By Assumption 2.1, b i , σ i , g i and φ 1 are bounded for i = 1, 2, 3. Then (3.5) holds by Theorem 2.2 in [7].…”

Section: The Dynamic Programming Principlementioning

confidence: 84%

“…It is easy to check that (5.2) satisfies the assumptions of Theorem 2.2 in [7]. Consequently, it has a unique…”

Section: The Smooth Casementioning

confidence: 88%

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The existence and uniqueness of viscosity solution to a kind of Hamilton-Jacobi-Bellman equations

Hu,

Ji,

Xue

2018

Preprint

Self Cite

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“…Besides, in the dynamic principal-agent problem with unobservable states and actions, the principal's problem can be formulated as a partial information optimal control problem of a FBSCS (see [24]). We refer to [7], [10], [11], [14], [22], [26], [28] for other works on optimization problems for FBSCSs.…”

Section: Introductionmentioning

confidence: 99%

Maximum principle for stochastic optimal control problem of finite state forward-backward stochastic difference systems

Ji¹,

Liu²

2019

Preprint

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In this paper, we study the maximum principle for stochastic optimal control problems of forward-backward stochastic difference systems (FBS∆Ss) where the uncertainty is modeled by a discrete time, finite state process, rather than white noises. Two types of FBS∆Ss are investigated. The first one is described by a partially coupled forward-backward stochastic difference equation (FBS∆E) and the second one is described by a fully coupled FBS∆E. By adopting an appropriate representation of the product rule and an appropriate formulation of the backward stochastic difference equation (BS∆E), we deduce the adjoint difference equation. Finally, the maximum principle for this optimal control problem with the control domain being convex is established.

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Stochastic maximum principle, dynamic programming principle, and their relationship for fully coupled forward-backward stochastic control systems

Hu¹,

Ji²,

Xue³

2018

Preprint

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Within the framework of viscosity solution, we study the relationship between the maximum principle (MP) in [9] and the dynamic programming principle (DPP) in [10] for a fully coupled forwardbackward 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. So this relationship becomes more complicated and almost no work involves this issue. 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 and second-order adjoint equations. Finally, we study the local case under the monotonicity conditions as in [14,27] and obtain the relationship between the MP in [27] and the DPP in [14].

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A global stochastic maximum principle for fully coupled forward-backward stochastic systems

Cited by 3 publications

References 0 publications

The existence and uniqueness of viscosity solution to a kind of Hamilton-Jacobi-Bellman equations

The existence and uniqueness of viscosity solution to a kind of Hamilton-Jacobi-Bellman equations

Maximum principle for stochastic optimal control problem of finite state forward-backward stochastic difference systems

Stochastic maximum principle, dynamic programming principle, and their relationship for fully coupled forward-backward stochastic control systems

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