Generalized Hamilton--Jacobi--Bellman Equations with Dirichlet Boundary Condition and Stochastic Exit Time Optimal Control Problem

Buckdahn, Rainer; Nie, Tianyang

doi:10.1137/140998160

Cited by 24 publications

(27 citation statements)

References 36 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Theorem 5.1 Suppose that assumption (H 1 ) and (H 2 ) hold. Then there exists a δ 0 > 0, which depends on k i , λ 1 , λ 2 , T , for i = 1, 4, 5, 6, 7, 8, 11, 12 such that when k 2 , k 3 , k 9 , 4,5,6,7,8,11,12 and is independent of T , such that when k 2 , k 3 , k 9 , k 10 ∈ [0, δ 1 ), there exists a unique adapted solution (X, Y, Z) to MF-FBSDE (29). Remark 5.1), by repeating the above discussion, one can show that if 2(λ 1 + λ 2 ) < −2 k 1 − k 2 6 − k 2 7 − k 2 8 , there exists a δ 1 > 0, which depends on k 1 , k i , λ 1 , λ 2 , for i = 4, 5, 6, 7, 8, 11, 12 and is independent of T , such that when k 2 , k 3 , k 9 , k 10 ∈ [0, δ 1 ), there exists a unique adapted solution (X, Y, Z) to MF-FBSDE (29).…”

Section: Existence and Uniqueness Of Consistency Condition Systemglobmentioning

confidence: 99%

See 1 more Smart Citation

Linear-Quadratic-Gaussian Mixed Mean-Field Games with Heterogeneous Input Constraints

Hu¹,

Huang²,

Nie³

2018

SIAM J. Control Optim.

Self Cite

View full text Add to dashboard Cite

We consider a class of linear-quadratic-Gaussian mean-field games with a major agent and considerable heterogeneous minor agents in the presence of mean-field interactions. The individual admissible controls are constrained in closed convex subsets Γ k of R m . The decentralized strategies for individual agents and consistency condition system are represented in an unified manner through a class of mean-field forward-backward stochastic differential equations involving projection operators on Γ k . The well-posedness of consistency system is established in both the local and global cases by the contraction mapping and discounting method respectively. Related ε−Nash equilibrium property is also verified.

show abstract

Section: Existence and Uniqueness Of Consistency Condition Systemglobmentioning

confidence: 99%

“…Thus, the proof for the major agent is completed by noticing (41). Let us now focus on the minor agents, for 1 ≤ i ≤ N , recalling (4), (12) and (17), we have…”

Section: ε-Nash Equilibrium For Problem (Cc)mentioning

confidence: 99%

Linear-Quadratic-Gaussian Mixed Mean-Field Games with Heterogeneous Input Constraints

Hu¹,

Huang²,

Nie³

2018

SIAM J. Control Optim.

Self Cite

View full text Add to dashboard Cite

show abstract

“…(ii) The representation theorem for generators of BSDEs can also be applied to prove the probabilistic interpretation for semilinear (or quasilinear) second order PDEs of both elliptic and parabolic types, just omit the control process v(·) in (6). So the representation theorem method can be regarded as a unified approach to the probabilistic interpretation for semilinear, quasilinear and HJB type PDEs.…”

Section: Probabilistic Interpretation For Hjb Equationsmentioning

confidence: 99%

“…Since then, many researchers began to investigate stochastic recursive optimal control problem induced by FBSDE systems. Buckdahn and Li [5] studied zero-sum two-player stochastic differential games via FBSDEs; recently, Buckdahn and Nie [6] considered a stochastic exit time optimal control problem.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Probabilistic interpretation of HJB equations by the representation theorem for generators of BSDEs

Xiao¹,

Fan²,

Tian³

2020

Electron. Commun. Probab.

View full text Add to dashboard Cite

The purpose of this note is to propose a new approach for the probabilistic interpretation of Hamilton-Jacobi-Bellman equations associated with stochastic recursive optimal control problems, utilizing the representation theorem for generators of backward stochastic differential equations. The key idea of our approach for proving this interpretation consists of transmitting the signs between the solution and generator via the identity given by representation theorem. Compared with existing methods, our approach seems to be more applicable for general settings. This can also be regarded as a new application of such representation theorem.

show abstract

A Neural Network-Based Policy Iteration Algorithm with Global $$H^2$$-Superlinear Convergence for Stochastic Games on Domains

Reisinger

Zhang

2020

Found Comput Math

View full text Add to dashboard Cite

In this work, we propose a class of numerical schemes for solving semilinear Hamilton-Jacobi-Bellman-Isaacs (HJBI) boundary value problems which arise naturally from exit time problems of diffusion processes with controlled drift. We exploit policy iteration to reduce the semilinear problem into a sequence of linear Dirichlet problems, which are subsequently approximated by a multilayer feedforward neural network ansatz. We establish that the numerical solutions converge globally in the H 2norm and further demonstrate that this convergence is superlinear, by interpreting the algorithm as an inexact Newton iteration for the HJBI equation. Moreover, we construct the optimal feedback controls from the numerical value functions and deduce convergence. The numerical schemes and convergence results are then extended to oblique derivative boundary conditions. Numerical experiments on the stochastic Zermelo navigation problem are presented to illustrate the theoretical results and to demonstrate the effectiveness of the method.

show abstract

Generalized Hamilton--Jacobi--Bellman Equations with Dirichlet Boundary Condition and Stochastic Exit Time Optimal Control Problem

Cited by 24 publications

References 36 publications

Linear-Quadratic-Gaussian Mixed Mean-Field Games with Heterogeneous Input Constraints

Linear-Quadratic-Gaussian Mixed Mean-Field Games with Heterogeneous Input Constraints

Probabilistic interpretation of HJB equations by the representation theorem for generators of BSDEs

A Neural Network-Based Policy Iteration Algorithm with Global $$H^2$$-Superlinear Convergence for Stochastic Games on Domains

Contact Info

Product

Resources

About