A maximum principle for fully coupled controlled forward–backward stochastic difference systems of mean-field type

Song, Teng; Liu, Bin

doi:10.1186/s13662-020-02640-x

Cited by 6 publications

(9 citation statements)

References 25 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Many results have been done on this topic for different kinds of continuoustime stochastic optimal control problems, for example [9,10,14,17,25,[27][28][29], and discrete-time stochastic optimal control problems, see [11,13,16,[18][19][20][21][22]26] and the references therein). The main difficulty of the stochastic maximum principle for an optimal control problem governed by continuous-time stochastic Itô equations is that the stochastic Itô integral is only of order ε ("hidden convexity" fails).…”

Section: Introductionmentioning

confidence: 99%

“…Song and Liu considered in [10] the optimal control problem for fully cou-pled forward-backward stochastic difference equations of mean-field type under weak convexity assumption. Note that the form of (3.6) as an adjoint equation which was introduced in [10] is one kind of backward stochastic difference equation. This adjoint equation is quite different from our adjoint equation (4) studied in this paper.…”

Section: Introductionmentioning

confidence: 99%

“…In [26], Wu and Zhang studied recently discrete-time stochastic optimal control problem with convex control domains, for which necessary condition in the form of Pontryagin's maximum principle and sufficient condition of optimality are derived. They also pointed out that how to overcome of integrability problem of the solution to the adjoint equation which was not taken into consideration in [10].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Stochastic maximum principle for discrete time mean-field optimal control problems

Ahmadova

Mahmudov

2023

Preprint

View full text Add to dashboard Cite

In this paper, we study the optimal control of a discrete-time stochastic differential equation (SDE) of mean-field type, where the coefficients can depend on both a function of the law and the state of the process. We establish a new version of the maximum principle for discrete-time mean-field type stochastic optimal control problems. Moreover, the cost functional is also of the mean-field type. This maximum principle differs from the classical principle one since we introduce new discrete-time mean-field backward (matrix) stochastic equations. Based on the discrete-time mean-field backward stochastic equations where the adjoint equations turn out to be discrete backward SDEs with mean field, we obtain necessary first-order and sufficient optimality conditions for the stochastic discrete mean-field optimal control problem. To verify, we apply the result to production and consumption choice optimization problem.

show abstract

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Stochastic maximum principle for discrete time mean-field optimal control problems

Ahmadova

Mahmudov

2023

Preprint

View full text Add to dashboard Cite

show abstract

“…A major technical challenge of such problems arises from the dependence of the (forward) diffusion term on the BSDE and the presence of jump diffusions. Previously, for such problems, only stochastic maximum principles have been established, which in general constitute only the necessary conditions for optimality in terms of variational inequalities (see References 34,39‐51 and the references therein). The notable result of the stochastic maximum principle for control of FBSDEs was obtained in Reference 41.…”

Section: Introductionmentioning

confidence: 99%

“…Hence, it is practically not implementable. The results of the work 41 were extended to the case of jump‐diffusion models, 47 the risk‐sensitive framework, 34 the problem with state constraints, 44 and the mean‐field type problems 42,48 . Moreover, when the FBSDE is not fully coupled, the stochastic maximum principle was also obtained in References 34,45.…”

Section: Introductionmentioning

confidence: 99%

A sufficient condition for optimal control problem of fully coupled forward‐backward stochastic systems with jumps: A state‐constrained control approach

Yang

Moon

2022

Optim Control Appl Methods

View full text Add to dashboard Cite

We study the stochastic optimal control problem for fully coupled forward-backward stochastic differential equations (FBSDEs) with jump diffusions. A major technical challenge of such problems arises from the dependence of the (forward) diffusion term on the backward SDE and the presence of jump diffusions. Previously, this class of problems has been solved via only the stochastic maximum principle, which guarantees only the necessary condition of optimality and requires identifying unknown parameters in the corresponding variational inequality. Our paper provides an alternative approach, which constitutes the sufficient condition for optimality. Specifically, the original fully coupled FBSDE control problem (referred to as (P)) is converted into the terminal state-constrained forward stochastic control problem (referred to as (P ′ )) that includes additional (possibly unbounded) control variables. Then (P ′ ) is solved via the backward reachability analysis, by which the value function of (P ′ ) is expressed as the zero-level set of the value function for the auxiliary unconstrained (forward) control problem (referred to as (P ′′ )). Unlike (P ′ ), (P ′′ ) is an unconstrained problem, which includes additional control variables as a consequence of the martingale representation theorem. We show that the value function for (P ′′ ) is the unique viscosity solution to the associated integro-type Hamilton-Jacobi-Bellman (HJB) equation. The viscosity solution analysis presented in our paper requires a new technique due to additional control variables in the Hamiltonian maximization and the presence of the nonlocal integral operator in terms of the (singular) Lévy measure. To solve the original problem (P), we reverse our approach. Specifically, we first solve (P ′′ ) to obtain the value function using the verification theorem and the viscosity solution of the HJB equation. Then (P ′ ) is solved by characterizing the zero-level set of the value function of (P ′′ ), from which the optimal solution of (P) can be constructed. To illustrate the theoretical results of this paper, applications to the linear-quadratic problem for fully coupled FBSDEs with jumps are also presented.

show abstract

Maximum principle for discrete-time stochastic control problem of mean-field type

Dong

Nie²,

Wu³

2022

Automatica

View full text Add to dashboard Cite

A maximum principle for fully coupled controlled forward–backward stochastic difference systems of mean-field type

Cited by 6 publications

References 25 publications

Stochastic maximum principle for discrete time mean-field optimal control problems

Stochastic maximum principle for discrete time mean-field optimal control problems

A sufficient condition for optimal control problem of fully coupled forward‐backward stochastic systems with jumps: A state‐constrained control approach

Maximum principle for discrete-time stochastic control problem of mean-field type

Contact Info

Product

Resources

About