Stochastic recursive optimal control problem with obstacle constraint involving diffusion type control

Xu, Zhenda

doi:10.1186/s13662-020-02844-1

Cited by 3 publications

(2 citation statements)

References 29 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In fact, by the recursivetype stochastic differential game, we are able to consider the general dynamic structure of the objective functional. For example, the wealth process of investors in mathematical finance, the utility maximization model in economics, and the (continuous-time) principalagent problem in economics can be formulated using the framework of recursive-type BSDE objective functionals, which describe the general dynamic behavior of the investors (agents); see [44,[47][48][49][50] and the references therein. (ii) Note that by (4), the objective functional of the SZSDG depends on the state and control path of the players.…”

Section: Remark 3 (I)mentioning

confidence: 99%

State and Control Path-Dependent Stochastic Zero-Sum Differential Games: Viscosity Solutions of Path-Dependent Hamilton–Jacobi–Isaacs Equations

Moon¹

2022

Mathematics

View full text Add to dashboard Cite

In this paper, we consider the two-player state and control path-dependent stochastic zero-sum differential game. In our problem setup, the state process, which is controlled by the players, is dependent on (current and past) paths of state and control processes of the players. Furthermore, the running cost of the objective functional depends on both state and control paths of the players. We use the notion of non-anticipative strategies to define lower and upper value functionals of the game, where unlike the existing literature, these value functions are dependent on the initial states and control paths of the players. In the first main result of this paper, we prove that the (lower and upper) value functionals satisfy the dynamic programming principle (DPP), for which unlike the existing literature, the Skorohod metric is necessary to maintain the separability of càdlàg (state and control) spaces. We introduce the lower and upper Hamilton–Jacobi–Isaacs (HJI) equations from the DPP, which correspond to the state and control path-dependent nonlinear second-order partial differential equations. In the second main result of this paper, we show that by using the functional Itô calculus, the lower and upper value functionals are viscosity solutions of (lower and upper) state and control path-dependent HJI equations, where the notion of viscosity solutions is defined on a compact κ-Hölder space to use several important estimates and to guarantee the existence of minimum and maximum points between the (lower and upper) value functionals and the test functions. Based on these two main results, we also show that the Isaacs condition and the uniqueness of viscosity solutions imply the existence of the game value. Finally, we prove the uniqueness of classical solutions for the (state path-dependent) HJI equations in the state path-dependent case, where its proof requires establishing an equivalent classical solution structure as well as an appropriate contradiction argument.

show abstract

Section: Remark 3 (I)mentioning

confidence: 99%

State and Control Path-Dependent Stochastic Zero-Sum Differential Games: Viscosity Solutions of Path-Dependent Hamilton–Jacobi–Isaacs Equations

Moon¹

2022

Mathematics

View full text Add to dashboard Cite

show abstract

“…Stochastic and fuzzy analysis methods are usually used to describe the imprecise or uncertain parameters in the study of uncertain optimal control problems. In this sense, there exist many works in the literature which deal with different theoretical and applied aspects of optimality conditions for stochastic optimal control [12,24,28,29] and fuzzy optimal control [9,10,26]. Specially when, due to the uncertainty or lack of knowledge around data, it is difficult for experts to provide an exact probability distribution or fuzzy membership function, stochastic optimal control and fuzzy optimal control are transformed into interval-valued optimal control.…”

Section: Introductionmentioning

confidence: 99%

Sufficient conditions for the optimal control problems with interval-valued objective function in admissible orders

Zhang

2023

Preprint

View full text Add to dashboard Cite

This paper addresses the optimal control problems with interval-valued objective function. We consider a kind of total order relationships ≤adm between two intervals. For each total order relationship, Kuhn-Tucker sufficient conditions for the optimal control problems with interval-valued objective function are obtained. And numerical example has been considered and solved. Kuhn-Tucker sufficient conditions given under total order relationship ≤adm are sufficient conditions under partial order relationship ≼LU for interval-valued optimal control problems. The results of this paper can be used to construct evolutionary algorithms for solving interval-valued optimal control problems.

show abstract

Sufficient conditions for interval-valued optimal control problems in admissible orders

Li,

Zhang

2024

Soft Comput

View full text Add to dashboard Cite

Stochastic recursive optimal control problem with obstacle constraint involving diffusion type control

Cited by 3 publications

References 29 publications

State and Control Path-Dependent Stochastic Zero-Sum Differential Games: Viscosity Solutions of Path-Dependent Hamilton–Jacobi–Isaacs Equations

State and Control Path-Dependent Stochastic Zero-Sum Differential Games: Viscosity Solutions of Path-Dependent Hamilton–Jacobi–Isaacs Equations

Sufficient conditions for the optimal control problems with interval-valued objective function in admissible orders

Sufficient conditions for interval-valued optimal control problems in admissible orders

Contact Info

Product

Resources

About