RECORD
Stochastic Recursive Optimal Control of McKean-Vlasov Type: A Viscosity Solution Approach
Optimization and Control · Submitted 3 Jul 2023
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We prove that the value function is the unique viscosity solutions of the associated generalized HJB equations in some separable Hilbert space.
The compiled record
The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
Open the compiled record
The abstract, sentence by sentence
In this paper, we study a kind of optimal control problem for forward-backward stochastic differential equations (FBSDEs for short) of McKean--Vlasov type via the dynamic programming principle (DPP for short) motivated by studying the infinite dimensional Hamilton--Jacobi--Bellman (HJB for short) equation derived from the decoupling field of the FBSDEs posed by Carmona and Delarue (\emph{Ann Probab}, 2015, \cite{cd15}). At the beginning, the value function is defined by the solution to the controlled BSDE alluded to earlier. On one hand, we can prove the value function is deterministic function with respect to the initial random variable; On the other hand, we can show that the value function is \emph{law-invariant}, i.e., depending on only via its distribution by virtue of BSDE property. Afterward, utilizing the notion of differentiability with respect to probability measures introduced by P.L. Lions \cite{Lions2012}, we are able to establish a DPP for the value function in the Wasserstein space of probability measures based on the application of BSDE approach, particularly, employing the notion of stochastic \emph{backward semigroups} associated with stochastic optimal control problems and It\^{o} formula along a flow property of the conditional law of the controlled forward state process. We prove that the value function is the unique viscosity solutions of the associated generalized HJB equations in some separable Hilbert space. Finally, as an application, we formulate an optimal control problem for linear stochastic differential equations with quadratic cost functionals of McKean-Vlasov type under nonlinear expectation, $g$-expectation introduced by Peng \cite{Peng04} and derive the optimal feedback control explicitly by means of several groups of Riccati equations.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | In this paper, we study a kind of optimal control problem for forward-backward stochastic differential equations (FBSDEs for short) of McKean--Vlasov type via the dynamic programming principle (DPP for short) motivated by studying the infinite dimensional Hamilton--Jacobi--Bellman (HJB for short) equation derived from the decoupling field of the FBSDEs posed by Carmona and Delarue (\emph{Ann Probab}, 2015, \cite{cd15}). | contribution 0.92 | 0.81 | 0.07 | asserted 0.96 |
| s1 | At the beginning, the value function is defined by the solution to the controlled BSDE alluded to earlier. | method 0.76 | 0.64 | 0.03 | asserted 1.00 |
| s2 | On one hand, we can prove the value function is deterministic function with respect to the initial random variable; On the other hand, we can show that the value function is \emph{law-invariant}, i.e., depending on only via its distribution by virtue of BSDE property. | result 0.97 | 0.91 | 0.06 | asserted 1.00 |
| s3 | Afterward, utilizing the notion of differentiability with respect to probability measures introduced by P.L. | method 0.97 | 0.39 | 0.02 | unclear 0.56 |
| s4 | Lions \cite{Lions2012}, we are able to establish a DPP for the value function in the Wasserstein space of probability measures based on the application of BSDE approach, particularly, employing the notion of stochastic \emph{backward semigroups} associated with stochastic optimal control problems and It\^{o} formula along a flow property of the conditional law of the controlled forward state process. | method 0.88 | 0.91 | 0.10 | asserted 0.99 |
| s5 | We prove that the value function is the unique viscosity solutions of the associated generalized HJB equations in some separable Hilbert space. | result 0.98 | 0.95 | 0.52 | asserted 1.00 |
| s6 | Finally, as an application, we formulate an optimal control problem for linear stochastic differential equations with quadratic cost functionals of McKean-Vlasov type under nonlinear expectation, $g$-expectation introduced by Peng \cite{Peng04} and derive the optimal feedback control explicitly by means of several groups of Riccati equations. | contribution 0.74 | 0.95 | 0.14 | asserted 1.00 |
What was examined, and what was not
- arXiv listing abstract v2
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
- C citations
- R cross-paper relations
The paper
Original abstract
In this paper, we study a kind of optimal control problem for forward-backward stochastic differential equations (FBSDEs for short) of McKean--Vlasov type via the dynamic programming principle (DPP for short) motivated by studying the infinite dimensional Hamilton--Jacobi--Bellman (HJB for short) equation derived from the decoupling field of the FBSDEs posed by Carmona and Delarue (\emph{Ann Probab}, 2015, \cite{cd15}). At the beginning, the value function is defined by the solution to the controlled BSDE alluded to earlier. On one hand, we can prove the value function is deterministic function with respect to the initial random variable; On the other hand, we can show that the value function is \emph{law-invariant}, i.e., depending on only via its distribution by virtue of BSDE property. Afterward, utilizing the notion of differentiability with respect to probability measures introduced by P.L. Lions \cite{Lions2012}, we are able to establish a DPP for the value function in the Wasserstein space of probability measures based on the application of BSDE approach, particularly, employing the notion of stochastic \emph{backward semigroups} associated with stochastic optimal control problems and It\^{o} formula along a flow property of the conditional law of the controlled forward state process. We prove that the value function is the unique viscosity solutions of the associated generalized HJB equations in some separable Hilbert space. Finally, as an application, we formulate an optimal control problem for linear stochastic differential equations with quadratic cost functionals of McKean-Vlasov type under nonlinear expectation, $g$-expectation introduced by Peng \cite{Peng04} and derive the optimal feedback control explicitly by means of several groups of Riccati equations.
Human review
How the literature uses it
What this paper does with the arXiv papers it cites
Discussion
Know something about this paper? Leave a comment, or highlight any passage above to comment on it.