Extended MF-FBSDEs with nonlinear domination-monotonicity conditions and stochastic optimal controls of Linear System with quadruple controls

TL;DR

This paper generalizes the well-posedness of extended mean-field forward-backward stochastic differential equations to nonlinear cases and applies these results to solve stochastic optimal control problems with quadruple controls.

Contribution

It extends domination-monotonicity conditions to nonlinear settings and derives explicit solutions for complex stochastic control problems.

Findings

Proved well-posedness of extended MF-FBSDEs under nonlinear conditions.

Established existence and uniqueness of optimal controls for specific stochastic problems.

Derived explicit closed-form solutions for the optimal controls.

Abstract

This paper extends the domination-monotonicity conditions, which guarantee the well-posedness of extended mean-filed forward-backward stochastic differential equations (extended MF-FBSDEs), from the previously studied linear framework to a nonlinear setting by incorporating nonlinear adjoint functions. Utilizing this generalized well-posedness result for extended MF-FBSDEs in conjunction with other refined analytical techniques, we address two classes of stochastic quadruple optimal controlled problems: a linear-convex problem and a linear-quadratic problem with input constraints that are permitted to be time-dependent and random. For each problem, we establish the existence and uniqueness of optimal controls and derive their explicit closed-form representations.