# A second-order stochastic maximum principle for generalized mean-field control problem

## TL;DR

This paper develops a second-order stochastic maximum principle for generalized mean-field control problems where the standard SMP fails due to Hamiltonian singularity, using advanced backward stochastic differential equations.

## Contribution

It introduces a second-order SMP for mean-field control problems with singular Hamiltonians, expanding the theoretical framework beyond first-order conditions.

## Key findings

- Derived a second-order SMP applicable to singular Hamiltonian cases
- Introduced a generalized mean-field backward stochastic differential equation for adjoint processes
- Provided mathematical tools for handling singularities in stochastic control

## Abstract

In this paper, we study the generalized mean-field stochastic control problem when the usual stochastic maximum principle (SMP) is not applicable due to the singularity of the Hamiltonian function. In this case, we derive a second order SMP. We introduce the adjoint process by the generalized mean-field backward stochastic differential equation. The keys in the proofs are the expansion of the cost functional in terms of a perturbation parameter, and the use of the range theorem for vector-valued measures.