# Risk-Sensitive Mean-Field-Type Control

## TL;DR

This paper develops a framework for risk-sensitive optimal control of mean-field type stochastic differential equations, deriving infinite-dimensional optimality equations and exploring connections with risk-neutral solutions.

## Contribution

It introduces a novel approach to risk-sensitive mean-field control, deriving new optimality equations and analyzing specific cost functionals.

## Key findings

- Derived infinite-dimensional optimality equations for risk-sensitive control
- Connected risk-sensitive solutions with risk-neutral cases
- Discussed linear-exponentiated quadratic cost functional

## Abstract

We study risk-sensitive optimal control of a stochastic differential equation (SDE) of mean-field type, where the coefficients are allowed to depend on some functional of the law as well as the state and control processes. Moreover the risk-sensitive cost functional is also of mean-field type. We derive optimality equations in infinite dimensions connecting dual functions associated with Bellman functional to the adjoint process of the Pontryagin maximum principle. The case of linear-exponentiated quadratic cost and its connection with the risk-neutral solution is discussed.