# `Controlled' versions of the Collatz-Wielandt and Donsker-Varadhan formulae

## TL;DR

This paper reviews how risk-sensitive costs and rewards can be characterized using abstract Collatz-Wielandt and Donsker-Varadhan formulas, providing linear and dynamic programming tools for finite state-action systems.

## Contribution

It introduces controlled versions of these formulas, enabling new linear and dynamic programming approaches for risk-sensitive decision-making in finite systems.

## Key findings

- Provides a unified framework for risk-sensitive costs and rewards
- Derives linear programming formulations for finite state-action systems
- Introduces controlled Donsker-Varadhan formula for rewards

## Abstract

This is an overview of the work of the authors and their collaborators on the characterization of risk sensitive costs and rewards in terms of an abstract Collatz-Wielandt formula and in case of rewards, also a controlled version of the Donsker-Varadhan formula. For the finite state and action case, this leads to useful linear and dynamic programming formulations in the reducible case.