# A Finite Horizon Optimal Switching Problem with Memory and Application to Controlled SDDEs

## TL;DR

This paper addresses a finite horizon optimal switching problem with memory, establishing existence of optimal controls, and applies it to controlled stochastic delay differential equations, including practical hydro-power revenue maximization.

## Contribution

It introduces a probabilistic approach to prove existence of optimal controls in switching problems with memory and applies it to stochastic delay differential equations with real-world relevance.

## Key findings

- Existence of optimal control established using Snell envelopes.
- Application to impulse control problems for SDDEs with jump processes.
- Relevance demonstrated in hydro-power revenue maximization.

## Abstract

We consider an optimal switching problem where the terminal reward depends on the entire control trajectory. We show existence of an optimal control by applying a probabilistic technique based on the concept of Snell envelopes. We then apply this result to solve an impulse control problem for stochastic delay differential equations driven by a Brownian motion and an independent compound Poisson process. Furthermore, we show that the studied problem arises naturally when maximizing the revenue from operation of a group of hydro-power plants with hydrological coupling.