# The Monotone Case Approach for the Solution of Certain Multidimensional Optimal Stopping Problems

## TL;DR

This paper introduces a monotone case approach to explicitly solve certain multidimensional optimal stopping problems in discrete and continuous time, providing solutions for classic problems like house-selling and investment.

## Contribution

It develops a unified method using Doob and Doob-Meyer decompositions for solving multidimensional optimal stopping problems explicitly.

## Key findings

- Explicit solutions for multidimensional house-selling and burglar's problem.
- Application of monotone case approach to Poisson disorder and investment problems.
- Provides a review of monotone case stopping using decomposition techniques.

## Abstract

This paper studies explicitly solvable multidimensional optimal stopping problems of sum- and product-type in discrete and continuous time using the monotone case approach. It gives a review on monotone case stopping using the Doob decomposition, resp. Doob-Meyer decomposition in continuous time, also in its multiplicative versions. The approach via these decompositions leads to explicit solutions for a variety of examples, including multidimensional versions of the house-selling and burglar's problem, the Poisson disorder problem, and an optimal investment problem.