A change of variable formula with applications to multi-dimensional optimal stopping problems

TL;DR

This paper develops a change of variable formula for functions with potentially unbounded second derivatives near a boundary, aiding in solving multi-dimensional optimal stopping problems where controlling derivatives is challenging.

Contribution

It introduces a novel change of variable formula applicable to functions with non-integrable second derivatives near a boundary, specifically for multi-dimensional optimal stopping problems.

Findings

Derived a change of variable formula for functions with exploding second derivatives.

Applicable to optimal stopping problems with complex boundary behavior.

Matches classical Itô's formula under certain conditions.

Abstract

We derive a change of variable formula for $C^1$ functions $U:\R_+\times\R^m\to\R$ whose second order spatial derivatives may explode and not be integrable in the neighbourhood of a surface $b:\R_+\times\R^{m-1}\to \R$ that splits the state space into two sets $\cC$ and $\cD$. The formula is tailored for applications in problems of optimal stopping where it is generally very hard to control the second order derivatives of the value function near the optimal stopping boundary. Differently to other existing papers on similar topics we only require that the surface $b$ be monotonic in each variable and we formally obtain the same expression as the classical It\^o's formula.