On a non-linear transformation between Brownian martingales

TL;DR

This paper explores a non-linear transformation between Brownian martingales, investigates the solvability of related systems, and introduces novel couplings and SPDE models for surface growth.

Contribution

It introduces a new non-linear transformation between Brownian martingales and develops models for surface growth using stochastic PDEs.

Findings

Established conditions for the solvability of the transformation systems

Developed monotone pathwise couplings for diffusion processes

Derived SPDEs modeling the growth of one-dimensional random surfaces

Abstract

The paper studies a non-linear transformation between Brownian martingales, which is given by the inverse of the pricing operator in the mathematical finance terminology. Subsequently, the solvability of systems of equations corresponding to such transformations is investigated. The latter give rise to novel monotone pathwise couplings of an arbitrary number of certain diffusion processes with varying diffusion coefficients. In the case that there is an uncountable number of these diffusion processes and that the index set is an interval such couplings can be viewed as models for the growth of one-dimensional random surfaces. With this motivation in mind, we derive the appropriate stochastic partial differential equations for the growth of such surfaces.