Matrix-valued SDEs arising from currency exchange markets

TL;DR

This paper develops a framework for matrix-valued stochastic differential equations inspired by currency exchange markets, establishing conditions for their existence and uniqueness using a trace-based Hilbert space structure.

Contribution

It introduces a novel Hilbert space framework for matrix-valued SDEs based on the trace of the Hadamard product, tailored for currency market modeling.

Findings

Established a Hilbert space structure for matrices using trace

Formulated stochastic integrals and SDEs in this framework

Provided conditions for existence and uniqueness of solutions

Abstract

In this paper, motivated by modelling currency exchange markets with matrix-valued stochastic processes, matrix-valued stochastic differential equations (SDEs) are formulated. This is done based on the matrix trace, as for the purpose of modelling currency exchange markets. To be more precise, we set up a Hilbert space structure for $n\times n$ square matrices via the trace of the Hadamard product of two matrices. With the help of this framework, one can then define stochastic integral of It\^o type and It\^o SDEs. Two types of sufficient conditions are discussed for the existence and uniqueness of solutions to the matrix-valued SDEs.