Optimal replication of random vectors by ordinary integrals

TL;DR

This paper addresses the problem of optimally replicating random vectors generated by Wiener processes through adapted integrals, aiming to minimize an integral norm at a fixed terminal time.

Contribution

It introduces a method for optimal replication of Wiener-generated random vectors using adapted processes minimizing an integral norm.

Findings

Derived explicit optimal adapted processes for vector replication

Established minimal integral norm conditions for the processes

Provided theoretical framework for Wiener process-based replication

Abstract

We consider a problem of replication of random vectors by ordinary integrals in the setting when a underlying random variable is generated by a Wiener process. The goal is to find an optimal adapted process such that its cumulative integral at a fixed terminal time matches this variable. The optimal process has to be minimal in an integral norm.