Optimal replication of random claims by ordinary integrals with applications in finance

TL;DR

This paper introduces a novel method for replicating random vectors using controlled ordinary differential equations, offering explicit solutions and potential applications in finance such as portfolio optimization and bond pricing.

Contribution

It presents a new approach to replication via ordinary integrals, expanding beyond traditional stochastic methods, and formulates an explicit optimal control solution.

Findings

Existence of solutions for the replication problem

Non-uniqueness of solutions

Explicit optimal control solution derived

Abstract

By the classical Martingale Representation Theorem, replication of random vectors can be achieved via stochastic integrals or solutions of stochastic differential equations. We introduce a new approach to replication of random vectors via adapted differentiable processes generated by a controlled ordinary differential equation. We found that the solution of this replication problem exists and is not unique. This leads to a new optimal control problem: find a replicating process that is minimal in an integral norm. We found an explicit solution of this problem. Possible applications to portfolio selection problems and to bond pricing models are suggested.