On stochastic calculus related to financial assets without semimartingales

TL;DR

This paper develops a stochastic calculus framework for financial assets without assuming semimartingale properties, exploring implications for arbitrage, hedging, and utility maximization under restricted strategies.

Contribution

It introduces the concept of -martingales and develops a related calculus, extending classical financial models to non-semimartingale asset price processes.

Findings

Non-arbitrage can hold with restricted strategies.

A calculus for -martingales is established.

Applications include hedging and utility maximization.

Abstract

This paper does not suppose a priori that the evolution of the price of a financial asset is a semimartingale. Since possible strategies of investors are self-financing, previous prices are forced to be finite quadratic variation processes. The non-arbitrage property is not excluded if the class $\mathcal{A}$ of admissible strategies is restricted. The classical notion of martingale is replaced with the notion of $\mathcal{A}$-martingale. A calculus related to $\mathcal{A}$-martingales with some examples is developed. Some applications to no-arbitrage, viability, hedging and the maximization of the utility of an insider are expanded. We finally revisit some no arbitrage conditions of Bender-Sottinen-Valkeila type.