Brownian semistationary processes and conditional full support

TL;DR

This paper investigates the conditional full support of Brownian semistationary processes, providing conditions under which these processes have this property, which has implications for financial modeling and arbitrage opportunities.

Contribution

It offers new sufficient conditions for Brownian semistationary processes to have conditional full support, extending understanding beyond semimartingale assumptions.

Findings

Processes can have conditional full support without being semimartingales.

No free lunches under proportional transaction costs for these processes.

They can be approximated by semimartingales with equivalent martingale measures.

Abstract

In this note, we study the infinite-dimensional conditional laws of Brownian semistationary processes. Motivated by the fact that these processes are typically not semimartingales, we present sufficient conditions ensuring that a Brownian semistationary process has conditional full support, a property introduced by Guasoni, R\'asonyi, and Schachermayer [Ann. Appl. Probab., 18 (2008) pp. 491--520]. By the results of Guasoni, R\'asonyi, and Schachermayer, this property has two important implications. It ensures, firstly, that the process admits no free lunches under proportional transaction costs, and secondly, that it can be approximated pathwise (in the sup norm) by semimartingales that admit equivalent martingale measures.