Stochastic integrals and conditional full support

TL;DR

This paper establishes conditions under which certain stochastic processes, including models with stochastic volatility and solutions to specific SDEs, possess the conditional full support property, extending previous results to broader settings.

Contribution

It provides new criteria for the conditional full support property for processes involving Brownian motion and extends these results to more general continuous processes.

Findings

Several stochastic volatility models have CFS.

Solutions of certain SDEs have CFS under new conditions.

Conditions for processes with finite variation K to have CFS when W is replaced by a general continuous process.

Abstract

We present conditions that imply the conditional full support (CFS) property, introduced by Guasoni, R\'asonyi, and Schachermayer [Ann. Appl. Probab., 18 (2008), pp. 491--520], for processes Z := H + K \cdot W, where W is a Brownian motion, H is a continuous process, and processes H and K are either progressive or independent of W. Moreover, in the latter case under an additional assumption that K is of finite variation, we present conditions under which Z has CFS also when W is replaced with a general continuous process with CFS. As applications of these results, we show that several stochastic volatility models and the solutions of certain stochastic differential equations have CFS.