Sticky processes, local and true martingales

TL;DR

This paper demonstrates that sticky processes can be approximated by true martingales under equivalent measures, with applications in finance, providing new insights into the structure of such stochastic processes.

Contribution

It establishes the existence of equivalent measures and true martingales close to sticky processes, extending the understanding of their structure and applications.

Findings

Existence of an equivalent measure under which a sticky process is close to a martingale.

For continuous processes, approximation can be in supremum norm.

Applications in mathematical finance demonstrate practical relevance.

Abstract

We prove that for a so-called sticky process $S$ there exists an equivalent probability $Q$ and a $Q$-martingale $\tilde{S}$ that is arbitrarily close to $S$ in $L^p(Q)$ norm. For continuous $S$, $\tilde{S}$ can be chosen arbitrarily close to $S$ in supremum norm. In the case where $S$ is a local martingale we may choose $Q$ arbitrarily close to the original probability in the total variation norm. We provide examples to illustrate the power of our results and present applications in mathematical finance.