Strict Local Martingales via Filtration Enlargement

TL;DR

This paper explores how initial filtration enlargement can transform true martingales into strict local martingales, with applications in mathematical finance, by establishing conditions under which this phenomenon occurs in stochastic differential equations.

Contribution

It provides a detailed analysis and sufficient conditions for the emergence of strict local martingales from true martingales through initial filtration expansion in various stochastic models.

Findings

Initial filtration enlargement can induce strict local martingales from true martingales.

Conditions are identified under which this transformation occurs in stochastic differential equations.

Applications are demonstrated in the context of mathematical finance.

Abstract

A strict local martingale is a local martingale that is not a martingale. We investigate how such a process might arise from a true martingale as a result of an enlargement of the filtration. We study and implement a particular type of enlargement, initial expansion of filtration, for various stochastic differential equations and provide sufficient conditions in each of these cases such that initial expansion can create a strict local martingale under an equivalent probability measure. Such situations arise in the theory of Mathematical Finance.