Strict Local Martingales via Filtration Enlargement

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.

Foreign Exchange Expectation Errors and Filtration Enlargements

“…We can now state Theorem 3 of Dandapani and Protter [1] (p.p. 10), which relates S local martingale under (F, P) to S strict local martingale under (G, Q).…”

“…We argue this puzzling phenomenon can be interpreted through strict local martingale theory-specifically through the mechanism detailed by Dandapani and Protter [1]. Dandapani and Protter [1] explain how enlarging filtration F to G changes the underlying measure (say from P to Q), which can induce a stochastic drift in the volatility of S, turning S from an F martingale to a G strict local martingale.…”

“…Dandapani and Protter [1] detail a mechanism through which new information introduced into the observed information set (the initial filtration F) induce a change in the measure, from P under F to Q under the enlarged filtration G, such that a F martingale process S can become a G strict local martingale for a random time interval.…”

“…∈ F. Dandapani and Protter [1] show that it is possible for, due to such change, process S to be a G strict local martingale. The mechanism that allows the construction of a strict local martingale is a class of stochastic volatility model, detailed below.…”

“…10), which relates S local martingale under (F, P) to S strict local martingale under (G, Q). [1]. Assume that k, H and J have right continuous paths almost surely, and Q(w :…”