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Martingales and strict local martingales

Oberseminar Finanz- und Versicherungsmathematik

Martingales and strict local martingales


When discussing the nature of nonnegative solutions of SDEs,
making the distinction between a strict local martingale and a true
martingale can be very important. The papers of Delbaen, Shirakawa,
Mijatovic, Urusov, Lions, Musiela, Andersen, Piterbarg, Bernard, Cui, and
finally McLeish have studied the case of a one dimensional SDE, with or
without stochastic volatility. We present two concepts: how a solution of an
SDE which is a martingale can become a strict local martingale by the
addition of new information to the underlying filtration, and how various
components of a vector of SDEs can be strict local martingales for some
components of the system, and martingales for others. This is based on joint
work with Philip Protter, Professor at Columbia University.