Local martingale

A local martingale is an adapted right continuous stochastic process on a filtered probability space , so that an increasing sequence of stop times exists with almost certain , so that there is a martingale for everyone . ${\displaystyle (X_{t})_{t\geq 0}}$ ${\displaystyle (\Omega ,{\mathcal {A}},P,({\mathcal {F}}_{t})_{t\geq 0})}$ ${\displaystyle (T_{n})_{n\in \mathbb {N} }}$ ${\displaystyle \lim _{n\to \infty }T_{n}=\infty }$ ${\displaystyle (X_{\min\{t,T_{n}\}}\mathrm {I} _{\{T_{n}>0\}})_{t\geq 0}}$ ${\displaystyle n}$

So it is a localization of the martingale term. Local martingales play a role in the theory of stochastic integration , more precisely the class of possible integrators corresponds to the semimartingals , sums of local martingales and adapted processes of finite variation .

The concept of the local martingale is a far-reaching generalization of the concept of martingale; there are examples of local martingales that can be integrated equally, which are not martingales.

literature

• Daniel Revuz, Marc Yor: Continuous Martingales and Brownian motion . Springer-Verlag, New York 1999, ISBN 3-540-64325-7 .