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In probability theory, Novikov's condition is the sufficient condition for a
stochastic process In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appea ...
which takes the form of the Radon–Nikodym derivative in Girsanov's theorem to be a
martingale Martingale may refer to: * Martingale (probability theory), a stochastic process in which the conditional expectation of the next value, given the current and preceding values, is the current value * Martingale (tack) for horses * Martingale (coll ...
. If satisfied together with other conditions, Girsanov's theorem may be applied to a Brownian motion stochastic process to change from the original measure to the new measure defined by the Radon–Nikodym derivative. This condition was suggested and proved by Alexander Novikov. There are other results which may be used to show that the Radon–Nikodym derivative is a martingale, such as the more general criterion
Kazamaki's condition In mathematics, Kazamaki's condition gives a sufficient criterion ensuring that the Doléans-Dade exponential of a local martingale is a true martingale. This is particularly important if Girsanov's theorem is to be applied to perform a change of ...
, however Novikov's condition is the most well-known result. Assume that (X_t)_ is a real valued adapted process on the probability space \left (\Omega, (\mathcal_t), \mathbb\right) and (W_t)_ is an adapted Brownian motion: If the condition : \mathbb\left ^ \right\infty is fulfilled then the process : \ \mathcal\left( \int_0^t X_s \; dW_s \right) \ = e^,\quad 0\leq t\leq T is a martingale under the probability measure \mathbb and the
filtration Filtration is a physical separation process that separates solid matter and fluid from a mixture using a ''filter medium'' that has a complex structure through which only the fluid can pass. Solid particles that cannot pass through the filter ...
\mathcal. Here \mathcal denotes the Doléans-Dade exponential.


References


External links

* {{cite web , title=Comments on Girsanov's Theorem , first=H. E. , last=Krogstad , work=IMF , year=2003 , url=http://www.math.ntnu.no/~hek/MA8101/GirsanovsTheorem.pdf , archiveurl=https://web.archive.org/web/20051201024314/http://www.math.ntnu.no/~hek/MA8101/GirsanovsTheorem.pdf , archivedate=December 1, 2005 Martingale theory