Novikov's Condition
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In
probability theory Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...
, 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 in a probability space, where the index of the family often has the interpretation of time. Sto ...
which takes the form of the Radon–Nikodym derivative in
Girsanov's theorem In probability theory, Girsanov's theorem or the Cameron-Martin-Girsanov theorem explains how stochastic processes change under changes in measure. The theorem is especially important in the theory of financial mathematics as it explains how to ...
to be a martingale. If satisfied together with other conditions, Girsanov's theorem may be applied to a
Brownian motion Brownian motion is the random motion of particles suspended in a medium (a liquid or a gas). The traditional mathematical formulation of Brownian motion is that of the Wiener process, which is often called Brownian motion, even in mathematical ...
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 Alexander Alexandrovich Novikov (; – 3 December 1976) was the chief marshal of aviation for the Soviet Air Forces during the Soviet Union's involvement in the World War II, Second World War. Lauded as "the man who has piloted the Red Air F ...
. 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, 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 Brownian motion is the random motion of particles suspended in a medium (a liquid or a gas). The traditional mathematical formulation of Brownian motion is that of the Wiener process, which is often called Brownian motion, even in mathematical ...
: 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 filte ...
\mathcal. Here \mathcal denotes the
Doléans-Dade exponential In stochastic calculus, the Doléans-Dade exponential or stochastic exponential of a semimartingale ''X'' is the unique strong solution of the stochastic differential equation dY_t = Y_\,dX_t,\quad\quad Y_0=1,where Y_ denotes the process of left lim ...
.


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* {{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