A process is said to be a local martingale if it is locally (http://planetmath.org/LocalPropertiesOfProcesses) a right-continuous martingale. That is, if there is a sequence of stopping times almost surely increasing to infinity and such that the stopped processes are martingales. Equivalently, is integrable and
for all . In the discrete-time case where then it can be shown that a local martingale is a martingale if and only if for every . More generally, in continuous-time where is an interval of the real numbers, then the stronger property that
is uniformly integrable for every gives a necessary and sufficient condition for a local martingale to be a martingale.
Local martingales form a very important class of processes in the theory of stochastic calculus. This is because the local martingale property is preserved by the stochastic integral, but the martingale property is not. Examples of local martingales which are not proper martingales are given by solutions to the stochastic differential equation
where is a nonnegative process, is a Brownian motion and is a fixed real number.
An alternative definition of local martingales which is sometimes used requires to be a martingale for each . This definition is slightly more restrictive, and is equivalent to the definition given above together with the condition that must be integrable.
|Date of creation||2013-03-22 15:12:43|
|Last modified on||2013-03-22 15:12:43|
|Last modified by||skubeedooo (5401)|