local properties of processes
Many properties of stochastic processes, such as the martingale property, can be generalized to a corresponding local property. The local properties can be more useful than the original property because they are often preserved under certain transformations of processes, such as random time changes and stochastic integration.
Let be a filtered probability space and be a property of stochastic processes with time index set . The property is said to hold locally for a process if there exists a sequence of stopping times taking values in and almost surely increasing to infinity, such that the stopped processes have property for each .
Often, the index set has a minimal element , in which case it is convenient to extend the concept of localization slightly so that holds locally if there is a sequence of stopping times almost surely increasing to infinity and such that have property .
The property of locally satisfying is often denoted as . Similarly, if is a class of processes then the processes which are locally in is denoted by . Letting be the stopping times taking the constant value shows that every process in is also locally in , so .
In most cases where localization is used, such as with the class of right-continuous martingales, for any process in and stopping time then is also in . If this is the case then it is easily shown that a process is locally in if and only if it is locally in . So, .
Examples of commonly used local properties are as follows.
A process is said to be a local martingale if it is locally a right-continuous martingale. That is, if there is a sequence of stopping times almost surely increasing to infinity and such that 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.
A local submartingale (resp. local supermartingale) is a right-continuous process which is locally a submartingale (resp. supermartingale). A local submartingale can be shown to be a submartingale if and only if is integrable and the set is locally integrable for every . In particular, every nonpositive local submartingale for which is integrable is a submartingale. Similarly every nonnegative supermartingale such that is integrable is a supermartingale.
An increasing and non-negative process is locally integrable if it is locally an integrable process. That is, there is a sequence of stopping times increasing to infinity and such that for every and . By monotonicity of , this is equivalent to . For example, the maximum process of a local martingale is locally integrable.
Similarly, in continuous-time, if is a property of stochastic processes and is a stochastic process such that the left limits of with respect to exist everywhere, then is said to prelocally satisfy if there is a sequence of stopping times almost surely increasing to infinity and such that the prestopped processes satisfy .
|Title||local properties of processes|
|Date of creation||2013-03-22 18:38:50|
|Last modified on||2013-03-22 18:38:50|
|Last modified by||gel (22282)|
|Defines||locally integrable process|
|Defines||locally bounded process|