ucp convergence of processes
in probability (http://planetmath.org/ConvergenceInProbability) as , for every . That is, if
for all . The notation is sometimes used, and is said to converge ucp to . This mode of convergence occurs frequently in the theory of continuous-time stochastic processes, and some examples are given below.
Note that the expression on the left hand side of (1) is a supremum of an uncountable set of random variables and, therefore, need not be a measurable quantity in general. If, however, the processes have left or right-continuous sample paths, then the supremum can be restricted to rational times
and is measurable. Typically, it is only required that the sample paths are left or right-continuous almost surely, so the above equality holds on a set of probability one. More generally, if the processes are jointly measurable, then the expression on the left hand side of (1) will be a measurable random variable in the completion (http://planetmath.org/CompleteMeasure) of the probability space (by the measurable projection theorem). This is enough to ensure that the definition above is meaningful, and gives a well defined topology on the space of jointly measurable processes.
then is a pseudometric on the space of measurable processes such that if and only if as . Furthermore, this becomes a metric under the identification of processes with almost surely identical sample paths.
If then we may pass to a subsequence satisfying . The Borel-Cantelli lemma then implies that uniformly on all compact intervals, with probability one. So, any sequence converging in the ucp topology has a subsequence converging uniformly on compacts, with probability one. Consequently, given any property of the sample paths which is preserved under uniform convergence on compacts, then it is also preserved under ucp convergence with probability one. In particular, ucp limits of cadlag processes are themselves cadlag.
Some examples of ucp convergence are given below.
Suppose that the stochastic differential equation
Setting for then these discrete approximations converge to in the ucp topology as the partition mesh goes to zero.
- 1 Philip E. Protter, Stochastic integration and differential equations. Second edition. Applications of Mathematics, 21. Stochastic Modelling and Applied Probability. Springer-Verlag, 2004.
|Title||ucp convergence of processes|
|Date of creation||2013-03-22 18:39:48|
|Last modified on||2013-03-22 18:39:48|
|Last modified by||gel (22282)|