ucp convergence of processes

Let (Ω,,(t)t+,) be a filtered probability space. Then a sequence of stochastic processesMathworldPlanetmath (Xtn)t+ is said to convergePlanetmathPlanetmath to the process (Xt) in the ucp topology (uniform convergence on compactsPlanetmathPlanetmath in probability) if

sups<t|Xsn-Xs|0 (1)

in probability (http://planetmath.org/ConvergenceInProbability) as n, for every t>0. That is, if


for all ϵ,t>0. The notation XnucpX is sometimes used, and Xn is said to converge ucp to X. 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 variablesMathworldPlanetmath 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 completionPlanetmathPlanetmath (http://planetmath.org/CompleteMeasure) of the probability spaceMathworldPlanetmath (by the measurable projection theorem). This is enough to ensure that the definition above is meaningful, and gives a well defined topologyMathworldPlanetmath on the space of jointly measurable processes.

The ucp topology can be generated by a pseudometric. For example, setting


then (X,Y)Ducp(X-Y) is a pseudometric on the space of measurable processes such that XnucpX if and only if Ducp(Xn-X)0 as n. Furthermore, this becomes a metric under the identification of processes with almost surely identical sample paths.

If XnucpX then we may pass to a subsequence satisfying D(Xnk-X)<2-k. The Borel-Cantelli lemmaMathworldPlanetmath then implies that XnkX 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.

  1. 1.

    If Xn,X are cadlag martingalesMathworldPlanetmath such that 𝔼[|Xtn-Xt|]0 for every t>0 then, Doob’s inequalityMathworldPlanetmath


    shows that Xn converges ucp to X.

  2. 2.

    Let X be a semimartingale and ξn be predictable processes converging pointwise to ξ such that supn|ξn| is X-integrable. Then, the dominated convergence theorem for stochastic integration (http://planetmath.org/DominatedConvergenceForStochasticIntegration) gives

  3. 3.

    Suppose that the stochastic differential equation


    for continuous functionsMathworldPlanetmathPlanetmath a,b: and Brownian motionMathworldPlanetmath W has a unique solution with X0=0. For any partitionPlanetmathPlanetmath 0=t0t1, the following discrete approximation can be constructed


    Setting X~t=X~tk for t(tk,tk+1) then these discrete approximations converge to X in the ucp topology as the partition mesh goes to zero.


  • 1 Philip E. Protter, Stochastic integration and differential equationsMathworldPlanetmath. Second edition. Applications of Mathematics, 21. Stochastic Modelling and Applied Probability. Springer-Verlag, 2004.
Title ucp convergence of processes
Canonical name UcpConvergenceOfProcesses
Date of creation 2013-03-22 18:39:48
Last modified on 2013-03-22 18:39:48
Owner gel (22282)
Last modified by gel (22282)
Numerical id 6
Author gel (22282)
Entry type Definition
Classification msc 60G05
Classification msc 60G07
Related topic CadlagProcess
Related topic SemimartingaleTopology
Defines ucp convergence
Defines ucp topology
Defines converges ucp