ucp convergence of processes
Let $(\mathrm{\Omega},\mathcal{F},{({\mathcal{F}}_{t})}_{t\in {\mathbb{R}}_{+}},\mathbb{P})$ be a filtered probability space. Then a sequence of stochastic processes^{} ${({X}_{t}^{n})}_{t\in {\mathbb{R}}_{+}}$ is said to converge^{} to the process $({X}_{t})$ in the ucp topology (uniform convergence on compacts^{} in probability) if
$$  (1) 
in probability (http://planetmath.org/ConvergenceInProbability) as $n\to \mathrm{\infty}$, for every $t>0$. That is, if
$$ 
for all $\u03f5,t>0$. The notation ${X}^{n}\stackrel{ucp}{\to}\mathrm{X}$ is sometimes used, and ${X}^{n}$ is said to converge ucp to $X$. This mode of convergence occurs frequently in the theory of continuoustime 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 rightcontinuous sample paths, then the supremum can be restricted to rational times
$$ 
and is measurable. Typically, it is only required that the sample paths are left or rightcontinuous almost surely, so the above equality holds on a set of probability one. More generally, if the processes are jointly measurable, then the expression on the left hand side of (1) will be a measurable random variable in the completion^{} (http://planetmath.org/CompleteMeasure) of the probability space^{} (by the measurable projection theorem). This is enough to ensure that the definition above is meaningful, and gives a well defined topology^{} on the space of jointly measurable processes.
The ucp topology can be generated by a pseudometric. For example, setting
$$ 
then $(X,Y)\mapsto {D}^{\mathrm{ucp}}(XY)$ is a pseudometric on the space of measurable processes such that ${X}^{n}\stackrel{ucp}{\to}\mathrm{X}$ if and only if ${D}^{\mathrm{ucp}}({X}^{n}X)\to 0$ as $n\to \mathrm{\infty}$. Furthermore, this becomes a metric under the identification of processes with almost surely identical sample paths.
If ${X}^{n}\stackrel{ucp}{\to}\mathrm{X}$ then we may pass to a subsequence satisfying $$. The BorelCantelli lemma^{} then implies that ${X}^{{n}_{k}}\to X$ 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.
If ${X}^{n},X$ are cadlag martingales^{} such that $\mathbb{E}[{X}_{t}^{n}{X}_{t}]\to 0$ for every $t>0$ then, Doob’s inequality^{}
$$\mathbb{P}(\underset{s\le t}{sup}{X}_{s}^{n}{X}_{s}>\u03f5)\le {\u03f5}^{1}\mathbb{E}[{X}_{t}^{n}{X}_{t}]$$ shows that ${X}^{n}$ converges ucp to $X$.

2.
Let $X$ be a semimartingale and ${\xi}^{n}$ be predictable processes converging pointwise to $\xi $ such that ${sup}_{n}{\xi}^{n}$ is $X$integrable. Then, the dominated convergence theorem for stochastic integration (http://planetmath.org/DominatedConvergenceForStochasticIntegration) gives
$$\int {\xi}^{n}\mathit{d}X\stackrel{ucp}{\to}\int \xi \mathrm{dX}.$$ 
3.
Suppose that the stochastic differential equation
$$dX=a(X)dW+b(X)dt$$ for continuous functions^{} $a,b:\mathbb{R}\to \mathbb{R}$ and Brownian motion^{} $W$ has a unique solution with ${X}_{0}=0$. For any partition^{} $0={t}_{0}\le {t}_{1}\le \mathrm{\cdots}\uparrow \mathrm{\infty}$, the following discrete approximation can be constructed
$${\stackrel{~}{X}}_{0}=0,{\stackrel{~}{X}}_{{t}_{k+1}}={\stackrel{~}{X}}_{{t}_{k}}+a({\stackrel{~}{X}}_{{t}_{k}})({W}_{{t}_{k+1}}{W}_{{t}_{k}})+b({\stackrel{~}{X}}_{{t}_{k}})({t}_{k+1}{t}_{k}).$$ Setting ${\stackrel{~}{X}}_{t}={\stackrel{~}{X}}_{{t}_{k}}$ for $t\in ({t}_{k},{t}_{k+1})$ then these discrete approximations converge to $X$ in the ucp topology as the partition mesh goes to zero.
References
 1 Philip E. Protter, Stochastic integration and differential equations^{}. Second edition. Applications of Mathematics, 21. Stochastic Modelling and Applied Probability. SpringerVerlag, 2004.
Title  ucp convergence of processes 
Canonical name  UcpConvergenceOfProcesses 
Date of creation  20130322 18:39:48 
Last modified on  20130322 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 