# completeness under ucp convergence

Let $(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\in\mathbb{R}_{+}},\mathbb{P})$ be a filtered probability space. Then, under the ucp topology, various classes of stochastic processes form complete (http://planetmath.org/Complete) vector spaces.

For a $\sigma$-algebra $\mathcal{M}$ on $\mathbb{R}_{+}\times\Omega$, we can look at the processes which are $\mathcal{M}$-measurable when regarded as a map from $\mathbb{R}_{+}\times\Omega$ to $\mathbb{R}$. As the ucp topology is only defined for jointly measurable processes, we restrict attention to sub-$\sigma$-algebras of the product $\mathcal{B}(\mathbb{R}_{+})\otimes\mathcal{F}$.

###### Theorem 1.

Let $\mathcal{M}$ be a sub-$\sigma$-algebra of $\mathcal{B}(\mathbb{R}_{+})\otimes\mathcal{F}$. Then, the set of $\mathcal{M}$-measurable processes is complete under ucp convergence.

That is, if $X^{n}$ is a sequence of $\mathcal{M}$-measurable processes such that $X^{n}-X^{m}\xrightarrow{\rm ucp}0$ as $m,n\rightarrow\infty$ then $X^{n}\xrightarrow{\rm ucp}X$ for some $\mathcal{M}$-measurable process $X$.

In particular, the spaces of jointly measurable, progressive, optional and predictable processes are each complete under ucp convergence. We can also look at the properties of the sample paths of the processes.

###### Theorem 2.

Let $S$ be any set of functions $\mathbb{R}_{+}\rightarrow\mathbb{R}$ which is closed (http://planetmath.org/Closed) under uniform convergence on compacts (the compact-open topology). Then, the set of jointly measurable processes whose sample paths are almost surely in $S$ is complete under ucp convergence.

So, for example, the continuous, right-continuous, left-continuous and cadlag processes are each complete under the ucp topology. Furthermore, combining theorems 1 and 2, and using the fact that a cadlag process is adapted if and only if it is jointly measurable (see measurability of stochastic processes (http://planetmath.org/MeasurabilityOfStochasticProcesses)), the following useful result is obtained.