dominated convergence for stochastic integration

The dominated convergence theorem for standard integration states that if a sequence of measurable functions converge to a limit, and are dominated by an integrable function, then their integrals converge to the integral of the limit. That is, the limit commutes with integration. A similar result holds for stochastic integration with respect to a semimartingale $X$, except the integrals are random variables, and the integrals converge in probability.

Theorem (Dominated convergence).

If ${\xi }^n$ are predictable processes converging pointwise to $\xi$, and $\mathrm{|}{\xi }^n\mathrm{|}\mathrm{\le }\alpha$ for every $n$ and some $X$-integrable process $\alpha$, then

$${\int }_0^t{\xi }^n𝑑X\to {\int }_0^t\xi 𝑑X$$(1)

in probability as $n\mathrm{\to }\mathrm{\infty }$. Furthermore, ucp convergence and semimartingale convergence hold.

Note that as $\xi$ and ${\xi }^n$ are bounded by an $X$-integrable process, they are guaranteed to also be $X$-integrable. Convergence in probability for each $t$ was taken as part of the definition of the stochastic integral, but the dominated convergence theorem stated here says that the stronger ucp and semimartingale convergence also hold.

If $\alpha$ is a locally bounded (http://planetmath.org/LocalPropertiesOfProcesses) predictable process, then it is automatically $X$-integrable for any semimartingale $X$. It follows that if ${\xi }^n$ are predictable processes converging to $\xi$ and if ${sup}_n|{\xi }^n|$ is locally bounded then the limit (1) holds. This result is sometimes known as the locally bounded convergence theorem.

To prove this result, it is enough to show that semimartingale convergence holds, as semimartingale convergence implies ucp convergence.So, let $|{\alpha }^n|\le 1$ be a sequence of simple predictable processes and set $Y^n=\int {\xi }^n𝑑X$, $Y=\int \xi 𝑑X$. Associativity of stochastic integration gives

$${\int }_0^t{\alpha }^n𝑑Y^n-{\int }_0^t{\alpha }^n𝑑Y={\int }_0^t{\alpha }^n({\xi }^n-\xi )𝑑X$$

However, $|{\alpha }^n({\xi }^n-\xi )|\le 2\alpha$, which is $X$-integrable. So, this converges to zero in probability by the definition of the stochastic integral, and $Y^n\to Y$ in the semimartingale topology.