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 , except the integrals are random variables, and the integrals converge in probability.
Theorem (Dominated convergence).
Note that as and are bounded by an -integrable process, they are guaranteed to also be -integrable. Convergence in probability for each 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 is a locally bounded (http://planetmath.org/LocalPropertiesOfProcesses) predictable process, then it is automatically -integrable for any semimartingale . It follows that if are predictable processes converging to and if 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 be a sequence of simple predictable processes and set , . Associativity of stochastic integration gives
However, , which is -integrable. So, this converges to zero in probability by the definition of the stochastic integral, and in the semimartingale topology.
|Title||dominated convergence for stochastic integration|
|Date of creation||2013-03-22 18:41:03|
|Last modified on||2013-03-22 18:41:03|
|Last modified by||gel (22282)|
|Defines||locally bounded convergence theorem|