stochastic integration as a limit of Riemann sums
As with the Riemann (http://planetmath.org/RiemannIntegral) and Riemann-Stieltjes integrals, the stochastic integral can be calculated as a limit of approximations computed on partitions (http://planetmath.org/Partition3), called Riemann sums.
Let be a sequence of partitions of ,
The stochastic integral of a process with respect to can be defined on each of the partitions,
Since the times tend to infinity as , all but finitely many terms are zero. Note that here, the process is sampled at , which are the left hand points of the intervals. For this reason, the stochastic integral is sometimes called the forward integral. Alternatively, the backward integral can be computed by sampling at time and the Stratonovich integral takes the average of and . However, these alternative integrals are less general and may not exist even when is a continuous and adapted process.
For left-continuous integrands, the approximations do indeed converge to the stochastic integral.
Similarly, convergence is also obtained for cadlag integrands. However, in this case, it is necessary to use the left limit in the integral. The integral of does not even exist when it is a general cadlag adapted process, as it might not be predictable.
Suppose that is a semimartingale and is a cadlag adapted process. Then,
in probability as . Furthermore, this converges ucp and in the semimartingale topology.
|Title||stochastic integration as a limit of Riemann sums|
|Date of creation||2013-03-22 18:41:33|
|Last modified on||2013-03-22 18:41:33|
|Last modified by||gel (22282)|