Russo–Vallois integral


In mathematical analysis, the Russo–Vallois integral is an extension to stochastic processes of the classical Riemann–Stieltjes integral
for suitable functions and. The idea is to replace the derivative by the difference quotient

Definitions

Definition: A sequence of stochastic processes converges uniformly on compact sets in probability to a process
if, for every and
One sets:
and
Definition: The forward integral is defined as the ucp-limit of
Definition: The backward integral is defined as the ucp-limit of
Definition: The generalized bracket is defined as the ucp-limit of
For continuous semimartingales and a càdlàg function H, the Russo–Vallois integral coincidences with the usual Itô integral:
In this case the generalised bracket is equal to the classical covariation. In the special case, this means that the process
is equal to the quadratic variation process.
Also for the Russo-Vallois Integral an Ito formula holds: If is a continuous semimartingale and
then
By a duality result of Triebel one can provide optimal classes of Besov spaces, where the Russo–Vallois integral can be defined. The norm in the Besov space
is given by
with the well known modification for. Then the following theorem holds:
Theorem: Suppose
Then the Russo–Vallois integral
exists and for some constant one has
Notice that in this case the Russo–Vallois integral coincides with the Riemann–Stieltjes integral and with the Young integral for functions with finite p-variation.