In mathematical analysis, the Russo–Vallois integral is an extension to stochastic processes of the classical Riemann–Stieltjes integral for suitable functions
by the difference quotient Definition: A sequence
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.
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