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 formula_2 and formula_3. The idea is to replace the derivative formula_4 by the difference quotient Definitions. Definition: A sequence formula_6 of stochastic processes converges uniformly on compact sets in probability to a process formula_7 if, for every formula_9 and formula_10 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 formula_21 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 formula_24 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 formula_29. Then the following theorem holds: Theorem: Suppose Then the Russo–Vallois integral exists and for some constant formula_34 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.