# Russo–Vallois integral

In mathematical analysis, the Russo–Vallois integral is an extension to stochastic processes of the classical Riemann–Stieltjes integral

$\int f\,dg=\int fg'\,ds$ for suitable functions $f$ and $g$ . The idea is to replace the derivative $g'$ by the difference quotient

$g(s+\varepsilon )-g(s) \over \varepsilon$ and to pull the limit out of the integral. In addition one changes the type of convergence.

## Definitions

Definition: A sequence $H_$ of stochastic processes converges uniformly on compact sets in probability to a process $H,$ $H={\text}\lim _H_,$ if, for every $\varepsilon >0$ and $T>0,$ $\lim _\mathbb (\sup _|H_(t)-H(t)|>\varepsilon )=0.$ One sets:

$I^{-}(\varepsilon ,t,f,dg)=\int _^f(s)(g(s+\varepsilon )-g(s))\,ds$ $I^{+}(\varepsilon ,t,f,dg)=\int _^f(s)(g(s)-g(s-\varepsilon ))\,ds$ and

$[f,g]_{\varepsilon }(t)=\int _^(f(s+\varepsilon )-f(s))(g(s+\varepsilon )-g(s))\,ds.$ Definition: The forward integral is defined as the ucp-limit of

$I^{-}$ : $\int _^fd^{-}g={\text}\lim _{\varepsilon \rightarrow \infty (0?)}I^{-}(\varepsilon ,t,f,dg).$ Definition: The backward integral is defined as the ucp-limit of

$I^{+}$ : $\int _^f\,d^{+}g={\text}\lim _{\varepsilon \rightarrow \infty (0?)}I^{+}(\varepsilon ,t,f,dg).$ Definition: The generalized bracket is defined as the ucp-limit of

$[f,g]_{\varepsilon }$ : $[f,g]_{\varepsilon }={\text}\lim _{\varepsilon \rightarrow \infty }[f,g]_{\varepsilon }(t).$ For continuous semimartingales $X,Y$ and a càdlàg function H, the Russo–Vallois integral coincidences with the usual Itô integral:

$\int _^H_\,dX_=\int _^H\,d^{-}X.$ In this case the generalised bracket is equal to the classical covariation. In the special case, this means that the process

$[X]:=[X,X]\,$ is equal to the quadratic variation process.

Also for the Russo-Vallois Integral an Ito formula holds: If $X$ is a continuous semimartingale and

$f\in C_(\mathbb ),$ then

$f(X_)=f(X_)+\int _^f'(X_)\,dX_+\int _^f''(X_)\,d[X]_.$ 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

$B_^{\lambda }(\mathbb ^)$ is given by

$||f||_^{\lambda }=||f||_}+\left(\int _^{\infty }}(||f(x+h)-f(x)||_})^\,dh\right)^$ with the well known modification for $q=\infty$ . Then the following theorem holds:

Theorem: Suppose

$f\in B_^{\lambda },$ $g\in B_^,$ $1/p+1/p'=1{\text{ and }}1/q+1/q'=1.$ Then the Russo–Vallois integral

$\int f\,dg$ exists and for some constant $c$ one has

$\left|\int f\,dg\right|\leq c||f||_^{\alpha }||g||_^.$ 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.