# 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 $H_{n}$ of stochastic processes converges uniformly on compact sets in probability to a process $H,$ $H={\text{ucp-}}\lim _{n\rightarrow \infty }H_{n},$ if, for every $\varepsilon >0$ and $T>0,$ $\lim _{n\rightarrow \infty }\mathbb {P} (\sup _{0\leq t\leq T}|H_{n}(t)-H(t)|>\varepsilon )=0.$ One sets:

$I^{-}(\varepsilon ,t,f,dg)={1 \over \varepsilon }\int _{0}^{t}f(s)(g(s+\varepsilon )-g(s))\,ds$ $I^{+}(\varepsilon ,t,f,dg)={1 \over \varepsilon }\int _{0}^{t}f(s)(g(s)-g(s-\varepsilon ))\,ds$ and

$[f,g]_{\varepsilon }(t)={1 \over \varepsilon }\int _{0}^{t}(f(s+\varepsilon )-f(s))(g(s+\varepsilon )-g(s))\,ds.$ Definition: The forward integral is defined as the ucp-limit of

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

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

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

$\int _{0}^{t}H_{s}\,dX_{s}=\int _{0}^{t}H\,d^{-}X.$ 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

$f\in C_{2}(\mathbb {R} ),$ then

$f(X_{t})=f(X_{0})+\int _{0}^{t}f'(X_{s})\,dX_{s}+{1 \over 2}\int _{0}^{t}f''(X_{s})\,d[X]_{s}.$ 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

$B_{p,q}^{\lambda }(\mathbb {R} ^{N})$ is given by

$||f||_{p,q}^{\lambda }=||f||_{L_{p}}+\left(\int _{0}^{\infty }{1 \over |h|^{1+\lambda q}}(||f(x+h)-f(x)||_{L_{p}})^{q}\,dh\right)^{1/q}$ with the well known modification for $q=\infty$ . Then the following theorem holds:

Theorem: Suppose

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

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

$\left|\int f\,dg\right|\leq c||f||_{p,q}^{\alpha }||g||_{p',q'}^{1-\alpha }.$ 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.