Russo–Vallois integral

In mathematical analysis, the Russo-Vallois integral is an extension of the classical Riemann-Stieltjes integral

$\int fdg=\int fg'ds$

for suitable functions $f$ and $g$. The idea is to replace the derivative $g'$ by the difference quotient

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

Definition: A sequence $H_{n}$ of processes converges uniformly on compact sets in probability to a process $H$,

$H={\text{ucp-}}\lim _{n\rightarrow \infty }H_{n}$,

if, for every $\epsilon >0$ and $T>0$,

$\lim _{n\rightarrow \infty }\mathbb {P} (\sup _{0\leq t\leq T}|H_{n}(t)-H(t)|>\epsilon )=0$.

On sets:

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

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

and

$[f,g]_{\epsilon }(t)={1 \over \epsilon }\in _{0}^{t}(f(s+\epsilon )-f(s))(g(s+\epsilon )-g(s))ds$.

Definition: The forward integral is defined as the ucp-limit of

$I^{-}$: $\int _{0}^{t}fd^{-}g={\text{ucp-}}\lim _{\epsilon \rightarrow \infty }I^{-}(\epsilon ,t,f,dg)$.

Definition: The backward integral is defined as the ucp-limit of

$I^{+}$: $\int _{0}^{t}fd^{+}g={\text{ucp-}}\lim _{\epsilon \rightarrow \infty }I^{+}(\epsilon ,t,f,dg)$.

Definition: The generalized bracked is defined as the ucp-limit of

$[f,g]_{\epsilon }$: $[f,g]_{\epsilon }={\text{ucp-}}\lim _{\epsilon \rightarrow \infty }[f,g]_{\epsilon }(t)$.

For continous semimartingales $X,Y$ and a cadlag function H, the Russo-Vallois integral coincidences with the usual Ito integral:

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

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-Vallios-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

$B_{p,q}^{\alpha }$

is given by

$||f||_{p,q}^{\alpha }=||f||_{L_{p}}+(\int _{-1}^{1}{1 \over |h|^{1+\alpha q}}(||f(.+h)-f(.)||_{L_{p}})^{q}dh)^{1/q}$

with the well known modification for $q=\infty$. Then the following theorem holds:

Theorem: Suppose

$f\in B_{p,q}^{\alpha }$,

$g\in B_{p',q'}^{1-\alpha }$,

$1/p+1/q=1$ and $1/p'+1/q'=1$.

Then the Russo-Vallois-integral

$\int fdg$

exists and for some constant $c$ one has

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

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.

References

• Russo, Vallois: Forward, backward and symmetric integrals, Prob. Th. and rel. fields 97 (1993)
• Russo, Vallois: The generalized covariation process and Ito-formula, Stoch. Proc. and Appl. 59 (1995)
• Zähle; Forward Integrals and SDE, Progress in Prob. Vol. 52 (2002)