# Russo-Vallois integral

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Russo-Vallois integral

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

:$int fdg=int fg\text{'}ds$

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

:$g\left(s+epsilon\right)-g\left(s\right)overepsilon$ 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 $H$,

:$H= ext\left\{ucp-\right\}lim_\left\{n ightarrowinfty\right\}H_n$,

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

:$lim_\left\{n ightarrowinfty\right\}mathbb\left\{P\right\}\left(sup_\left\{0leq tleq T\right\}|H_n\left(t\right)-H\left(t\right)|>epsilon\right)=0$.

On sets::$I^-\left(epsilon,t,f,dg\right)=\left\{1overepsilon\right\}int_0^tf\left(s\right)\left(g\left(s+epsilon\right)-g\left(s\right)\right)ds$, :$I^+\left(epsilon,t,f,dg\right)=\left\{1overepsilon\right\}int_0^tf\left(s\right)\left(g\left(s\right)-g\left(s-epsilon\right)\right)ds$

and

:$\left[f,g\right] _epsilon \left(t\right)=\left\{1overepsilon\right\}in_0^t\left(f\left(s+epsilon\right)-f\left(s\right)\right)\left(g\left(s+epsilon\right)-g\left(s\right)\right)ds$.

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

:$I^-$: $int_0^t fd^-g= ext\left\{ucp-\right\}lim_\left\{epsilon ightarrowinfty\right\}I^-\left(epsilon,t,f,dg\right)$.

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

:$I^+$: $int_0^t fd^+g= ext\left\{ucp-\right\}lim_\left\{epsilon ightarrowinfty\right\}I^+\left(epsilon,t,f,dg\right)$.

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

:$\left[f,g\right] _epsilon$: $\left[f,g\right] _epsilon= ext\left\{ucp-\right\}lim_\left\{epsilon ightarrowinfty\right\} \left[f,g\right] _epsilon \left(t\right)$.

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

:$int_0^t H_sdX_s=int_0^t Hd^-X$.

In this case the generalised bracket is equal to the classical covariation. In the special case, this means that the process

:$\left[X\right] := \left[X,X\right]$

is equal to the quadratic variation process.

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

:$fin C_2\left(mathbb\left\{R\right\}\right)$,

then

:$f\left(X_t\right)=f\left(X_0\right)+int_0^t f\text{'}\left(X_s\right)dX_s+\left\{1over 2\right\}int_0^t f"\left(X_s\right)d \left[X\right] _s$.

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

:$B_\left\{p,q\right\}^lambda\left(mathbb\left\{R\right\}^N\right)$

is given by

:$||f||_\left\{p,q\right\}^lambda=||f||_\left\{L_p\right\}+\left(int_\left\{0\right\}^\left\{infty\right\}\left\{1over |h|^\left\{1+lambda q\left(||f\left(x+h\right)-f\left(x\right)||_\left\{L_p\right\}\right)^q dh\right)^\left\{1/q\right\}$

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

Theorem: Suppose

:$fin B_\left\{p,q\right\}^lambda$, :$gin B_\left\{p\text{'},q\text{'}\right\}^\left\{1-lambda\right\}$, :$1/p+1/q=1$ and $1/p\text{'}+1/q\text{'}=1$.

Then the Russo-Vallois-integral

:$int fdg$

exists and for some constant $c$ one has

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

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)
*Fournier, Adams: Sobolev Spaces, Elsevier, second edition (2003)

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