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

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

:$H=\; ext\{ucp-\}lim\_\{n\; ightarrowinfty\}H\_n$,

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

:$lim\_\{n\; ightarrowinfty\}mathbb\{P\}(sup\_\{0leq\; tleq\; T\}|H\_n(t)-H(t)|>epsilon)=0$.

On sets::$I^-(epsilon,t,f,dg)=\{1overepsilon\}int\_0^tf(s)(g(s+epsilon)-g(s))ds$, :$I^+(epsilon,t,f,dg)=\{1overepsilon\}int\_0^tf(s)(g(s)-g(s-epsilon))ds$

and

:$[f,g]\; \_epsilon\; (t)=\{1overepsilon\}in\_0^t(f(s+epsilon)-f(s))(g(s+epsilon)-g(s))ds$.

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

:$I^-$: $int\_0^t\; fd^-g=\; ext\{ucp-\}lim\_\{epsilon\; ightarrowinfty\}I^-(epsilon,t,f,dg)$.

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

:$I^+$: $int\_0^t\; fd^+g=\; ext\{ucp-\}lim\_\{epsilon\; ightarrowinfty\}I^+(epsilon,t,f,dg)$.

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

:$[f,g]\; \_epsilon$: $[f,g]\; \_epsilon=\; ext\{ucp-\}lim\_\{epsilon\; ightarrowinfty\}\; [f,g]\; \_epsilon\; (t)$.

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

:$[X]\; :=\; [X,X]$

is equal to the quadratic variation process.

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

:$fin\; C\_2(mathbb\{R\})$,

then

:$f(X\_t)=f(X\_0)+int\_0^t\; f\text{'}(X\_s)dX\_s+\{1over\; 2\}int\_0^t\; f"(X\_s)d\; [X]\; \_s$.

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

:$||f||\_\{p,q\}^lambda=||f||\_\{L\_p\}+(int\_\{0\}^\{infty\}\{1over\; |h|^\{1+lambda\; q(||f(x+h)-f(x)||\_\{L\_p\})^q\; dh)^\{1/q\}$

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

Theorem: Suppose

:$fin\; B\_\{p,q\}^lambda$, :$gin\; B\_\{p\text{'},q\text{'}\}^\{1-lambda\}$, :$1/p+1/q=1$ and $1/p\text{'}+1/q\text{'}=1$.

Then the Russo-Vallois-integral

:$int\; fdg$

exists and for some constant $c$ one has

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

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