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 cadlag function H, the Russo–Vallois integral coincidences with the usual Ito 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.

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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