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 $T>0,$

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

On 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

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

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

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

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-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

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

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)