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:

$I^-(\varepsilon,t,f,dg)={1\over\varepsilon}\int_0^tf(s)(g(s+\varepsilon)-g(s))\,ds$

$I^+(\varepsilon,t,f,dg)={1\over\varepsilon}\int_0^t f(s)(g(s)-g(s-\varepsilon)) \, ds$

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)