Summation of Grandi's series

In mathematical analysis, the Russo–Vallois integral is an extension to stochastic processes of the classical Riemann–Stieltjes integral

$\int fdg=\int f{g}^{\prime }ds$

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

$\frac{g(s+\epsilon )-g(s)}{\epsilon }$ 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 $H,$

$H=\text{ucp-}\underset{n\to \mathrm{\infty }}{\lim }{H}_n,$

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

$\underset{n\to \mathrm{\infty }}{\lim }\mathbb{\mathbb{P}}({\sup }_{0\le t\le T}|H_n(t)-H(t)|>\epsilon )=0.$

On sets:

$I^{-}(\epsilon ,t,f,dg)=\frac{1}{\epsilon }\underset{0}{\overset{t}{\int }}f(s)(g(s+\epsilon )-g(s))ds$

$I^{+}(\epsilon ,t,f,dg)=\frac{1}{\epsilon }\underset{0}{\overset{t}{\int }}f(s)(g(s)-g(s-\epsilon ))ds$

and

$[f,g{]}_{\epsilon }(t)=\frac{1}{\epsilon }\underset{0}{\overset{t}{\int }}(f(s+\epsilon )-f(s))(g(s+\epsilon )-g(s))ds.$

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

$I^{-}$: $\underset{0}{\overset{t}{\int }}f{d}^{-}g=\text{ucp-}\underset{\epsilon \to \mathrm{\infty }}{\lim }{I}^{-}(\epsilon ,t,f,dg).$

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

$I^{+}$: $\underset{0}{\overset{t}{\int }}f{d}^{+}g=\text{ucp-}\underset{\epsilon \to \mathrm{\infty }}{\lim }{I}^{+}(\epsilon ,t,f,dg).$

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

$[f,g{]}_{\epsilon }$: $[f,g{]}_{\epsilon }=\text{ucp-}\underset{\epsilon \to \mathrm{\infty }}{\lim }[f,g{]}_{\epsilon }(t).$

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

$\underset{0}{\overset{t}{\int }}{H}_{s}d{X}_s=\underset{0}{\overset{t}{\int }}H{d}^{-}X.$

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

$f(X_t)=f(X_0)+\underset{0}{\overset{t}{\int }}{f}^{\prime }(X_s)d{X}_s+\frac{1}{2}\underset{0}{\overset{t}{\int }}{f}^{″}(X_s)d[X{]}_s.$

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

$\left|\right|f|{|}_{p,q}^{\lambda }=|\left|f\right|{|}_{L_p}+{(\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{1}{|h{|}^{1+\lambda q}}(\left|\right|f(x+h)-f(x)\left|{|}_{L_p}{)}^{q}dh\right)}^{1/q}$

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

Theorem: Suppose

$f\in B_{p,q}^{\lambda },$

$g\in B_{p^{\prime },q^{\prime }}^{1-\lambda },$

$1/p+1/p^{\prime }=1\text{ and }1/q+1/q^{\prime }=1.$

Then the Russo–Vallois integral

$\int fdg$

exists and for some constant $c$ one has

$|\int fdg|\le c\left|\right|f\left|{|}_{p,q}^{\alpha }\right|\left|g\right|{|}_{p^{\prime },q^{\prime }}^{1-\alpha }.$

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)