Russo-Vallois integral

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In mathematical analysis, the Russo-Vallois integral is an extension of the classical Riemann-Stieltjes integral

\int fdg=\int fg'ds

for suitable functions f and g. The idea is to replace the derivative g' by the difference quotient

g(s+\epsilon)-g(s)\over\epsilon and to pull the limit out of the integral. In addition one changes the type of convergence.

Definition: A sequence Hn of processes converges uniformly on compact sets in probability to a process H,

H=\text{ucp-}\lim_{n\rightarrow\infty}H_n,

if, for every ε > 0 and T > 0,

\lim_{n\rightarrow\infty}\mathbb{P}(\sup_{0\leq t\leq T}|H_n(t)-H(t)|>\epsilon)=0.

On sets:

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

and

[f,g]_\epsilon (t)={1\over\epsilon}\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=\text{ucp-}\lim_{\epsilon\rightarrow\infty}I^-(\epsilon,t,f,dg).

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

I + : \int_0^t fd^+g=\text{ucp-}\lim_{\epsilon\rightarrow\infty}I^+(\epsilon,t,f,dg).

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

[f,g]ε: [f,g]_\epsilon=\text{ucp-}\lim_{\epsilon\rightarrow\infty}[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

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}+(\int_{0}^{\infty}{1\over |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

f\in B_{p,q}^\lambda,
g\in B_{p',q'}^{1-\lambda},
1 / p + 1 / q = 1 and 1 / p' + 1 / q' = 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',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.

[edit] 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)