- Russo-Vallois integral
mathematical analysis, the Russo-Vallois integral is an extension of the classical Riemann-Stieltjes integral
for suitable functions and . The idea is to replace the
derivativeby the difference quotient
: and to pull the limit out of the integral. In addition one changes the type of convergence.
Definition: A sequence of processes converges uniformly on compact sets in probability to a process ,
if, for every and ,
On sets::, :
Definition: The forward integral is defined as the ucp-limit of
Definition: The backward integral is defined as the ucp-limit of
Definition: The generalized bracked is defined as the ucp-limit of
semimartingales and a cadlag functionH, the Russo-Vallois integral coincidences with the usual Ito integral:
In this case the generalised bracket is equal to the classical covariation. In the special case, this means that the process
is equal to the
quadratic variation process.
Also for the Russo-Vallios-Integral an
Ito formulaholds: If is a continuous semimartingale and
By a duality result of
Triebelone can provide optimal classes of Besov spaces, where the Russo-Vallois integral can be defined. The norm in the Besov-space
is given by
with the well known modification for . Then the following theorem holds:
:, :, : and .
Then the Russo-Vallois-integral
exists and for some constant one has
Notice that in this case the Russo-Vallois-integral coincides with the
Riemann-Stieltjes integraland with the Young integral for functions with finite p-variation.
*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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