Russo–Vallois integral explained

\intfdg=\intfg'ds

for suitable functions

and

. The idea is to replace the derivative

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=ucp-\lim_{n → infty}H_n,

if, for every

\varepsilon>0

and

T>0,

\lim_{n → infty}P(\sup_0\leq|H_{n(t)-H(t)|>\varepsilon)=0.}

One sets:

	tf(s)(g(s+\varepsilon)-g(s))ds
I
	0

	t
I
	0

f(s)(g(s)-g(s-\varepsilon))ds

and

[f,g]_\varepsilon(t)={1\over

	t(f(s+\varepsilon)-f(s))(g(s+\varepsilon)-g(s))ds.
\varepsilon}\int
	0

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

I^-

	t
\int
	0

	-g=ucp-\lim
fd
	\varepsilon → infty(0?)

I^{-(\varepsilon,t,f,dg).}

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

I⁺

	t
\int
	0

fd^+g=ucp-\lim_{\varepsilon → infty}I^{+(\varepsilon,t,f,dg).}

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

[f,g]_\varepsilon

[f,g]_{\varepsilon=ucp-\lim}_{\varepsilon → infty}[f,g]_\varepsilon(t).

For continuous semimartingales

X,Y

and a càdlàg function H, the Russo–Vallois integral coincidences with the usual Itô integral:

	t
\int
	0

H_sdX_s=\int

	t

	0

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-Vallois Integral an Ito formula holds: If

is a continuous semimartingale and

f\inC_2(R),

then

f(X_t)=f(X_0)+\int

	t

	0

f'(X_s)dX_s+{1\over

	t
2}\int
	0

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

	λ(R
B
	p,q

^N)

is given by

	λ=\|\|f\|\|
\|\|f\|\|
	L_p

	infty
\left(\int
	0

{1\over|h|^1+λ

}(||f(x+h)-f(x)||_)^q \, dh\right)^

with the well known modification for

q=infty

. Then the following theorem holds:

Theorem: Suppose

f\in

	λ,
B
	p,q

g\in

	1-λ
B
	p',q'

1/p+1/p'=1and1/q+1/q'=1.

Then the Russo–Vallois integral

\intfdg

exists and for some constant

one has

\left|\intfdg\right|\leqc

	\alpha
\|\|f\|\|
	p,q

	1-\alpha
\|\|g\|\|
	p',q'

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, Francesco. Vallois, Pierre. Forward, backward and symmetric integration. Prob. Th. And Rel. Fields. 97. 1993. 403–421. 10.1007/BF01195073. free.
Russo, F.. Vallois, P.. The generalized covariation process and Ito-formula. Stoch. Proc. And Appl.. 59. 1. 81–104. 1995. 10.1016/0304-4149(95)93237-A. free.
Book: Zähle, Martina. Forward Integrals and Stochastic Differential Equations. In: Seminar on Stochastic Analysis, Random Fields and Applications III. Progress in Prob. Vol. 52. 2002. 293–302. Birkhäuser, Basel. 10.1007/978-3-0348-8209-5_20. 978-3-0348-9474-6 .
Book: Adams, Robert A.. Fournier, John J. F.. Sobolev Spaces. Elsevier. second. 2003. 9780080541297 .