Russo-Vallois integral


Russo-Vallois integral

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)overepsilon and to pull the limit out of the integral. In addition one changes the type of convergence.

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

:H= ext{ucp-}lim_{n ightarrowinfty}H_n,

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

:lim_{n ightarrowinfty}mathbb{P}(sup_{0leq tleq T}|H_n(t)-H(t)|>epsilon)=0.

On sets::I^-(epsilon,t,f,dg)={1overepsilon}int_0^tf(s)(g(s+epsilon)-g(s))ds, :I^+(epsilon,t,f,dg)={1overepsilon}int_0^tf(s)(g(s)-g(s-epsilon))ds

and

: [f,g] _epsilon (t)={1overepsilon}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= ext{ucp-}lim_{epsilon ightarrowinfty}I^-(epsilon,t,f,dg).

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

:I^+: int_0^t fd^+g= ext{ucp-}lim_{epsilon ightarrowinfty}I^+(epsilon,t,f,dg).

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

: [f,g] _epsilon: [f,g] _epsilon= ext{ucp-}lim_{epsilon ightarrowinfty} [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

:fin C_2(mathbb{R}),

then

:f(X_t)=f(X_0)+int_0^t f'(X_s)dX_s+{1over 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}{1over |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

:fin B_{p,q}^lambda, :gin 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.

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)


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