Russo–Vallois integral - Alchetron, The Free Social Encyclopedia

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

∫ f d g = ∫ f g ′ d s

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

g ( s + ε ) − g ( s ) ε 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 = ucp- lim n → ∞ H n ,

if, for every ε > 0 and T > 0 ,

lim n → ∞ P ( sup 0 ≤ t ≤ T | H n ( t ) − H ( t ) | > ε ) = 0.

One sets:

I − ( ε , t , f , d g ) = 1 ε ∫ 0 t f ( s ) ( g ( s + ε ) − g ( s ) ) d s I + ( ε , t , f , d g ) = 1 ε ∫ 0 t f ( s ) ( g ( s ) − g ( s − ε ) ) d s

and

[ f , g ] ε ( t ) = 1 ε ∫ 0 t ( f ( s + ε ) − f ( s ) ) ( g ( s + ε ) − g ( s ) ) d s .

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

I − : ∫ 0 t f d − g = ucp- lim ε → ∞ ( 0 ? ) I − ( ε , t , f , d g ) .

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

I + : ∫ 0 t f d + g = ucp- lim ε → ∞ ( 0 ? ) I + ( ε , t , f , d g ) .

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

[ f , g ] ε : [ f , g ] ε = ucp- lim ε → ∞ [ f , g ] ε ( t ) .

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

∫ 0 t H s d X s = ∫ 0 t 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-Vallois Integral an Ito formula holds: If X is a continuous semimartingale and

f ∈ C 2 ( R ) ,

then

f ( X t ) = f ( X 0 ) + ∫ 0 t f ′ ( X s ) d X s + 1 2 ∫ 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 λ ( R N )

is given by

| | f | | p , q λ = | | f | | L p + ( ∫ 0 ∞ 1 | h | 1 + λ q ( | | f ( x + h ) − f ( x ) | | L p ) q d h ) 1 / q

with the well known modification for q = ∞ . Then the following theorem holds:

Theorem: Suppose

f ∈ B p , q λ , g ∈ B p ′ , q ′ 1 − λ , 1 / p + 1 / p ′ = 1 and 1 / q + 1 / q ′ = 1.

Then the Russo–Vallois integral

∫ f d g

exists and for some constant c one has

| ∫ f d g | ≤ c | | f | | p , q α | | g | | p ′ , q ′ 1 − α .

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 integral Wikipedia

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