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.
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.
