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 {\displaystyle \int f\,dg=\int fg'\,ds}
for suitable functions f {\displaystyle f} and g {\displaystyle g} . The idea is to replace the derivative g ′ {\displaystyle g'} by the difference quotient
g ( s + ε ) − g ( s ) ε {\displaystyle 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 {\displaystyle H_{n}} of stochastic processes converges uniformly on compact sets in probability to a process H , {\displaystyle H,}
H = ucp- lim n → ∞ H n , {\displaystyle H={\text{ucp-}}\lim _{n\rightarrow \infty }H_{n},}
if, for every ε > 0 {\displaystyle \varepsilon >0} and T > 0 , {\displaystyle T>0,}
lim n → ∞ P ( sup 0 ≤ t ≤ T | H n ( t ) − H ( t ) | > ε ) = 0. {\displaystyle \lim _{n\rightarrow \infty }\mathbb {P} (\sup _{0\leq t\leq T}|H_{n}(t)-H(t)|>\varepsilon )=0.}
One sets:
I − ( ε , t , f , d g ) = 1 ε ∫ 0 t f ( s ) ( g ( s + ε ) − g ( s ) ) d s {\displaystyle I^{-}(\varepsilon ,t,f,dg)={1 \over \varepsilon }\int _{0}^{t}f(s)(g(s+\varepsilon )-g(s))\,ds}
I + ( ε , t , f , d g ) = 1 ε ∫ 0 t f ( s ) ( g ( s ) − g ( s − ε ) ) d s {\displaystyle I^{+}(\varepsilon ,t,f,dg)={1 \over \varepsilon }\int _{0}^{t}f(s)(g(s)-g(s-\varepsilon ))\,ds}
and
[ f , g ] ε ( t ) = 1 ε ∫ 0 t ( f ( s + ε ) − f ( s ) ) ( g ( s + ε ) − g ( s ) ) d s . {\displaystyle [f,g]_{\varepsilon }(t)={1 \over \varepsilon }\int _{0}^{t}(f(s+\varepsilon )-f(s))(g(s+\varepsilon )-g(s))\,ds.}
Definition: The forward integral is defined as the ucp-limit of
I − {\displaystyle I^{-}} : ∫ 0 t f d − g = ucp- lim ε → ∞ ( 0 ? ) I − ( ε , t , f , d g ) . {\displaystyle \int _{0}^{t}fd^{-}g={\text{ucp-}}\lim _{\varepsilon \rightarrow \infty (0?)}I^{-}(\varepsilon ,t,f,dg).}
Definition: The backward integral is defined as the ucp-limit of
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