Russo–Vallois integral

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

${\displaystyle \int f\,dg=\int fg'\,ds}$

for suitable functions ${\displaystyle f}$ and ${\displaystyle g}$. The idea is to replace the derivative ${\displaystyle g'}$ by the difference quotient

${\displaystyle g(s+\varepsilon )-g(s) \over \varepsilon }$ and to pull the limit out of the integral. In addition one changes the type of convergence.

Definitions

Definition: A sequence ${\displaystyle H_{n}}$ of stochastic processes converges uniformly on compact sets in probability to a process ${\displaystyle H,}$

${\displaystyle H={\text{ucp-}}\lim _{n\rightarrow \infty }H_{n},}$

if, for every ${\displaystyle \varepsilon >0}$ and ${\displaystyle T>0,}$

${\displaystyle \lim _{n\rightarrow \infty }\mathbb {P} (\sup _{0\leq t\leq T}|H_{n}(t)-H(t)|>\varepsilon )=0.}$

One sets:

${\displaystyle I^{-}(\varepsilon ,t,f,dg)={1 \over \varepsilon }\int _{0}^{t}f(s)(g(s+\varepsilon )-g(s))\,ds}$

${\displaystyle I^{+}(\varepsilon ,t,f,dg)={1 \over \varepsilon }\int _{0}^{t}f(s)(g(s)-g(s-\varepsilon ))\,ds}$

and

${\displaystyle [f,g]_{\varepsilon }(t)={1 \over \varepsilon }\int _{0}^{t}(f(s+\varepsilon )-f(s))(g(s+\varepsilon )-g(s))\,ds.}$

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

${\displaystyle I^{-}}$: ${\displaystyle \int _{0}^{t}fd^{-}g={\text{ucp-}}\lim _{\varepsilon \rightarrow \infty (0?)}I^{-}(\varepsilon ,t,f,dg).}$

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

${\displaystyle I^{+}}$: ${\displaystyle \int _{0}^{t}f\,d^{+}g={\text{ucp-}}\lim _{\varepsilon \rightarrow \infty (0?)}I^{+}(\varepsilon ,t,f,dg).}$

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

${\displaystyle [f,g]_{\varepsilon }}$: ${\displaystyle [f,g]_{\varepsilon }={\text{ucp-}}\lim _{\varepsilon \rightarrow \infty }[f,g]_{\varepsilon }(t).}$

For continuous semimartingales ${\displaystyle X,Y}$ and a càdlàg function H, the Russo–Vallois integral coincidences with the usual Itô integral:

${\displaystyle \int _{0}^{t}H_{s}\,dX_{s}=\int _{0}^{t}H\,d^{-}X.}$

In this case the generalised bracket is equal to the classical covariation. In the special case, this means that the process

${\displaystyle [X]:=[X,X]\,}$

is equal to the quadratic variation process.

Also for the Russo-Vallois Integral an Ito formula holds: If ${\displaystyle X}$ is a continuous semimartingale and

${\displaystyle f\in C_{2}(\mathbb {R} ),}$

then

${\displaystyle f(X_{t})=f(X_{0})+\int _{0}^{t}f'(X_{s})\,dX_{s}+{1 \over 2}\int _{0}^{t}f''(X_{s})\,d[X]_{s}.}$

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

${\displaystyle B_{p,q}^{\lambda }(\mathbb {R} ^{N})}$

is given by

${\displaystyle ||f||_{p,q}^{\lambda }=||f||_{L_{p}}+\left(\int _{0}^{\infty }{1 \over |h|^{1+\lambda q}}(||f(x+h)-f(x)||_{L_{p}})^{q}\,dh\right)^{1/q}}$

with the well known modification for ${\displaystyle q=\infty }$. Then the following theorem holds:

Theorem: Suppose

${\displaystyle f\in B_{p,q}^{\lambda },}$

${\displaystyle g\in B_{p',q'}^{1-\lambda },}$

${\displaystyle 1/p+1/p'=1{\text{ and }}1/q+1/q'=1.}$

Then the Russo–Vallois integral

${\displaystyle \int f\,dg}$

exists and for some constant ${\displaystyle c}$ one has

${\displaystyle \left|\int f\,dg\right|\leq c||f||_{p,q}^{\alpha }||g||_{p',q'}^{1-\alpha }.}$

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