Russo–Vallois integral

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

[math]\displaystyle{ \int f \, dg=\int fg' \, ds }[/math]

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

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

Definitions

Definition: A sequence [math]\displaystyle{ H_n }[/math] of stochastic processes converges uniformly on compact sets in probability to a process [math]\displaystyle{ H, }[/math]

[math]\displaystyle{ H=\text{ucp-}\lim_{n\rightarrow\infty}H_n, }[/math]

if, for every [math]\displaystyle{ \varepsilon\gt 0 }[/math] and [math]\displaystyle{ T\gt 0, }[/math]

[math]\displaystyle{ \lim_{n\rightarrow\infty}\mathbb{P}(\sup_{0\leq t\leq T}|H_n(t)-H(t)|\gt \varepsilon)=0. }[/math]

One sets:

[math]\displaystyle{ I^-(\varepsilon,t,f,dg)={1\over\varepsilon}\int_0^tf(s)(g(s+\varepsilon)-g(s))\,ds }[/math]

[math]\displaystyle{ I^+(\varepsilon,t,f,dg)={1\over\varepsilon}\int_0^t f(s)(g(s)-g(s-\varepsilon)) \, ds }[/math]

and

[math]\displaystyle{ [f,g]_\varepsilon (t)={1\over \varepsilon}\int_0^t(f(s+\varepsilon)-f(s))(g(s+\varepsilon)-g(s))\,ds. }[/math]

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

[math]\displaystyle{ I^- }[/math]: [math]\displaystyle{ \int_0^t fd^-g=\text{ucp-}\lim_{\varepsilon\rightarrow\infty (0?)}I^-(\varepsilon,t,f,dg). }[/math]

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

[math]\displaystyle{ I^+ }[/math]: [math]\displaystyle{ \int_0^t f \, d^+g = \text{ucp-}\lim_{\varepsilon\rightarrow\infty (0?)}I^+(\varepsilon,t,f,dg). }[/math]

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

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

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

[math]\displaystyle{ \int_0^t H_s \, dX_s=\int_0^t H \, d^-X. }[/math]

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

[math]\displaystyle{ [X]:=[X,X] \, }[/math]

is equal to the quadratic variation process.

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

[math]\displaystyle{ f\in C_2(\mathbb{R}), }[/math]

then

[math]\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. }[/math]

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

[math]\displaystyle{ B_{p,q}^\lambda(\mathbb{R}^N) }[/math]

is given by

[math]\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} }[/math]

with the well known modification for [math]\displaystyle{ q=\infty }[/math]. Then the following theorem holds:

Theorem: Suppose

[math]\displaystyle{ f\in B_{p,q}^\lambda, }[/math]

[math]\displaystyle{ g\in B_{p',q'}^{1-\lambda}, }[/math]

[math]\displaystyle{ 1/p+1/p'=1\text{ and }1/q+1/q'=1. }[/math]

Then the Russo–Vallois integral

[math]\displaystyle{ \int f \, dg }[/math]

exists and for some constant [math]\displaystyle{ c }[/math] one has

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

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. ISBN 978-3-0348-9474-6. 
• Adams, Robert A.; Fournier, John J. F. (2003). Sobolev Spaces (second ed.). Elsevier. ISBN 9780080541297. https://books.google.com/books?id=R5A65Koh-EoC.