Russo–Vallois integral

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

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

and to pull the limit out of the integral. In addition one changes the type of convergence.


Definition: A sequence of stochastic processes converges uniformly on compact sets in probability to a process

if, for every and

One sets:


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


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


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


For continuous semimartingales and a càdlàg function H, the Russo–Vallois integral coincidences with the usual Itô integral:

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

is equal to the quadratic variation process.

Also for the Russo-Vallois Integral an Ito formula holds: If is a continuous semimartingale and


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

is given by

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

Theorem: Suppose

Then the Russo–Vallois integral

exists and for some constant one has

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.


  • 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.
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