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.
if, for every and
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
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
is given by
with the well known modification for . Then the following theorem holds:
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.