J Theor Probab (2017) 30:932–960
A Discrete-Time Clark–Ocone Formula and its
Application to an Error Analysis
· Takafumi Amaba
Received: 9 March 2015 / Revised: 12 November 2015 / Published online: 18 January 2016
© Springer Science+Business Media New York 2016
Abstract In this paper, we will establish a discrete-time version of Clark(–Ocone–
Haussmann) formula, which can be seen as an asymptotic expansion in a weak sense.
The formula is applied to the estimation of the error caused by the martingale rep-
resentation. Throughout, we use another distribution theory with respect to Gaussian
rather than Lebesgue measure, which can be seen as a discrete Malliavin calculus.
Keywords Discrete Clark–Ocone formula · Discrete Malliavin calculus · Sobolev
differentiability-index is the rate of convergence
Mathematics Subject Classiﬁcation (2010) Primary 60H07; secondary 60F05 ·
Let T > 0, (W
be a Brownian motion starting from 0, and (F
natural ﬁltration. Let X ∈ L
) be differentiable in the sense of Malliavin, for
which we may write X ∈ D
(see e.g., ). Then, it holds that
Jirô Akahori: The First author was supported by JSPS KAKENHI Grant Number 23330109, 24340022,
23654056, 25285102 and the project RARE-318984 (an FP7 Marie Curie IRSES).
Takafumi Amaba: This work was supported by JSPS KAKENHI Grant Number 24·5772.
Ritsumeikan University, 1-1-1 Nojihigashi, Kusatsu, Shiga 525-8577, Japan