Let me define $$ J^f_{n}(t) = \, \int_0^t \int_0^{t_1} \ldots \int_0^{t_{n-1}} f(t, t_1, \ldots, t_n) \; dB_{t_n} ...dB_{t_1} $$ where $f:[0,1]^{n+1} \to \mathbb{R}$ is a nice deterministic function. Note that the upper limit of the outer integral appears in the integrand. I would like an estimate on the tail of $\sup_{t \in (0,1]} |J^f_n(t)|$ i.e. $$ \mathbb{P} \left( \sup_{t \in (0,1]} |J^f_n(t)| \geq K\right) \leq \; ? $$ Note that we can write $J^f_n(t)$ in the following form $$ J^f_n(t) = \int_0^t J^f_{n-1}(t,t_1) \; dB_{t_1} $$ which is precisely the form of a Gaussian Volterra process (see e.g. http://www.infres.enst.fr/~decreuse/recherche/volterra.pdf ) except here we have a stochastic kernel.

  • Has anyone encountered a process like this before? Does it have a name?

In the case of a constant integrand, such an estimate is fairly easy to find. For the integrand $f \equiv 1$, I define $$ I_{n}(t):= J^f_n(t) = \int_0^t \int_0^{t_1} \ldots \int_0^{t_{n-1}} 1 \; dB_{t_n} ...dB_{t_1} $$ then we can use the martingale inequality (found e.g. in proposition 4.24 here statslab.cam.ac.uk/~beresty/teach/StoCal/sc3.pdf ) $$ \mathbb{P} \left( \sup_{t \in (0,T]} |M_t| \geq K_1, \langle M_. \rangle _T \leq K_2\ \right) \leq 2 \exp \left( - \frac{K_1^2}{2K_2} \right) $$ recursively applied to $I_n(t)$, then $I_{n-1}(t)$ etc. to get a bound of the type $$ \mathbb{P} \left( \sup_{t \in (0,1]} |I_n(t)| \geq K\right) \leq C_n \exp \left( - K^{d_n} \right) $$ with $C_n$ and $d_n$ positive constants.

  • The problem for $J^f_n(t)$ is that, in general, it is not a martingale or a Guassian process

With this in mind, what tools can we use to analyse $J^f_{n}(t)$

If the function $f(t, t_1, \ldots, t_n)$ splits into $f_0(t)f_1(t_1)\ldots f_n(t_n)$ then we can pass each function outside one of the integrals and we are left with a martingale inside so this makes things considerably easier. This motivates the following question

  • Is there a theory for approximating general $f(t, t_1, \ldots, t_n)$ by products of functions $f_0(t) f_1(t_1) \ldots f_n(t_n)$ ?
  • 3
    $\begingroup$ Your $I_n$ is basically a Hermite function, by Ito's classical results on the "multiple Wiener integral". See, e.g. section 5.4.1 of books.google.com/books?id=V2BS_Dmp0XoC $\endgroup$ Commented Apr 9, 2013 at 20:40
  • $\begingroup$ Hi, thanks for the reference. It seems iterated/multiple Ito integrals with constant integrand are quite well understood. It is really when the integrand depends on the upper limit that things become harder. $\endgroup$
    – user31090
    Commented Apr 10, 2013 at 13:25

3 Answers 3


The estimate you are interested in has already been studied. The tail bound

$ \mathbb{P} \left( \sup_{t \in [0,1]} | J_n^f (t)| \ge K \right) \le C_1 \exp( -C_2 K^{2/n}) $

was proved by C. Borell. The main tools are infinite dimensional isoperimetric inequalities. It was later proved by M. Ledoux that we even have

$ \lim_{K \to \infty} \frac{1}{K^{2/n}}\log \mathbb{P} \left( \sup_{t \in [0,1]} | J_n^f (t)| \ge K \right) =-I_n(f) $

where $I_n(f)$ is known explicitly.

Michel Ledoux, A note on large deviations for Wiener chaos


enter link description here

Here is my book (385 p) about approximation of multiple Ito and Stratonovich stochastic integrals




Here is 2 my books (1000p in Russian and 385p in English) from 2017, where iterated Ito and Stratonovich stochastic integrals approximation is systematically considered by multiple Fourier-Legendre and trigonometric Fourier series. Expansions and exact mean-square errors of approximations is derived for integrals of multiplicity 2,3,4,5. General case (k, k in N) is also considered... The books are FREE.


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