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 $f_0(t) f_1(t_1) \ldots f_n(t_n)$ ?**