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 $J^f_{n}(t)$

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

I would like an estimate on the tail of the supremum of an iterated Ito integral.

In the case of a constant integrand, this is fairly easy to find. For the integrand identically equal to 1, I define $$ I_{n}(t) = \int_0^t \int_0^{t_1} \ldots \int_0^{t_{n-1}} 1 \; dB_{t_n} ...dB_{t_1} $$ 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 integrand I am interested in is actually of the type $f(t, t_1, \ldots, t_n)$, so the upper limit in the integral appears in the integrand. That is, I want an estimate on the tail of. $$ \sup_{t \in (0,1]} J_{n}(t) = \sup_{t \in (0,1]} \, \int_0^t \int_0^{t_1} \ldots \int_0^{t_{n-1}} f(t, t_1, \ldots, t_n) \; dB_{t_n} ...dB_{t_1} $$

This can be written as $$ J_{n}(t) = \int_0^t J_{n-1}(t,t_1) \; dB_{t_1} $$ which is exactly like a Guassian Volterra process, but with a stochastic kernel. (Does this process have a name?) This process is not a martingale in general, nor (I think) is it a Gaussian process. I am therefore not sure what tools I can apply.

Has anyone encountered a process like this before? Can an inequality like the classical martingale one above be shown to apply in this situation?

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 $J^f_{n}(t)$