Stochastic Integral with Time-Dependent Integrand

Question

The Ito integral $\int_0^t H_s dX_s$ is typically defined for predictable, locally bounded processes $H$ and continuous semimartingales $X$. I'm wondering whether one can make sense of a "stochastic convolution":

$$ \int_0^t H_{t - s} dX_s. \tag{1} $$

More generally, let $H\colon \Omega\times[0,\infty)^2\to \mathbb{R}$ be such that $(\omega, t) \mapsto H(\omega, r, t)$ is predictable and locally bounded for every $r \in [0, \infty)$. Then, for every such $r$,

$$ (\omega, t) \mapsto \int_0^t H(\omega, r, s) d X(\omega, s) $$

is a continuous semimartingale, and it remains to plug in $r=t$. What conditions do I need on $H$ such that

$$ (\omega, t) \mapsto \int_0^t H(\omega, t, s) d X(\omega, s) $$

is an adapted, continuous process? (Semimartingale?) Furthermore, can I adapt useful results, like Ito's Formula, to such integrals? If the general two-parameter case brings additional complications, for my particular use case it would be sufficient to be able to work with the convolutions in (1).

Any help or pointers would be greatly appreciated.

Answer 1

Under suitable assumptions, this stochastic convolution is well-defined if one undertands the integral in the rough path sense. For the idea of a situation where this applies, see for instance Theorem 3.5 in the paper by M. Hairer and A. Gerasimovics: https://arxiv.org/pdf/1811.06339.pdf

While the rough integral will generally keep some properties of Ito's, like continuity and adaptedness, other are lost like the martingale property. However, the rough integral has very nice continuity properties.