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.