Consider this equation (convolution of two functions)
$dy(t)=f(\tau)g(t-\tau)d\tau$.
For the case when $g$ and $f$ are continuous and continuously differentiable functions, we can say that it is well-defined. However if one function say $f$ is a random variable/random noise (continuous but nondifferentiable) then can we write the above equation in the form of a controlled differential equation in higher dimension $d$:
$dY_t^i=\sum_{j = 1}^{d}V_j^i(Y_t)dX_t^j$.
In other words, are we solving some Controlled differential equation when one of the functions of a convolution is a random variable? The unknown noise can we model as a contribution from additional dimensions.
Edit: I realized that the question is not clear and it can be very specific. In many applications $y(t)$ is known and we need to estimate $f$. But generally, $f$ is added with Gaussian Noise so, I am wondering if we can formulate the equation like stochastic differential so that we can have a solution option.
EDIT Changed from Stochastic to Controlled as it is more appropriate.
EDIT It may be equivalent to say can we formulate each independent time sample as an additional space dimension?
The equation for $y$ is equivalent to $$ y(t)=y(0)+\int_0^t f(\tau)\,g(t-\tau)\,d\tau\,. $$ Differentiating with respect to $t$ yields $$ y'(t)=f(t)\,g(0)+\int_0^tf(\tau)\,g'(t-\tau)\,d\tau\,. $$ If you want to bring this into an SDE/ODE form $dy(t)=\mu(y(t),t)\,dt+\sigma(y(t),t)\,dB_t$ it is clear than you can choose $\sigma\equiv 0$ (which makes it an ODE). When $$ g(s)=e^{as}\,,\quad g'(s)=a\,g(s) $$ then clearly $$ y'(t)=f(t)+a\int_0^tf(\tau)\,g(t-\tau)\,d\tau=f(t)+a\,(y(t)-y(0))\,. $$ Therefore, for this particular $g$ you take $\mu(y,t)=f(t)+ay-ay(0)\,.$
Are these enough hints ? Perhaps a general $g$ can be handled using the Laplace transform.