Volterra integral equation

Question

I am studying the following Volterra equation \begin{equation} f(t) = g(t) +\int_0^t \mathrm d \tau\;g(\tau) f(t-\tau) \end{equation} which I would like to solve for $f(t)$, $g(t)$ being a known bounded and continuous function. I tackled the problem by substituting recursively the equation onto itself, to get: \begin{align} f(t) &= g(t) +\int_0^t\mathrm d t_1 \;g(t_1)g(t-t_1)+\int_0^t\mathrm d t_1\; g(t_1)\int_0^{t-t_1}\mathrm d t_2 \;g(t_2)f(t_1-t_2)=\cdots=\\ &=g(t) +\left.\sum_{n=1}^{+\infty}\int_0^t \mathrm d t_1\cdots \int_0^{t_{n-1}} \mathrm d t_n\; g(t_1)\cdots g(t_n)g(t_{n-1}-t_n)\right|_{t_0=t}= \\ &=g(t) +\left.\sum_{n=1}^{+\infty}\int_0^t\mathrm d t_1 \; \cdots \int_0^t\mathrm d t_n \;\left[\prod_{m=1}^n\theta(t_{m-1}-t_m) g(t_m)\right]g(t_{n-1}-t_n)\right|_{t_0=t}\label{T-} \end{align} At this point, I am not sure on how to proceed, as the last factor on the right does not allow to straightforwardly define a time-ordered exponential. Is it actually possible to sum the series, to get a closed form for the solution of $f(t)$? If so, any advise pointing to the right direction is very much appreciated.