Sometimes I see stochastic integrals of the form: $$\int_{-\infty}^t f(t-u)dL_u,$$ where $L$ is a Lévy process. But I do not understand how this is defined. $L$ is just defined for positive times and $L_0=0$ so the filtration is such that $\mathcal{F}_0$ is trivial.
I have encountered this in many settings: Ornstein-Uhlenbeck processes, CARMA processes, Hawkes processes: $N_t$ with intensity $\lambda_t = \lambda + \int_{-\infty}^t g(t-u)dN_u$. Here, how is the latter integral to be understood if $N_u$ is just defined for $u\geq 0$?
Thank you very much in advance.