I believe the Hawkes process conditional intensity $\lambda_t$ can be interpreted as an Ornstein-Uhlenbeck type process, but with jumps instead of the usual Brownian motion. Is this a correct interpretation? I cannot find this connection made explicit in the Hawkes literature.
For a very verbose derivation: Given the OU SDE $$ d\lambda_t = -\beta(\lambda_t - \mu) \ dt + \alpha \ dN_t $$ where $N_t$ is a jump process with unit jumps, multiply through by $e^{\beta t}$, apply Ito's lemma, and integrate $$ \begin{align*} e^{\beta t} \ d\lambda_t + \beta(\lambda_t - \mu) e^{\beta t} \ dt &= \alpha e^{\beta t} \ dN_t \\ \int_0^t d[e^{\beta t}(\lambda_t - \mu)] &= \int_0^t \alpha e^{\beta s} \ dN_s \\ \lambda_t &= e^{-\beta t}\lambda_0 + \mu + \int_0^t \alpha e^{-\beta (t-s)} \ dN_s \end{align*} $$ The integral can be rewritten as an explicit sum over the jumps, which are known, $$ \lambda_t = e^{-\beta t}\lambda_0 + \mu + \sum_{i: \ t_i < t} \alpha e^{-\beta(t-t_i)} $$ and assuming no initial impulse, $\lambda_0 = 0$, we recover the typical statement of a univariate Hawkes intensity.