Suppose $(I_t)_{t \in \mathbb{R}^+}$ and $(D_t)_{t \in \mathbb{R}^+}$ and $(C_t)_{t \in \mathbb{R}^+}$ are three dependent Poisson processes with parameters $\lambda_I$, $\lambda_D$, and $\lambda_C$ and with $I_0=0$, $D_0=0$, and $C_0=0$. Hence, the distributions of $I_t$, $D_t$, and $C_t$ are respectively Poisson($\lambda_I t$), Poisson($\lambda_D t$), Poisson($\lambda_C t$).
I have daily observations data on all three of them. Is there a way to model dependence between these continuous-time stochastic processes? Specifically, I need to find $P(I_t \leq n_1, D_t \leq n_2, C_t \leq n_3)$ for all $t \in \mathbb{R}^+$ and some fixed values of $n_1$, $n_2$, and $n_3$ (which are given).
I was thinking about somehow applying copula theory to model this dependence. With three random variable $X$, $Y$, and $Z$, we could calibrate some generic copula (e.g., Gaussian copula) to the data, or even estimate the empirical copula from the data. But here we don't have random variables but stochastic processes; the distributions of $I_t$, $D_t$, and $C_t$ change with $t$. Is there some way to generalize dependence modelling through copulas from random variables to stochastic processes?