Let $X_t$ be a compound Poisson process defined as $X_t = \sum_{i=1}^{N_t} D_i$, where $D_i$ are i.i.d. and $D_i \sim Exp(\mu)$ and $N_t$ is a Poisson process with parameter $\lambda$. As usual the Poisson process is independent from $D_i$. Let $\{t_0=0, t_1, \dots, t_N=T \}$ a time grid with constant step, i.e. $t_k = \frac{T k}{N}$. Is it possible to write in a closed form the cumulative distribution function of a single increment, i.e.

$F(x) = \mathbb{P}(X_{t_{k+1}} - X_{t_k} \leq x)$?

Thanks for every suggestion!


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