Suppose I have a stochastic process
$dS_t= rS_t dt + \sigma S_t dW_t + dJ_t$
where $W_t$ is a brownian motion and $J_t$ a compound poisson process of parameter $\lambda$ with lognormal jump size, $Y_i$~ $ \mathcal{LN}(\mu, \sigma_j)$.
Is there a way to find an analytical formula for the hitting time? The aim would be finding the best parameters for the process in order to match some data (calibration)
[EDIT] Here is where I stopped
$\tau = \inf\{t>0 : S_t = x\}$
where x is some barrier $\in R$
$Pr\left\{S_0 e^{(r-(1/2) \sigma^2)t + \sigma W_t + \sum_{i=0}^{N(t)} Y_i} = x \right\} =\\Pr \left\{(r-(1/2)\sigma^2)t + \sigma W_t + \sum_{i=0}^{N(t)}Y_i =\ln(x/S_0) \right\} = \\ Pr\left\{\sigma W_t + \sum_{i=0}^{N(t)}Y_i =\ln(x/S_0) - (r-(1/2)\sigma^2)t \right\}$
The problem is here...I don't know which distribution comes out in the left hand side