# jump-diffusion hitting time

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

• @ Vittorio Apicella : I don't think there is a closed form formula for this because the characteristic function doesn't look so good to me (maybe there might be some hope using some special functions but I am not versed enough in this of stuff to tell if it's worth the time), anyway by the Monte Carlo simulation it should be quite easy to get a good approximation in reasonable time. Best regards – TheBridge Jun 2 '15 at 9:06