Take a Poisson process $N_t$, a Brownian motion $W_t$ and constants $T > 0$ and $a > 0$. Suppose $N$ and $W$ are independent. I'm interested in the probability that $W$ does not cross over $a + N_t$ during the interval $[0,T]$. To this end, I want to first calculate the above probability conditioned on $N_T = n$ and $W_T = x$, $(x < n)$.

Thus I want to calculate $$ P\bigl(W_t < a + N_t , \forall t \leq T \; \bigl| \; W_T = x, N_T = n \bigl). $$

Given that the conditioned on $N_T$, the jump times of $N$ are uniformly distributed over the interval $[0,T]$, and that the average path of the Poisson bridge is simply the straight line joining the points $(0,a)$ and $(T,n)$, my intuition is that this conditional probability will end up being the same as the probability that the Brownian bridge between $(0,0)$ and $(T,x)$ remains below the straight line joining $(0,a)$ and $(T,n)$.

Can someone suggest a way to approach this problem?

fluctuation theory of Lévy processes. Usually, things get pretty complicated here, and unless you are extremely lucky, explicit expressions are not available. A general approach is as follows. The bi-variate Laplace transform of $q(a,t)$ is given by the Pecherskii–Rogozin identity in terms of Wiener–Hopf factors. These are usually given in an integral form only. Nevertheless, it is sometimes possible... $\endgroup$5more comments