# Intersection of a Poisson bridge and a Brownian bridge

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?

• Is it correct that you assume that $N$ and $W$ are independent processes? Otherwise nothing can be concluded. Sep 21, 2020 at 17:45
• @DieterKadelka yes, I forgot to mention that. I'll put that in now. Thank you.
– zab
Sep 21, 2020 at 17:53
• I found a result by Perry, Stadje and Zacks that calculates an integral equation satisfied by the distribution of first meeting time of a Brownian motion and an independent compound Poisson process (not bridges). I can't easily deduce my conjecture from there. I guess I could adapt their argument to come up with a new integral equation, but I foresee difficulties.
– zab
Sep 21, 2020 at 18:18
• (1/2) What you are after is the probability $q(a,t)$ that the Lévy process $X_t := W_t - N_t$ does not exceed level $a$ up to time $t$. This is the fundamental question studied in 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... Sep 21, 2020 at 19:53
• (2/2) ...to invert the Laplace transform and get a reasonable expression for $q(t,a)$. If you are interested, I can provide some entry points to literature, and, time permitting, have a look and see if there is any hope for workable expressions for this particular process. Sep 21, 2020 at 19:53