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?

  • 1
    $\begingroup$ Is it correct that you assume that $N$ and $W$ are independent processes? Otherwise nothing can be concluded. $\endgroup$ Sep 21, 2020 at 17:45
  • $\begingroup$ @DieterKadelka yes, I forgot to mention that. I'll put that in now. Thank you. $\endgroup$
    – zab
    Sep 21, 2020 at 17:53
  • $\begingroup$ 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. $\endgroup$
    – zab
    Sep 21, 2020 at 18:18
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    $\begingroup$ (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... $\endgroup$ Sep 21, 2020 at 19:53
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    $\begingroup$ (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. $\endgroup$ Sep 21, 2020 at 19:53


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