Assume a BM in 3d domain (infinite) with a small absorbing subdomain (cube, sphere, ect), centered at point $p_s=(x_s,y_s,z_s)$ . BM starts at point $p_0=(x_0,y_0,z_0)$ and when it riches the subdomain it vanishes.
I am interested in computing a pdf of first hitting times to this subdomain, assuming that there is a side killing process (independent of BM) with some given killing rate $λ$.
My steps to solve this problem are :
I can simulate my random walk and find discrete pdf of the hitting times to the sub domain $f_t(t)$. On the other hand, a probability that event $T(E)$ ( $T(E)=E_n−E_{n−1}$ -the inter-arrival time between two consecutive occurrences of the event) is not occurring by time t is $P(T(E)>t)=e^{−λt}$
In this case the probability that BM will rich the target and will not be killed is $f_t(t)e^{−λt}$ (?)
But, can I say this is a density function of the process I am interested in?
Thanks!
