I have just took my first course of stochastic calculus and I am having trouble. The problem is the following:
Consider a firm that pays a coupon $\delta$ until the cashflow $d\pi_t = \mu \pi_t dt + \sigma \pi_t dZ_t$ hits a $\pi_{min}$ where it decides to close. We are interested in finding the value of the bond $P(\pi_0) = \mathbb{E}\Biggl[\int_0^\tau e^{-rs}\delta ds|\pi_0\Biggr]$ where $\tau= \min_{\{t\}} \{t>0:\pi_t = \pi_{min}\}$.
I read many chapters of the course bibliography but they all talk about theorems and cases where we are interested in finding an optimal stopping time or integrals where one of the limits is a stopping time but what we are integrating is a stochastic process. Here, I understand that we are integrating a deterministic function and the only stochastic object is the limit of the integral. I've been thinking about it for several days and I haven't been able to solve it. Another thing that confused me is that the title of the exercise is Feyman-Kac, but from what we learned in the course, Feyman-Kac equation allows us to find distributions of stochastic processes, and here what we actually want is to price a bond.