Optimal local first exit time (Q1037017)

The authors study the properties of local first exit times from an interval \((-\infty, b)\) in the line by a generic class of processes on the real line. They relate such times to upcrossing times of the process of intervals of the type \((b-r, b)\), and show how to compute the expected value of a class of functionals that appear naturally in the theory of system reliability. Typical functionals are of the type \[ C -v E_x[\int_0^\tau\exp(-\int_0^t(X_s)\,ds)\,dt]-wE_x[\exp(-\int_0^\tau(X_s)\,ds)]. \] The problem of studying the nature of the local first exit time is the first step toward the computation of these functionals. Then comes the problem of determining the properties of the minimizing \(\tau\). They do this in some particular cases.