First Exit and Optimization Problems for a CIR Diffusion Process

Let {X(t), t ≥ 0} be a CIR diffusion process, and τ (x) be the first time that X(t) = 0 or c, given that X(0) = x ∈ (0, c). First, we compute the moment-generating function and the expected value of τ (x). Then, an optimal control problem is considered for {X(t), t ≥ 0}. Finally, we add jumps to the diffusion process and we calculate in a particular case the probability that X(τ (x)) = 0, as well as the expected time needed to leave the interval (0, c). Explicit and exact results are obtained.