Optimal hitting time and perpetual option in a non-Lévy model: Application to Real options

We consider a stochastic process S, the underlying dynamics of which involve a diffusion and a point process with a stochastic intensity: St = s0 (1 +Φ) Dt exp α− 1 2 σ t+ σWt where (Wt; t ≥ 0) is a Brownian motion, Φ is a random variable and the process (Dt; t ≥ 0) is defined by Dt = 1t≥T , with T is a random variable of exponential law. In such a framework, we aim at studying the following problem: Ct = ess sup τ∈Υt E e−μ(τ−t) (Sτ − 1) / Ft , (1) where (Ft) denotes the filtration specified by the available information, Υt denotes the set of Ft−stopping times taking values in (t,+∞[ and μ is a positive real number. This issue comes from real options, and more precisely from the investor’s problem to finance a project, when a (unique) sudden and drastic change can occur in the environment. No assumption is made on this shock that can seriously impact, positively or negatively, the project’s future cash flows and therefore the investment decision. Many situations can lead to a major brutal change in the project environment: for instance, a currency devaluation, an increase in interest rates, the finalization of brand-new technologies or the entry of a new competing firm. Mathematically speaking, all these situations are equivalent to consider a single jump diffusion process and this problem is similar to the study of a perpetual American call option written on the project profit/cost ratio S and with a strike equal to 1.