function $H(t)$ at the same time he or she evaluates the option price.

Perpetual American options {#S4}
==========================

We analyse now the problem of valuating perpetual American options, i.e. when we have $T -t_0\rightarrow \infty$. It can be objected that perpetual American options have limited practical interest since, in general, actual derivatives expire. In this sense, one can argue that they represent the limiting value of a far from maturity contract, and therefore they may help in the pricing process if the theoretical price cannot be computed [@BAW87]. On the other hand, the existence of a fixed expiration time is a feature which is not shared by systems coming from other branches of science. The results we introduce in this section may be thus of interest in different fields. We will stress this interpretation later.

The major simplification that perpetual American options bring is that the value of the boundary must be piece-like constant, $$H(t)=H_a {\mathbf 1}_{t\leq \tau} +H_b {\mathbf 1}_{t>\tau},$$ given that the problem is stationary. Then, since the process $S(t)$ is continuous, we will have: $$\begin{aligned}
P^{\pm}(t_0,{S_0};{\mathbb{K}})&=&X^{\pm}(H_a;{\mathbb{K}}) {\mathbb{E}}[e^{-r({t^{\pm}}_0-t_0)}{\mathbf 1}_{{t^{\pm}}_0\leq \tau}|{\mathcal{F}\/}(t_0)]\nonumber \\&+&X^{\pm}(H_b;{\mathbb{K}}) {\mathbb{E}}[e^{-r({t^{\pm}}_0-t_0)}{\mathbf 1}_{{t^{\pm}}_0 > \tau}|{\mathcal{F}\/}(t_0)], \label{Ppm}\end{aligned}$$ with ${t^{\pm}}_0={t^{\pm}}(t_0)$ and ${S_0}=S(t_0)$. The problem is simpler in the case of 