# Perpetual American Cancellable Standard Options in Models with Last Passage Times
## 2 Preliminaries
### The Optimal Stopping Problems  
In order to compute the expectations in (1), let us now introduce the conditional survival processes \(Z=(Z_{t})_{t\geq 0}\) and \(Y=(Y_{t})_{t\geq 0}\) defined by \(Z_{t}\ =\ P(\theta\ >\ t\,|\,\mathcal{F}_{t})\) and \(Y_{t}\ =\ P(\eta\ >\ t\,|\,\mathcal{F}_{t})\), for all \(t\geq 0\), respectively. Note that the processes \(Z\) and \(Y\) are called the Azema supermartingales of the random times \(\theta\) and \(\eta\) (see, e.g., [13] ([Section 1.2.1])). By using the fact that the running maximum of a Brownian motion with the drift coefficient \(r\ -\ \delta\ -\ \sigma^{2}/2\ <\ 0\) has an exponential distribution with the mean \(1/(\sigma^{2}\ +\ 2(\delta\ -\ r))\), while the running minimum of a Brownian motion with the drift coefficient \(r\ -\ \delta\ -\ \sigma^{2}/2\ >\ 0\)has an exponential distribution with the mean \(1/\left(2(r\ -\ \delta)\ -\ \sigma^{2}\right)\) (see, e.g., [14] ([Chapter II, Exercise 3.12])), we have:  
\[Z_{t}= \begin{cases}\left(h/X_{t}\right)^{\alpha}\wedge 1,&\text{if}\ \ \alpha\ <\ 0\\ 1,&\text{if}\ \ \alpha\geq 0\end{cases} \quad\text{and}\quad Y_{t}= \begin{cases} \left(g/X_{t}\right)^{\alpha}\wedge 1,&\text{if}\ \ \alpha\ >\ 0\\ 1,&\text{if}\ \ \alpha\leq 0\end{cases}  \tag{5}\]  
where we set \(\alpha\ =\ 2(r-\delta)/\sigma^{2}-1\). The representations in (5) can also be obtained by applying [PERSON]'s maximal equality (see, e.g., [13] ([Lemma 0.1]) or [14] ([Chapter II, Exercise 3.12])) to the process \(X^{-\alpha}\ =\ (X_{t}^{-\alpha})_{t\geq 0}\), which is a strictly positive continuous local martingale, and thus, a supermartingale converging to zero at infinity. Then, it follows from a direct application of the tower property for conditional expectations that the first terms in the right-hand sides of the expressions in (1) have the form:  
\[E\big{[}e^{-r\tau}\left(L-X_{\tau}\right)I(\tau\ <\ \theta)\big{]}\ =\ E\big{[}e^{-r\tau}\left(L-X_{\tau}\right) \left((h/X_{\tau})^{\alpha}\wedge 1\right)\big{]} \tag{6}\]  
when \(\alpha\ <\ 0\), and  
\[E\big{[}e^{-r\xi}\left(X_{\xi}-K\right)I(\zeta\ <\ \eta)\big{]}\ =\ E\big{[}e^{-r\xi}\left(X_{\xi}-K \right)\left((g/X_{\xi})^{\alpha}\wedge 1\right)\big{]} \tag{7}\]  
when \(\alpha\ >\ 0\), for any stopping times \(\tau\) and \(\zeta\) of the process \(X\). It is seen from the structure of the rewards in (1) with (6) and (7) that it is not optimal to exercise the perpetual American calculable put or call options, when \(X_{t}\ >\ L\) or \(X_{t}\ <\ K\) holds, for any \(t\geq 0\), respectively. By applying the change-of-variable formula from [12] ([Theorem 3.1]) (see also [4] ([Chapter II, Section 3.5]) for a summary of the related results and further references) to the process \(e^{-rt}G_{1}(X_{t})\) with \(G_{1}(x)\ =\ (L-x)((h/x)^{\alpha}\wedge 1)\), for \(x\ >\ 0\), we obtain the representation:  
\[e^{-rt}\,G_{1}(X_{t})\ =\ G_{1}(x) \tag{8}\] \[+\int_{0}^{t}e^{-ru}\left(G_{1}^{\prime}(X_{u})\left(r-\delta \right)X_{u}+G_{1}^{\prime\prime}(X_{u})\,\frac{\sigma^{2}X_{u}^{2}}{2}-r\,G_ {1}(X_{u})\right)I(X_{u}\
eq h)\,du\] \[+\int_{0}^{t}e^{-ru}\left(G_{1}^{\prime}(h+)-G_{1}^{\prime}(h-) \right)I(X_{u}\ =\ h)\,d\ell_{u}^{h}(X)+N_{t}^{1}\]  
when \(\alpha\ <\ 0\), for each \(x\ >\ 0\) and all \(t\geq 0\). Here, the process \(N^{1}=(N_{t}^{1})_{t\geq 0}\) given by:  
\[N_{t}^{1}=\int_{0}^{t}e^{-ru}\,G_{1}^{\prime}(X_{u})\left(h/X_{u}\right)^{ \alpha}I(X_{u}\
eq h)\,\sigma\,dB_{u} \tag{9}\]  
is a continuous uniformly integrable martingale under the probability measure \(P\), when \(\alpha\ <\ 0\), while the process \(\ell^{h}(X)\ =\ (\ell_{t}^{h}(X))_{t\geq 0}\) defined as the limit in probability by:  
\[\ell_{t}^{h}(X)\ =\ P-\lim_{\varepsilon\downarrow 0}\frac{1}{2\varepsilon} \int_{0}^{t}I\big{(}h-\varepsilon\ <\ X_{u}\ <\ h+\varepsilon\big{)}\,\sigma^{2}\,X_{u}^{2}\,du \tag{10}\]  
is the local time of the process \(X\) at the point \(h\).