# Perpetual American Cancellable Standard Options in Models with Last Passage Times
## 4 Main Results and Proofs  
In this section, based on the previous computations, we formulate and prove our main results. **Theorem 1**. : _Let the processes \(X\) be given by (2), with some \(r\ >\ 0\), \(\delta\ >\ 0\), and \(\sigma\ >\ 0\), and the inequality \(\delta^{\prime}\equiv 2r-\delta-\sigma^{2}\ >\ 0\) be satisfied. Suppose that the random times \(\theta\) and \(\eta\) are defined by (4), which are honest times but not stopping times with respect to the natural filtration \((\mathcal{F}_{t})_{t\geq 0}\) of the process \(X\). Then, the value functions of the perpetual American cancellable standard put and call options from (20) and (21) admit the expressions:_  
\[V^{*}(x)= \begin{cases}V(x;a^{*}),&\text{if}\quad x\ >\ a^{*}\\ 0,&\text{if}\quad 0\ <\ x\leq a^{*}\end{cases}  \tag{45}\]  
_whenever \(\alpha\equiv 2(r-\delta)/\sigma^{2}-1\ <\ 0\), and_  
\[U^{*}(x)= \begin{cases}U(x;b^{*}),&\text{if}\quad 0\ <\ x\ <\ b^{*}\\ 0,&\text{if}\quad x\geq b^{*}\end{cases}  \tag{46}\]  
_whenever \(\alpha\ >\ 0\). Here, the function \(V(x;a)\) admits the representation of (37) and the optimal exercise boundary \(a^{*}\) is given by (43), whenever \(\alpha\ <\ 0\), while \(U(x;b)\) admits the representation of (38) and the optimal exercise boundary \(b^{*}\) is given by (43), whenever \(\alpha\ >\ 0\)._  
Since both parts of the assertion stated above are proved using similar arguments, we only give a proof for the case of the one-dimensional optimal stopping problem of (21) related to the perpetual American cancellable standard call options. Proof. : In order to verify the assertion stated above, it remains for us to show that the function defined in (46) coincides with the value function in (21) and that the stopping time \(\zeta^{*}\) in (22) is optimal with the boundary \(b^{*}\) specified above. Let us denote by \(U(x)\) the right-hand side of the expression in (46). Then, it is shown by means of straightforward calculations from the previous section that the function \(U(x)\) solves the right-hand system of (24)-(29). Observe that the function \(U(x)\) is \(C^{2}\) on the closures \(\{x\ >\ 0\,|\ 0\ <\ x\leq g\}\) and \(\{x\ >\ 0\,|\ g\leq x\leq b^{*}\}\), while it is equal to zero on the closed set \(\{x\ >\ 0\,|\ x\geq b^{*}\}\). Hence, by applying the change-of-variable formula from [12] ([Theorem 3.1]) to the process \(e^{-rt}U(X_{t})\), we obtain the expression:  
\[e^{-rt}\,U(X_{t}) =U(x)+\int_{0}^{t}e^{-ru}\,(\mathbb{L}U-rU)(X_{u})\,I(X_{u}\
eq g,X_{u}\
eq b)\,du+M_{t} \tag{47}\] \[\quad+\int_{0}^{t}e^{-ru}\,\left(U^{t}(g+)-U^{t}(g-)\right)\,I(X _{u}\ =\ g)\,d\mathcal{E}_{u}^{\mathcal{E}}(X)\]  
for all \(t\geq 0\). Here, the process \(M\ =\ (M_{t})_{t\geq 0}\) defined by:  
\[M_{t}\ =\ \int_{0}^{t}e^{-ru}\,U^{t}(X_{u})\,I(X_{u}\
eq g)\,\sigma\,X_{u}\, dB_{u} \tag{48}\]  
is a continuous local martingale with respect to the probability measure \(P_{x}\). Note that, since the time spent by the process \(X\) at the constant boundary \(b\) is of the Lebesgue measure zero (see, e.g., [16] ([Chapter II, Section 1])), the indicators in the first line of the expression of (47) as well as in the expression of (48) can be set equal to one. It follows from straightforward calculations and the arguments of the previous section that the function \(U(x)\) satisfies the second-order ordinary differential equation in (24), which together with the right-hand conditions of (25)-(27) as well as the fact that the right-hand inequality in (29) holds imply that the inequality \((\mathbb{L}U-rU)(x)\leq-H_{2}(x)\) is satisfied with \(H_{2}(x)\) given by (19), for all \(x\ >\ 0\) such that \(x\
eq b\). Moreover, we observe directly from the expressions in (38), (41) and (42) that the function \(U(x)\) is convex and decreasesto zero, because its first-order derivative \(U^{\prime}(x)\) is negative and increases to zero, while its second-order derivative \(U^{\prime\prime}(x)\) is positive, on the interval \(0\;<\;x\;<\;b\), under \(\alpha\;>\;0\).