Denote
Thanks to (3.26), we can define the free boundary of problem (1.4), at which it is optimal for the firm to call the bond, where
(see Figure 1). It is clear that
Theorem 4.1.
Denote . If , then , which means that
Proof.
Define
We claim that possess the following four properties:

(i)
,

(ii)
, for all ,

(iii)
, for all ,

(iv)
, a.e. in .
In fact, from the definition of , we have that
then property (i) is obvious.
Moreover, if , then we deduce
Combining , we have property (ii). It is easy to check property (iii) from the definition of . Next, we manifest property (iv) according to the following two cases. In the case of ,
In the other case of ,
So, we testify properties (i)–(iv). In the following, we utilize the properties to prove .
Otherwise, is nonempty; then we have that
Moreover, on the parabolic boundary of . According to the ABP maximum principle (see [20]), we have that
which contradicts the definition of . So, we achieve that .
Combining for any , it is clear that
which means that , , and for any
Theorem 4.2.
The free boundary is decreasing in the interval . Moreover, . And .
Proof.
(3.26) and (3.27) imply that
Hence, for any unit vector satisfying , the directional derivative of function along admits
that is, is increasing along the director . Combining the condition in , we know that is monotonically decreasing. Hence, exists, and we can define
Since , so . On the other hand, if , then
It is impossible because is continuous on .
In the following, we prove that is continuous in . If it is false, then there exists , such that (see Figure 4)
Moreover,
Differentiating (4.18) with respect to , then
On the other hand, for any in this case, and we know that by (3.26). Applying the strong maximum principle to (4.19), we obtain
So, we can define in . Considering and , we see that in , which contradicts that for any . Therefore .
Theorem 4.3.
There exists some such that for any and is strictly decreasing on .
Proof.
Define . In the first, we prove that . Otherwise, and for .
Recalling the initial value, we see that
Meanwhile, in the domain implies that for any (see Figure 4); then is not continuous at the point , which contradicts .
In the second, we prove that . In fact, according to Lemma 3.1, for any , hence, . If , then the free boundary includes a horizontal line , . Repeating the method in the proof of Theorem 4.2, then we can obtain a contradiction. So, .
At last, we prove that is strictly decreasing on . Otherwise, has a vertical part. Suppose that the vertical line is , then for any . Since is continuous across the free boundary, then for any . In this case, we infer that
On the other hand, in the domain , and satisfy, respectively,
Then the strong maximum principle implies that , which contradicts the second equality in (4.22).
Theorem 4.4.
for any with (see Figure 1), where
where is the positive characteristic root of , that is, the positive root of the algebraic equation
Proof.
Define
We claim that and possess the following three properties:
(i) for and for ,
(ii) in ,
(iii) in .
In fact, if we notice that , then we have that
It is obvious that . So, we obtain property (i).
Moreover, we compute
Hence, we have property (ii).
It is not difficult to check that, for any ,
Then we show property (iii).
Repeating the method in the proof of Theorem 3.2, we can derive that in from properties (i)(ii). And property (iii) implies that in the domain , which means that for any .
Next, we prove that . Otherwise, ; then the free boundary includes a horizontal line , . Repeating the method in the proof of Theorem 4.2, then we can obtain a contradiction. So, .
Theorem 4.5.
The free boundary .
Proof.
Fix and , and denote . According to Theorem 4.4, the free boundary while lies in the domain (see Figure 5).
In the first, we prove that there exists an such that
In fact, satisfy the equations
then the interior estimate of the parabolic equation implies that there exists a positive constant such that
here
On the other hand, we see that in from (3.27), and . Applying the strong maximum principle to , we deduce that
It means that is strictly decreasing on . It follows that, by ,
Moreover,
Employing the strong maximum principle, we see that there is a , such that
Provided that is small enough. Combining (4.32), there exists a positive such that
Next, we concentrate on problem (1.4) in the domain . It is clear that satisfies
And we can use the following problem to approximate the above problem:
Recalling (4.38), we see that, if is small enough, on the parabolic boundary of . Moreover, satisfies
Applying the comparison principle, we obtain
As the method in the proof of Theorem 3.3, we can show that weakly converges to in and (4.30) is obvious.
On the other hand, we see that in for any positive number from (3.26) and (3.27). So,
which means that there exists a uniform cone such that the free boundary should lies in the cone. As the method in [9], it is easy to derive that . Moreover can be deduced by the bootstrap method. Since is arbitrary and the free boundary is a vertical line while , then .