2.1. The optimal stopping problem

It is seen that the problem of (1.1) can be embedded into the optimal stopping problem for the (time-homogeneous strong) Markov process $(X, S, \Theta)=(X_t, S_t, \Theta_t)_{t \ge 0}$ with the value function

(2.1) \begin{equation} V^*_i(x, s) = \sup_{\tau} {\mathbb E}_{x, s, i} \big[e^{- r \tau} \, \big( (1 - \Theta_{\tau}) \, (L_0 \, S_{\tau} - K_0) + \Theta_{\tau} \, (L_1 \, S_{\tau} - K_1) \big) \big] \end{equation}

for some fixed $L_{i} > 0$ and $K_{i} \ge 0$, $i = 0, 1$, where the supremum is taken over all stopping times $\tau$ with respect to the natural filtration $(\mathcal{G}_t)_{t \ge 0}$ of the process $(X, \Theta)$. Here, $S=(S_t)_{t \ge 0}$ is running maximum process the associated with X, defined by

(2.2) \begin{equation} S_t = s \vee \Big( \max_{0 \le u \le t} X_u \Big) \end{equation}

for $s \ge x > 0$. We denote by ${\mathbb E}_{x, s, i}$ the expectation with respect to the probability measure ${\mathbb P}_{x, s, i}$ under which the three-dimensional (strong Markov) process $(X, S, \Theta)$ starts at $(x, s, i) \in E \times \{ 0, 1 \}$, and by $E = \{ (x, s) \in {\mathbb R}^2 \, | \, 0 < x \le s \}$ the state space of the process (X, S). We further assume that $\Theta$ has the transition-probability matrix $\{ (\lambda_0 e^{-(\lambda_0+\lambda_1) t} + \lambda_1)/(\lambda_0 + \lambda_1), \lambda_0 (1 - e^{-(\lambda_0+\lambda_1) t})/(\lambda_0 + \lambda_1); \lambda_1(1 - e^{-(\lambda_0+\lambda_1) t})/(\lambda_0 + \lambda_1), (\lambda_1 e^{-(\lambda_0+\lambda_1) t} + \lambda_0)/(\lambda_0 + \lambda_1) \}$, and the intensity matrix $\{ - \lambda_0, \lambda_0; \lambda_1, - \lambda_1 \}$, for all $t \ge 0$, and some $\lambda_i > 0$, $i = 0, 1$, fixed. In other words, the Markov chain $\Theta$, called a telegraphic signal process, changes its state from i to $1-i$ at exponentially distributed times of intensity $\lambda_i$, $i = 0, 1$ (see e.g. [31, Chapter IX, Section 4] or [14, Chapter VIII]). It can be assumed without loss of generality that $\lambda_i$ is the government subsidy implementation intensity and $\lambda_{1-i}$ is the abolishment intensity, for any $i = 0, 1$ fixed.

2.2. Structure of the optimal stopping time

Let us first determine the structure of the optimal stopping time in the problem of (2.1).

1. (i) By applying the arguments from [13, Subsection 3.2] and [35, Proposition 2.1] to the optimal stopping problem in (2.1), we see that it is never optimal to stop when $X_t = S_t$, for each $t \ge 0$ (this fact will be also re-proved independently in Part (iii) below). It also follows directly from the structure of (2.1) that it is never optimal to stop when $S_t \le K_i/L_i$, for all $t \ge 0$ and any $i = 0, 1$ fixed. In other words, this fact shows that all points (x, s) from the set

   (2.3) \begin{align} C_i^{\prime} &= \{ (x, s) \in E \, | \, 0 < x \le s \le K_i/L_i \} \end{align}
   and from the diagonal $\{ (x, s) \in {\mathbb R}^2 \, | \, 0 < x = s \}$ belong to the continuation region
   (2.4) \begin{align} C^*_i &= \{ (x, s) \in E \, | \, V^*_i(x, s) > L_i \, s - K_{i} \} \end{align}
   for every $i = 0, 1$. It is seen from the solution below that $V^*_i(x, s)$ is continuous, so that $C^*_i$ is open, for $i = 0, 1$.

2. (ii) It follows from the definition of the process (X, S) in (1.3)–(1.4) and (2.2) and the structure of the reward in (2.1) that, for each $s > K_i/L_i$ fixed, there exists a sufficiently small $x > 0$ such that the point (x, s) belongs to the stopping region

   (2.5) \begin{align} {D}^*_i &= \{ (x, s) \in E \, | \, V^*_i(x, s) = L_i \, s - K_{i} \} \end{align}
   for every $i = 0, 1$. This property can be explained by the fact that the costs of waiting until the process X coming from such a small $x > 0$ increases to the current value of the running maximum process S may be too large, because of the presence of the discounting factor in the reward functional in (2.1). Let us now denote by $\tau^* = \tau^*(x, s, i)$ the optimal stopping time in the problem of (2.1) and take another starting point ($x'$, s, i) for the process $(X, S, \Theta)$ such that $0 < x' < x < s$ and $s > K_i/L_i$, for any $i = 0, 1$ fixed. Then, using the fact that the running maximum S from (2.2) of the process X from (1.3) started at the point x’ is less than or equal to the running maximum started at x, by virtue of the linear structure of the reward functional in (2.1), we get $V^*_i(x', s) \le V^*_i(x, s) = L_i s - K_i$, so that $(x', s) \in {D}^*_i$, for $i = 0, 1$. On the other hand, we can take a point $(x, s) \in C^*_i$ and consider the associated optimal stopping time $\tau^* = \tau^*(x, s, i)$, for every $i = 0, 1$. Hence, for any other starting point ($x, s', i$) of the process $(X, S, \Theta)$ such that $0 < K_i/L_i \le x \le s' < s$, we obtain
   (2.6) \begin{align} &V^*_i(x, s') - (L_i \, s' - K_i)\\ \notag &\quad \ge {\mathbb E}_{x, s', i} \big[e^{- r \tau^*} \, \big( (1 - \Theta_{\tau^*}) \, (L_0 \, S_{\tau^*} - K_0) + \Theta_{\tau^*} \, (L_1 \, S_{\tau^*} - K_1) \big) \big] - (L_i \, s' - K_i) \\ \notag &\quad \ge {\mathbb E}_{x, s, i} \big[e^{- r \tau^*} \, \big( (1 - \Theta_{\tau^*}) \, (L_0 \, S_{\tau^*} - K_0) + \Theta_{\tau^*} \, (L_1 \, S_{\tau^*} - K_1) \big) \big] - (L_i \, s - K_i) \\ \notag &\quad = V^*_i(x, s) - (L_i \, s - K_i) > 0, \end{align}
   so that $(x, s') \in {C}^*_i$, for $i = 0, 1$. Thus, combining these arguments with the comments in [13, Subsection 3.3] and [35, Subsection 3.3], we may conclude that there exists a nondecreasing function $g^*_i(s)$, for $s > K_{i}/L_i$ and $i = 0, 1$, such that the continuation region $C^*_i$ in (2.4) is an open set consisting of $C_i^{\prime}$ from (2.3) and the set
   (2.7) \begin{align} &C^*_i \setminus C_i^{\prime} = \{ (x, s) \in E \, | \, g^*_i(s) < x \le s, s > K_{i}/L_i \}, \end{align}
   while the stopping region ${D}^*_i$ in (2.5) is a closed set of the form
   (2.8) \begin{align} &D^*_i = \{ (x, s) \in E \, | \, 0 < x \le g^*_i(s), s \ge K_{i}/L_i \} \end{align}
   for $i = 0, 1$. Note that the existence of such a boundary $g^*_i(s)$ can also be deduced from the convexity of the function $x \mapsto V^*_i(x, s)$ on (0, s), for each $s > K_i/L_i$ and every $i = 0, 1$.

3. (iii) Let us now determine the locations of the optimal stopping boundaries $g^*_i(s)$, $i = 0, 1$. For this purpose, we denote by $U^*_i(x, s)$ the value function of the optimal stopping problem in (5.1) below, which can be obtained from that of (2.1) by setting $K_{i} = 0$, so that $U^*_i(x, s) = W^*_i(x, s;\ 0, 0)$, with $W^*_i(x, s) \equiv W^*_i(x, s;\ K_0, K_1)$ defined in (5.1), for every $i = 0, 1$. The resulting problem reflects the extreme scenario in which the change of policy affects only the revenue of the firm, while the costs always remain constant at a level of zero. It is shown in [Reference Guo23] that the function $U^*_i(x, s) = W^*_i(x, s;\ 0, 0)$ with $W^*_i(x, s) \equiv W^*_i(x, s;\ K_0, K_1)$ from (5.1) admits the explicit expressions in (5.29) and (5.30) below, and the optimal stopping time has the structure $\eta^* = \inf \{t \ge 0 \, | \, X_t \le a^*_{\Theta_t} S_t \}$, where $a^*_i$ is defined by $a^*_i = b^*_i(0, 0)$, with $b^*_i \equiv b^*_i(K_0, K_1)$ determined from the expressions in (5.15) with (5.16), or (5.19) and (5.22), or (5.25) and (5.28), for every $i = 0, 1$ (see Theorem 5.1 below). Suppose that $a^*_i s < g^*_i(s)$ holds, for some $s > K_{i}/L_i$, $i = 0, 1$. Then, for each $x \in (a^*_i s, g^*_i(s))$ given and fixed, we would have $U^*_i(x, s) - K_{i} > L_i s - K_{i} = V^*_i(x, s)$, contradicting the obvious fact that $U^*_i(x, s) - K_{i} \le V^*_i(x, s)$ for all $(x, s) \in E$ with $s > K_{i}/L_i$, $i = 0, 1$, as is clearly seen from the structure of the payoff in (2.1). Thus, we may conclude that $g^*_i(s) \le a^*_i s < s$ should hold for all $s > K_{i}/L_i$ and every $i = 0, 1$ (see Figure 1 below for a computer drawing of the optimal stopping boundaries $g^*_i(s)$, $i = 0, 1$).

   Figure 1:

   A computer drawing of the optimal stopping boundaries $g^*_j(s)$, $j = 0, 1$.

   

4. (iv) In order to further characterise the asymptotic behaviour of the boundaries $g^*_i(s)$, $i = 0, 1$, we observe that for the value functions $V^*_i(x, s)$ from (2.1) and $U^*_i(x, s) = W^*_i(x, s;\ 0, 0)$ with $W^*_i(x, s) \equiv W^*_i(x, s;\ K_0, K_1)$ from (5.1), the inequalities

   (2.9) \begin{align} &0 \le U^*_i(x, s) - K_{i} \le V^*_i(x, s) \le U^*_i(x, s) \end{align}
   are satisfied, for all $(x, s) \in E$ such that $s > K_{i}/L_i$, $i = 0, 1$, so that the inequalities
   (2.10) \begin{equation} 0 \le \frac{U^*_i(s, s)}{s} - \frac{K_i}{s} \le \frac{V^*_i(s, s)}{s} \le \frac{U^*_i(s, s)}{s} \end{equation}
   hold for all $s > K_i/L_i$ and every $i = 0, 1$. Therefore, letting s go to infinity in (2.10), we obtain
   (2.11) \begin{equation} \liminf_{s \to \infty} \frac{V^*_i(s, s)}{s} = \limsup_{s \to \infty} \frac{V^*_i(s, s)}{s} = \lim_{s \to \infty} \frac{U^*_i(s, s)}{s} \end{equation}
   for $i = 0, 1$.

2.3. The free-boundary problem

By means of standard arguments based on an application of Itô’s formula, it is shown that the infinitesimal operator ${\mathbb L}$ of the process (X, S) from (1.3)–(1.4) and (2.2) acts on an arbitrary function F(x, s) from the class $C^{2,1}$ on E according to the rule

(2.12) \begin{align}({\mathbb L} F)(x, s) &= \mu \, x \, \partial_x F(x, s)+ \frac{\sigma^2 x^2}{2} \, \partial_{xx} F(x, s)\end{align}

for all $0 < x < s$. In order to find analytic expressions for the unknown value functions $V^*_i(x, s)$ from (2.1) and the unknown boundaries $g^*_i(s)$, $i = 0, 1$, from (2.7)–(2.8), let us use the results of the general theory of optimal stopping problems for Markov processes (see e.g. [39, Chapter IV, Section 8]) and for maximum processes (see e.g. [39, Chapter V, Sections 15–20] and references therein). We can therefore reduce the optimal stopping problem of (2.1) to the equivalent free-boundary problem:

(2.13) \begin{align} &({{\mathbb L}} V_i - (r + \lambda_i) \, V_i)(x, s) = - \lambda_i \, V_{1-i}(x, s) \quad \text{for } (x, s) \in C_i \text{ such that } x \neq s, \end{align}

(2.14) \begin{align} &V_i(x, s) \big|_{x = g_i(s)+} = L_i \, s - K_{i} \quad \text{({\it instantaneous stopping})}, \end{align}

(2.15) \begin{align} &\partial_x V_i(x, s) \big|_{x = g_i(s)+} = 0 \quad \text{({\it smooth fit})}, \end{align}

(2.16) \begin{align} &\partial_s V_i(x, s) \big|_{x = s-} = 0 \quad \text{({\it normal reflection})}, \end{align}

(2.17) \begin{align} &V_i(x, s) = L_i \, s - K_{i} \quad \text{for } (x, s) \in D_i, \end{align}

(2.18) \begin{align} &V_i(x, s) > L_i \, s - K_{i} \quad \text{for } (x, s) \in C_i, \end{align}

(2.19) \begin{align} &({{\mathbb L}} V_i - (r + \lambda_i) \, V_i)(x, s) < - \lambda_i \, V_{1-i}(x, s) \quad \text{for } (x, s) \in D_i, \end{align}

where $C_i$ and $D_i$ are defined as $C^*_i$ and $D^*_i$ in (2.3) and (2.7)–(2.8) with $g_i(s)$ instead of $g^*_i(s)$, $i = 0, 1$. Observe that the superharmonic characterisation of the value function (see e.g. [39, Chapter IV, Section 9]) implies that the $V^*_i(x, s)$ are the smallest functions satisfying (2.13)–(2.14) and (2.17)–(2.18) with the boundaries $g^*_i(s)$, $i = 0, 1$. Here, the condition of (2.14) is satisfied for all $s \ge K_{i}/L_i$, while the conditions of (2.15)–(2.16) are satisfied for all $s > K_{i}/L_i$ and any $i = 0, 1$ fixed.

3.1. The solution to the problem in the case $\lambda_i > 0$, $i = 0, 1$

Let us first assume that $\lambda_i > 0$, $i = 0, 1$, holds. In this case, following the arguments of [12, pp. 303–309], we introduce the functions ${\overline V}_0(x, s) = V_0(x, s)/\lambda_0 + V_1(x, s)/\lambda_1$ and ${\overline V}_1(x, s) = V_{1}(x, s) - V_0(x, s)$, for all $0 < x \le s$.

1. (i) It is shown by means of straightforward calculations that the functions ${\overline V}_{j}(x, s)$, $j = 0, 1$, satisfy the system of second-order ordinary differential equations

   (3.1) \begin{equation} \mu \, x \, \partial_x {\overline V}_j(x, s) + \frac{\sigma^2 x^2}{2} \, \partial_{xx} {\overline V}_j(x, s) - (r + (\lambda_0 + \lambda_1) \, j) \, {\overline V}_j(x, s) = 0 \end{equation}
   for all $g_i(s) < x < s$ and $s > K_{1-i}/L_{1-i}$, and any $i = 0, 1$ fixed. It follows that the equations in (3.1) admit the general solutions
   (3.2) \begin{equation} {\overline V}_j(x, s) = {\overline C}_{j,1}(s) \, x^{\gamma_{j,1}} + {\overline C}_{j,2}(s) \, x^{\gamma_{j,2}} \end{equation}
   for $g_i(s) < x < s$ and $s > K_{1-i}/L_{1-i}$, where the ${\overline C}_{j,k}(s)$ are some arbitrary (continuously differentiable) functions, and the $\gamma_{j, k}$, $k = 1, 2$, are explicitly given by
   (3.3) \begin{equation} \gamma_{j, k} = \frac{1}{2} - \frac{\mu}{\sigma^2} - (-1)^k \sqrt{ \Big( \frac{1}{2} - \frac{\mu}{\sigma^2} \Big)^2 + \frac{2(r + (\lambda_0 + \lambda_1) j)}{\sigma^2}}, \end{equation}
   so that $\gamma_{j, 2} < 0 < 1 < \gamma_{j, 1}$ holds, for $j = 0, 1$. Then the general solution for the system of second-order ordinary differential equations in (2.13) with (2.12) has the form
   (3.4) \begin{equation} V_j(x, s) = \sum_{k = 1}^2 \big( \lambda_{j} \, C_{j, k}(s) \, x^{\gamma_{j, k}} - (-1)^j \, C_{1-j, k}(s) \, x^{\gamma_{1-j, k}} \big) \end{equation}
   for all $g_i(s) < x < s$ and $s > K_{1-i}/L_{1-i}$, where the ${C}_{j,k}(s)$ are some arbitrary (continuously differentiable) functions, for every $j = 0, 1$ and $k = 1, 2$. By virtue of the fact that $V_i(x, s) = L_{i} s - K_{i}$ for $g_{1-i}(s) < x \le g_i(s)$, it is seen that the general solution of the equation in (2.13) with (2.12) for $V_{1-i}(x, s)$ takes the form
   (3.5) \begin{equation} V_{1-i}(x, s) = D_{1-i,1}(s) \, x^{\beta_{1-i, 1}} + D_{1-i,2}(s) \, x^{\beta_{1-i, 2}} + \lambda_{1-i} \, (L_i \, s - K_i)/(r + \lambda_{1-i}) \end{equation}
   for $g_{1-i}(s) < x < g_i(s)$ and $s > K_{1-i}/L_{1-i}$, where the $D_{1-i,k}(s)$ are some arbitrary (continuously differentiable) functions, and the $\beta_{1-i, k}$, $k = 1, 2$, are explicitly given by
   (3.6) \begin{equation} \beta_{1-i, k} = \frac{1}{2} - \frac{\mu}{\sigma^2} - (-1)^k \sqrt{ \Big( \frac{1}{2} - \frac{\mu}{\sigma^2} \Big)^2 + \frac{2(r + \lambda_{1-i})}{\sigma^2}}, \end{equation}
   so that $\beta_{1-i, 2} < 0 < 1 < \beta_{1-i, 1}$ holds, for $i = 0, 1$. Hence, by applying the conditions of (2.14)–(2.15) to the functions in (3.4) and (3.5), we obtain the equalities
   (3.7) \begin{align}&\sum_{k = 1}^2 \Big( \lambda_{i} \, {C}_{i,k}(s) \, g^{\gamma_{i, k}}_{i}(s) - (-1)^{i}\, {C}_{1-i,k}(s) \, g^{\gamma_{1-i, k}}_{i}(s) \Big) = L_i \, s - K_i, \end{align}
   (3.8) \begin{align}&\sum_{k = 1}^2 \Big( \lambda_{i} \, {C}_{i,k}(s) \, \gamma_{i, k} \, g^{\gamma_{i, k}}_{i}(s)- (-1)^{i} \, {C}_{1-i,k}(s) \, \gamma_{1-i, k} \, g^{\gamma_{1-i, k}}_{i}(s) \Big) = 0, \end{align}
   (3.9) \begin{align}\notag&D_{1-i,1}(s) \, g^{\beta_{1-i, 1}}_{1-i}(s) + D_{1-i,2}(s) \, g^{\beta_{1-i, 2}}_{1-i}(s)+ \lambda_{1-i} \, (L_i \, s - K_i)/(r + \lambda_{1-i}) \\&\quad = L_{1-i} \, s - K_{1-i}, \end{align}
   (3.10) \begin{align}&D_{1-i,1}(s) \, \beta_{1-i, 1} \, g^{\beta_{1-i, 1}}_{1-i}(s)+ D_{1-i,2}(s) \, \beta_{1-i, 2} \, g^{\beta_{1-i, 2}}_{1-i}(s) = 0,\end{align}

   for $s > K_{1-i}/L_{1-i}$ and any $i = 0, 1$ fixed. Moreover, since the function in (3.5) should be (at least) twice continuously differentiable at the boundary $g_i(s)$, we get the equalities

   (3.11) \begin{align}\notag&\sum_{k = 1}^2 \Big( \lambda_{1-i} \, {C}_{1-i,k}(s) \, g^{\gamma_{1-i, k}}_i(s)- (-1)^{1-i} \, {C}_{i,k}(s) \, g^{\gamma_{i, k}}_i(s) \Big) \\&\qquad = D_{1-i,1}(s) \, g^{\beta_{1-i, 1}}_i(s) + D_{1-i,2}(s) \, g^{\beta_{1-i, 2}}_i(s) +\lambda_{1-i} \, (L_i \, s - K_i)/(r + \lambda_{1-i}), \end{align}
   (3.12) \begin{align}\notag&\sum_{k = 1}^2 \Big( \lambda_{1-i} \, {C}_{1-i,k}(s) \, \gamma_{1-i, k} \, g^{\gamma_{1-i, k}}_i(s) -(-1)^{1-i} \, {C}_{i,k}(s) \, \gamma_{i, k} \, g^{\gamma_{i, k}}_i(s) \Big), \\&\qquad = D_{1-i,1}(s) \, \beta_{1-i, 1} \, g^{\beta_{1-i, 1}}_i(s) + D_{1-i,2}(s) \, \beta_{1-i, 2} \,g^{\beta_{1-i, 2}}_i(s), \end{align}
   (3.13) \begin{align}\notag&\sum_{k = 1}^2 \Big( \lambda_{1-i} \, {C}_{1-i,k}(s) \, \gamma_{1-i, k} (\gamma_{1-i, k}-1) \, g^{\gamma_{1-i, k}}_i(s) - (-1)^{1-i} \, {C}_{i,k}(s) \, \gamma_{i, k} (\gamma_{i, k}-1) \, g^{\gamma_{i, k}}_i(s) \Big) \\\notag&\qquad = D_{1-i,1}(s) \, \beta_{1-i, 1} (\beta_{1-i, 1}-1) \, g^{\beta_{1-i, 1}}_i(s)\\& \qquad\quad + D_{1-i,2}(s) \, \beta_{1-i, 2} (\beta_{1-i, 2}-1) \, g^{\beta_{1-i, 2}}_i(s),\end{align}

   for $s > K_{1-i}/L_{1-i}$. Finally, by applying the condition of (2.16) to the functions in (3.4), we conclude that the equalities

   (3.14) \begin{align}&\sum_{k = 1}^2 \Big( \lambda_{j} \, {C}_{j,k}^{\prime}(s) \, s^{\gamma_{j, k}}- (-1)^{j} \, {C}_{1-j,k}^{\prime}(s) \, s^{\gamma_{1-j, k}} \Big) = 0\end{align}
   hold for all $s > K_{1-i}/L_{1-i}$ and every $j = 0, 1$.

   Solving the system of equations in (3.9)–(3.10), we obtain that the candidate value function admits the representation

   (3.15) \begin{align} &V_{1-i}(x, s;\ g_{1-i}(s)) = \sum_{k = 1}^2 D_{1-i,k}(s;\ g_{1-i}(s)) \, x^{\beta_{1-i, k}} + \frac{\lambda_{1-i} (L_i s - K_i)}{r+\lambda_{1-i}}\nonumber\\[-18pt] \end{align}
   for $g_{1-i}(s) < x \le g_i(s)$ and $s > K_{1-i}/L_{1-i}$, with
   (3.16) \begin{align} &D_{1-i,k}(s;\ g_{1-i}(s)) = \frac{\beta_{1-i,3-k} (L_{1-i}^{\prime} s - K_{1-i}^{\prime})}{(\beta_{1-i,3-k} - \beta_{1-i,k}) g_{1-i}^{\beta_{1-i,k}}(s)}\nonumber\\[-18pt] \end{align}
   for $k = 1, 2$, where we set $L_{1-i}^{\prime} = L_{1-i} - {\lambda}_{1-i} L_i/(r + {\lambda}_{1-i})$ and $K_{1-i}^{\prime} = K_{1-i} - {\lambda}_{1-i} K_i/(r + {\lambda}_{1-i})$, for any $i = 0, 1$ fixed. Then, solving the system of equations in (3.7)–(3.8) and (3.11)–(3.12), we obtain that the candidate value functions admit the representations
   (3.17) \begin{align} \notag &{\hspace*{-4pt}}V_j(x, s;\ g_i(s), g_{1-i}(s)) \\ &{\hspace*{-4pt}}\quad = \sum_{k = 1}^2 \Big( \lambda_{j} \, {C}_{j, k}(s;\ g_i(s), g_{1-i}(s)) \, x^{\gamma_{j, k}} - (-1)^{j} \, {C}_{1-j, k}(s;\ g_{i}(s), g_{1-i}(s)) \, x^{\gamma_{1-j, k}} \Big)\end{align}
   for $g_i(s) < x \le s$ and $s > K_{1-i}/L_{1-i}$, and every $j = 0, 1$, where the functions ${C}_{j,k}(s;\ g_i(s), g_{1-i}(s))$, $j = 0, 1$, $k = 1, 2$, admit the representations
   (3.18) \begin{align} \notag C_{i, k}(s;\ g_i(s), g_{1-i}(s)) &= (-1)^i \big/ \big( (\lambda_0 \lambda_1 + 1) (\gamma_{i, 3-k} - \gamma_{i, k}) g_i^{\gamma_{i, k}}(s) \big) \\ \notag &\quad \times \bigg( \sum_{l = 1}^2 D_{1-i, l}(s;\ g_{1-i}(s)) (\gamma_{i, 3-k} - \beta_{1-i, l}) g_i^{\beta_{1-i, l}}(s) \\ &\phantom{\times \bigg( \;\:} + \frac{\lambda_{1-i} (1 + (-1)^i (r + \lambda_{1-i})) \gamma_{i, 3-k} (L_i s - K_i)}{r + \lambda_{1-i}} \bigg)\nonumber\\[-18pt] \end{align}
   and
   (3.19) \begin{align} \notag &C_{1-i, k}(s;\ g_i(s), g_{1-i}(s)) = \lambda_i \big/ \big( (\lambda_0 \lambda_1 + 1) (\gamma_{1-i, 3-k} - \gamma_{1-i, k}) g_i^{\gamma_{1-i, k}}(s) \big) \\ \notag &\quad \times \bigg( \sum_{l = 1}^2 D_{1-i, l}(s;\ g_{1-i}(s)) (\gamma_{1-i, 3-k} - \beta_{1-i, l}) g_i^{\beta_{1-i, l}}(s) \\ &\phantom{\times \bigg( \;\:} + \frac{(\lambda_0 \lambda_1 - (-1)^i (r + \lambda_{1-i})) \gamma_{1-i, 3-k} (L_i s - K_i)}{\lambda_i (r + \lambda_{1-i})} \bigg)\nonumber\\[-18pt] \end{align}
   with $D_{1-i, k}(s;\ g_{1-i}(s))$, $k = 1, 2$, from (3.16). Hence, we may conclude that the candidate boundaries $g_j(s)$, $j = 0, 1$, satisfy the coupled system of first-order nonlinear ordinary differential equations in (3.14) with
   (3.20) \begin{align} \notag &C_{i, k}^{\prime}(s;\ g_i(s), g_i^{\prime}(s), g_{1-i}(s), g_{1-i}^{\prime}(s)) = - C_{i, k}(s;\ g_i(s), g_{1-i}(s)) \gamma_{i, k} \big( g_i^{\prime}(s) \big/ g_i(s) \big) \\ \notag &\quad +\frac{(-1)^i}{(\lambda_0 \lambda_1 + 1) (\gamma_{i, 3-k} - \gamma_{i, k}) g_i^{\gamma_{i, k}}(s)} \bigg( \frac{(1 + (-1)^i (r + \lambda_{1-i})) \gamma_{i, 3-k} L_i}{r + \lambda_{1-i}} \\ \notag &\quad + \sum_{l = 1}^2 \bigg( D_{1-i, l}^{\prime}(s;\ g_{1-i}(s), g_{1-i}^{\prime}(s)) + D_{1-i, l}(s;\ g_{1-i}(s)) \beta_{1-i, l} \frac{g_i^{\prime}(s)}{g_i(s)} \bigg) \\ &\phantom{+ \sum_{l = 1}^2 \bigg( \;\:} \times (\gamma_{i, 3-k} - \beta_{1-i, l}) g_i^{\beta_{1-i, l}}(s) \bigg) \end{align}
   and
   (3.21) \begin{align} \notag &C_{1-i, k}^{\prime}(s;\ g_i(s), g_i^{\prime}(s), g_{1-i}(s), g_{1-i}^{\prime}(s)) \\ \notag &\quad = - C_{1-i, k}(s;\ g_i(s), g_{1-i}(s)) \gamma_{1-i, k} \big( g_i^{\prime}(s) \big/ g_i(s) \big) \\ \notag &\quad +\frac{\lambda_i}{(\lambda_0 \lambda_1 + 1) (\gamma_{1-i, 3-k} - \gamma_{1-i, k}) g_i^{\gamma_{1-i, k}}(s)} \bigg( \frac{(\lambda_0 \lambda_1 - (-1)^i (r + \lambda_{1-i})) \gamma_{1-i, 3-k} L_i}{\lambda_i (r + \lambda_{1-i})} \\ \notag &\quad+ \sum_{l = 1}^2 \bigg( D_{1-i, l}^{\prime}(s;\ g_{1-i}(s), g_{1-i}^{\prime}(s)) + D_{1-i, l}(s;\ g_{1-i}(s)) \beta_{1-i, l} \frac{g_i^{\prime}(s)}{g_i(s)} \bigg) \\ &\quad\phantom{+ \sum_{l = 1}^2 \bigg( \;\:} \times (\gamma_{1-i, 3-k} - \beta_{1-i, l}) g_i^{\beta_{1-i, l}}(s) \bigg), \end{align}
   as well as
   (3.22) \begin{align} \notag &D_{1-i,k}^{\prime}(s;\ g_{1-i}(s), g_{1-i}^{\prime}(s)) \\[3pt] &= \frac{\beta_{1-i,3-k} (L_{1-i}^{\prime} - \beta_{1-i, k} (L_{1-i}^{\prime} s - K_{1-i}^{\prime}) (g_{1-i}^{\prime}(s)/g_{1-i}(s)))} {(\beta_{1-i,3-k} - \beta_{1-i,k}) g_{1-i}^{\beta_{1-i,k}}(s)} \end{align}
   for $s > K_{1-i}/L_{1-i}$, where $D_{1-i, k}(s;\ g_{1-i}(s))$, and $C_{j, k}(s;\ g_i(s), g_{1-i}(s))$, for $i, j = 0, 1$ and $k = 1, 2$, are given by the expressions in (3.16) and (3.18)–(3.19), respectively. It is seen that the system of equations in (3.14) is degenerated in the sense that the candidate boundary $g_{i}(s)$ solves the appropriate single first-order nonlinear ordinary differential equation, where the boundary $g_{1-i}(s)$ can be uniquely expressed in terms of $g_{i}(s)$, by means of the arithmetic power equation in (3.13), for any $i = 0, 1$ fixed. In order to characterise the asymptotic behaviour of the candidate boundaries $g_j(s)$, $j = 0, 1$, we observe that the expressions in (3.15) and (3.17) imply the representations
   (3.23) \begin{align} &\frac{V_{1-i}(s, s;\ g_{1-i}(s))}{s} = \sum_{k = 1}^2 {\widetilde D}_{1-i, k}(s;\ h_{1-i}(s)) \, h^{- \beta_{1-i,k}}_{1-i}(s) + \frac{\lambda_{1-i} (L_i - K_i/s)}{r+\lambda_{1-i}} \end{align}
   and
   (3.24) \begin{align} \notag &{V_{i}(s, s;\ g_i(s), g_{1-i}(s))}/{s} \\ &\quad = \sum_{k=1}^2 \Big( \lambda_{i} \, {\widetilde C}_{i, k}(s;\ h_i(s), h_{1-i}(s)) \, h^{-\gamma_{i, k}}_i(s) - (-1)^{i} \, {\widetilde C}_{1-i, k}(s;\ h_{i}(s), h_{1-i}(s)) \, h^{-\gamma_{1-i, k}}_i(s) \Big) \end{align}
   with some appropriately chosen ${\widetilde D}_{1-i, k}(s;\ h_{1-i}(s))$ and ${\widetilde C}_{j, k}(s;\ h_i(s), h_{1-i}(s))$, for $i, j = 0, 1$ and $k = 1, 2$, which are of constant signs as $s \to \infty$, where we set $h_j(s) = g_j(s)/s$, for all $s > K_j/L_j$ and every $j = 0, 1$. Then, letting s go to infinity in (3.23) and (3.24) and taking into account the assumption that $h_j(s) \le a^*_j$ for all $s > K_j/L_j$, so that ${\underline h}_j \le {\overline h}_j \le a^*_j < 1$, we may conclude from the expressions in (2.11) that ${\underline h}_j = {\overline h}_j = a^*_j$ should hold, where we set ${\underline h}_j = {\liminf}_{s \to \infty} h_j(s)$ and ${\overline h}_j = {\limsup}_{s \to \infty} h_j(s)$ for every $j = 0, 1$. Hence, we obtain that the candidate boundaries $g_j(s)$, $j = 0, 1$, should satisfy the property
   (3.25) \begin{equation} \lim_{s \to \infty} \frac{g_j(s)}{s} = a^*_j, \end{equation}
   which gives the initial conditions (at infinity) for the equations in (3.14) with (3.20)–(3.21) and (3.22). We also note that the inequalities in (2.9) applied to the functions in (3.15) and (3.17) yield that the candidate boundary $g_{1-i}(s)$ should satisfy the properties
   (3.26) \begin{align} & g_{1-i}((K_{1-i}/L_{1-i})+) = 0 \quad \text{and}\nonumber\\ &\quad g_{1-i}(s) \sim A_{1-i} (L_{1-i} s - K_{1-i})^{1/\beta_{1-i,1}} \quad \text{as} \quad s \downarrow K_{1-i}/L_{1-i} \!\!\! \end{align}
   with some constant $A_{1-i} > 0$ which is also specified by means of the condition of (3.25) above.

   We further consider the maximal admissible solutions $(g^*_i(s), g^*_{1-i}(s))$ of the system of first-order nonlinear ordinary differential equations in (3.14) with (3.20)–(3.21) and (3.22), as the largest possible solutions which satisfy the inequalities $0 < g^*_j(s) \le a^*_j s$, $j = 0, 1$, for all $s > K_{1-i}/L_{1-i}$ and every $i = 0, 1$. By virtue of the classical results on the existence and uniqueness of solutions for systems of first-order nonlinear ordinary differential equations, we may conclude that the system of equations in (3.14) with (3.20)–(3.21) and (3.22) admits (locally) unique solutions, in view of the facts that the Jacobian matrix associated with this implicit system is nonsingular and the right-hand sides in the resulting normal (canonical) form corresponding to this system are (locally) continuous in $(s, g_i(s), g_{1-i}(s))$ and (locally) Lipschitz in $g_i(s)$ and $g_{1-i}(s)$, for each $s > K_{1-i}/L_{1-i}$ fixed, and every $i = 0, 1$ (see e.g. [35, Subsection 3.9]). It is then shown by means of technical arguments based on Picard’s method of successive approximations that there exists a unique solution $(g_i(s), g_{1-i}(s))$ to the system of equations in (3.14) with (3.20)–(3.21) and (3.22), for $s > K_{1-i}/L_{1-i}$, started at some point $(s_{i,0}, a^*_i s_{i,0}, a^*_{1-i} s_{i,0})$ such that $s_{i,0} > K_{1-i}/L_{1-i}$, for any $i = 0, 1$ fixed (see also [22, Subsection 3.2] and [35, Example 4.4] for similar arguments based on the analysis of other single first-order nonlinear ordinary differential equations). Hence, to construct the appropriate functions $g^*_j(s)$, $j = 0, 1$, which satisfy the system of equations in (3.14) with (3.20)–(3.21) and (3.22) and stay strictly below the lines $a^*_j s$, $j = 0, 1$, we can follow the arguments from [38, Subsection 3.5] (among others), which are based on the construction of sequences of so-called bad-good solutions which intersect those lines. For this purpose, for any sequences $(s_{j,l})_{l \in {\mathbb N}}$ such that $s_{j,l} > K_{1-i}/L_{1-i}$ and $s_{j,l} \uparrow \infty$ as $l \to \infty$, for every $j = 0, 1$, we can construct the sequence of solutions $(g_{i,l}(s), g_{1-i,l}(s))$, $l \in {\mathbb N}$, to the system in (3.14) with (3.20)–(3.21) and (3.22), for all $s > K_{1-i}/L_{1-i}$ and any $i = 0, 1$, such that $g_{j,l}(s_{j,l}) = a^*_j s_{j,l}$ holds for each $l \in {\mathbb N}$ and every $j = 0, 1$. It follows from the form of the arithmetic equations for $a^*_i = b^*_i(0, 0)$, with $b^*_i \equiv b^*_i(K_0, K_1)$ in (5.15) with (5.13)–(5.14) and (5.16) below, that for each such solution $(g_{i,l}(s), g_{1-i,l}(s))$, the inequality $g_{j,l}^{\prime}(s_{j,l}) > a^*_j$ should hold for each $l \in {\mathbb N}$ and every $i, j = 0, 1$ (see also [34, pp. 979–982] for the analysis of solutions of the single first-order nonlinear differential equation corresponding to the case $L_0 = L_1$ and $K_0 = K_1$). Observe that, by virtue of the uniqueness of solutions mentioned above, we know that no two curves $s \mapsto g_{j,l}(s)$ and $s \mapsto g_{j,m}(s)$, for $l, m \in {\mathbb N}$, $l \neq m$, can intersect, and thus, we see that the sequence $(g_{j,l}(s))_{l \in {\mathbb N}}$ is decreasing, so that the limit $g^*_j(s) = \lim_{l \to \infty} g_{j,l}(s)$ exists, for each $s > K_{1-i}/L_{1-i}$ and every $i, j = 0, 1$ fixed. We may therefore conclude that $(g^*_i(s), g^*_{1-i}(s))$ provides the maximal solution to the system of equations in (3.14) with (3.20)–(3.21) and (3.22) such that $0 < g^*_j(s) \le a^*_j s$ holds, for all $s > K_{1-i}/L_{1-i}$ and every $i, j = 0, 1$. Moreover, since the right-hand sides of the corresponding normal form of the system of first-order nonlinear ordinary differential equations in (3.14) with (3.20)–(3.21) and (3.22) are (locally) Lipschitz in s, for $s > K_{1-i}/L_{1-i}$ and every $i = 0, 1$, one can deduce by means of Gronwall’s inequality that the functions $g_{j,l}(s)$, $l \in {\mathbb N}$, are continuous, so that the functions $g^*_j(s)$, $j = 0, 1$, are continuous too. The corresponding maximal admissible solutions of first-order nonlinear ordinary differential equations and the associated maximality principle for solutions of optimal stopping problems, which is equivalent to the superharmonic characterisation of the payoff functions, were established in [Reference Peskir35] and further developed in [Reference Graversen and Peskir22], [Reference Pedersen34], [Reference Guo and Shepp24], [Reference Gapeev16], [Reference Baurdoux and Kyprianou6], [Reference Guo and Zervos25], [Reference Peskir37]–[Reference Peskir38], [Reference Glover, Hulley and Peskir21], [Reference Ott33], [Reference Kyprianou and Ott30], [Reference Baurdoux and van Schaik7], [Reference Gapeev and Rodosthenous17]–[Reference Gapeev and Rodosthenous19], and [Reference Rodosthenous and Zervos40], as well as other subsequent papers (see [39, Chapter I; Chapter V, Section 17] for further references).

2. (ii) In order to find the candidate value functions $V_j(x, s)$ and the boundaries $g_j(s)$, $j = 0, 1$, in the region $K_i/L_i < x \le s \le K_{1-i}/L_{1-i}$, we recall that the general solution of (2.13) for $V_j(x, s)$, $j = 0, 1$, takes the form of (3.4), for all $g_i(s) < x < s$, while the general solution for $V_{1-i}(x, s)$ is given by (3.5), for all $0 < x < g_i(s)$ and $K_i/L_i < s \le K_{1-i}/L_{1-i}$, for any $i = 0, 1$ fixed. Then, by applying the conditions of (2.14)–(2.16) to the function in (3.4), we obtain that the equalities (3.7)–(3.8) and (3.14) hold, for $K_{i}/L_{i} < s \le K_{1-i}/L_{1-i}$ under $j = i$. In this case, the candidate value function ${V}_{1-i}(x, s;\ g_i(s))$ takes the form

   (3.27) \begin{equation} {V}_{1-i}(x, s;\ g_i(s)) = {D}_{1-i,1}(s;\ g_i(s)) \, x^{\beta_{1-i,1}} + {\lambda_{1-i} \, (L_i \, s - K_i)}/({r + \lambda_{1-i}}) \end{equation}
   for $0 < x \le g_i(s)$, where ${D}_{1-i,2}(s;\ g_i(s)) = 0$ should hold, since otherwise ${V}_{1-i}(x, s;\ g_i(s)) \to \pm \infty$ as $x \downarrow 0$, which must be excluded since the function ${V}^*_{1-i}(x, s)$ in (2.1) is bounded. Moreover, since the function $V_{1-i}(x, s)$ should be twice continuously differentiable at the boundary $g_i(s)$, we get that the equalities in (3.11)–(3.13) are satisfied with ${D}_{1-i,1}(s) = D_{1-i,1}(s;\ g_i(s))$ and ${D}_{1-i,2}(s) = 0$. Hence, solving the system of equations in (3.7)–(3.8) and (3.11)–(3.13) with ${D}_{1-i,1}(s) = D_{1-i,1}(s; g_i(s))$ and ${D}_{1-i,2}(s) = 0$, we obtain that the candidate value function $V_i(x, s;\ g_i(s))$ admits the representation of (3.17) under $j = i$, for $g_i(s) < x \le s$ and $K_i/L_i < s \le K_{1-i}/L_{1-i}$, where the functions $C_{i,k}(s;\ g_i(s))$, $k = 1, 2$, admit the representations in (3.18)–(3.19) with $D_{1-i,1}(s;\ g_i(s))$ given by
   (3.28) \begin{align} \notag &D_{1-i,1}(s;\ g_i(s)) = \big( (L_i s - K_i)/g_i^{\beta_{1-i, 1}}(s) \big) \\ \notag &\quad \times \bigg( \sum_{k = 1}^2 \bigg( \frac{\lambda_0 \lambda_1 \gamma_{1-i, k} (\gamma_{1-i, k}-1)} {(\lambda_0 \lambda_1 + 1) (\gamma_{1-i, 3-k} - \gamma_{1-i, k})} \, \frac{(\lambda_0 \lambda_1 - (-1)^i (r + \lambda_{1-i})) \gamma_{1-i, 3-k}} {\lambda_{i}(r + \lambda_{1-i})} \\ \notag &\quad \phantom{\times \bigg( \sum_{k = 1}^2 \bigg(\;\:} + \frac{\gamma_{i, k} (\gamma_{i, k}-1)} {(\lambda_0 \lambda_1 + 1) (\gamma_{i, 3-k} - \gamma_{i, k})} \, \frac{\lambda_{1-i} (1 + (-1)^i (r + \lambda_{1-i})) \gamma_{i, 3-k}} {r + \lambda_{1-i}} \bigg) \bigg) \bigg/ \\ \notag &\quad\bigg( \sum_{k = 1}^2 \bigg( \frac{\lambda_0 \lambda_1 \gamma_{1-i, k} (\gamma_{1-i, k}-1)(\beta_{1-i,1} - \gamma_{1-i, 3-k})} {(\lambda_0 \lambda_1 + 1) (\gamma_{1-i, 3-k} - \gamma_{1-i, k})} + \frac{\gamma_{i, k} (\gamma_{i, k}-1)(\beta_{1-i,1} - \gamma_{i, 3-k})} {(\lambda_0 \lambda_1 + 1) (\gamma_{i, 3-k} - \gamma_{i, k})} \bigg) \\ &\phantom{\bigg( \sum_{k = 1}^2 \;\:}- \beta_{1-i,1} (\beta_{1-i,1} - 1) \bigg) \end{align}
   and ${D}_{1-i,2}(s;\ g_i(s)) = 0$, for any $i = 0, 1$ fixed. Therefore, we may conclude that the candidate boundary $g_i(s)$ satisfies the ordinary differential equation in (3.14) under $j = i$ with ${C}_{i,k}(s;\ g_i(s))$, $k = 1, 2$, determined as in (3.20)–(3.21),
   (3.29) \begin{align} &D_{1-i, 1}^{\prime}(s;\ g_i(s)) = D_{1-i, 1}(s;\ g_i(s)) \, \big( {L_i}/({L_i s - K_i}) - \beta_{1-i,1} \big( {g_i^{\prime}(s)}/{g_i(s)} \big) \big), \end{align}
   and $D_{1-i, 2}^{\prime}(s;\ g_i(s)) = 0$, for $K_i/L_i < s < K_{1-i}/L_{1-i}$, for any $i = 0, 1$ fixed. Observe that it also follows from the properties
   \begin{equation*}{V}_{1-i}((K_{1-i}/L_{1-i})-, (K_{1-i}/L_{1-i})-) = {V}_{1-i}(K_{1-i}/L_{1-i}, K_{1-i}/L_{1-i})\end{equation*}
   and
   \begin{equation*}{V}_{1-i}^{\prime}((K_{1-i}/L_{1-i})-, (K_{1-i}/L_{1-i})-) = {V}_{1-i}^{\prime}((K_{1-i}/L_{1-i})+, (K_{1-i}/L_{1-i})+)\end{equation*}
   that the candidate boundary should satisfy the condition
   \begin{equation*}g_i((K_{1-i}/L_{1-i})-) = g_i((K_{1-i}/L_{1-i})+),\end{equation*}
   for any $i = 0, 1$ fixed. We also note that, by arguments similar to the ones from the end of Part (i) above, the candidate boundary $g_{i}(s)$ should satisfy the properties
   (3.30) \begin{equation} g_{i}((K_{i}/L_{i})+) = 0 \quad \text{and} \quad g_{i}(s) \sim A_{i} \, (L_{i} \, s - K_{i})^{1/\gamma_{1,1}} \quad \text{as} \quad s \downarrow K_{i}/L_{i} \end{equation}
   with some constant $A_{i} > 0$ which is specified by means of the initial condition of (3.25) above.

3. (iii) In order to find the candidate value functions $V_j(x, s)$, $j = 0, 1$, in the region $0 < x \le s \le K_i/L_i$, for any $i = 0, 1$ fixed, we observe that the expression

   (3.31) \begin{align} V^*_{j}(x, s) &= {\mathbb E}_{x, j} \big[ e^{- r {\tau}_{i}^{\prime}} \big] \, V^*_j(K_{i}/L_{i}, K_{i}/L_{i}) \end{align}
   holds, for the first hitting time
   (3.32) \begin{equation} {\tau}_{1-i}^{\prime} = \inf \{ t \ge 0 \, | \, X_t \ge K_{1-i}/L_{1-i} \}, \end{equation}
   where the candidate solutions for $V^*_j(K_{i}/L_{i}, K_{i}/L_{i})$, $j = 0, 1$, are determined from Parts (i)–(ii) above, for any $i = 0, 1$ fixed. We recall that the general solution of (2.13) for $V_j(x, s)$, $j = 0, 1$, takes the form of (3.4), for all $0 < x < s \le K_i/L_i$. Note that, in this case, the candidate value functions ${V}_{j}(x, s;\ K_{i}/L_{i})$, $j = 0, 1$, are given by (3.17), for $0 < x \le s \le K_i/L_i$, where ${C}_{j,2}(s;\ g_i(s), g_{1-i}(s)) = 0$ should hold for every $j = 0, 1$, since otherwise ${V}_{j}(x, s;\ K_{i}/L_{i}) \to \pm \infty$ as $x \downarrow 0$, which must be excluded since the functions ${V}^*_{j}(x, s)$, $j = 0, 1$, in (3.31) are bounded. Then, by applying the boundary conditions ${V}_{j}((K_{i}/L_{i})-, (K_{i}/L_{i})-) ={V}_{j}(K_{i}/L_{i}, K_{i}/L_{i})$ to the function in (3.4) with ${C}_{j,2}(s) = 0$, $j = 0, 1$, we get that the candidate value function takes the form
   (3.33) \begin{equation} {V}_{j}(x;\ K_{i}/L_{i}) = \lambda_{j} \, {C}_{j,1}(K_{i}/L_{i}) \, x^{\gamma_{j, 1}} - (-1)^{j} \, {C}_{1-j, 1}(K_{i}/L_{i}) \, x^{\gamma_{1-j, 1}} \end{equation}
   for $0 < x \le s \le K_{i}/L_{i}$, with
   (3.34) \begin{align} &{C}_{j,1}(K_{i}/L_{i}) = \frac{{\lambda}_{1-j} V_j(K_{i}/L_{i}, K_{i}/L_{i}) + (-1)^j V_{1-j}(K_{i}/L_{i}, K_{i}/L_{i})} {({\lambda}_0 {\lambda}_1 + 1)(K_{i}/L_{i})^{\gamma_{j,1}}}, \end{align}
   where the function $V_j(K_{i}/L_{i}, K_{i}/L_{i}) = V_j((K_{i}/L_{i})+, (K_{i}/L_{i})+)$, $j = 0, 1$, is determined in Parts (i)–(ii) above, for any $i = 0, 1$ fixed.

3.2. The solution to the problem in the case $\lambda_i > \lambda_{1-i} = 0$

Let us now assume that $\lambda_i > \lambda_{1-i} = 0$ holds, for any $i = 0, 1$ fixed.

1. (i) In this case, the general solution of the second-order ordinary differential equation in (2.13) with (2.12) for $V_{1-i}(x, s)$ has the form

   (3.35) \begin{equation} V_{1-i}(x, s) = C_{1-i,1}(s) \, x^{\gamma_{0, 1}} + C_{1-i,2}(s) \, x^{\gamma_{0, 2}} \end{equation}
   for all $g_{1-i}(s) < x < s$, where $C_{1-i,k}(s)$, $k = 1, 2$, are some arbitrary functions, and $\gamma_{0, 2} < 0 < 1 < \gamma_{0, 1}$ are defined in (3.3). Then, by applying the conditions of (2.14)–(2.16) to the function in (3.35), we obtain the equalities
   (3.36) \begin{align} &C_{1-i,1}(s) \, g_{1-i}^{\gamma_{0, 1}}(s) + C_{1-i,2}(s) \, g_{1-i}^{\gamma_{0, 2}}(s) = L_{1-i} \, s - K_{1-i}, \end{align}
   (3.37) \begin{align} &C_{1-i,1}(s) \, \gamma_{0, 1} \, g^{\gamma_{0, 1}}_{1-i}(s) + C_{1-i,2}(s) \, \gamma_{0, 2} \, g_{1-i}^{\gamma_{0, 2}}(s) = 0, \end{align}
   (3.38) \begin{align} &C_{1-i,1}^{\prime}(s) \, s^{\gamma_{0, 1}} + C_{1-i,2}^{\prime}(s) \, s^{\gamma_{0, 2}} = 0 \end{align}

   for $s > K_{1-i}/L_{1-i}$ (see [Reference Pedersen34] and [Reference Guo and Shepp24]). Hence, solving the system of equations in (3.36)–(3.37), we obtain that the candidate value function admits the representation

   (3.39) \begin{equation} V_{1-i}(x, s;\ g_{1-i}(s)) = C_{1-i,1}(s;\ g_{1-i}(s)) \, x^{\gamma_{0,1}} + C_{1-i,2}(s;\ g_{1-i}(s)) \, x^{\gamma_{0,2}} \end{equation}
   for $g_{1-i}(s) < x \le s$, with
   (3.40) \begin{align} &C_{1-i,k}(s;\ g_{1-i}(s)) = {\gamma_{0,3-k} (L_{1-i} s - K_{1-i})} \big/ \big({(\gamma_{0,3-k} - \gamma_{0,k}) g^{\gamma_{0,k}}_{1-i}(s)} \big) \end{align}
   for $k = 1, 2$. Thus, by applying the condition of (3.38) to the functions in (3.40), we conclude that the candidate boundary satisfies the ordinary differential equation
   (3.41) \begin{align} &g_{1-i}^{\prime}(s) = \frac{L_{1-i} g_{1-i}(s)}{L_{1-i} s - K_{1-i}} \, \frac{\gamma_{0,2}(s/g_{1-i}(s))^{\gamma_{0,1}} - \gamma_{0,1}(s/g_{1-i}(s))^{\gamma_{0,2}}} {\gamma_{0,1} \gamma_{0,2}((s/g_{1-i}(s))^{\gamma_{0,1}}-(s/g_{1-i}(s))^{\gamma_{0,2}})} \end{align}
   for $s > K_{1-i}/L_{1-i}$ (see [Reference Pedersen34] and [Reference Guo and Shepp24]). It is easily seen that the right-hand side of the expression in (3.41) is strictly positive, so that the candidate boundary $g_{1-i}(s)$ is a strictly increasing function on $(K_{1-i}/L_{1-i}, \infty)$. We also note that the functions in (3.40) do not depend on the boundary $g_{1-i}(s)$ directly, since the Markov chain $\Theta$ cannot leave the state $1-i$ when $\lambda_{1-i} = 0$, for any $i = 0, 1$ fixed.

   Substituting the expression for $V_{1-i}(x, s;\ g_{1-i}(s))$ from (3.39) with (3.40) into the equation of (2.13) for $V_i(x, s)$, we obtain that the general solution has the form

   (3.42) \begin{align} V_i(x, s) &= C_{i,1}(s) \, x^{\beta_{i, 1}} + C_{i,2}(s) \, x^{\beta_{i, 2}} + V_{1-i}(x, s;\ g_{1-i}(s)) \end{align}
   for all $g_{i}(s) < x < s$, where $C_{i,k}(s)$, $k = 1, 2$, are some arbitrary functions, and $\beta_{i, 2} < 0 < 1 < \beta_{i, 1}$ are defined as in (3.6). Then, by applying the conditions of (2.14)–(2.16) to the function in (3.42), we obtain the equalities
   (3.43) \begin{align} &C_{i,1}(s) \, g_i^{\beta_{i, 1}}(s) + C_{i,2}(s) \, g_i^{\beta_{i, 2}}(s) + V_{1-i}(g_i(s), s;\ g_{1-i}(s)) = L_i \, s - K_i, \end{align}
   (3.44) \begin{align} &C_{i,1}(s) \, \beta_{i, 1} \, g_i^{\beta_{i, 1}}(s) + C_{i,2}(s) \, \beta_{i, 2} \, g_i^{\beta_{i, 2}}(s) + g_i(s) \, \partial_x V_{1-i}(g_i(s), s;\ g_{1-i}(s)) = 0, \end{align}
   (3.45) \begin{align} &C_{i,1}^{\prime}(s) \, s^{\beta_{i, 1}} + C_{i,2}^{\prime}(s) \, s^{\beta_{i, 2}} + (\partial_s + g_{1-i}^{\prime}(s) \partial_{g_{1-i}}) V_{1-i}(s, s;\ g_{1-i}(s)) = 0 \end{align}

   for all $s > K_{1-i}/L_{1-i}$. In this case, solving the system of equations in (3.43)–(3.44), we obtain that the candidate value function admits the representation

   (3.46) \begin{align} &V_i(x, s;\ g_i(s), g_{1-i}(s)) = \sum_{k = 1}^2 C_{i,k}(s;\ g_i(s), g_{1-i}(s)) \, x^{\beta_{i, k}} + V_{1-i}(x, s;\ g_{1-i}(s)) \end{align}
   for $g_i(s) < x \le s$, with
   (3.47) \begin{align} \notag &C_{i,k}(s;\ g_i(s), g_{1-i}(s)) \\[3pt] &\quad = \frac{\partial_x V_{1-i}(g_i(s), s;\ g_{1-i}(s)) - \beta_{i,3-k} \, (V_{1-i}(g_i(s), s;\ g_{1-i}(s)) - L_i s + K_i)} {(\beta_{i,3-k} - \beta_{i,k}) g_i^{\beta_{i,k}}(s)} \end{align}
   for $k = 1, 2$. Thus, by applying the condition of (3.45) to the functions in (3.47), we conclude that the candidate boundary satisfies the ordinary differential equation
   (3.48) \begin{align} g_i^{\prime}(s) &= - \big( (\partial_s + g_{1-i}^{\prime}(s) \partial_{g_{1-i}}) V_{1-i}(s, s;\ g_{1-i}(s))\\[3pt] \notag &\quad + (\partial_s + g_{1-i}^{\prime}(s) \partial_{g_{1-i}}) C_{i,1}(s;\ g_i(s), g_{1-i}(s)) s^{\beta_{i,1}} \\[3pt] \notag &\quad + (\partial_s + g_{1-i}^{\prime}(s) \partial_{g_{1-i}}) C_{i,2}(s;\ g_i(s), g_{1-i}(s)) s^{\beta_{i,2}} \big) \big/ \\[3pt] \notag &\quad \big( \partial_{g_{i}} C_{i,1}(s;\ g_i(s), g_{1-i}(s)) s^{\beta_{i,1}} + \partial_{g_{i}} C_{i,2}(s;\ g_i(s), g_{1-i}(s)) s^{\beta_{i,2}} \big) \end{align}
   for $s > K_{1-i}/L_{1-i}$ and any $i = 0, 1$ fixed. It is seen that the system of equations in (3.41) and (3.48) is degenerated in the sense that the candidate boundary $g_{1-i}(s)$ solves the single first-order nonlinear ordinary differential equation in (3.41), while the boundary $g_{i}(s)$ can be expressed in terms of $g_{1-i}(s)$, by means of the first-order nonlinear ordinary differential equation in (3.48), for any $i = 0, 1$ fixed. In order to characterise the asymptotic behaviour of the candidate boundaries $g_j(s)$, $j = 0, 1$, we observe that the expressions in (3.39)–(3.40) and (3.46)–(3.47) imply the representations
   (3.49) \begin{align} &{V_{1-i}(s, s;\ g_{1-i}(s))}/{s}\\ \notag &= {\widetilde C}_{1-i, 1}(s;\ h_{1-i}(s)) \, h^{- \gamma_{0,1}}_{1-i}(s) + {\widetilde C}_{1-i, 2}(s;\ h_{1-i}(s)) \, h^{- \gamma_{0,2}}_{1-i}(s) \end{align}
   and
   (3.50) \begin{align} &{V_{i}(s, s;\ g_i(s), g_{1-i}(s))}/{s}\\ \notag &= \sum_{k=1}^2 \Big( {\widetilde C}_{i, k}(s;\ h_i(s), h_{1-i}(s)) \, h^{- \beta_{i,k}}_i(s) + {\widetilde C}_{1-i, k}(s;\ h_i(s), h_{1-i}(s)) \, h^{- \gamma_{0,k}}_i(s) \Big), \end{align}
   with some appropriately chosen ${\widetilde C}_{j, k}(s;\ h_i(s), h_{1-i}(s))$, for $i, j = 0, 1$ and $k = 1, 2$, which are of constant signs as $s \to \infty$, where we recall that $h_j(s) = g_j(s)/s$ for all $s > K_j/L_j$ and every $j = 0, 1$. Therefore, following the arguments from the end of Part (i) of Subsection 3.1, we may conclude that the candidate boundary $g_{1-i}(s)$ should satisfy the conditions of (3.25), as well as the properties
   (3.51) \begin{align} & g_{1-i}((K_{1-i}/L_{1-i})+) = 0 \quad \text{and}\nonumber\\ &\quad g_{1-i}(s) \sim A_{1-i} \, (L_{1-i} \, s - K_{1-i})^{1/\gamma_{0,1}} \quad \text{as} \quad s \downarrow K_{1-i}/L_{1-i} \end{align}
   with some constant $A_{1-i} > 0$ which is also specified by means of the condition of (3.25) above. It also follows from the arguments of the end of Part (i) of Subsection 3.1the solution above that there exist maximal solutions $g^*_j(s)$, $j = 0, 1$, of the first-order nonlinear ordinary differential equations in (3.41) and (3.48) such that the inequalities $0 < g^*_j(s) \le a^*_j s$ hold, for $j = 0, 1$.

2. (ii) In order to find the candidate value functions $V_j(x, s)$, $j = 0, 1$, and the boundary $g_i(s)$ in the region $K_i/L_i < x \le s \le K_{1-i}/L_{1-i}$, we recall that the general solution of (2.13) is given by (3.35), for all $0 < x < g_i(s)$ and any $i = 0, 1$ fixed. Note that, in this case, the candidate value function ${V}_{1-i}(x, s;\ K_{1-i}/L_{1-i})$ takes the form of (3.39), for $0 < x \le g_i(s)$, where ${C}_{1-i,2}(s;\ K_{1-i}/L_{1-i}) = 0$ should hold, since otherwise ${V}_{1-i}(x, s;\ K_{1-i}/L_{1-i}) \to \pm \infty$ as $x \downarrow 0$, which must be excluded since the function ${V}^*_{1-i}(x, s)$ in (2.1) is bounded. Hence, by applying the boundary condition ${V}_{1-i}((K_{1-i}/L_{1-i})-, (K_{1-i}/L_{1-i})-) ={V}_{1-i}(K_{1-i}/L_{1-i}, K_{1-i}/L_{1-i})$ to the function in (3.39) with $C_{1-i,2}(s;\ K_{1-i}/L_{1-i}) = 0$, we get that the candidate value function takes the form

   (3.52) \begin{equation} {V}_{1-i}(x;\ K_{1-i}/L_{1-i}) = (L_{1-i} \, x / K_{1-i})^{\gamma_{0,1}} \, V_{1-i}(K_{1-i}/L_{1-i}, K_{1-i}/L_{1-i}) \end{equation}
   for $0 < x \le g_i(s)$, $i = 0, 1$, where the candidate solution
   \begin{equation*}V_{1-i}(K_{1-i}/L_{1-i}, K_{1-i}/L_{1-i}) = V_{1-i}((K_{1-i}/L_{1-i})+, (K_{1-i}/L_{1-i})+)\end{equation*}
   is determined in Part (i) above, for any $i = 0, 1$ fixed.

   Substituting the expression for ${V}_{1-i}(x;\ K_{1-i}/L_{1-i})$ from (3.52) into the equation of (2.13) for $V_i(x, s)$, we obtain that its general solution has the form of (3.35), for $g_i(s) < x < s$ and any $i = 0, 1$ fixed. Then, by applying the conditions of (2.14)–(2.16) to the function in (3.35), we obtain the equalities (3.43)–(3.45), with ${V}_{1-i}(x;\ K_{1-i}/L_{1-i})$ given by (3.52), for $K_i/L_i < x \le s \le K_{1-i}/L_{1-i}$. In this case, solving the system of equations in (3.43)–(3.44), we conclude that the candidate value function admits the representation

   (3.53) \begin{align} &V_i(x, s;\ g_i(s)) = C_{i,1}(s;\ g_i(s)) \, x^{\beta_{i, 1}} + C_{i,2}(s;\ g_i(s)) \, x^{\beta_{i, 2}} + V_{1-i}(x;\ K_{1-i}/L_{1-i}) \end{align}
   for $g_i(s) < x \le s$, with
   (3.54) \begin{align} &C_{i,k}(s;\ g_i(s)) \\[3pt] \notag &= \frac{V_{1-i}^{\prime}(g_i(s);\ K_{1-i}/L_{1-i}) - \beta_{i,3-k} (V_{1-i}(g_i(s);\ K_{1-i}/L_{1-i}) - L_i s + K_i)} {(\beta_{i,3-k} - \beta_{i,k}) g_i^{\beta_{i,k}}(s)} \end{align}
   for $k = 1, 2$. Thus, by applying the condition of (3.45) to the functions in (3.54), we conclude that the candidate boundary satisfies the ordinary differential equation
   (3.55) \begin{align} &g_i^{\prime}(s) = - \frac{\partial_s C_{i,1}(s;\ g_i(s)) s^{\beta_{i,1}} + \partial_s C_{i,2}(s;\ g_i(s)) s^{\beta_{i,2}}} {\partial_{g_{i}} C_{i,1}(s;\ g_i(s)) s^{\beta_{i,1}} + \partial_{g_{i}} C_{i,2}(s;\ g_i(s)) s^{\beta_{i,2}}} \end{align}
   for $K_i/L_i < s < K_{1-i}/L_{1-i}$. We also note that, by arguments similar to the ones from the end of Part (i) above, the candidate boundary $g_{i}(s)$ should satisfy the properties
   (3.56) \begin{equation} g_{i}((K_{i}/L_{i})+) = 0 \quad \text{and} \quad g_{i}(s) \sim A_{i} \, (L_{i} \, s - K_{i})^{1/\beta_{i,1}} \quad \text{as} \quad s \downarrow K_{i}/L_{i} \end{equation}
   with some constant $A_{i} > 0$ which is specified by means of the initial condition of (3.25) above.

3. (iii) In order to find the candidate value functions $V_j(x, s)$, $j = 0, 1$, in the region $0 < x \le s \le K_i/L_i$, we recall that the general solution of (2.13) is given by (3.35), for all $0 < x < s \le K_i/L_i$ and any $i = 0, 1$ fixed. Note that, in this case, the candidate value functions ${V}_{j}(x, s;\ K_{i}/L_{i})$ take the form of (3.39), for $0 < x \le s \le K_i/L_i$, where ${C}_{j,2}(s;\ K_{i}/L_{i}) = 0$ should hold for every $j = 0, 1$, since otherwise ${V}_{j}(x, s;\ K_{i}/L_{i}) \to \pm \infty$ as $x \downarrow 0$, which must be excluded since the functions ${V}^*_{j}(x, s)$, $j = 0, 1$, in (3.31) are bounded. Hence, by applying the boundary condition ${V}_j((K_{i}/L_{i})-, (K_{i}/L_{i})-) = {V}_j(K_{i}/L_{i}, K_{i}/L_{i})$ to the function in (3.39) with ${C}_{j,2}(s;\ K_{i}/L_{i}) = 0$, $j = 0, 1$, we get that the candidate value functions take the form

   (3.57) \begin{equation} {V}_{j}(x;\ K_{i}/L_{i}) = (L_i x/K_i)^{\gamma_{0, 1}} \, {V}_{j}(K_{i}/L_{i}, K_{i}/L_{i}) \end{equation}
   for $0 < x \le s \le K_{i}/L_{i}$, where the functions ${V}_{j}(K_{i}/L_{i}, K_{i}/L_{i}) = {V}_{j}((K_{i}/L_{i})+, (K_{i}/L_{i})+)$, $j = 0, 1$, are determined in Part (ii) above, for any $i = 0, 1$ fixed.

3.3. The solution to the problem in the case $\lambda_{1-i} > \lambda_i = 0$

Let us now assume that $\lambda_{1-i} > \lambda_i = 0$ holds, for any $i = 0, 1$ fixed.

1. (i) In this case, the general solution of the second-order ordinary differential equation in (2.13) with (2.12) for $V_i(x, s)$ has the form

   (3.58) \begin{equation} V_i(x, s) = C_{i,1}(s) \, x^{\gamma_{0, 1}} + C_{i,2}(s) \, x^{\gamma_{0, 2}} \end{equation}
   for all $g_i(s) < x \le s$, where $C_{i,k}(s)$, $k = 1, 2$, are some arbitrary functions and $\gamma_{0, 2} < 0 < 1 < \gamma_{0, 1}$ are given by (3.3) above. Then, by applying the conditions of (2.14)–(2.16) to the function in (3.58), we obtain the equalities
   (3.59) \begin{align} &C_{i,1}(s) \, g_i^{\gamma_{0, 1}}(s) + C_{i,2}(s) \, g_i^{\gamma_{0, 2}}(s) = L_i \, s - K_i, \end{align}
   (3.60) \begin{align} &C_{i,1}(s) \, \gamma_{0, 1} \, g_i^{\gamma_{0, 1}}(s) + C_{i,2}(s) \, \gamma_{0, 2} \, g_i^{\gamma_{0, 2}}(s) = 0, \end{align}
   (3.61) \begin{align} &C_{i,1}^{\prime}(s) \, s^{\gamma_{0, 1}} + C_{i,2}^{\prime}(s) \, s^{\gamma_{0, 2}} = 0 \end{align}

   for $s > K_{i}/L_{i}$ (see [Reference Guo and Shepp24,Reference Pedersen34]). Hence, solving the system of equations in (3.59)–(3.60), we obtain that the candidate value function admits the representation

   (3.62) \begin{equation} V_{i}(x, s;\ g_{i}(s)) = C_{i,1}(s;\ g_{i}(s)) \, x^{\gamma_{0,1}} + C_{i,2}(s;\ g_{i}(s)) \, x^{\gamma_{0,2}} \end{equation}
   for $g_i(s) < x \le s$, with
   (3.63) \begin{align} &C_{i,k}(s;\ g_{i}(s)) = {\gamma_{0,3-k} (L_{i} s - K_{i})} \big/ \big( {(\gamma_{0,3-k} - \gamma_{0,k}) g^{\gamma_{0,k}}_{i}(s)} \big) \end{align}
   for $k = 1, 2$. Thus, by applying the condition of (3.61) to the functions in (3.63), we conclude that the candidate boundary satisfies the ordinary differential equation
   (3.64) \begin{align} &g_{i}^{\prime}(s) = \frac{L_{i} g_{i}(s)}{L_{i} s - K_{i}} \, \frac{\gamma_{0,2}(s/g_{i}(s))^{\gamma_{0,1}} - \gamma_{0,1}(s/g_{i}(s))^{\gamma_{0,2}}} {\gamma_{0,1} \gamma_{0,2}((s/g_{i}(s))^{\gamma_{0,1}}-(s/g_{i}(s))^{\gamma_{0,2}})} \end{align}
   for $s > K_{i}/L_{i}$ (see [Reference Pedersen34] and [Reference Guo and Shepp24]). It is easily seen that the right-hand side of the expression in (3.64) is strictly positive, so that the candidate boundary $g_{i}(s)$ is a strictly increasing function on $(K_{i}/L_{i}, \infty)$. We also note that the functions in (3.63) do not depend on the boundary $g_{1-i}(s)$ directly, since the Markov chain $\Theta$ cannot leave the state i when $\lambda_i = 0$, for any $i = 0, 1$ fixed.

   Substituting the expression $V_i(x, s) = L_i s - K_i$ into the equation of (2.13) for $V_{1-i}(x, s)$, we obtain that its general solution has the form of (3.5), for all $g_{1-i}(s) < x < g_i(s)$, where $D_{1-i,k}(s)$ are some arbitrary functions, and $\beta_{1-i, 2} < 0 < 1 < \beta_{1-i, 1}$ are explicitly given by (3.6) above. Then, by applying the conditions of (2.14)–(2.15) to the function in (3.5), we obtain the equalities of (3.9)–(3.10), for $s > K_{1-i}/L_{1-i}$. In this case, solving the system of equations in (3.9)–(3.10), we obtain that the candidate value function $V_{1-i}(x, s;\ g_{1-i}(s))$ admits the representation of (3.15), for $g_{1-i}(s) < x \le g_i(s)$, with $D_{1-i,k}(s;\ g_{1-i}(s))$, $k = 1, 2$, given by (3.16), where we set $L_{1-i}^{\prime} = L_{1-i} - {\lambda}_{1-i} L_i/(r + {\lambda}_{1-i})$ and $K_{1-i}^{\prime} = K_{1-i} - {\lambda}_{1-i} K_i/(r + {\lambda}_{1-i})$. We now recall that the general solution of the equation in (2.13) for $V_{1-i}(x, s)$ has the form

   (3.65) \begin{align} &V_{1-i}(x, s) = C_{1-i,1}(s) \, x^{\beta_{1-i, 1}} + C_{1-i,2}(s) \, x^{\beta_{1-i, 2}} + V_{i}(x, s;\ g_{i}(s)) \end{align}
   for $g_i(s) < x < s$, where $V_{i}(x, s;\ g_{i}(s))$ is given by (3.62) with (3.63). Moreover, using the fact that the function $V_{1-i}(x, s)$ in (3.65) should be twice continuously differentiable at $g_i(s)$, for any $i = 0, 1$ fixed, and taking into account the conditions of (3.59)–(3.60), we get the equalities
   (3.66) \begin{align} &C_{1-i,1}(s) \, g_{i}^{\beta_{1-i, 1}}(s) + C_{1-i,2}(s) \, g_{i}^{\beta_{1-i, 2}}(s) + L_i \, s - K_i = V_{1-i}(g_i(s), s;\ g_{1-i}(s)), \end{align}
   (3.67) \begin{align} \notag &C_{1-i,1}(s) \, \beta_{1-i, 1} \, g^{\beta_{1-i, 1}}_{i}(s) + C_{1-i,2}(s) \, \beta_{1-i, 2} \, g_{i}^{\beta_{1-i, 2}}(s) \\ &\quad = g_i(s) \, \partial_x V_{1-i}(g_i(s), s;\ g_{1-i}(s)), \end{align}
   (3.68) \begin{align} \notag &C_{1-i,1}(s) \, \beta_{1-i, 1} (\beta_{1-i, 1} - 1) \, g^{\beta_{1-i, 1}}_{i}(s) + C_{1-i,2}(s) \, \beta_{1-i, 2} (\beta_{1-i, 2} - 1) \, g_{i}^{\beta_{1-i, 2}}(s) \\ &\quad = g^2_i(s) \, \partial_{xx} V_{1-i}(g_i(s), s;\ g_{1-i}(s)) \end{align}

   for all $s > K_{1-i}/L_{1-i}$, where $V_{1-i}(x, s;\ g_{1-i}(s))$ is given by (3.15) with (3.16). Thus, solving the system of equations in (3.66)–(3.67), we obtain that the candidate value function admits the representation

   (3.69) \begin{align} &V_{1-i}(x, s;\ g_i(s), g_{1-i}(s)) = \sum_{k = 1}^2 C_{1-i,k}(s;\ g_i(s), g_{1-i}(s)) \, x^{\beta_{1-i, k}} + V_{i}(x, s;\ g_{i}(s)) \end{align}
   for $g_i(s) < x \le s$, with
   (3.70) \begin{align} &C_{1-i,k}(s;\ g_i(s), g_{1-i}(s))\\ \notag &= \frac{\beta_{1-i,3-k} (V_{1-i}(g_i(s), s;\ g_{1-i}(s)) - L_i s + K_i) - g_i(s) \partial_x V_{1-i}(g_i(s), s;\ g_{1-i}(s))} {(\beta_{1-i,3-k} - \beta_{1-i,k}) g^{\beta_{1-i,k}}_{i}(s)} \end{align}
   for $k = 1, 2$. Then, by applying the condition of (2.16) to the function in (3.65), we obtain the equalities
   (3.71) \begin{align} &C_{1-i,1}^{\prime}(s) \, s^{\beta_{1-i, 1}} + C_{1-i,2}^{\prime}(s) \, s^{\beta_{1-i, 2}} + (\partial_s + g_i^{\prime}(s) \, \partial_{g_i} V_i)(s, s;\ g_i(s)) = 0 \end{align}
   for $s > K_{1-i}/L_{1-i}$. Hence, by applying the condition of (3.71) to the functions in (3.70), we conclude that the candidate boundary satisfies the ordinary differential equation
   (3.72) \begin{align} g_{1-i}^{\prime}(s) &= - \big( (\partial_s + g_{i}^{\prime}(s) \partial_{g_{i}}) V_{i}(s, s;\ g_i(s))\\ \notag &\quad+ (\partial_s + g_{i}^{\prime}(s) \partial_{g_{i}}) C_{1-i,1}(s;\ g_i(s), g_{1-i}(s)) s^{\beta_{1-i,1}} \\ \notag &\quad + (\partial_s + g_{i}^{\prime}(s) \partial_{g_{i}}) C_{1-i,2}(s;\ g_i(s), g_{1-i}(s)) s^{\beta_{1-i,2}} \big) \big/ \\ \notag &\qquad\big( \partial_{g_{1-i}} C_{1-i,1}(s;\ g_i(s), g_{1-i}(s)) s^{\beta_{1-i,1}} + \partial_{g_{1-i}} C_{1-i,2}(s;\ g_i(s), g_{1-i}(s)) s^{\beta_{1-i,2}} \big) \end{align}
   for $s > K_{1-i}/L_{1-i}$. It is seen that the system of equations in (3.64) and (3.72) is degenerated in the sense that the candidate boundary $g_{i}(s)$ solves the single first-order nonlinear ordinary differential equation in (3.64), while the boundary $g_{1-i}(s)$ can be either determined from the first-order nonlinear ordinary differential equation in (3.72) or expressed in terms of $g_i(s)$, by means of the arithmetic power equation in (3.68), for any $i = 0, 1$ fixed.

   In order to characterise the asymptotic behaviour of the candidate boundaries $g_j(s)$, $j = 0, 1$, we observe that the expressions in (3.15)–(3.16) and (3.69)–(3.70) imply the representations of (3.23) and

   (3.73) \begin{align} V_{i}(s, s;\ g_{i}(s))/s = {\widetilde C}_{i,1}(s;\ h_{i}(s)) \, h^{-\gamma_{0,1}}_i(s) + {\widetilde C}_{i,2}(s;\ h_{i}(s)) \, h^{-\gamma_{0,2}}_i(s), \end{align}
   as well as
   (3.74) \begin{align} &{V_{1-i}(s, s;\ g_i(s), g_{1-i}(s))}/{s}\\ \notag &= \sum_{k=1}^2 \Big( {\widetilde C}_{1-i, k}(s;\ h_i(s), h_{1-i}(s)) \, h^{- \beta_{1-i,k}}_{1-i}(s) +{\widetilde C}_{i, k}(s;\ h_i(s)) \, h^{- \gamma_{0,k}}_{1-i}(s) \Big), \end{align}
   with some appropriately chosen ${\widetilde C}_{i, k}(s;\ h_{i}(s))$ and ${\widetilde C}_{1-i, k}(s;\ h_i(s), h_{1-i}(s))$, for $i = 0, 1$ and $k = 1, 2$, which are of constant signs as $s \to \infty$, where we recall that $h_j(s) = g_j(s)/s$, for all $s > K_j/L_j$ and every $j = 0, 1$. Therefore, following the arguments from the end of Part (i) of Subsection 3.1, we may conclude that the candidate boundary $g_{1-i}(s)$ should satisfy the conditions of (3.25), as well as the properties
   (3.75) \begin{align} &g_{1-i}((K_{1-i}/L_{1-i})+) = 0 \quad \text{and}\nonumber\\ &\quad g_{1-i}(s) \sim A_{1-i} \, (L_{1-i} \, s - K_{1-i})^{1/\beta_{1-i,1}} \!\!\!\!\! \quad \text{as} \quad s \downarrow K_{1-i}/L_{1-i} \end{align}
   with some constant $A_{1-i} > 0$ which is also specified by means of the condition of (3.25) above. It also follows from the arguments of the end of Part (i) of Subsection 3.1 above that there exist maximal solutions $g^*_j(s)$, $j = 0, 1$, of the first-order nonlinear ordinary differential equations in (3.64) and (3.72) such that the inequalities $0 < g^*_j(s) \le a^*_j s$ hold, for $j = 0, 1$.

2. (ii) In order to find the candidate value function $V_{1-i}(x, s)$ in the region $K_i/L_i < x \le s \le K_{1-i}/L_{1-i}$, we recall that the general solution of (2.13) is given by (3.5), for all $0 < x < g_i(s)$ and any $i = 0, 1$ fixed. In this case, following the arguments of Part (ii) of Subsection 3.1 above and solving the equation in (3.68) with $C_{1-i,k}(s;\ g_i(s), g_{1-i}(s))$, $k = 1, 2$, from (3.70) and $V_{i}(x, s;\ g_{i}(s))$ from (3.62) with (3.63), we obtain that the candidate solution ${V}_{1-i}(x, s;\ g_i(s))$ takes the form of (3.27) with $D_{1-i,1}(s;\ g_i(s))$ given by

   (3.76) \begin{equation}D_{1-i,1}(s;\ g_i(s)) = {r (L_i s - K_i)} \big/ \big( ({r + \lambda_{1-i}})(1 - \beta_{1-i,1}(\beta_{1-i,1}-1))) g_i^{\beta_{1-i,1}}(s) \big)\end{equation}
   for $0 < x \le g_i(s)$, $i = 0, 1$. We also note that, by arguments similar to the ones from the end of Part (i) above, the candidate boundary $g_{i}(s)$ should satisfy the properties
   (3.77) \begin{equation} g_{i}((K_{i}/L_{i})+) = 0 \quad \text{and} \quad g_{i}(s) \sim A_{i} \, (L_{i} \, s - K_{i})^{1/\gamma_{0,1}} \quad \text{as} \quad s \downarrow K_{i}/L_{i} \end{equation}
   with some constant $A_{i} > 0$ which is specified by means of the condition of (3.25) above.

3. (iii) By means of the same arguments as in Part (iii) of Subsection 3.2, it is shown that the candidate value functions ${V}_{j}(x, s;\ K_{i}/L_{i})$, $j = 0, 1$, take the form of (3.57), for all $0 < x \le s \le K_{i}/L_{i}$ and any $i = 0, 1$ fixed, where the functions ${V}_{j}(K_{i}/L_{i}, K_{i}/L_{i}) = {V}_{j}(K_{i}/L_{i}+, K_{i}/L_{i}+)$, $j = 0, 1$, are determined in Parts (i)–(ii) above.

5.1. The optimal stopping and free-boundary problem

It is seen that the problem of (1.2) can be embedded into the optimal stopping problem for the (time-homogeneous strong) Markov process $(X, S, \Theta)=(X_t, S_t, \Theta_t)_{t \ge 0}$ with the value function

(5.1) \begin{equation} W^*_i(x, s) = \sup_{\zeta} {\mathbb E}_{x, s, i} \big[e^{- r \zeta} \, \big( (1 - \Theta_{\zeta}) \, (L_0 \, S_{\zeta} - K_0 \, X_{\zeta}) + \Theta_{\zeta} \, (L_1 \, S_{\zeta} - K_1 \, X_{\zeta}) \big) \big] \end{equation}

for some $L_{i} > 0$ and $K_{i} \ge 0$, $i = 0, 1$, fixed, where the supremum is taken over all stopping times $\zeta$ with respect to $(\mathcal{G}_t)_{t \ge 0}$. It can be shown by means of the same arguments as in Subsection 2.2 above that the optimal stopping time in the problem of (5.1) has the structure

(5.2) \begin{equation} \zeta^* = \inf \{ t \ge 0 \, | \, X_t \le b^*_{\Theta_t} \, S_t \}, \end{equation}

with some numbers $0 < b^*_i < 1$, $i = 0, 1$, to be determined.

In order to find analytic expressions for the unknown value functions $W^*_i(x, s)$ from (5.1) and the unknown boundaries $b^*_i s$, $i = 0, 1$, from (5.2), we can formulate the following free-boundary problem:

(5.3) \begin{align} &({{\mathbb L}} W_i - (r + \lambda_i) \, W_i)(x, s) = - \lambda_i \, W_{1-i}(x, s) \quad \text{for } b_i \, s < x < s, \end{align}

(5.4) \begin{align} &W_i(x, s) \big|_{x = (b_i s)+} = L_i \, s - K_i \, b_i \, s \quad \text{({\it instantaneous stopping})}, \end{align}

(5.5) \begin{align} &\partial_x W_i(x, s) \big|_{x = (b_i s)+} = - K_i \quad \text{({\it smooth fit})}, \end{align}

(5.6) \begin{align} &\partial_s W_i(x, s) \big|_{x = s-} = 0 \quad \text{({\it normal reflection})}, \end{align}

(5.7) \begin{align} &W_i(x, s) = L_i \, s - K_i \, x \quad \text{for } 0 < x < b_i \, s, \end{align}

(5.8) \begin{align} &W_i(x, s) > L_i \, s - K_i \, x \quad \text{for } b_i \, s < x \le s, \end{align}

(5.9) \begin{align} &({{\mathbb L}} W_i - (r + \lambda_i) \, W_i)(x, s) < - \lambda_i \, W_{1-i}(x, s) \quad \text{for } 0 < x < b_i \, s, \end{align}

where the conditions of (5.4)–(5.6) are satisfied for all $s > 0$ and any $i = 0, 1$ fixed.

5.2. Solution to the free-boundary problem

Let us now present the solution of the free-boundary problem in (5.3)–(5.9).

1. (i) Let us first assume that $\lambda_i > 0$, $i = 0, 1$, holds. In this case, by using straightforward calculations from Part (i) of Subsection 3.1, it can be shown that the solution of the system in (5.3)–(5.9) takes the form

   (5.10) \begin{align} &W_{1-i}(x, s;\ b^*_{1-i}) = \sum_{k = 1}^2 {\widehat D}_{1-i,k}(s;\ b^*_{1-i}) \, x^{\beta_{1-i, k}} + \frac{\lambda_{1-i} L_i s}{r + \lambda_{1-i}} - \frac{\lambda_{1-i} K_i x}{r + \lambda_{1-i} - \mu} \end{align}
   for $0 < b^*_{1-i} s < x \le b^*_i s$, with
   (5.11) \begin{align} &{\widehat D}_{1-i,k}(s;\ b^*_{1-i}) = \frac{\beta_{1-i,3-k} L_{1-i}^{\prime} s + (1-\beta_{1-i,3-k}) K_{1-i}^{\prime\prime} b^*_{1-i} s}{(\beta_{1-i,3-k} - \beta_{1-i,k}) (b^*_{1-i} s)^{\beta_{1-i,k}}} \end{align}
   for $k = 1, 2$, where we recall that $L_{1-i}^{\prime} = L_{1-i} - \lambda_{1-i} L_i/(r + \lambda_{1-i})$ and set $K_{1-i}^{\prime\prime} = K_{1-i} - \lambda_{1-i} K_i/(r + \lambda_{1-i} - \mu)$, and
   (5.12) \begin{align} \notag &W_j(x, s;\ b^*_i, b^*_{1-i}) \\ &= \sum_{k = 1}^2 \Big( \lambda_j \, {\widehat C}_{j,k}(s;\ b^*_i, b^*_{1-i}) \, x^{\gamma_{j, k}} - (-1)^{j} \, {\widehat C}_{1-j, k}(s;\ b^*_i, b^*_{1-i}) \, x^{\gamma_{1-j, k}} \Big) \end{align}
   for $0 < b^*_i s < x < s$, $i = 0, 1$, where the functions ${\widehat C}_{j,k}(s;\ b^*_i, b^*_{1-i})$, for $j = 0, 1$ and $k = 1, 2$, admit the representations
   (5.13) \begin{align} \notag {\widehat C}_{i, k}(s;\ b^*_i, b^*_{1-i}) &= (-1)^i \big/ \big( (\lambda_0 \lambda_1 + 1) (\gamma_{i, 3-k} - \gamma_{i, k}) (b^*_i s)^{\gamma_{i, k}} \big) \\[3pt] \notag &\quad \times \bigg( \sum_{l = 1}^2 {\widehat D}_{1-i, l}(s;\ b^*_{1-i}) (\gamma_{i, 3-k} - \beta_{1-i, l}) (b^*_i s)^{\beta_{1-i, l}} \\[3pt] \notag &\quad \phantom{\times \bigg(\;\:}+ \frac{\lambda_{1-i} (1 + (-1)^i (r + \lambda_{1-i})) \gamma_{i, 3-k} L_i s}{r + \lambda_{1-i}} \\[3pt] &\quad\phantom{\times \bigg(\;\:} - \frac{\lambda_{1-i} (1 + (-1)^i (r + \lambda_{1-i} - \mu)) (\gamma_{i, 3-k} - 1) K_i b^*_i s}{r + \lambda_{1-i} - \mu} \bigg) \end{align}
   and
   (5.14) \begin{align} \notag {\widehat C}_{1-i, k}(s;\ b^*_i, b^*_{1-i}) &= \lambda_i \big/ \big( (\lambda_0 \lambda_1 + 1) (\gamma_{1-i, 3-k} - \gamma_{1-i, k}) (b^*_{1-i} s)^{\gamma_{1-i, k}} \big) \\[3pt] \notag &\quad \times \bigg( \sum_{l = 1}^2 {\widehat D}_{1-i, l}(s;\ b^*_{1-i}) (\gamma_{1-i, 3-k} - \beta_{1-i, l}) (b^*_i s)^{\beta_{1-i, l}} \\[3pt] \notag &\quad \phantom{\times \bigg(\;\:}+ \frac{(\lambda_0 \lambda_1 - (-1)^i (r + \lambda_{1-i})) \gamma_{1-i, 3-k} L_i s}{\lambda_i (r + \lambda_{1-i})} \\[3pt] &\phantom{\times \bigg(\;\:} - \frac{(\lambda_0 \lambda_1 - (-1)^i (r + \lambda_{1-i} - \mu)) (\gamma_{1-i, 3-k} - 1) K_i b^*_i s}{\lambda_i (r + \lambda_{1-i} - \mu)} \bigg) \end{align}
   with ${\widehat D}_{1-i, k}(s;\ b^*_{1-i})$, $k = 1, 2$, from (5.11). Then, by applying the condition of (5.6) to the functions in (5.12), we conclude that the values $b^*_j$, $j = 0, 1$, are determined from the system of arithmetic equations
   (5.15) \begin{align}&\sum_{k = 1}^2 \Big( \lambda_j \, {\widehat C}_{j,k}^{\prime}(s;\ b_i, b_{1-i}) \, s^{\gamma_{j, k}}- (-1)^{j} \, {\widehat C}_{1-j,k}^{\prime}(s;\ b_i, b_{1-i}) \, s^{\gamma_{1-j, k}} \Big) = 0\end{align}
   with
   (5.16) \begin{align} {\widehat C}_{j, k}^{\prime}(s;\ b_i, b_{1-i}) &= (1 - \gamma_{j,k}) \, {\widehat C}_{j, k}(s;\ b_i, b_{1-i})/s, \end{align}
   where ${\widehat C}_{j, k}(s;\ b^*_i, b^*_{1-i})$, for $i, j = 0, 1$ and $k = 1, 2$, are given by the expressions in (5.13)–(5.14).

2. (ii) Let us now assume that $\lambda_i > {\lambda}_{1-i} = 0$ holds for any $i = 0, 1$ fixed. In this case, by using straightforward calculations from Part (i) of Subsection 3.2, it can be shown that the candidate solution of the system in (5.3)–(5.9) takes the form

   (5.17) \begin{equation} W_{1-i}(x, s;\ b^*_{1-i}) = {\widehat C}_{1-i,1}(s;\ b^*_{1-i}) \, x^{\gamma_{0,1}} + {\widehat C}_{1-i,2}(s;\ b^*_{1-i}) \, x^{\gamma_{0,2}} \end{equation}
   for $0 < b^*_{1-i} s < x \le s$, with
   (5.18) \begin{align} &{\widehat C}_{1-i,k}(s;\ b^*_{1-i}) = {\gamma_{0,3-k} (L_{1-i} \, s - K_{1-i} \, b^*_{1-i} s)} \big/ \big( {(\gamma_{0,3-k} - \gamma_{0,k}) (b^*_{1-i} s)^{\gamma_{0,k}}} \big) \end{align}
   for $k = 1, 2$, where the value $b^*_{1-i}$ is determined by the equation
   (5.19) \begin{align} &b^{\gamma_{0,1} - \gamma_{0,2}}_{1-i} = \frac{(\gamma_{0,1} - 1) (\gamma_{0,2} (L_{1-i} - K_{1-i} b_{1-i}) + K_{1-i} b_{1-i})}{(\gamma_{0,2} - 1) (\gamma_{0,1} (L_{1-i} - K_{1-i} b_{1-i}) + K_{1-i} b_{1-i})} \end{align}
   for $i = 0, 1$ (see [Reference Beibel and Lerche8]).

   Substituting the expression for $W_{1-i}(x, s;\ b^*_{1-i})$ from (5.17) into the equation of (5.3) for $W_i(x, s)$ and applying the conditions of (5.4)–(5.6), we obtain that the candidate value function admits the representation

   (5.20) \begin{align} &W_i(x, s;\ b^*_i, b^*_{1-i}) = \sum_{k = 1}^2 {\widehat C}_{i,k}(s;\ b^*_i, b^*_{1-i}) \, x^{\beta_{i, k}} + W_{1-i}(x, s;\ b^*_{1-i}) \end{align}
   for $0 < b^*_i s < x \le s$, with
   (5.21) \begin{align} \notag &{\widehat C}_{i,k}(s;\ b^*_i, b^*_{1-i}) \\[3pt] &= \frac{\partial_x W_{1-i}(b^*_i s, s;\ b^*_{1-i}) - \beta_{i,3-k} \, (W_{1-i}(b^*_i s, s;\ b^*_{1-i}) - L_i s + K_i b^*_i s)} {(\beta_{i,3-k} - \beta_{i,k}) (b^*_i s)^{\beta_{i,k}}} \end{align}
   for $k = 1, 2$, and the value $b^*_i$ is determined by the arithmetic equation
   (5.22) \begin{align} &{\widehat C}_{i,1}^{\prime}(s;\ b_i, b^*_{1-i}) \, s^{\gamma_{0,1}} + {\widehat C}_{i,2}^{\prime}(s;\ b_i, b^*_{1-i}) \, s^{\gamma_{0,2}} + \partial_s {W}_{1-i}(s, s;\ b^*_{1-i}) = 0, \end{align}
   where ${W}_{1-i}(s, s;\ b^*_{1-i})$ is given by (5.17), for $i = 0, 1$.

3. (iii) Let us finally assume that $\lambda_{1-i} > \lambda_i = 0$ holds for any $i = 0, 1$ fixed. In this case, by using straightforward calculations from Part (i) of Subsection 3.3, it can be shown that the candidate solution of the system in (5.3)–(5.9) takes the form

   (5.23) \begin{equation} W_{i}(x, s;\ b^*_i) = {\widehat C}_{i,1}(s;\ b^*_{i}) \, x^{\gamma_{0,1}} + {\widehat C}_{i,2}(s;\ b^*_{i}) \, x^{\gamma_{0,2}} \end{equation}
   for $0 < b^*_i s < x \le s$, with
   (5.24) \begin{align} &{\widehat C}_{i,k}(s;\ b^*_{i}) = \big( {\gamma_{0,3-k} (L_{i} \, s - K_{i} \, b^*_i s)} + K_i b^*_i s \big) \big/ \big( {(\gamma_{0,3-k} - \gamma_{0,k}) (b^*_i s)^{\gamma_{0,k}}} \big) \end{align}
   for $k = 1, 2$, where the value $b^*_{i}$ is determined by the equation
   (5.25) \begin{align} &b^{\gamma_{0,1} - \gamma_{0,2}}_{i} = \frac{(\gamma_{0,1} - 1) (\gamma_{0,2} (L_i - K_i b_i) + K_i b_i)} {(\gamma_{0,2} - 1) (\gamma_{0,1} (L_i - K_i b_i) + K_i b_i)} \end{align}
   for $i = 0, 1$ (see [Reference Beibel and Lerche8]).

   Substituting the function $W_i(x, s) = L_i s - K_i x$ into the equation of (5.3) for $W_{1-i}(x, s)$, by applying the conditions of (5.4)–(5.5), we obtain that the candidate value function $W_{1-i}(x, s; b^*_{1-i})$ admits the representation of (5.10) for $0 < b^*_{1-i} s < x \le b^*_i s$, with ${\widehat D}_{1-i, k}(s;\ b^*_{1-i})$, $k = 1, 2$, given by (5.11), where we recall that $L_{1-i}^{\prime} = L_{1-i} - \lambda_{1-i} L_i/(r + \lambda_{1-i})$ and $K_{1-i}^{\prime\prime} = K_{1-i} - \lambda_{1-i} K_i/(r + \lambda_{1-i} - \mu)$, for $i = 0, 1$, fixed. Then, using the fact that the function $W_{1-i}(x, s)$ should be continuously differentiable at $b^*_i s$, we obtain the representation

   (5.26) \begin{align} &W_{1-i}(x, s;\ b^*_i, b^*_{1-i}) = \sum_{k = 1}^2 {\widehat C}_{1-i,k}(s;\ b^*_i, b^*_{1-i}) \, x^{\beta_{1-i, k}} + W_i(x, s;\ b^*_i) \end{align}
   for $0 < b^*_{i} s < x \le s$, with
   (5.27) \begin{align} \notag &{\widehat C}_{1-i,k}(s;\ b^*_i, b^*_{1-i}) \\[3pt] &= \frac{\beta_{1-i,3-k} (W_{1-i}(b^*_i s, s;\ b^*_{1-i}, b^*_i) - L_i s + K_i b^*_i s) - (b^*_i s) \partial_x W_{1-i}(b^*_i s, s;\ b^*_{1-i}, b^*_i) - K_i b^*_i s} {(\beta_{1-i,3-k} - \beta_{1-i,k}) (b^*_i s)^{\beta_{1-i,k}}} \end{align}
   for $k = 1, 2$, where $W_{1-i}(x, s;\ b^*_{1-i}, b^*_i)$ is given by (5.10) with (5.11), and the value $b^*_{1-i}$ is determined by the arithmetic equation
   (5.28) \begin{align} &{\widehat C}_{1-i,1}^{\prime}(s;\ b^*_i, b_{1-i}) \, s^{\beta_{1-i, 1}} + {\widehat C}_{1-i,2}^{\prime}(s;\ b^*_i, b_{1-i}) \, s^{\beta_{1-i, 2}} + \partial_s W_i(s, s;\ b^*_i) = 0, \end{align}
   where $W_i(s, s;\ b^*_i)$ is given by (5.23), for $i = 0, 1$.

Summarising the facts proved above, we formulate the following result, which can be proved by means of the same arguments as Theorem 4.1 above.

Theorem 5.1. Suppose that the assumptions of Theorem 4.1 hold. Then the value function of the optimal stopping problem in (2.1) admits the representations

(5.29) \begin{equation} W^*_i(x, s)= \begin{cases} W_i(x, s;\ b^*_i, b^*_{1-i}) & \text{if} \quad b^*_i s < x \le s, \\[3pt] W_i(x, s;\ b^*_i) & \text{if} \quad b^*_i s < x \le s, \\[3pt] L_i \, s - K_i \, x & \text{if} \quad 0 < x \le b^*_{i} s, \end{cases} \end{equation}

and

(5.30) \begin{equation} W^*_{1-i}(x, s)= \begin{cases} W_{1-i}(x, s;\ b^*_i, b^*_{1-i}) & \text{if} \quad b^*_i s < x \le s, \\[3pt] W_{1-i}(x, s;\ b^*_{1-i}) & \text{if} \quad b^*_{1-i} s < x \le b^*_i s, \\[3pt] L_{1-i} \, s - K_{1-i} \, x & \text{if} \quad 0 < x \le b^*_{1-i} s, \end{cases} \end{equation}

and the optimal stopping time has the form of (5.2), where the candidate value functions and linear boundaries are specified as follows:

1. (i) If $\lambda_i > 0$ for every $i = 0, 1$, then $W_j(x, s;\ b^*_i, b^*_{1-i})$, $j = 0, 1$, are given by (5.12), for $b^*_i s < x \le s$, with ${\widehat C}_{j, k}(s;\ b^*_i, b^*_{1-i})$, $j = 0, 1$, $k = 1, 2$, from (5.13)–(5.14). $W_{1-i}(x, s; b^*_{1-i})$ is given by (5.10), for $b^*_{1-i} s < x \le b^*_i s$, with ${\widehat D}_{1-i,k}(s;\ b^*_{1-i})$, $k = 1, 2$, from (5.11), while $b^*_j$, $j = 0, 1$, are determined by the arithmetic equations in (5.15) with (5.16).

2. (ii) If $\lambda_i > \lambda_{1-i} = 0$ for any $i = 0, 1$ fixed, then $W_{1-i}(x, s;\ b^*_{1-i})$ is given by (5.17), for $b^*_{1-i} s < x \le s$, with ${\widehat C}_{1-i,k}(s;\ b^*_{1-i})$, $k = 1, 2$, from (5.18). $W_i(x, s;\ b^*_i, b^*_{1-i})$ is given by (5.20), for $b^*_i s < x \le s$, with ${\widehat C}_{i,k}(s;\ b^*_i, b^*_{1-i})$, $k = 1, 2$, from (5.21), while $b^*_j$, $j = 0, 1$, are determined by the arithmetic equations in (5.19) and (5.22).

3. (iii) If $\lambda_{1-i} > \lambda_i = 0$ for any $i = 0, 1$ fixed, then $W_i(x, s;\ b^*_i)$ is given by (5.23), for $b^*_i s < x \le s$, with ${\widehat C}_{i,k}(s;\ b^*_i)$, $k = 1, 2$, from (5.24); $W_{1-i}(x, s;\ b^*_{1-i})$ is given by (5.10), for $b^*_{1-i} s < x \le b^*_i s$, with ${\widehat D}_{1-i,k}(s;\ b^*_{1-i})$, $k = 1, 2$, from (5.11); and $W_{1-i}(x, s;\ b^*_i, b^*_{1-i})$ is given by (5.26), for $b^*_i s < x \le s$, with ${\widehat C}_{1-i,k}(s;\ b^*_i, b^*_{1-i})$, $k = 1, 2$, from (5.27), while $b^*_j$, $j = 0, 1$, are determined by the arithmetic equations in (5.25) and (5.28).