2.1. The optimal stopping game

It is seen that the problem of (1.3) and (1.4) can be embedded into the optimal stopping zero-sum game for the two-dimensional continuous (time-homogeneous) strong Markov process $(Y, Q)=(Y_t, Q_t)_{t \, \ge \,0}$ with the upper and lower value functions $V^*(y, q) \ge V_*(y, q)$ given by

(2.1) \begin{equation}V^*(y, q) = \inf_{\zeta} \sup_{\tau} {\mathbb E} \bigg[ \int_0^{\tau \wedge \zeta} e^{- {r}\, s} \,Y^{(y, q)}_s \, ds + e^{- {r} \tau} \, {K} \, I(\tau < \zeta) + e^{- {r} \zeta} \,{L} \, I(\zeta \le \tau) \bigg]\end{equation}

and

(2.2) \begin{equation}V_*(y, q) = \sup_{\tau} \inf_{\zeta} {\mathbb E} \bigg[ \int_0^{\tau \wedge \zeta} e^{- {r}\, s} \,Y^{(y, q)}_s \, ds + e^{- {r} \tau} \, {K} \, I(\tau < \zeta) + e^{- {r} \zeta} \,{L} \, I(\zeta \le \tau) \bigg]\end{equation}

for some $r > 0$ and $L > K > 0$ fixed, where the suprema and infima are taken over all stopping times $\tau$ and $\zeta$ of the process (Y, Q). Hereafter, we indicate by $(Y^{(y, q)}, Q^{(q)})$ the dependence of the solution (Y, Q) of the system of stochastic differential equations in (1.5) and (1.6) on the starting point $(y, q) \in {\mathbb R}_{++} \times [0, 1]$ . We may conclude from the results of [Reference Ekström and Peskir24, Theorem 2.1] and [Reference Peskir68, Theorem 2.1] that the optimal stopping game of (2.1) and (2.2) has a value providing a Stackelberg equilibrium, so that the equality $V^*(y, q) = V_*(y, q)$ holds, for each $(y, q) \in {\mathbb R}_{++} \times [0, 1]$ fixed. It follows from the explicit form of the process Y in (1.1) and (1.2) and the representations for the processes Y and Q in (1.5) and (1.6) that the discounted process $(e^{- {r} t} Y_t)_{t \, \ge \,0}$ is a strict supermartingale closed at zero, under the assumptions that ${r} > 0$ and $\eta_0 > \eta_1 > 0$ . Thus, the value functions in (2.1) and (2.2) are restricted between K and L, for each $(y, q) \in {\mathbb R}_{++} \times [0, 1]$ fixed.

Observe that the arguments of the proofs from [Reference Ekström and Peskir24, Theorem 2.1] and [Reference Peskir68, Theorem 2.1], which are also applicable to strong Markov time-space processes with constant killing rates, can be naturally extended to the case of the discounted optimal stopping zero-sum game of (2.1) and (2.2) for the process (Y, Q) solving the system of stochastic differential equations in (1.5) and (1.6) (see also [Reference Peskir69, Theorem 3.1] for the related result in a model based on a standard Brownian motion in the interval [0,Reference Abramovitz and Stegun1] absorbed at either 0 or 1). Note that the natural analogues of the conditions of [Reference Ekström and Peskir24, Formula (2.1)] and [Reference Peskir68, Formulae (2.9) and (2.12)] are clearly satisfied for the discounted payoffs $e^{- r t} K$ and $e^{- r t} L$ from (2.1) and (2.2). Hence, by applying the resulting extensions of the results on optimal stopping zero-sum games for continuous (time-homogeneous) strong Markov processes of [Reference Ekström and Peskir24, Theorem 2.1] and [Reference Peskir68, Theorem 2.1], we obtain from the structure of the reward functional in the game of (2.1) and (2.2) with infinite time horizon that the stopping times forming a Nash equilibrium exist and have the form

(2.3) \begin{equation}\tau^* = \inf \big\{ t \, \ge \,0 \; \big| \; V^*(Y_t, Q_t) = K \big\}\quad \text{and} \quad\zeta_* = \inf \big\{ t \, \ge \,0 \; \big| \; V_*(Y_t, Q_t) = L \big\}\end{equation}

so that the appropriate continuation and stopping regions $C_*$ and $D_*$ are given by

(2.4) \begin{equation}C_* = \big\{ (y, q) \in {\mathbb R}_{++} \times [0, 1] \; \big| \; K < V^*(y, q) = V_*(y, q) < L \big\}\end{equation}

and

(2.5) \begin{equation}D_* = \big\{ (y, q) \in {\mathbb R}_{++} \times [0, 1] \; \big| \;\text{either} \;\;V^*(y, q) = V_*(y, q) = K \;\; \text{or} \;\; V^*(y, q) = V_*(y, q) = L \big\},\end{equation}

respectively. We show in parts (i) and (ii) of Section 2.3 below that the sets $C_*$ and $D_*$ in (2.4) and (2.5) are non-empty. We will prove in Lemma 3.1 below that $V^*(y, q) = V_*(y, q)$ forms a continuous function, so that the set $C_*$ is open and the set $D_*$ is closed.

2.2. The structure of optimal stopping times

Let us now show the form of the optimal stopping times $\tau^*$ and $\zeta_*$ in (2.3) and clarify the structure of the associated continuation and stopping regions $C_*$ and $D_*$ in (2.4) and (2.5), respectively.

(i) In order to provide the upper and lower bounds for the value functions and the existence of the optimal stopping boundaries in the optimal stopping zero-sum game of (2.1) and (2.2), let us consider the optimal stopping problems with the value functions

(2.6) \begin{equation}{\underline W}(y, j) = \sup_{\tau'} {\mathbb E} \bigg[ \int_0^{\tau'} e^{- {r}\, s} \, Y^{(y, j)}_s \, ds+ e^{- {r} \tau'} \, {K} \bigg]\end{equation}

and

(2.7) \begin{equation}{\overline W}(y, j) = \inf_{\zeta'} {\mathbb E} \bigg[ \int_0^{\zeta'} e^{- {r}\, s} \, Y^{(y, j)}_s \, ds+ e^{- {r} \zeta'} \, {L} \bigg]\end{equation}

for some ${r} > 0$ and $L > K > 0$ fixed, where the supremum and infimum are taken over all stopping times $\tau'$ and $\zeta'$ with respect to the natural filtration ${\mathcal H} = ({\mathcal H}_t)_{t \, \ge \,0}$ of the process $(Y, \boldsymbol{\Theta})$ with the first component explicitly given by (1.1) and (1.2). Hereafter, we indicate by $(Y^{(y, j)}, \boldsymbol{\Theta}^{(j)})$ the dependence of the process $(Y, \boldsymbol{\Theta})$ on its starting point $(y, j) \in {\mathbb R}_{++} \times \{ 0, 1 \}$ .

It is shown by means of arguments similar to those used in [Reference Gapeev36, Appendix] as well as in [Reference Guo44, Section 2] and [Reference Guo and Zhang45, Section 2] applied to solving other optimal stopping problems in models with observable continuous-time Markov chains (or using the easily shown convexity and concavity of the value functions) that the optimal stopping times in the problems of (2.6) and (2.7) have the form

(2.8) \begin{equation}{\underline \tau} = \inf \big\{ t \, \ge \,0 \; \big| \; Y_t \le {\underline g}(\boldsymbol{\Theta}_t) \big\}\quad \text{and} \quad{\overline \zeta} = \inf \big\{ t \, \ge \,0 \; \big| \; Y_t \ge {\overline h}(\boldsymbol{\Theta}_t) \big\}\end{equation}

for some numbers $0 < {\underline g}(j) \le {r} K < {r} L \le {\overline h}(j)$ , for $j = 0, 1$ , to be determined. It follows from the arguments in [Reference Gapeev36, Appendix] and Section 5.5 below resulting at the statements of Corollaries 5.2 and 5.3 below (see also [Reference Gapeev36; Corollary A.2] as well as [Reference Guo44, Theorem 2.1] and [Reference Guo and Zhang45, Theorem 2]) that the value functions ${\underline W}(y, j)$ and ${\overline W}(y, j)$ of the auxiliary optimal stopping problems from (2.6) and (2.7) admit the explicit expressions in (5.56) and (5.57), and the numbers ${\underline g}(j)$ and ${\overline h}(j)$ , for $j = 0, 1$ , in the associated optimal stopping times ${\underline \tau}$ and ${\overline \zeta}$ from (2.8) are determined from the expressions in (5.49)–(5.50) and (5.54)–(5.55) with (5.15) as well as in (5.68)–(5.69), respectively.

Note that, since the suprema and infima in (2.1) and (2.2) are taken over all stopping times $\tau$ and $\zeta$ with respect to the filtration $({\mathcal G}_t)_{t \, \ge \,0}$ , which is smaller than the appropriate filtration $({\mathcal H}_t)_{t \, \ge \,0}$ for the stopping times $\tau'$ and $\zeta'$ over which the supremum and infimum are taken in (2.6) and (2.7), respectively, the inequalities $V^*(y, j) \le {\underline W}(y, j)$ and $V_*(y, j) \ge {\overline W}(y, j)$ should hold, for all $y \in {\mathbb R}_{++} \times \{ 0, 1 \}$ . The latter inequalities also yield the fact that the points $((0, {\underline g}(1)] \cup [{\overline h}(0), \infty)) \times [0, 1]$ surely belong to the stopping region $D_*$ in (2.5).

(ii) We now observe that, by means of straightforward computations, it is seen that the expressions

(2.9) \begin{align}&{\mathbb E} \bigg[ \int_0^{\tau \wedge \zeta} e^{- {r}\, s} \, Y^{(y, q)}_s \, ds+ e^{- {r} \tau} \, {K} \, I(\tau < \zeta) + e^{- {r} \zeta} \, {L} \, I(\zeta \le \tau) \bigg] \nonumber\\ &= {K} + {\mathbb E} \bigg[ \int_0^{\tau \wedge \zeta} e^{- {r}\, s} \, \big( Y^{(y, q)}_s - {r} K \big) \, ds+ e^{- {r} \zeta} \, ({L - K}) \, I(\zeta \le \tau) \bigg]\end{align}

and

(2.10) \begin{align}&{\mathbb E} \bigg[ \int_0^{\tau \wedge \zeta} e^{- {r}\, s} \, Y^{(y, q)}_s \, ds+ e^{- {r} \tau} \, {K} \, I(\tau < \zeta) + e^{- {r} \zeta} \, {L} \, I(\zeta \le \tau) \bigg] \nonumber\\&= {L} + {\mathbb E} \bigg[ \int_0^{\tau \wedge \zeta} e^{- {r}\, s} \, \big( Y^{(y, q)}_s - {r} L \big) \, ds- e^{- {r} \tau} \, ({L - K}) \, I(\tau < \zeta) \bigg]\end{align}

hold, for any stopping times $\tau$ and $\zeta$ . Then it follows from (2.9) and (2.10) and the structure of the stopping times in (2.3) that the (actually coinciding) value functions of the optimal stopping game in (2.1) and (2.2) admit the representations

(2.11) \begin{align}V^*(y, q) &= {K} + {\mathbb E} \bigg[ \int_0^{\tau^* \wedge \zeta_*} e^{- {r}\, s} \, \big( Y^{(y, q)}_s - {r} K \big) \, ds + e^{- {r} \zeta_*} \, ({L - K}) \, I(\zeta_* \le \tau^*) \bigg]\end{align}

and

(2.12) \begin{align}V_*(y, q) &= {L} + {\mathbb E} \bigg[ \int_0^{\tau^* \wedge \zeta_*} e^{- {r}\, s} \, \big( Y^{(y, q)}_s - {r} L \big) \, ds - e^{- {r} \tau^*} \, ({L - K}) \, I(\tau^* < \zeta_*) \bigg]\end{align}

for all $(y, q) \in {\mathbb R}_{++} \times [0, 1]$ . Here, we denote by $\tau^* = \tau^*(y, q)$ and $\zeta_* = \zeta_*(y, q)$ the stopping times forming a Nash equilibrium in (2.1) and (2.2) associated with the starting point $(y, q) \in {\mathbb R}_{++} \times [0, 1]$ of the process (Y, Q). On the one hand, it is seen from the structure of the integrands in (2.11) and (2.12) that it is not optimal for the holder of the bond (maximizer of the expected reward) to exercise the contract (convert the bond) earlier than the writer may withdraw it when $Y_t > {r} K$ , while it is not optimal for the writer of the bond (minimizer of the expected reward) to withdraw the contract (redeem the bond) earlier than the holder may exercise it when $Y_t < {r} L$ , for $t \, \ge \,0$ . These facts mean that the points $(y, q) \in ({r} K, {r} L) \times [0, 1]$ , for which both inequalities $y < {r} L$ and $y > {r} K$ hold simultaneously, belong to the continuation region $C_*$ in (2.4). On the other hand, the structure of the integrands and payoffs in (2.11) and (2.12) also implies that the holder of the bond should exercise the contract at some time when $Y_t \le {r} K$ , while the writer of the bond should withdraw the contract at some time when $Y_t \ge {r} L$ , for $t \, \ge \,0$ . These facts mean that the points $(y, q) \in ((0, {r} K] \cup [{r} L, \infty)) \times [0, 1]$ cover the stopping region $D_*$ in (2.5).

(iii) Let us now fix some $(y, q) \in C_*$ such that either $y < {r} K$ or $y > {r} L$ holds, and consider the optimal stopping times $\tau^* = \tau^*(y, q)$ and $\zeta_* = \zeta_*(y, q)$ for the writer and the holder of the game-type contingent claim considered. Then, by means of the results of general optimal stopping theory for Markov processes (cf., for example, [Reference Peskir and Shiryaev71, Chapter I, Section 2.2]), we conclude from the structure of the continuation region $C_*$ in (2.4) and the form of the stopping times in (2.3) as well as from the equalities in (2.11) and (2.12) that either

(2.13) \begin{align}&V^*(y, q) - {K}= {\mathbb E} \bigg[ \int_0^{\tau^* \wedge \zeta_*} e^{- {r}\, s} \, \big( Y^{(y, q)}_s- {r} K \big) \, ds + e^{- {r} \zeta_*} \, ({L - K}) \, I(\zeta_* \le \tau^*) \bigg] > 0\end{align}

or

(2.14) \begin{align}&V_*(y, q) - {L}= {\mathbb E} \bigg[ \int_0^{\tau^* \wedge \zeta_*} e^{- {r}\, s} \, \big( Y^{(y, q)}_s - {r} L \big) \, ds- e^{- {r} \tau^*} \, ({L - K}) \, I(\tau^* < \zeta_*) \bigg] < 0\end{align}

holds. Moreover, it follows from the comparison results for solutions of (one-dimensional time-homogeneous) stochastic differential equations of [Reference Ferreyra and Sundar32, Theorem 1] applied to the processes Y and Q with the representations in (1.5) and (1.6) that the inequality $Y^{(y', q')}_t \le Y^{(y'', q'')}_t$ holds under $\eta_0 > \eta_1 > 0$ , for each $0 < y' \le y''$ and $0 < q' \le q'' < 1$ as well as all $t \, \ge \,0$ . Then the inequality

(2.15) \begin{align}&{\mathbb E} \bigg[ \int_0^{\tau \wedge \zeta} e^{- {r}\, s} \, Y^{(y', q')}_s \, ds+ e^{- {r} \tau} \, {K} \, I(\tau < \zeta) + e^{- {r} \zeta} \, {L} \, I(\zeta \le \tau) \bigg] \nonumber\\&\le {\mathbb E} \bigg[ \int_0^{\tau \wedge \zeta} e^{- {r}\, s} \, Y^{(y'', q'')}_s \, ds+ e^{- {r} \tau} \, {K} \, I(\tau < \zeta) + e^{- {r} \zeta} \, {L} \, I(\zeta \le \tau) \bigg]\end{align}

holds for any stopping times $\tau$ and $\zeta$ of the process (Y, Q). Hence, taking the supremum and infimum over all stopping times $\tau$ and $\zeta$ in either of the order in (2.15), we may conclude that the (coinciding) upper and lower functions $V^*(y, q)$ and $V_*(y, q)$ defined in (2.1) and (2.2) are increasing in the variables y and q on ${\mathbb R}_{++}$ and [0, 1], respectively.

Figure 1. Computer plots of the optimal stopping boundaries $g^*(q)$ and $h_*(q)$ .

On the one hand, taking any y’ such that either $y < y' \le {r} K$ and $q < q'$ , or ${r} L \le y' < y$ and $q' < q$ holds, and using the property that the functions $V^*(y, q) = V_*(y, q)$ are increasing in y and q on ${\mathbb R}_{++}$ and [0, 1], respectively, we obtain from and (2.14) that either the inequalities $V^*(y', q') - K \ge V^*(y, q) - K > 0$ or $V_*(y', q') - L \le V_*(y, q) - L < 0$ are satisfied, so that $(y', q') \in C_*$ too. On the other hand, if we assume that $(y, q) \in D_*$ such that either $y \le {r} K$ or $y \ge {r} L$ holds, then using arguments similar to those above, we obtain that either the inequalities $V^*(y'', q'') - K \le V^*(y, q) - K = 0$ hold, for all $y'' \le y < {r} K$ and $q'' \le q$ , or the ones $V_*(y'', q'') - L \ge V_*(y, q) - L = 0$ hold, for all ${r} L < y \le y''$ and $q \le q''$ , so that $(y'', q'') \in D_*$ . Thus, we may conclude that the stopping times $\tau^*$ and $\zeta_*$ have the form of (2.16), where the left-and right-hand boundary functions $g^*(q)$ and $h_*(q)$ are decreasing on [0, 1].

Observe that, if we suppose that $g^*(j) < {\underline g}(j)$ or $h_*(j) > {\overline h}(j)$ holds, for some $j = 0, 1$ , then, for each $y \in (g^*(j), {\underline g}(j))$ or $y \in ({\overline h}(j), h_*(j))$ fixed, we would have $V^*(y, j) > K = {\underline W}(y, j)$ or $V_*(y, j) < L = {\overline W}(y, j)$ , respectively, which contradicts the obvious facts mentioned above that the inequalities $V^*(y, j) \le {\underline W}(y, j)$ and $V_*(y, j) \ge {\overline W}(y, j)$ should hold, for all $y \in {\mathbb R}_{++} \times \{ 0, 1 \}$ . Thus, we may conclude that the inequalities ${\underline g}(j) \le g^*(j)$ and $h_*(j) \le {\overline h}(j)$ should hold, for every $j = 0, 1$ (see Figure 1 for computer plots of the optimal stopping boundaries $g^*(q)$ and $h_*(q)$ , for $q \in [0, 1]$ , as well as ${\underline g}(j)$ and ${\overline h}(j)$ , for $j = 0, 1$ ).

Summarizing the arguments shown above, let us formulate the following assertion.

Lemma 2.1. Suppose that the processes Y and Q are defined by (1.1)–(1.2) and (1.5)–(1.6) with (1.7), for some ${r} > 0$ , $\eta_0 > \eta_1 > 0$ , $\sigma > 0$ , and $\lambda_j \ge 0$ for $j = 0, 1$ . Then the stopping times $\tau^*$ and $\zeta_*$ from (2.3) forming a Nash equilibrium in the optimal stopping zero-sum game of (2.1) and (2.2), for some $L > K > 0$ fixed, admit the representations

(2.16) \begin{equation}\tau^* = \inf \big\{ t \, \ge \,0 \; \big| \; Y_t \le g^*(Q_t) \big\}\quad \text{and} \quad\zeta_* = \inf \big\{ t \, \ge \,0 \; \big| \; Y_t \ge h_*(Q_t) \big\}\end{equation}

so that the continuation and stopping regions $C_*$ and $D_*$ in (2.4) and (2.5) take the form

(2.17) \begin{equation}C_* = \big\{ (y, q) \in {\mathbb R}_{++} \times [0, 1] \; \big| \; g^*(q) < y < h_*(q) \big\}\end{equation}

and

(2.18) \begin{equation}D_* = \big\{ (y, q) \in {\mathbb R}_{++} \times [0, 1] \; \big| \;\text{either} \;\; y \le g^*(q) \;\; \text{or} \;\; y \ge h_*(q) \big\},\end{equation}

respectively. Here, $g^*(q)$ and $h_*(q)$ are functions satisfying the properties

(2.19) \begin{equation}g^*(q): [0, 1] \rightarrow (0, {r} K] \;\; \text{is decreasing and} \;\;{\underline g}(j) \le g^*(j) \le {r} K \;\; \text{holds, for} \;\; j \in \{ 0, 1 \},\end{equation}

and

(2.20) \begin{equation}h_*(q): [0, 1] \rightarrow [{r} L, \infty) \;\; \text{is decreasing and} \;\;{r} L \le h_*(j) \le {\overline h}(j) \;\; \text{holds, for} \;\; j \in \{ 0, 1 \},\end{equation}

where ${\underline g}(j)$ and ${\overline h}(j)$ , for $j = 0, 1$ , are determined from (5.49)–(5.50) and (5.54)–(5.55) with (5.15) as well as (5.68)–(5.69).

3.1. The free-boundary problem

By means of standard arguments based on an application of Itô’s formula (cf., for example, [Reference Liptser and Shiryaev61, Theorem 4.4], [Reference Karatzas and Shreve52, Chapter V, Section 5.1], [Reference Revuz and Yor72, Chapter IV, Theorem 3.3] and [Reference Øksendal64, Theorem 7.5.4]), it is shown that the infinitesimal operator ${\mathbb L}_{(Y, Q)}$ of the process (Y, Q) solving the stochastic differential equations (1.5) and (1.6) has the structure

(3.1) \begin{align} {\mathbb L}_{(Y, Q)} &= \big( r - \eta_0 - (\eta_1 - \eta_0) \, q \big) \, y \, \partial_y+ \frac{\sigma^2 y^2}{2} \, \partial_{yy} - (\eta_1 - \eta_0) \, y \, q (1-q) \, \partial_{y q} \nonumber\\&\phantom{\;\: =}+ \big( \lambda_0 \, (1 - q) - \lambda_1 \, q \big) \, \partial_{q} + \frac{1}{2} \,\bigg( \frac{\eta_0 - \eta_1}{\sigma} \bigg)^2 \, q^2 (1-q)^2 \, \partial_{qq} \end{align}

for all $(y, q) \in {\mathbb R}_{++} \times (0, 1)$ . In order to characterize the unknown value functions $V^*(y, q) = V_*(y, q)$ from (2.1) and (2.2) and the unknown boundaries $g^*(q)$ and $h_*(q)$ from (2.16), we may use the results of general theory of optimal stopping problems for continuous-time Markov processes (cf., for example, [Reference Peskir and Shiryaev71, Chapter IV, Section 8]) and formulate the associated free-boundary problem

(3.2) \begin{align}({\mathbb L}_{(Y, Q)} V - {r} V)(y, q) = - y \quad &\text{for} \ g(q) < y < h(q),\end{align}

(3.3) \begin{align}{V}(y, q) \big|_{y = g(q)} = K \quad &\text{and} \quad {V}(y, q) \big|_{y = h(q)} = L,\end{align}

(3.4) \begin{align}V_y(y, q) \big|_{y = g(q)} = 0 \quad &\text{and} \quad V_{y}(y, q) \big|_{y = h(q)} = 0,\end{align}

(3.5) \begin{align}V_{q}(y, q) \big|_{y = g(q)} = 0 \quad &\text{and} \quad V_{q}(y, q) \big|_{y = h(q)} = 0,\end{align}

(3.6) \begin{align}V(y, q) = K \quad \text{for} \ y < g(q) \quad &\text{and} \quad V(y, q) = L \quad \text{for} \ y > h(q),\end{align}

(3.7) \begin{align}V(y, q) > K \quad \text{for} \ y > g(q) \quad &\text{and} \quad V(y, q) < L \quad \text{for} \ y < h(q),\end{align}

(3.8) \begin{align}({\mathbb L}_{(Y, Q)} V - {r} V)(y, q) < - y \quad &\text{for} \ y < g(q),\end{align}

(3.9) \begin{align}({\mathbb L}_{(Y, Q)} V - {r} V)(y, q) > - y \quad &\text{for} \ y > h(q),\end{align}

for $q \in (0, 1)$ . Observe that the semiharmonic characterization of the value function proved in [Reference Peskir68, Theorem 2.1] implies that $V^*(y, q) = V_*(y, q)$ forms the smallest function satisfying the left-hand part of the system in (3.2)–(3.3) and (3.6)–(3.7) with the boundary $g^*(q)$ and the largest function satisfying the right-hand part of the system in (3.2)–(3.3) and (3.6)–(3.7) with the boundary $h_*(q)$ . Note that inequalities (3.8) and (3.9) follow directly from the assertion of Lemma 2.1 which was proved in parts (i)–(iii) of Section 2.3 above.

3.2. Continuity of the value function

Let us now show that the lower value function $V_*(y, q)$ (which coincides with the upper value function $V^*(y, q)$ ) of the optimal stopping zero-sum game in (2.1) and (2.2) is continuous at any point $(y, q) \in {\mathbb R}_{++} \times [0, 1]$ . In order to deduce this property, it is enough to prove the following assertion.

Lemma 3.1. The lower value function $V_*(y, q)$ (which coincides with the upper value function $V^*(y, q)$ ) of the optimal stopping zero-sum game in (2.1) and (2.2) has the properties

(3.10) \begin{align} &y \mapsto V_*(y, q) \ \text{is continuous at} \ y' \ \text{uniformly over} \ q \in [q'-{\delta}, q'+{\delta}], \end{align}

(3.11) \begin{align} &q \mapsto V_*(y', q) \ \text{is continuous at} \ q', \end{align}

for every $(y', q') \in {\mathbb R}_{++} \times (0, 1)$ fixed, with some ${\delta} > 0$ small enough.

Proof. In order to deduce property (3.10), let us fix some $y_1 \le y_2$ in $[y' - {\delta}, y' + {\delta}]$ and $q \in [q' - {\delta}, q' + {\delta}]$ such that the associated square belongs to ${\mathbb R}_{++} \times [0, 1]$ . We consider $\tau^* = \tau^*(y_2, q)$ and $\zeta_* = \zeta_*(y_1, q)$ the stopping times from the Nash equilibria in (2.1) and (2.2) associated with the starting points $(y_2, q)$ and $(y_1, q)$ of the process (Y, Q), respectively. Then, taking into account the structure of the integrand in (2.2) based on the explicit form of the process Y in (1.1) and (1.2) and the representations for the components of the process (Y, Q) in (1.5) and (1.6) as well as the fact that the left-hand boundary $g^*(q)$ and the right-hand boundary $h_*(q)$ from (2.16) are decreasing functions, under the assumption that $\eta_0 > \eta_1 > 0$ , we get

(3.12) \begin{align} 0 &\le V_*(y_2, q) - V_*(y_1, q) \nonumber\\ &\le {\mathbb E} \bigg[ \int_0^{\tau^* \wedge \zeta_*} e^{- {r}\, s} \, Y^{(y_2, q)}_s \, ds + e^{- {r} \tau^*} \, {K} \, I(\tau^* < \zeta_*) + e^{- {r} \zeta_*} \, {L} \, I(\zeta_* \le \tau^*) \bigg] \nonumber\\ \notag &\phantom{\le \;\:} - {\mathbb E} \bigg[ \int_0^{\tau^* \wedge \zeta_*} e^{- {r}\, s} \, Y^{(y_1, q)}_s \, ds + e^{- {r} \tau^*} \, {K} \, I(\tau^* < \zeta_*) + e^{- {r} \zeta_*} \, {L} \, I(\zeta_* \le \tau^*) \bigg] \nonumber\\ &= {\mathbb E} \bigg[ \int_0^{\tau^* \wedge \zeta_*} e^{- {r}\, s} \, \big( Y^{(y_2, q)}_s - Y^{(y_1, q)}_s \big) \, ds \bigg] = (y_2 - y_1) \, {\mathbb E} \bigg[ \int_0^{\tau^* \wedge \zeta_*} e^{- {r}\, s} \, Y^{(1, q)}_s \, ds \bigg], \end{align}

where the last expectation is finite, for all $y_1 \le y_2$ from $[y' - {\delta}, y' + {\delta}]$ . Hence, we see that the right-hand side in (3.12) converges monotonically to zero as $y_1$ approaches $y_2$ , independently of $q \in [q'-{\delta}, q'+{\delta}]$ , for any ${\delta} > 0$ fixed, and thus property (3.10) holds.

In order to deduce property (3.11), we fix $q_1 \le q_2$ in $[q' - {\delta}, q' + {\delta}]$ such that the associated interval belongs to [0,Reference Abramovitz and Stegun1]. We now denote by $\tau^* = \tau^*(y', q_2)$ and $\zeta_* = \zeta_*(y', q_1)$ the stopping times from the Nash equilibria in (2.1) and (2.2) associated with the starting points $(y', q_2)$ and $(y', q_1)$ of the process (Y, Q), respectively. Then, taking into account the explicit form of the process Y in (1.1) and (1.2) and the representations for the components of the process (Y, Q) in (1.5) and (1.6), we get

(3.13) \begin{align} 0 &\le V_*(y', q_2) - V_*(y', q_1) \nonumber\\ &\le {\mathbb E} \bigg[ \int_0^{\tau^* \wedge \zeta_*} e^{- {r}\, s} \, Y^{(y', q_2)}_s \, ds + e^{- {r} \tau^*} \, {K} \, I(\tau^* < \zeta_*) + e^{- {r} \zeta_*} \, {L} \, I(\zeta_* \le \tau^*) \bigg] \nonumber\\ &\phantom{\le \;\:}- {\mathbb E} \bigg[ \int_0^{\tau^* \wedge \zeta_*} e^{- {r}\, s} \, Y^{(y', q_1)}_s \, ds + e^{- {r} \tau^*} \, {K} \, I(\tau^* < \zeta_*) + e^{- {r} \zeta_*} \, {L} \, I(\zeta_* \le \tau^*) \bigg] \nonumber\\ &= {\mathbb E} \bigg[ \int_0^{\tau^* \wedge \zeta_*} e^{- {r}\, s} \, \big( Y^{(y', q_2)}_s - Y^{(y', q_1)}_s \big) \, ds \bigg] = y' \, {\mathbb E} \bigg[ \int_0^{\tau^* \wedge \zeta_*} e^{- {r}\, s} \, \big( Y^{(1, q_2)}_s - Y^{(1, q_1)}_s \big) \, ds \bigg], \end{align}

where the last expectation is finite like tat in (3.12), for all $q_1 \le q_2$ from $[q' - {\delta}, q' + {\delta}]$ , and, under the assumption that $\eta_0 > \eta_1 > 0$ , we have

(3.14) \begin{align} &Y^{(1, q_2)}_t - Y^{(1, q_1)}_t = Y^{(1, q_1)}_t \, \bigg( \frac{Y^{(1, q_2)}_t}{Y^{(1, q_1)}_t} - 1 \bigg) \nonumber\\ &= Y^{(1, q_1)}_t \, \bigg( \exp \bigg( \int_0^t (\eta_0 - \eta_1) \, \big( Q^{(q_2)}_s - Q^{(q_1)}_s \big) \, ds \bigg) - 1 \bigg) \end{align}

for all $t \, \ge \,0$ . Hence, by using the comparison results for strong solutions of stochastic differential equations from [Reference Ferreyra and Sundar32, Theorem 1] and applying the Lebesgue dominated convergence theorem, we may conclude that the right-hand side in (3.13) converges to zero as $q_1$ approaches $q_2$ , for any ${\delta} > 0$ fixed, and thus property (3.11) holds. This fact, together with property (3.10), means in particular that the instantaneous-stopping conditions of (3.3) above are satisfied.

3.3. Regularity of the optimal stopping boundaries

Let us now prove the probabilistic regularity of the boundary $\partial C_*$ of the continuation region in (2.17).

Proof. By virtue of sample path and distributional properties of the two-dimensional diffusion process (Y, Q) from (1.5) and (1.6) starting at some point $(y', q') \in {\mathbb R}_{++} \times (0, 1)$ fixed, on an infinitesimally small deterministic time interval of length $\varepsilon$ , we observe that the representations

(3.15) \begin{align} &Y_{\varepsilon} = y' + \big( r - \eta_0 - (\eta_1 - \eta_0) \, q' \big) \, y' \, \varepsilon + \sigma \, y' \, {\overline B}_{\varepsilon} + o(\varepsilon) \quad \text{as} \ \varepsilon \downarrow 0 \ \text{(${\mathbb P}$-almost surely (a.s.))} \end{align}

and

(3.16) \begin{align} &Q_{\varepsilon} = q' + \big( \lambda_0 \, (1 - q') - \lambda_1 \, q' \big) \, \varepsilon + \frac{\eta_0 - \eta_1}{\sigma} \, q' (1-q') \, {\overline B}_{\varepsilon} + o(\varepsilon) \quad \text{as} \ \varepsilon \downarrow 0 \ \text{(${\mathbb P}$-a.s.)} \end{align}

hold, where $o(\varepsilon)$ denotes a random function satisfying $o(\varepsilon)/\varepsilon \to 0$ as $\varepsilon \downarrow 0$ ( ${\mathbb P}$ -a.s.). Recall that ${\overline B}_{\varepsilon} \sim \sqrt{\varepsilon} \Psi$ as $\varepsilon \downarrow 0$ ( ${\mathbb P}$ -a.s.), where $\Psi$ is a standard normal random variable. Then, starting from the point (y’, q’) and letting the process (Y, Q) evolve for an infinitesimal amount of time $\varepsilon$ , we see that the process moves infinitesimally in the direction

(3.17) \begin{align} &\big( Y_{\varepsilon} - y', Q_{\varepsilon} - q' \big) \sim \bigg( \sqrt{\varepsilon} \, \sigma \, y' \, \Psi, \sqrt{\varepsilon} \, \frac{\eta_0 - \eta_1}{\sigma} \, q' (1-q') \, \Psi \bigg) \quad \text{as} \ \varepsilon \downarrow 0 \ \text{(${\mathbb P}$-a.s.)} \end{align}

for any point $(y', q') \in {\mathbb R}_{++} \times (0, 1)$ fixed. In other words, the two-dimensional process (Y, Q) moves in either the south-west or the north-east direction in the plane, under the assumption that $\eta_0 > \eta_1 > 0$ . Hence, combining this fact with the fact that the boundaries $g^*(q)$ and $h_*(q)$ are decreasing and taking into account the fact that the local drift of the process (Y, Q) has the order of $\varepsilon$ , we may conclude that the first hitting time $\tau^*(y', q') \wedge \zeta_*(y', q')$ in the stopping set $D_*$ converges to zero ( ${\mathbb P}$ -a.s.) as the point (y’, q’) approaches the point (y, q) such that $y = g^*(q)$ or $y = h_*(q)$ . This fact means precisely that all the points of the boundary $\partial C_*$ are probabilistically regular for the stopping region $D_*$ relative to the process (Y, Q) (cf., for example, [Reference Øksendal64, Section 9.2] for an extensive discussion on this point and other references to the related literature). Note that we can also deduce the assertion stated above by means of arguments similar to those applied in the proofs in [Reference Johnson and Peskir49, Section 10] and [Reference Gapeev and Peskir40, Theorem 12], with the difference with respect to the arguments used in the latter reference that we have the same standard Brownian motion driving the stochastic differential equations (1.5) and (1.6) for the processes Y and Q, and not the independent ones. More precisely, the desired assertion follows directly from the structure of the regions $C_*$ and $D_*$ in (2.17) and (2.18) having the left- and right-hand decreasing boundary functions $g^*(q)$ and $h_*(q)$ and an application of Blumenthal’s zero–one law (cf., for example, [Reference Øksendal64, Lemma 9.2.6] and subsequent results and examples).

3.4. Smooth-fit conditions for the value function

Let us now show that the lower value function $V_*(y, q)$ (which coincides with the upper value function $V^*(y, q)$ ) of the optimal stopping zero-sum game in (2.1) and (2.2) satisfies the smooth-fit conditions of (3.4) above.

Proof. Let us now consider a point $(y, q) \in [{r} L, \infty) \times (0, 1)$ at the boundary $\partial C_*$ , so that $y = h_*(q)$ and $V_*(y, q) = L$ holds. In order to derive the right-hand property in (3.4), we first observe directly from the structure of the continuation region $C_*$ in (2.4) and (2.17) that the inequality

(3.18) \begin{align} &\liminf_{{\delta} \downarrow 0} \frac{V_*(y - {\delta}, q) - V_*(y, q)}{- {\delta}} \ge 0 \end{align}

is satisfied, due to the fact that the value function $V_*(y, q)$ from (2.2) is increasing in y on ${\mathbb R}_{++}$ (see part (i) of Section 2.3 above). Let us now denote by $\tau^1_{{\delta}} = \tau^*(y - {\delta}, q)$ and $\zeta^1_{{\delta}} = \zeta_*(y - {\delta}, q)$ the stopping times which form a Nash equilibrium in (2.1) and (2.2) associated with the starting point $(y - {\delta}, q)$ of the process (Y, Q), with some ${\delta} > 0$ small enough. Then, taking into account the structure of the integrand in (2.2) based on the explicit form of the process Y in (1.1) and the representations for the components of the process (Y, Q) in (1.5) and (1.6) as well as the fact that the left-hand boundary $g^*(q)$ and the right-hand boundary $h_*(q)$ from (2.16) are decreasing functions, we get

(3.19) \begin{align} &V_*(y - {\delta}, q) - V_*(y, q) \nonumber\\ &\ge {\mathbb E} \bigg[ \int_0^{\tau^1_{{\delta}} \wedge \zeta^1_{{\delta}}} e^{- {r}\, s} \, Y^{(y - {\delta}, q)}_s \, ds + e^{- {r} \tau^1_{{\delta}}} \, {K} \, I(\tau^1_{{\delta}} < \zeta^1_{{\delta}}) + e^{- {r} \zeta^1_{{\delta}}} \, {L} \, I(\zeta^1_{{\delta}} \le \tau^1_{{\delta}}) \bigg] \nonumber\\ &\phantom{\le \;\:} - {\mathbb E} \bigg[ \int_0^{\tau^1_{{\delta}} \wedge \zeta^1_{{\delta}}} e^{- {r}\, s} \, Y^{(y, q)}_s \, ds + e^{- {r} \tau^1_{{\delta}}} \, {K} \, I(\tau^1_{{\delta}} < \zeta^1_{{\delta}}) + e^{- {r} \zeta^1_{{\delta}}} \, {L} \, I(\zeta^1_{{\delta}} \le \tau^1_{{\delta}}) \bigg] \nonumber\\ &= {\mathbb E} \bigg[ \int_0^{\tau^1_{{\delta}} \wedge \zeta^1_{{\delta}}} e^{- {r}\, s} \, \big( Y^{(y - {\delta}, q)}_s - Y^{(y, q)}_s \big) \, ds \bigg] = - {\delta} \, {\mathbb E} \bigg[ \int_0^{\tau^1_{{\delta}} \wedge \zeta^1_{{\delta}}} e^{- {r}\, s} \, Y^{(1, q)}_s \, ds \bigg], \end{align}

where the last expectation is positive and finite, for any ${\delta} > 0$ small enough. Hence, by using the fact that $\tau^1_{{\delta}} \wedge \zeta^1_{{\delta}} \to 0$ ( ${\mathbb P}$ -a.s.) as ${\delta} \downarrow 0$ , due to the probabilistic regularity of the boundary $\partial C_*$ for the region $D_*$ relative to (Y, Q), and applying the Lebesgue dominated convergence theorem, we obtain that the inequality

(3.20) \begin{align} &\limsup_{{\delta} \downarrow 0} \frac{V_*(y - {\delta}, q) - V_*(y, q)}{- {\delta}} \le 0 \end{align}

holds. Thus, taking inequalities (3.18) and (3.20) together, we conclude that the smooth-fit condition in the right-hand part of (3.4) is satisfied, while the condition in the left-hand part is deduced in the same way.

In order to derive the property in the right-hand part of (3.5), we first observe directly from the structure of the continuation region $C_*$ in (2.4) and (2.17) that the inequality

(3.21) \begin{align} &\liminf_{{\delta} \downarrow 0} \frac{V_*(y, q - {\delta}) - V_*(y, q)}{-{\delta}} \ge 0 \end{align}

is satisfied, due to the fact, proved in part (ii) of Section 2.3 above, that the value function $V_*(y, q)$ from (2.2) is increasing in q on (0, 1). Let us finally denote by $\tau^2_{{\delta}} = \tau^*(y, q - {\delta})$ and $\zeta^2_{{\delta}} = \zeta_*(y, q - {\delta})$ the stopping times which form a Nash equilibrium in (2.1) and (2.2) associated with the starting point $(y, q - {\delta})$ of the process (Y, Q), with some ${\delta} > 0$ small enough. Then, taking into account the explicit form of the process Y in (1.1) and the representations for the components of the process (Y, Q) in (1.5) and (1.6), by applying Itô’s formula to the process $(Q_t/(1 - Q_t))_{t \, \ge \,0}$ , we get

(3.22) \begin{align} &V_*(y, q - {\delta}) - V_*(y, q) \nonumber\\ &\ge {\mathbb E} \bigg[ \int_0^{\tau^2_{{\delta}} \wedge \zeta^2_{{\delta}}} e^{- {r}\, s} \, Y^{(y, q - {\delta})}_s \, ds + e^{- {r} \tau^2_{{\delta}}} \, {K} \, I(\tau^2_{{\delta}} < \zeta^2_{{\delta}}) + e^{- {r} \zeta^2_{{\delta}}} \, {L} \, I(\zeta^2_{{\delta}} \le \tau^2_{{\delta}}) \bigg] \nonumber\\ &\phantom{\le \;\:} - {\mathbb E} \bigg[ \int_0^{\tau^2_{{\delta}} \wedge \zeta^2_{{\delta}}} e^{- {r}\, s} \, Y^{(y, q)}_s \, ds + e^{- {r} \tau^2_{{\delta}}} \, {K} \, I(\tau^2_{{\delta}} < \zeta^2_{{\delta}}) + e^{- {r} \zeta^2_{{\delta}}} \, {L} \, I(\zeta^2_{{\delta}} \le \tau^2_{{\delta}}) \bigg] \nonumber\\ &= {\mathbb E} \bigg[ \int_0^{\tau^2_{{\delta}} \wedge \zeta^2_{{\delta}}} e^{- {r}\, s} \, \big( Y^{(y, q - {\delta})}_s - Y^{(y, q)}_s \big) \, ds \bigg] = y \, {\mathbb E} \bigg[ \int_0^{\tau^2_{{\delta}} \wedge \zeta^2_{{\delta}}} e^{- {r}\, s} \, \big( Y^{(1, q - {\delta})}_s - Y^{(1, q)}_s \big) \, ds \bigg], \end{align}

where the last expectation is negative and finite, for any $q \in (0, 1)$ and ${\delta} > 0$ small enough, under the assumption that $\eta_0 > \eta_1 > 0$ , and we have

(3.23) \begin{align} &Y^{(1, q - {\delta})}_t - Y^{(1, q)}_t = Y^{(1, q)}_t \, \bigg( \exp \bigg( \int_0^t (\eta_0 - \eta_1) \, \big( Q^{(q - {\delta})}_s - Q^{(q)}_s \big) \, ds \bigg) - 1 \bigg) \nonumber\\ &= Y^{(1, q)}_t \, \bigg( \exp \bigg( \int_0^t (\eta_0 - \eta_1) \, Q^{(q)}_s \big( 1 - Q^{(q - {\delta})}_s \big) \, \bigg( \frac{Q^{(q - {\delta})}_s (1 - Q^{(q)}_s)}{(1 - Q^{(q - {\delta})}_s) Q^{(q)}_s} - 1 \bigg) \, ds \bigg) - 1 \bigg) \end{align}

and

(3.24) \begin{align} \frac{Q^{(q - {\delta})}_t (1 - Q^{(q)}_t)}{(1 - Q^{(q - {\delta})}_t) Q^{(q)}_t} &= \frac{(q - {\delta})(1 - q)}{(1 - q + {\delta}) q} \, \exp \bigg( \int_0^t \bigg( \frac{\lambda_0 (1 - Q^{(q - {\delta})}_s) - \lambda_1 Q^{(q - {\delta})}_s}{Q^{(q - {\delta})}_s (1 - Q^{(q - {\delta})}_s)} \nonumber\\ &\phantom{=\;\:}- \frac{\lambda_0 (1 - Q^{(q)}_s) - \lambda_1 Q^{(q)}_s}{Q^{(q)}_s (1 - Q^{(q)}_s)} + \frac{(\eta_0 - \eta_1)^2}{\sigma^2} \, \big( Q^{(q - {\delta})}_s - Q^{(q)}_s \big) \bigg) \, ds \bigg) \end{align}

for all $t \, \ge \,0$ . Hence, we can take into account the dependence structure of the solution $Q^{(q)}$ of the stochastic differential equation (1.6) on its starting point $q \in (0, 1)$ and provide the appropriate Taylor’s expansions for the exponential functions in (3.23) and (3.24) to get that the inequality

(3.25) \begin{align} &\int_0^{\tau^2_{{\delta}} \wedge \zeta^2_{{\delta}}} e^{- {r}\, s} \, \big( Y^{(1, q - {\delta})}_s - Y^{(1, q)}_s \big) \, ds \ge - {\delta} \, (\eta_0 - \eta_1) \, (\tau^2_{{\delta}} \wedge \zeta^2_{{\delta}}) + o({\delta}) \quad \text{as} \ {\delta} \downarrow 0 \ \text{(${\mathbb P}$-a.s.)} \end{align}

holds, where $o({\delta})$ denotes a random bounded function satisfying $o({\delta})/{\delta} \to 0$ as ${\delta} \downarrow 0$ ( ${\mathbb P}$ -a.s.). Thus, by using the fact that $\tau^2_{{\delta}} \wedge \zeta^2_{{\delta}} \to 0$ ( ${\mathbb P}$ -a.s.) as ${\delta} \downarrow 0$ , due to the probabilistic regularity of the boundary $\partial C_*$ for the region $D_*$ relative to (Y, Q), and applying the Lebesgue dominated convergence theorem, we combine (3.22)–(3.25) to obtain that the inequality

(3.26) \begin{align} &\limsup_{{\delta} \downarrow 0} \frac{V_*(y, q - {\delta}) - V_*(y, q)}{- {\delta}} \le 0 \end{align}

is satisfied. Therefore, taking inequalities (3.21) and (3.26) together, we conclude that the smooth-fit condition in the right-hand part of (3.5) above is satisfied, while the condition in the left-hand part is deduced in the same way.

3.5. Continuous differentiability of the value function

Let us now show that the lower value function $V_*(y, q)$ (which coincides with the upper value function $V^*(y, q)$ ) of the optimal stopping zero-sum game in (2.1) and (2.2) is continuously differentiable on the whole state space ${\mathbb R}_{++} \times [0, 1]$ of the process (Y, Q).

Proof. We first recall that, by means of the arguments related to the proof of the strong Markov property of the process (Y, Q), which means the subsequent applications of Itô’s formula and Doob’s optional sampling theorem for the resulting local martingale taken at the time of the exit of the process (Y, Q) from a (bounded) open subset of the continuation region $C_*$ from (2.17) followed by a passage to the limit in the resulting quotient as in [Reference Peskir and Shiryaev71, Chapter III, Sections 7.1--7.3], it is shown that the value function $V_*(y, q)$ solves the parabolic-type partial differential equation of (3.1) and (3.2) (which has degenerate coefficients), and thus the function $V_*(y, q)$ surely belongs at least to the class $C^{1,1}$ (which coincides with the class $C^1$ ) on the continuation region $C_*$ in (2.17). In this respect, it remains for us to prove that the partial derivatives $(V_*)_y(y, q)$ and $(V_*)_{q}(y, q)$ are continuous functions at the boundary $\partial C_*$ . For this purpose, we will show the existence of other directional derivatives of $V_*(y, q)$ along the boundary $\partial C_*$ following the schema of arguments used in [Reference Johnson and Peskir49, Section 11] combined with those of Lemma 3.3 above. Note that the method applied below allow us to prove the assertion for any point (y, q) of the state space ${\mathbb R}_{++} \times (0, 1)$ of the process (Y, Q).

On the one hand, we need to show that the property

(3.27) \begin{align} &\lim_{n \to \infty} (V_*)_y(y_n, q_n) = 0 \end{align}

holds, for any sequence $(y_n, q_n)_{n \in {\mathbb N}}$ tending to (y, q) as $n \to \infty$ such that $y = g^*(q)$ or $y = h_*(q)$ . Since we have either $V_*(y_n, q_n) = K$ or $V_*(y_n, q_n) = L$ , for $(y_n, q_n) \in D_*$ , and the conditions of (3.4) and (3.5) hold at either $y = g^*(q)$ or $y = h_*(q)$ , respectively, there is no restriction in fixing some (y, q) such that $y = h_*(q)$ and assuming that $(y_n, q_n) \in C_*$ , for every $n \in {\mathbb N}$ . Then, because of the fact, proved in part (i) of Section 2.3 above, that the function $V_*(y, q)$ from (2.2) is increasing in y on ${\mathbb R}_{++}$ , we may conclude that the inequality

(3.28) \begin{align} &\liminf_{n \to \infty} (V_*)_y (y_n, q_n) = \liminf_{n \to \infty} \lim_{{\delta} \downarrow 0} \frac{V_*(y_n - {\delta}, q_n) - V_*(y_n, q_n)}{- {\delta}} \ge 0 \end{align}

holds. Thus, it remains for us to show that the inequality

(3.29) \begin{align} &\limsup_{n \to \infty} (V_*)_y (y_n, q_n) = \limsup_{n \to \infty} \lim_{{\delta} \downarrow 0} \frac{V_*(y_n - {\delta}, q_n) - V_*(y_n, q_n)}{- {\delta}} \le 0 \end{align}

holds too. For this purpose, we observe from the first identity in (3.29) that one can choose subsequences $(y_{n_k}, q_{n_k})_{k \in {\mathbb N}}$ and $({\delta}_k)_{k \in {\mathbb N}}$ such that

(3.30) \begin{align} &\limsup_{n \to \infty} (V_*)_y (y_n, q_n) = \lim_{k \to \infty} \frac{V_*(y_{n_k} - {\delta}_k, q_{n_k}) - V_*(y_{n_k}, q_{n_k})}{- {\delta}_k} \end{align}

with $(y_{n_k} - {\delta}_k, q_{n_k})_{k \in {\mathbb N}}$ tending to (y, q) as $k \to \infty$ . Let us consider the stopping times $\tau^1_k = \tau^*(y_{n_k} - {\delta}_k, q_{n_k})$ and $\zeta^1_k = \zeta_*(y_{n_k} - {\delta}_k, q_{n_k})$ which form a Nash equilibrium for the starting points $(y_{n_k} - {\delta}_k, q_{n_k})$ of the process (Y, Q), for every $k \in {\mathbb N}$ . Hence, taking into account the explicit form of the process Y in (1.1) and the representations for the components of the process (Y, Q) in (1.5) and (1.6), we find in the same way as in (3.19) above that

(3.31) \begin{align} &V_*(y_{n_k} - {\delta}_k, q_{n_k}) - V_*(y_{n_k}, q_{n_k}) \ge - {\delta}_k \, {\mathbb E} \bigg[ \int_0^{\tau^1_k \wedge \zeta^1_k} e^{- {r}\, s} \, Y^{(1, q_{n_k})}_s \, ds \bigg], \end{align}

where the last expectation is positive and finite, for every $k \in {\mathbb N}$ . Hence, letting $k \to \infty$ and recalling the fact that $\tau^1_k \wedge \zeta^1_k \to 0$ ( ${\mathbb P}$ -a.s.) as $k \to \infty$ , due to the probabilistic regularity of the boundary $\partial C_*$ for the region $D_*$ relative to (Y, Q), we see by the Lebesgue dominated convergence theorem that (3.31), combined with (3.30), implies the desired (3.29). Therefore, taking inequalities (3.28) and (3.29) together, we obtain property (3.27) at $y = h_*(q)$ , while the same property at $y = g^*(q)$ is deduced similarly.

On the other hand, we need to show that the property

(3.32) \begin{align} &\lim_{n \to \infty} (V_*)_{q}(y_n, q_n) = 0 \end{align}

holds, for any sequence $(y_n, q_n)_{n \in {\mathbb N}}$ tending to (y, q) as $n \to \infty$ . Since we have either $V_*(y_n, q_n) = K$ or $V_*(y_n, q_n) = L$ , for $(y_n, q_n) \in D_*$ , and conditions (3.4) hold at either $y = g^*(q)$ or $y = h_*(q)$ , respectively, we may fix some (y, q) such that $y = h_*(q)$ and assume again that $(y_n, q_n) \in C_*$ , for every $n \in {\mathbb N}$ . Then, because of the fact, proved in part (ii) of Section 2.3 above, that the function $V_*(y, q)$ from (2.2) is decreasing in q on (0, 1), we may conclude that the inequality

(3.33) \begin{align} &\liminf_{n \to \infty} (V_*)_{q}(y_n, q_n) = \liminf_{n \to \infty} \lim_{{\delta} \downarrow 0} \frac{V_*(y_n, q_n - {\delta}) - V_*(y_n, q_n)}{- {\delta}} \ge 0 \end{align}

holds. Thus, it remains for us to show that the inequality

(3.34) \begin{align} &\limsup_{n \to \infty} (V_*)_{q}(y_n, q_n) = \limsup_{n \to \infty} \lim_{{\delta} \downarrow 0} \frac{V_*(y_n, q_n - {\delta}) - V_*(y_n, q_n)}{- {\delta}} \le 0 \end{align}

holds too. In this case, we observe from the first identity in (3.34) that one can choose subsequences $(y_{n_l}, q_{n_l})_{l \in {\mathbb N}}$ and $({\delta}_l)_{l \in {\mathbb N}}$ such that

(3.35) \begin{align} &\limsup_{n \to \infty} (V_*)_{q}(y_n, q_n) = \lim_{l \to \infty} \frac{V_*(y_{n_l}, q_{n_l} - {\delta}_l) - V_*(y_{n_l}, q_{n_l})}{- {\delta}_l} \end{align}

with $(y_{n_l}, q_{n_l} - {\delta}_l)_{l \in {\mathbb N}}$ tending to (y, q) as $l \to \infty$ . Let us consider the stopping times $\tau^2_l = \tau^*(y_{n_l}, q_{n_l} - {\delta}_l)$ and $\zeta^2_l = \zeta_*(y_{n_l}, q_{n_l} - {\delta}_l)$ which form a Nash equilibrium for the starting points $(y_{n_l}, q_{n_l} - {\delta}_l)$ of the process (Y, Q), for every $l \in {\mathbb N}$ . Then, taking into account the explicit form of the process Y in (1.1) and the representations for the components of the process (Y, Q) in (1.5) and (1.6), by applying Itô’s formula to the process $Q/(1 - Q)$ , we find in the same way as in (3.22) with (3.23) and (3.24) above that

(3.36) \begin{align} V_*(y_{n_l}, q_{n_l} - {\delta}_l) - V_*(y_{n_l}, q_{n_l}) &\ge y_{n_l} \, {\mathbb E} \bigg[ \int_0^{\tau^2_l \wedge \zeta^2_l} e^{- {r}\, s} \, \big( Y^{(1, q_{n_k} - {\delta}_k)}_s - Y^{(1, q_k)}_s \big) \, ds \bigg] \\ \notag &\ge - {\delta}_l \, y_{n_l} \, (\eta_0 - \eta_1) \, {\mathbb E} \big[ \tau^2_l \wedge \zeta^2_l \big] + {\mathbb E} \big[ o({\delta}_l) \big] \end{align}

holds, where the first term on the last line is positive and finite, under $\eta_0 > \eta_1 > 0$ , and $o({\delta})$ denotes a random bounded function $o({\delta})/{\delta} \to 0$ as ${\delta} \downarrow 0$ ( ${\mathbb P}$ -a.s.), for every $l \in {\mathbb N}$ . Hence, letting $l \to \infty$ and recalling the fact that $\tau^2_l \wedge \zeta^2_l \to 0$ ( ${\mathbb P}$ -a.s.) as $l \to \infty$ , due to the probabilistic regularity of the boundary $\partial C_*$ for the region $D_*$ relative to (Y, Q), by means of arguments similar to those used in the end of the previous subsection, we see by the Lebesgue dominated convergence theorem that (3.36), combined with (3.35), implies the desired (3.34). Therefore, taking inequalities (3.33) and (3.34) together, we obtain property (3.32) at $y = h_*(q)$ , while the same property at $y = g^*(q)$ is deduced similarly.

4.1. The change of variables

In order to provide the analysis of the free-boundary problem in (3.1) and (3.2)–(3.9) and be able to apply the change-of-variable formula from [Reference Peskir67, Theorem 3.1] in the verification assertion below, we introduce an appropriate change of variables to reduce the infinitesimal operator ${\mathbb L}_{(Y, Q)}$ of the process (Y, Q) from (3.1) to the normal (or canonical) form and reformulate the initial optimal stopping problem within the new coordinates. For this purpose, let us define the process $Z = (Z_t)_{t \, \ge \,0}$ by

(4.1) \begin{equation} Z_t = \frac{Y^{- \rho}_t Q_t}{1-Q_t} \quad \text{with} \quad \rho = \frac{\eta_0 - \eta_1}{\sigma^2} \quad \text{and thus} \quad Q_t = \frac{Y^{\rho}_t Z_t}{1+Y^{\rho}_t Z_t}, \end{equation}

for all $t \, \ge \,0$ . Then, by applying Itô’s formula to (4.1), we get from the representations in (1.5) and (1.6) that the process (Y, Z) solves the (two-dimensional) system of stochastic differential equations

(4.2) \begin{equation} dY_t = \bigg( r - \eta_0 - (\eta_1 - \eta_0) \, \frac{Y^{\rho}_t Z_t}{1+ Y^{\rho}_t Z_t} \bigg) \, Y_t \, dt + \sigma \, Y_t \, d{\overline B}_t \quad (Y_0=y) \end{equation}

and

(4.3) \begin{align} &dZ_t = \bigg( \frac{(\lambda_0 - \lambda_1 Y^{\rho}_t Z_t)(1 + Y^{\rho}_t Z_t)}{Y^{\rho}_t Z_t} - \frac{\rho}{2} \, \big( 2r - \eta_0 - \eta_1 - \sigma^2 \big) \bigg) \, Z_t \, dt \quad \bigg( Z_0 = z \equiv \frac{y^{- \rho} q}{1-q} \bigg) \end{align}

for any $(y, z) \in {\mathbb R}^2_{++}$ (cf., for example, [Reference Gapeev35, Section 3], [Reference Gapeev and Shiryaev43, Section 3.5] and [Reference Johnson and Peskir49, Section 6] among others for similar transformations of variables). It is seen from the form of the stochastic differential equation (4.3) that the process Z starting at $z \in {\mathbb R}_{++}$ is of bounded variation. More precisely, if the inequality $Y^{\rho}_t Z_t < \nu$ holds with

(4.4) \begin{align} \nu &= \frac{1}{2 \lambda_1} \, \bigg( \sqrt{ \Big(\lambda_1 - \lambda_0 + \frac{\rho}{2} \, \big( 2r - \eta_0 - \eta_1 - \sigma^2 \big) \Big)^2 + 4 \lambda_0 \lambda_1} \nonumber\\ &\phantom{= \frac{1}{2 \lambda_1} \, \bigg(\;\:} - \Big( \lambda_1 - \lambda_0 + \frac{\rho}{2} \, \big( 2r - \eta_0 - \eta_1 - \sigma^2 \big) \Big) \bigg) > 0 \end{align}

for $t \, \ge \,0$ , then the process Y is increasing, while if the inequality $Y^{\rho}_t Z_t > \nu$ holds, for $t \, \ge \,0$ , then the process Z is decreasing.

Observe that, for any $(y, q) \in {\mathbb R}_{++} \times (0, 1)$ fixed, the value function of the optimal stopping zero-sum game in (2.1) and (2.2) takes the form $V^*(y, q) = V_*(y, q) = U_*(y, y^{- \rho} q/(1-q))$ with

(4.5) \begin{align} U_*(y, z) &= \sup_{\tau} \inf_{\zeta} {\mathbb E}_{y, z} \bigg[ \int_0^{\tau \wedge \zeta} e^{- {r}\, s} \, Y_s \, ds + e^{- {r} \tau} \, {K} \, I(\tau < \zeta) + e^{- {r} \zeta} \, {L} \, I(\zeta \le \tau) \bigg] \nonumber\\ &= \inf_{\zeta} \sup_{\tau} {\mathbb E}_{y, z} \bigg[ \int_0^{\tau \wedge \zeta} e^{- {r}\, s} \, Y_s \, ds + e^{- {r} \tau} \, {K} \, I(\tau < \zeta) + e^{- {r} \zeta} \, {L} \, I(\zeta \le \tau) \bigg] \end{align}

for some $\lambda > 0$ and $L > K > 0$ fixed, where ${\mathbb E}_{y, z}$ denotes the expectation taken with respect to the probability measure ${\mathbb P}$ under the assumption that the two-dimensional continuous (time-homogeneous) strong Markov process (Y, Z) satisfying the system of stochastic differential equations (4.2) and (4.3) starts at some $(y, z) \in {\mathbb R}^2_{++}$ (with probability one under ${\mathbb P}$ ). We note from the relations in (4.1) that there exists a one-to-one correspondence between the processes (Y, Q) and (Y, Z), and thus the suprema and infima in (4.5) are equivalently taken over all stopping times $\tau$ and $\zeta$ with respect to the natural filtration of (Y, Z) which coincides with $({\mathcal G}_t)_{t \, \ge \,0}$ . Therefore, it follows from (2.16) that the stopping times $\tau^*$ and $\zeta_*$ in (4.5) can be expressed as

(4.6) \begin{equation}\tau^* = \inf \big\{ t \, \ge \,0 \; \big| \; Y_t \le a^*(Z_t) \big\}\quad \text{and} \quad\zeta_* = \inf \big\{ t \, \ge \,0 \; \big| \; Y_t \ge b_*(Z_t) \big\},\end{equation}

and thus the continuation region $C_*$ in (2.4) is associated with the set

(4.7) \begin{equation}F_* = \big\{ (y, z) \in {\mathbb R}^2_{++} \; \big| \; a^*(z) < y < b_*(z) \big\}\end{equation}

with some functions $a^*(z)$ and $b_*(z)$ such that the inequalities $0 < a^*(z) \le {r} K < {r} L \le b_*(z)$ hold, for $z \in {\mathbb R}_{++}$ . In order to provide relations between the functions $g^*(q)$ , $h_*(q)$ and $a^*(z)$ , $b_*(z)$ , we follow the sequence of arguments around [Reference Johnson and Peskir49, formula (11.20)] to see that, for each $z \in {\mathbb R}_{++}$ fixed, there exists a unique $q \in (0, 1)$ such that the equalities

(4.8) \begin{equation}y = g^*(q) = \bigg( \frac{q}{z(1-q)} \bigg)^{1/\rho} = a^*(z)\quad \text{and} \quad y = h_*(q) = \bigg( \frac{q}{z(1-q)} \bigg)^{1/\rho} = b_*(z)\end{equation}

hold. In this case, we observe that the first equalities in (4.8) take the form $q = (g^*)^{-1}(y)$ and $q = (h_*)^{-1}(y)$ , where $(g^*)^{-1}(y)$ and $(h_*)^{-1}(y)$ are understood as the generalized inverses of the functions $g^*(q)$ and $h_*(q)$ . Then it follows from the change of variables introduced in (4.1) that the latter equalities are equivalent to

(4.9) \begin{equation}\frac{y^{\rho} z}{1+ y^{\rho} z} = (g^*)^{-1}(y), \quad \text{and thus}\quad z = \frac{(g^*)^{-1}(y)}{y^{\rho} (1 - (g^*)^{-1}(y))} \equiv {\widetilde g}(y),\end{equation}

and

(4.10) \begin{equation}\frac{y^{\rho} z}{1+ y^{\rho} z} = (h_*)^{-1}(y), \quad \text{and thus}\quad z = \frac{(h_*)^{-1}(y)}{y^{\rho} (1 - (h_*)^{-1}(y))} \equiv {\widetilde h}(y),\end{equation}

for each $z \in {\mathbb R}_{++}$ fixed. Hence, we may conclude that the functions ${\widetilde g}(y)$ and ${\widetilde h}(y)$ from (4.9) and (4.10) are positive and decreasing, and thus we can define $a^*(z) = {\widetilde g}^{-1}(z)$ and $b_*(z) = {\widetilde h}^{-1}(z)$ , for each $z \in {\mathbb R}_{++}$ fixed, where ${\widetilde g}^{-1}(z)$ and ${\widetilde h}^{-1}(z)$ are the generalized inverses of the functions ${\widetilde g}(y)$ and ${\widetilde h}(y)$ . In this view, by virtue of the fact, proved in Lemma 2.1, that the boundaries $g^*(q)$ and $h_*(q)$ in (2.16) and (2.17)–(2.18) are decreasing, we may conclude that the boundaries $a^*(z)$ and $b_*(z)$ in (4.6) and (4.7) are of bounded variation. On the other hand, we obtain from the change of variables in (4.1) that the last equality in (4.8) takes the form

(4.11) \begin{equation}\frac{y^{-\rho} q}{1 - q} = (a^*)^{-1}(y), \quad \text{and thus}\quad \textit{q} = \frac{y^{\rho} (a^*)^{-1}(y)}{1 + y^{\rho} (a^*)^{-1}(y)} \equiv {\widehat g}(y),\end{equation}

and

(4.12) \begin{equation}\frac{y^{-\rho} q}{1 - q} = (b_*)^{-1}(y), \quad \text{and thus}\quad \textit{q} = \frac{y^{\rho} (b_*)^{-1}(y)}{1 + y^{\rho} (b_*)^{-1}(y)} \equiv {\widehat h}(y),\end{equation}

for each $q \in (0, 1)$ fixed. Therefore, we may conclude that the functions ${\widehat a}(y)$ and ${\widehat b}(y)$ from (4.11) and (4.12) are positive and decreasing, and thus we can define $g^*(q) = {\widehat g}^{-1}(q)$ and $h_*(q) = {\widehat h}^{-1}(q)$ , for each $q \in (0, 1)$ fixed, where ${\widehat g}^{-1}(q)$ and ${\widehat h}^{-1}(q)$ are the generalized inverses of the functions ${\widehat g}(y)$ and ${\widehat h}(y)$ , respectively.

By means of standard arguments based on an application of Itô’s formula, it is shown that the infinitesimal operator ${\mathbb L}_{(Y, Z)}$ of the process (Y, Z) solving the stochastic differential equations (4.2) and (4.3) has the structure

(4.13) \begin{align} {\mathbb L}_{(Y, Z)} &= \bigg( r - \eta_0 - (\eta_1 - \eta_0) \, \frac{y^{\rho} z}{1 + y^{\rho} z} \bigg) \, y \, \partial_y + \frac{\sigma^2 y^2}{2} \, \partial_{yy} \nonumber\\ &\phantom{=\;\:}+ \bigg( \frac{(\lambda_0 - \lambda_1 y^{\rho} z)(1 + y^{\rho} z)}{y^{\rho} z} - \frac{\rho}{2} \, \big(2 r - \eta_0 - \eta_1 - \sigma^2 \big) \bigg) \, z \, \partial_z \end{align}

for all $(y, z) \in {\mathbb R}^2_{++}$ . In this case, according to system (3.2)–(3.8) above, the unknown value function $U_*(y, z)$ from (4.5) and the unknown boundaries $a^*(z)$ and $b_*(z)$ from (4.6) can be characterized by the associated free-boundary problem

(4.14) \begin{align}({\mathbb L}_{(Y, Z)} U - {r} U)(y, z) = - y \quad &\text{for} \ a(z) < y < b(z),\end{align}

(4.15) \begin{align}U(y, z) \big|_{y = a(z)} = K \quad &\text{and} \quad U(y, z) \big|_{y = b(z)} = L,\end{align}

(4.16) \begin{align}U_y(y, z) \big|_{y = a(z)} = 0 \quad &\text{and} \quad U_y(y, z) \big|_{y = b(z)} = 0,\end{align}

(4.17) \begin{align}U_z(y, z) \big|_{y = a(z)} = 0 \quad &\text{and} \quad U_z(y, z) \big|_{y = b(z)} = 0,\end{align}

(4.18) \begin{align}U(y, z) = K \quad \text{for} \ y < a(z) \quad &\text{and}\quad U(y, z) = L \quad \text{for} \ y > b(z),\end{align}

(4.19) \begin{align}U(y, z) > K \quad \text{for} \ y > a(z) \quad &\text{and}\quad U(y, z) < L \quad \text{for} \ y < b(z),\end{align}

(4.20) \begin{align}({\mathbb L}_{(Y, Z)} U - r U)(y, z) < - y \quad &\text{for} \ y < a(z),\end{align}

(4.21) \begin{align}({\mathbb L}_{(Y, Z)} U - r U)(y, z) > - y \quad &\text{for} \ y > b(z),\end{align}

for $z \in {\mathbb R}_{++}$ . We recall that the existence of the value of the optimal stopping zero-sum game in (4.5) follows from the results of [Reference Ekström and Peskir24, Theorem 2.1] and [Reference Peskir68, Theorem 2.1] and the one-to-one correspondence between the processes (Y, Q) and (Y, Z). Then, by means of the arguments of the proof of the strong Markov property of the process (Y, Z), which means the subsequent standard applications of Itô’s formula and Doob’s optional sampling theorem for the resulting local martingale taken at the time of the exit of the process (Y, Z) from a (bounded) open subset of $F_*$ from (4.7) followed by a passage to the limit in the resulting quotient as in [Reference Peskir and Shiryaev71, Chapter III, Sections 7.1--7.3], it is shown that the value function $U_*(y, z)$ from (4.5) solves the parabolic-type partial differential equation in (4.13) and (4.14). Hence, taking into account the probabilistic regularity of the points of the optimal exercise boundary $\partial C_*$ for the stopping region $D_*$ relative to the process (Y, Q), we may conclude from the results on parabolic-type partial differential equations from [Reference Liberman60, Chapter V], combined with standard techniques related to the proof of the strong Markov property of the process (Y, Z) mentioned above, that the value function $U_*(y, z)$ belongs to the class $C^{2, 1}$ in the regions $F_* \cap ({\mathbb R}^2_{++} \setminus E)$ , which naturally coincides with the closure of $F_*$ , because the set $E = \{ (y, z) \in {\mathbb R}^2_{++} \, | \, y^{\rho} z = \nu \}$ is thin, where the region $F_*$ has the form of (4.7) and $\nu$ is given by (4.4). Moreover, by virtue of the regularity of the value function proved in Lemmas 3.1–3.4 above and the bijective and smooth change of variables introduced in (4.1), it follows that the instantaneous-stopping and smooth-fit conditions of (4.15) and (4.16)–(4.17) hold for the value function $U_*(y, z)$ too.

4.2. The verification lemma

In order to formulate and prove the main results of the paper, taking into account the structure of the partial differential equation in (4.13) and (4.14) as well as the instantaneous-stopping and smooth-fit conditions in (4.15) and (4.16)–(4.17), we observe that the second-order derivative $(U_*)_{yy}(y, z)$ admits a continuous extension to the closure of the appropriate continuation region $F_*$ from (4.7). This fact means that, by virtue of the probabilistic regularity of the boundary $\partial C_*$ for $D_*$ relative to (Y, Q) proved in Lemma 3.2 above, the function $U_*(y, z)$ admits a natural extension in the class $C^{2, 1}$ in the closure of the region $F_* \cap ({\mathbb R}^2_{++} \setminus E)$ , which naturally coincides with the closure of $F_*$ , and in the class $C^{2, 0}$ in the closure of the appropriate region $F_* \cap E$ , where $E = \{ (y, z) \in {\mathbb R}^2_{++} \, | \, y^{\rho} z = \nu \}$ and $\nu$ is given by (4.4). Moreover, by virtue of the results of [Reference Peskir70, Theorem 3], it follows from the regularity of the value function $U_*(y, z)$ and the expressions (4.9) and (4.11) that the boundaries $a^*(z)$ and $b_*(z)$ in (4.7) are continuous functions (of bounded variation). Recall the property that the process Y is monotone outside the curve $E = \{ (y, z) \in {\mathbb R}^2_{++} \, | \, y^{\rho} z = \nu \}$ with $\nu$ given by (4.4). In this case, it can be shown by means of arguments similar to those applied in part 1 of the proof of [Reference Johnson and Peskir49, Theorem 19] that there exists a sequence of piecewise-monotone processes $Z^n = (Z^n_t)_{t \, \ge \,0}$ , for $n \in {\mathbb N}$ , which converges to Z in ${\mathbb P}$ -probability on compact time intervals from ${\mathbb R}_{+} \equiv [0, \infty)$ , and the sequence of total variations of $Z^n$ , for $n \in {\mathbb N}$ , also converge in ${\mathbb P}$ -probability to that of Z, on each interval [0, T], for any $T > 0$ fixed, as $n \to \infty$ . Note that, without loss of generality, the processes $Z^n$ , for $n \in {\mathbb N}$ , can be assumed to be continuous, by virtue of possible applications of standard straight-line approximations. Since each of the resulting continuous processes $a^*(Z^n)$ and $b_*(Z^n)$ are of bounded variation, so that they are continuous semimartingales, the change-of-variable formula from [Reference Peskir67, Theorem 3.1] can be applied to the process $e^{- {r} t} U(Y, Z^n)$ , for every $n \in {\mathbb N}$ , and thus to the process $e^{- {r} t} U(Y, Z)$ , by virtue of the appropriate convergence relations mentioned above and the assumed regularity of the candidate value function U(y, z) which is defined by the right-hand side of (4.27) below (see part 1 of the proof of [Reference Johnson and Peskir49, Theorem 19] for further arguments).

In order to formulate the assertion below, let us denote by $p(t; y, z; y', z')$ the transition density of the continuous (time-homogeneous) strong Markov process (Y, Z) under ${\mathbb P}_{y, z}$ in the sense that

(4.22) \begin{equation}{\mathbb P}_{y, z}( Y_t \le y'', Z_t \le z'') = \int_0^{y''} \int_0^{z''} p(t; y, z; y', z') \, dy' \, dz'\end{equation}

for $t > 0$ with (y, z) and $(y'', z'') \in {\mathbb R}^2_{++}$ . The function $p(t; y, z; y', z')$ is characterized as the unique non-negative solution to the Kolmogorov backward equation

(4.23) \begin{equation}p_t(t; y, z; y', z') = ({\mathbb L}_{(Y, Z)} p)(t; y, z; y', z')\end{equation}

with the initial condition

(4.24) \begin{equation}p(0+; y, z; y', z') = \delta_{(y, z)}(y', z') \quad \textrm{(weakly)}\end{equation}

satisfying

(4.25) \begin{equation}\int_0^{\infty} \int_0^{\infty} p(t; y, z; y', z') \, dy' dz' = 1\end{equation}

for $t > 0$ and (y, z) as well as (y’, z’) in ${\mathbb R}^2_{++}$ (cf. [Reference Feller30]), where we recall that the infinitesimal operator ${\mathbb L}_{(Y, Z)}$ of the process (Y, Z) is given in (4.13) above and $\delta_{(y, z)}$ denotes the Dirac measure at the point $(y, z) \in {\mathbb R}^2_{++}$ . The initial value problem (4.22)–(4.25) can be used to determine the transition density $p(t; y, z; y', z')$ . Having found the function $p(t; y, z; y', z')$ , we can evaluate the expression

(4.26) \begin{align}&{\mathbb E}_{y, z} \big[ Y_t \, I(a(Z_t) < Y_t < b(Z_t) \big]= \int_0^{\infty} \int_{a(z')}^{b(z')} y' \, p(t; y, z; y', z') \, dy' \, dz',\end{align}

for $t > 0$ and $(y, z) \in {\mathbb R}^2_{++}$ , which appears in the formulation of the verification assertion below.

We now formulate the following verification assertion related to the free-boundary problem in (4.13) and (4.14)–(4.21). The proof is based on the essentially non-trivial development of the arguments from [Reference Johnson and Peskir49, Theorem 19] to the case of the considered optimal stopping zero-sum game of (4.5) for the two-dimensional diffusion process (4.2) and (4.3).

Lemma 4.1. Suppose that the processes Y and Z solve the stochastic differential equations (4.2) and (4.3) with ${r} > 0$ , $\eta_0 > \eta_1 > 0$ , $\sigma > 0$ , and $\lambda_j \ge 0$ for $j = 0, 1$ . Then the value function $U_*(y, z)$ of the optimal stopping zero-sum game in (4.5), with some $L > K > 0$ fixed, admits the representation

(4.27) \begin{align}U_*(y, z)&= \int_0^{\infty} \int_0^{\infty} \int_{a^*(z')}^{b_*(z')}e^{- {r} t} \, y' \, p(t; y, z; y', z') \, dy' \, dz' \, dt\end{align}

for $(y, z) \in {\mathbb R}^2_{++}$ , where the continuous boundaries of bounded variation $a^*(z)$ and $b_*(z)$ such that $0 < a^*(z) \le {r} K < {r} L \le b_*(z)$ holds, for $z \in {\mathbb R}_{++}$ , are determined as a unique solution to the coupled system of nonlinear Fredholm integral equations

(4.28) \begin{align}&K = \int_0^{\infty} \int_0^{\infty} \int_{a(z')}^{b(z')}e^{- {r} t} \, y' \, p(t; a(z), z; y', z') \, dy' \, dz' \, dt\end{align}

and

(4.29) \begin{align}&L = \int_0^{\infty} \int_0^{\infty} \int_{a(z')}^{b(z')}e^{- {r} t} \, y' \, p(t; b(z), z; y', z') \, dy' \, dz' \, dt\end{align}

for $z \in {\mathbb R}_{++}$ , among the couples of continuous functions of bounded variation a(z) and b(z) satisfying the inequalities $0 < a(z) \le {r} K < {r} L \le b(z)$ , while the stopping times $\tau^*$ and $\zeta_*$ from (4.6) form a Nash equilibrium.

4.3. The result

Taking the assertions of Lemmas 2.1, 3.1–3.4, and 4.1 together, we conclude this paper by formulating its main result concerning the optimal stopping problem related to the pricing of perpetual redeemable convertible bonds in the hidden Markov model considered.

4.4. The solution in a particular case

In order to underline the complexity in the structure of solutions to the optimal stopping zero-sum game of (2.1) and (2.2) in the two-dimensional diffusion model defined by (1.1)–(1.2) and (1.5)–(1.6) with (1.7), we present a specific choice of parameters under which the original problem admits an analytic solution. More precisely, let us assume for the rest of this section that $\lambda_0 = \lambda_1 = 0$ and $\eta_0 + \eta_1 = 2r - \sigma^2$ hold in (4.3). The first equality means that $\boldsymbol{\Theta}_t = \boldsymbol{\Theta}_0$ , for all $t \, \ge \,0$ , where ${\mathbb P}(\boldsymbol{\Theta}_0=1)=q$ and ${\mathbb P}(\boldsymbol{\Theta}_0=0)=1-q$ , for $q \in [0, 1]$ (cf., for example, [Reference Peskir and Shiryaev71, Chapter VI, Section 21] and [Reference Gapeev35, Section 4]). Such a situation occurs when the unknown economic state of the firm is not changed during the whole infinite time interval (see also [Reference Décamps, Mariotti and Villeneuve19, Reference Ekström and Lu23, Reference Ekström and Vaicenavicius25, Reference Gapeev35] for solutions of other optimal stopping problems in the models with random drift rates). In this case, the parabolic-type partial differential equation in (4.13) and (4.14) is degenerated into an ordinary one, and the general solution to that equation which is finite at the infinity takes the form

(4.56) \begin{equation}U(y, z) = G_1(z) \, H_1(y, z) + G_2(z) \, H_2(y, z) + H_0(y, z)\end{equation}

where $G_k(z)$ , for $k = 1, 2$ , are arbitrary continuous functions. Here, we assume that the functions $H_k(y, z)$ , for $k = 1, 2$ , represent fundamental solutions to the homogeneous second-order ordinary differential equation related to that of (4.13) and (4.14) under $\lambda_0 = \lambda_1 = 0$ and $\eta_0 + \eta_1 = 2 r - \sigma^2$ given by

(4.57) \begin{equation}H_k(y, z) = y^{\chi_{0,k}} \, F (\psi_{k,1}, \psi_{k,2};\, \varphi_k; - y^{\rho} z)\end{equation}

for every $k = 1, 2$ and each $z \in {\mathbb R}_{++}$ fixed. We denote by $F(\alpha, \beta; \gamma; x)$ the Gauss hypergeometric function which is defined by means of the expansion

(4.58) \begin{equation}F(\alpha, \beta; \gamma; x) = 1 + \sum_{m = 1}^{\infty} \frac{(\alpha)_m (\beta)_m}{(\gamma)_m} \, \frac{x^m}{m!}\end{equation}

for $\alpha, \beta, \gamma \in {\mathbb R}$ such that $\gamma \neq 0, -1, -2, \ldots,$ where $(\alpha)_m = \alpha (\alpha+1) \cdots (\alpha+m-1)$ , for $m \in {\mathbb N}$ , and $(\alpha)_0 = 1$ (cf., for example, [Reference Abramovitz and Stegun1, Chapter XV] and [Reference Bateman and Erdélyi2, Chapter II]), and additionally set

(4.59) \begin{equation}\psi_{k,l} = \frac{\chi_{0,k} - \chi_{1,l}}{\rho}\quad \text{and} \quad \varphi_k = 1 + \frac{2}{\rho} \,\bigg( \chi_{0,k} - \frac{1}{2} + \frac{r - \eta_0}{\sigma^2} \bigg)\end{equation}

as well as

(4.60) \begin{equation}\chi_{j, i} = \frac{1}{2} - \frac{r - \eta_j}{\sigma^2} - (-1)^j\sqrt{ \bigg( \frac{1}{2} - \frac{r - \eta_j}{\sigma^2} \bigg)^2+ \frac{2r}{\sigma^2}}\end{equation}

so that $\chi_{j,2} < 0 < 1 < \chi_{j,1}$ holds, for every $k, l, i = 1, 2$ and $j = 0, 1$ . Thus, $H_k(y, z)$ , for $k = 1, 2$ , in (4.57) are (strictly) increasing and decreasing (convex) functions satisfying the properties $H_1(0+, z) = + 0$ , $H_1(\infty, z) = \infty$ and $H_2(0+, z) = \infty$ , $H_2(\infty, z) = + 0$ , for each $z \in {\mathbb R}_{++}$ fixed (cf., for example, [Reference Rogers and Williams73, Chapter V, Section 50] for further details). Moreover, the function $H_0(y, z)$ in (4.56) represents a particular solution to the equation in (4.13) and (4.14) given by

(4.61) \begin{equation}H_{0}(y, z) = H_{1}(y, z) \int^y \frac{2 H_{2}(x, z)}{\sigma^2 x G_{0}(x, z)} \, dx+ H_{2}(y, z) \int_y \frac{2 H_{1}(x, z)} {\sigma^2 x G_{0}(x, z)} \, dx\end{equation}

where $G_0(y, z)$ is the appropriate Wronskian determinant which admits the representation

(4.62) \begin{equation}G_0(y, z) = H_1(y, z) \, \partial_y H_2(y, z) - \partial_y H_1(y, z) \, H_2(y, z)\end{equation}

for each $z \in {\mathbb R}_{++}$ fixed.

Finally, by applying conditions (4.15) and (4.16) to the function in (4.56), we obtain that the equalities

(4.63) \begin{align}&G_1(z) \, H_1(a(z), z) + G_2(z) \, H_2(a(z), z) + H_0(a(z), z) = K,\end{align}

(4.64) \begin{align}&G_1(z) \, \partial_y H_1(a(z), z) + G_2(z) \, \partial_y H_2(a(z), z) + \partial_y H_0(a(z), z) = 0,\end{align}

(4.65) \begin{align}&G_1(z) \, H_1(b(z), z) + G_2(z) \, H_2(b(z), z) + H_0(b(z), z) = L ,\end{align}

(4.66) \begin{align}&G_1(z) \, \partial_y H_1(b(z), z) + G_2(z) \, \partial_y H_2(b(z), z) + \partial_y H_0(b(z), z) = 0\end{align}

should hold, for each $z \in {\mathbb R}_{++}$ fixed. Then, solving the system of equations in (4.63) and (4.65), we obtain that the function

(4.67) \begin{align}&U(y, z; a(z), b(z))= G_1(z; a(z), b(z)) \, H_1(y, z) + G_2(z; a(z), b(z)) \, H_2(y, z) + H_0(y, z)\end{align}

for $a(z) < y < b(z)$ , satisfies the system in (4.13) and (4.14)–(4.15), when we set

(4.68) \begin{align}&G_i(z; a(z), b(z))= \frac{(K - H_{0}(a(z), z)) H_{3-i}(b(z), z) - (L - H_{0}(b(z), z)) H_{3-i}(a(z), z)}{H_{i}(a(z), z) H_{3-i}(b(z), z) - H_{i}(b(z), z) H_{3-i}(a(z), z)}\end{align}

for each $z \in {\mathbb R}_{++}$ fixed, and every $i = 1, 2$ . Hence, putting the expressions from (4.68) into the system of equations in (4.64) and (4.66), we obtain that the function in (4.67) satisfies conditions (4.16) when the equalities

(4.69) \begin{align}&\frac{(K - H_0(a(z), z)) \partial_y H_i(a(z), z) + \partial_y H_0(a(z), z) H_i(a(z), z)}{(L - H_0(b(z), z)) \partial_y H_i(b(z), z) + \partial_y H_0(h(z), z) H_i(b(z), z)}= \frac{G_0(a(z), z)}{G_0(b(z), z)}\end{align}

hold, for each $z \in {\mathbb R}_{++}$ fixed, and every $i = 1, 2$ , where the function $G_0(y, z)$ is given by (4.62). Observe that it can be shown by means of straightforward computations based on applications of the implicit function theorem that the candidate value function in (4.67) with (4.68) and such that the equalities in (4.69) hold inherently satisfies the conditions of (4.17) (see also [Reference Gapeev35, Remark 4.1] for similar arguments applied for another optimal stopping and free-boundary problem in this model). Note that the uniqueness of the pair of solutions $a^*(z)$ and $b_*(z)$ to the system of higher transcendental equations of (4.69) follows from the uniqueness of the solution to the system in (4.13) and (4.14)–(4.17) and (4.19)–(4.21) which is proved in Lemma 4.1 above. Recall that the existence of the solution to the system in (4.13) and (4.14)–(4.17) and (4.19)–(4.21) follows from the results of [15, Theorem 4.1].

Let us now summarize the above results in the following assertion.

5.1. The optimal stopping game and free-boundary problem

Let us now consider an optimal stopping zero-sum game with the upper and value functions $W^*(y, j) \ge W_*(y, j)$ given by

(5.1) \begin{align} &{W}^*(y, j) = \inf_{\zeta'} \sup_{\tau'} {\mathbb E}_{y, j} \bigg[ \int_0^{\tau' \wedge \zeta'} e^{- {r}\, s} \, Y_s \, ds + e^{- {r} \tau'} \, {K} \, I(\tau' < \zeta') + e^{- {r} \zeta'} \, {L} \, I(\zeta' \le \tau') \bigg] \end{align}

and

(5.2) \begin{align} &{W}_*(y, j) = \sup_{\tau'} \inf_{\zeta'} {\mathbb E}_{y, j} \bigg[ \int_0^{\tau' \wedge \zeta'} e^{- {r}\, s} \, Y_s \, ds + e^{- {r} \tau'} \, {K} \, I(\tau' < \zeta') + e^{- {r} \zeta'} \, {L} \, I(\zeta' \le \tau') \bigg] \end{align}

for some ${r} > 0$ and $L > K > 0$ fixed, where the suprema and infima are taken over all stopping times $\tau'$ and $\zeta'$ with respect to the natural filtration $({\mathcal H}_t)_{t \, \ge \,0}$ of the process $(Y, \boldsymbol{\Theta})$ with the first component given by (1.1) and (1.2). Here, ${\mathbb E}_{y, j}$ denotes the expectation taken with respect to the probability measure ${\mathbb P}_{y, j}$ under which the two-dimensional right-continuous (time-homogeneous) strong Markov process $(Y, \boldsymbol{\Theta})$ , which is also left-continuous over stopping times with respect to $({\mathcal H}_t)_{t \, \ge \,0}$ , that is, quasi-left-continuous (cf., for example, [Reference Revuz and Yor72, Chapter III, Section 2, Exercise 2.33]), starts at some $(y, j) \in {\mathbb R}_{++} \times \{ 0, 1 \}$ . Then, by referring to arguments similar to those used in Section 2.2 above and applying the results of [Reference Ekström and Peskir24, Theorem 2.1] and [Reference Peskir68, Theorem 2.1], we conclude that the optimal stopping game of (5.1) and (5.2) admits both a Stackelberg and a Nash equilibrium, so that in particular the equality ${W}^*(y, j) = {W}_*(y, j)$ holds, for each $(y, j) \in {\mathbb R}_{++} \times \{ 0, 1 \}$ fixed. Since the continuous-time Markov chain $\boldsymbol{\Theta}$ is observable in this formulation, being based on arguments similar to those presented in parts (ii) and (iii) of Section 2.3 and taking into account the arguments used in [Reference Gapeev36, Appendix] as well as in [Reference Guo44, Section 2] and [Reference Guo and Zhang45, Section 2], we may assume that the stopping times forming a Nash equilibrium in the game of (5.1) and (5.2) are of the form

(5.3) \begin{equation} {\overline \tau} = \inf \big\{ t \, \ge \,0 \; \big| \; Y_t \le {\overline g}(\boldsymbol{\Theta}_t) \big\} \quad \text{and} \quad {\underline \zeta} = \inf \big\{ t \, \ge \,0 \; \big| \; Y_t \ge {\underline h}(\boldsymbol{\Theta}_t) \big\} \end{equation}

for some numbers $0 < {\overline g}(j) \le {r} K < {r} L \le {\underline h}(j)$ , for $j = 0, 1$ , to be determined.

In order to specify the location of the optimal stopping boundaries ${\overline g}(j)$ and ${\underline h}(j)$ , for $j = 0, 1$ , we use comparison arguments similar to those applied in part (iii) of Section 2.3 above. More precisely, if we suppose that ${\overline g}(j) < {\underline g}(j)$ or ${\underline h}(j) > {\overline h}(j)$ holds, for some $j = 0, 1$ , then, for each $y \in ({\overline g}(j), {\underline g}(j))$ or $y \in ({\overline h}(j), {\underline h}(j))$ fixed, we would have $W^*(y, j) > K = {\underline W}(y, j)$ or $W_*(y, j) < L = {\overline W}(y, j)$ , respectively, contradicting the obvious facts that the equalities $W^*(y, j) \le {\underline W}(y, j)$ and $W_*(y, j) \ge {\overline W}(y, j)$ hold, for all $(y, j) \in {\mathbb R}_{++} \in \{ 0, 1 \}$ . Here, we recall that the functions ${\underline W}(y, j)$ and ${\overline W}(y, j)$ from (2.6) and (2.7) admit the explicit expressions in (5.56) and (5.57) and the associated optimal stopping times ${\underline \tau}$ and ${\overline \zeta}$ are given by (2.8), where the numbers ${\underline g}(j)$ and ${\overline h}(j)$ , for $j = 0, 1$ , are determined from the expressions (5.49)–(5.50) and (5.54)–(5.55) with (5.15), respectively. Thus, we may conclude that the inequalities ${\underline g}(j) \le {\overline g}(j)$ and ${\underline h}(j) \le {\overline h}(j)$ should hold, for every $j = 0, 1$ .

By means of standard arguments based on an application of Itô’s formula, it is shown that the infinitesimal operator ${\mathbb L}_{(Y, \boldsymbol{\Theta})}$ of the process $(Y, \boldsymbol{\Theta})$ from (1.1)–(1.2) acts on an arbitrary locally bounded function F(y, j), which is of the class $C^2$ in y on ${\mathbb R}_{++}$ under $\boldsymbol{\Theta} = j$ , for any $j = 0, 1$ fixed, according to the rule

(5.4) \begin{align} ({\mathbb L}_{(Y, \boldsymbol{\Theta})}F)(y, j) &= \big( r - \eta_0 - (\eta_1 - \eta_0) \, j \big) \, y \, F_y(y, j) \nonumber\\ &\phantom{=\;\:}+ \frac{\sigma^2 y^2}{2} \, \, F_{yy}(y, j) + \eta_j \, \big( F(y, 1-j) - F(y, j) \big) \end{align}

for all $(y, j) \in {\mathbb R}_{++} \times \{ 0, 1 \}$ . Following the way of arguments from [Reference Guo44, Section 2] and [Reference Guo and Zhang45, Section 2] (see also [Reference Jiang and Pistorius47] for the case of optimal stopping problems in more general models with regime switching), we conclude that the unknown value functions $W^*(y, j) = W_*(y, j)$ from (5.1) and (5.2) and the unknown numbers ${\overline g}(j)$ and ${\underline h}(j)$ , for $j = 0, 1$ , from (5.3) solve the following free-boundary problem for a coupled system of second-order ordinary differential equations:

(5.5) \begin{align} ({\mathbb L}_{(Y, \boldsymbol{\Theta})} W - {r} W)(y, j) = - y \quad &\text{for} \ g(j) < y < h(j), \end{align}

(5.6) \begin{align} W(y, j) \big|_{y = g(j)} = 0 \quad &\text{and} \quad W(y, j) \big|_{y = h(j)} = 0, \end{align}

(5.7) \begin{align} W_y(y, j) \big|_{y = g(j)} = 0 \quad &\text{and} \quad W_y(y, j) \big|_{y = h(j)} = 0, \end{align}

(5.8) \begin{align} W(y, j) = K \quad \text{for} \ y < g(j) \quad &\text{and} \quad W(y, j) = L \quad \text{for} \ y > h(j), \end{align}

(5.9) \begin{align} W(y, j) > K \quad \text{for} \ y > g(j) \quad &\text{and} \quad W(y, j) < L \quad \text{for} \ x < h(j), \end{align}

(5.10) \begin{align} ({\mathbb L}_{(Y, \boldsymbol{\Theta})}W - {r} W)(y, j) < - y \quad &\text{for} \ y < g(j), \end{align}

(5.11) \begin{align} ({\mathbb L}_{(Y, \boldsymbol{\Theta})}W - {r} W)(y, j) > - y \quad &\text{for} \ y > h(j), \end{align}

for $j = 0, 1$ .

5.2. Solutions to the free-boundary problems

By means of straightforward computations, we obtain that the general solutions to the two-dimensional system of second-order ordinary differential equations in (5.4) and (5.5) are given by

(5.12) \begin{equation} W(y, 1) = D_1(1) \, y^{\alpha_{1}} + D_2(1) \, y^{\alpha_{2}} + B(y, 1) \end{equation}

for $g(1) < y \le g(0)$ , and

(5.13) \begin{equation} W(y, j) = C_{1}(j) \, y^{\beta_1} + C_{2}(j) \, y^{\beta_2} + C_{3}(j) \, y^{\beta_3} + C_{4}(j) \, y^{\beta_4} + A(y, j) \end{equation}

for $g(0) < y < h(1)$ and $j = 0, 1$ , as well as

(5.14) \begin{equation} W(y, 0) = D_1(0) \, y^{\gamma_{1}} + D_2(0) \, y^{\gamma_{2}} + B(y, 0) \end{equation}

for $h(1) \le y < h(0)$ , with

(5.15) \begin{equation} A(y, j) = \frac{(\eta_{1-j} + \lambda_0 + \lambda_1) y} {\eta_0 \eta_1 + \eta_0 \lambda_1 + \eta_1 \lambda_0} \quad \text{and} \quad B(y, j) = \frac{y}{\eta_j + \lambda_j} \end{equation}

for $y \in {\mathbb R}_{++}$ and $j = 0, 1$ , where $C_k(j)$ and $D_i(j)$ , for $j = 0, 1$ , $k = 1, 2, 3, 4$ , and $i = 1, 2$ , are arbitrary constants. Here, the $\alpha_i$ , for $i = 1, 2$ , are explicitly given by

(5.16) \begin{equation} \alpha_i = \frac{1}{2} - \frac{r - \eta_1}{\sigma^2} - (-1)^i \sqrt{ \bigg( \frac{1}{2} - \frac{r - \eta_1}{\sigma^2} \bigg)^2 + \frac{2(r + \lambda_1)}{\sigma^2}} \end{equation}

so that $\alpha_{2} < 0 < 1 < \alpha_{1}$ , $\beta_k$ , for $k = 1, 2, 3, 4$ , are the roots of the appropriate characteristic equation

(5.17) \begin{equation} P_0(\beta) \, P_1(\beta) = \lambda_0 \lambda_1, \quad \text{with} \ P_j(\beta) = r + \lambda_j - (r - \eta_j) \, \beta - \frac{\sigma^2}{2} \, \beta (\beta - 1), \end{equation}

so that $\beta_{4} < \beta_{3} < 0 < 1 < \beta_{2} < \beta_{1}$ holds, and the $\gamma_i$ , for $i = 1, 2$ , are explicitly given by

(5.18) \begin{equation} \gamma_i = \frac{1}{2} - \frac{r - \eta_0}{\sigma^2} - (-1)^i \sqrt{ \bigg( \frac{1}{2} - \frac{r - \eta_0}{\sigma^2} \bigg)^2 + \frac{2(r + \lambda_0)}{\sigma^2}} \end{equation}

so that $\gamma_{2} < 0 < 1 < \gamma_{1}$ holds (cf., for example, [Reference Guo44, Section 2] and [Reference Guo and Zhang45, Section 2] for similar arguments related to other optimal stopping problems in models with observable continuous-time Markov chains).

Then, using the structure of the coupled system of ordinary differential equations in (5.4) and (5.5) and applying the conditions of (5.6) and (5.7) to the functions in (5.12)–(5.14) with (5.15), we obtain that the equalities

(5.19) \begin{align} &D_{1}(1) \, g^{\gamma_{1}}(1) + D_{2}(1) \, g^{\gamma_{2}}(1) + B(g(1), 1) = K, \end{align}

(5.20) \begin{align} &D_{1}(1) \, \gamma_1 \, g^{\gamma_{1}}(1) + D_{2}(1) \, \gamma_2 \, g^{\gamma_{2}}(1) + B_y(g(1), 1) \, g(1) = 0, \end{align}

(5.21) \begin{align} &C_1(0) \, g^{\beta_{1}}(0) + C_{2}(0) \, g^{\beta_{2}}(0) + C_3(0) \, g^{\beta_{3}}(0) + C_{4}(0) \, g^{\beta_{4}}(0) + A(g(0), 0) = K, \end{align}

(5.22) \begin{align} &C_{1}(0) \, \beta_{1} \, g^{\beta_{1}}(0) + C_{2}(0) \, \beta_{2} \, g^{\beta_{2}}(0) + C_{3}(0) \, \beta_{3} \, g^{\beta_{3}}(0) + C_{4}(0) \, \beta_{4} \, g^{\beta_{4}}(0), \! + A_y(g(0), 0) \, g(0) = 0, \end{align}

(5.23) \begin{align} &C_k(0) \, P_0(\beta_k) = C_k(1) \, \eta_{1} \quad \text{for} \quad k = 1, 2, 3, 4 \end{align}

(5.24) \begin{align} &C_1(1) \, h^{\beta_{1}}(1) + C_{2}(1) \, h^{\beta_{2}}(1) + C_3(1) \, h^{\beta_{3}}(1) + C_{4}(1) \, h^{\beta_{4}}(1) + A(h(1), 1) = L, \end{align}

(5.25) \begin{align} &C_{1}(1) \, \beta_{1} \, h^{\beta_{1}}(1) + C_{2}(1) \, \beta_{2} \, h^{\beta_{2}}(1) + C_{3}(1) \, \beta_{3} \, h^{\beta_{3}}(1) + C_{4}(1) \, \beta_{4} \, h^{\beta_{4}}(1) + A_y(h(1), 1) \, h(1) = 0, \end{align}

(5.26) \begin{align} &D_{1}(0) \, h^{\alpha_{1}}(0) + D_{2}(0) \, h^{\alpha_{2}}(0) + B(h(0), 0) = L, \end{align}

(5.27) \begin{align} &D_{1}(0) \, \alpha_1 \, h^{\alpha_{1}}(0) + D_{2}(0) \, \alpha_2 \, h^{\alpha_{2}}(0) + B_y(h(0), 0) \, h(0) = 0 \end{align}

should hold, where we have $P_0(\beta_k)/\lambda_1 = \lambda_0/P_1(\beta_k)$ , for $k = 1, 2, 3, 4$ , according to (5.17). Observe that, since the points ${\overline g}(0)$ and ${\underline h}(1)$ belong to the appropriate continuation region when $\boldsymbol{\Theta} = 1$ and $\boldsymbol{\Theta} = 0$ , respectively, the function in (5.13) should be (at least) continuously differentiable, and thus the equalities

(5.28) \begin{align} &C_{1}(1) \, g^{\beta_{1}}(0) + C_{2}(1) \, g^{\beta_{2}}(0) + C_{3}(1) \, g^{\beta_{3}}(0) + C_{4}(1) \, g^{\beta_{4}}(0) + A(g(0), 1), \nonumber\\ &\quad= D_{1}(1) \, g^{\gamma_1}(0) + D_{2}(1) \, g^{\gamma_2}(0) + B(g(0), 1), \end{align}

(5.29) \begin{align} &C_{1}(1) \, \beta_1 \, g^{\beta_{1}}(0) + C_{2}(1) \, \beta_2 \, g^{\beta_{2}}(0) + C_{3}(1) \, \beta_3 \, g^{\beta_{3}}(0) + C_{4}(1) \, \beta_4 \, g^{\beta_{4}}(0), \nonumber\\ &\quad = D_{1}(1) \, \gamma_1 \, g^{\gamma_1}(0) + D_{2}(1) \, \gamma_2 \, g^{\gamma_2}(0) + B_y(g(0), 1) \, g(0) - A_y(g(0), 1) \, g(0) \end{align}

(5.30) \begin{align} &C_{1}(0) \, h^{\beta_{1}}(1) + C_{2}(0) \, h^{\beta_{2}}(1) + C_{3}(0) \, h^{\beta_{3}}(1) + C_{4}(0) \, h^{\beta_{4}}(1) + A(h(1), 0),\nonumber\\ &\quad = D_{1}(0) \, h^{\alpha_1}(1) + D_{2}(0) \, h^{\alpha_2}(1) + B(h(1), 0), \end{align}

(5.31) \begin{align} &C_{1}(0) \, \beta_1 \, h^{\beta_{1}}(1) + C_{2}(0) \, \beta_2 \, h^{\beta_{2}}(1) + C_{3}(0) \, \beta_3 \, h^{\beta_{3}}(1) + C_{4}(0) \, \beta_4 \, h^{\beta_{4}}(1),\nonumber\\ &\quad= D_{1}(0) \, \alpha_1 \, h^{\alpha_1}(1) + D_{2}(0) \, \alpha_2 \, h^{\alpha_2}(1) + B_y(h(1), 0) \, h(1) - A_y(h(1), 0) \, h(1) \end{align}

should be satisfied. Hence, solving the systems of arithmetic equations in (5.19)–(5.20) and (5.26)–(5.27) as well as those of the Vandermonde type in (5.21)–(5.22) and (5.28)–(5.29), and (5.24)–(5.25) and (5.24)–(5.25) with (5.23), by means of straightforward computations, we obtain that the functions

(5.32) \begin{equation} W(y, 1; g(1)) = \sum_{j = 1}^2 \frac{B_y(g(1), 1) g(1) - \gamma_{3-j} (B(g(1), 1) - K)} {\gamma_{3-j} - \gamma_{j}} \, \bigg( \frac{y}{g(1)} \bigg)^{\gamma_{j}} + B(y, 1) \end{equation}

for $g(1) < y \le g(0)$ , and

(5.33) \begin{equation} W(y, j; g(0), g(1), h(0), h(1)) = \sum_{k = 1}^4 C_k(j; g(0), g(1), h(0), h(1)) \, y^{\beta_{k}} + A(y, j) \end{equation}

for $g(0) < y < h(1)$ and $j = 0, 1$ , as well as

(5.34) \begin{equation} W(y, 0; h(0)) = \sum_{j = 1}^2 \frac{B_y(h(0), 0) h(0) - \alpha_{3-j} (B(h(0), 0) - L)} {\alpha_{3-j} - \alpha_{j}} \, \bigg( \frac{y}{h(0)} \bigg)^{\alpha_{j}} + B(y, 0) \end{equation}

for $h(1) \le y < h(0)$ , satisfy the system in (5.4) and (5.5) and (5.6)–(5.7), when the equalities

(5.35) \begin{align} &\sum_{k = 1}^4 C_k(1; g(0), g(1), h(0), h(1)) \, h^{\beta_k}(1) + A(h(1), 1) = L \end{align}

and

(5.36) \begin{align} &\sum_{k = 1}^4 C_k(1; g(0), g(1), h(0), h(1)) \, \beta_k \, h^{\beta_k}(1) + A_y(h(1), 1) \, h(1) = 0 \end{align}

as well as

(5.37) \begin{align} &\sum_{k = 1}^4 C_k(0; g(0), g(1), h(0), h(1)) \, h^{\beta_k}(1) + A(h(1), 0), \\ \notag &= \sum_{j = 1}^2 \frac{B_y(h(0), 0) h(0) - \alpha_{3-j} (B(h(0), 0) - L)}{\alpha_{3-j} - \alpha_{j}} \, \bigg( \frac{h(1)}{h(0)} \bigg)^{\alpha_{j}} + B(h(1), 0) \end{align}

and

(5.38) \begin{align} &\sum_{k = 1}^4 C_k(0; g(0), g(1), h(0), h(1)) \, \beta_k \, h^{\beta_k}(1) + A_y(h(1), 0) \, h(1) \\ \notag &= \sum_{j = 1}^2 \frac{B_y(h(0), 0) h(0) - \alpha_{3-j} (B(h(0), 0) - L)}{\alpha_{3-j} - \alpha_{j}} \, \alpha_j \, \bigg( \frac{h(1)}{h(0)} \bigg)^{\alpha_{j}} + B_y(h(1), 0) \, h(1) \end{align}

hold. Here, the coefficients $C_k(0; g(0), g(1), h(0), h(1))$ , for $k = 1, 2, 3, 4$ , are explicitly given by

(5.39) \begin{align} C_1(0) &= \big( \big( (W_y(g(0), 1) - A_y(g(0), 1)) \, g(0) - \beta_4 \, (W(g(0), 1) - A(g(0), 1)) \nonumber\\ &\phantom{= \big( \big( \;\:}+ \delta_3 \, \big( A_y(g(0), 0) \, g(0) + \beta_4 \, (K - A(g(0), 0)) \big) \big) \, (\beta_2 - \beta_3) (\delta_2 - \delta_4) \nonumber\\ &\phantom{= \big( \;\:}- \big( (W_y(g(0), 1) - A_y(g(0), 1)) \, g(0) + \delta_4 \, A_y(g(0), 0) \, g(0) \nonumber\\ &\phantom{= \big( \big( \;\:} - \beta_3 \, \big( W(g(0), 1) - A(g(0), 1) - \delta_4 \, (K - A(g(0), 0)) \big) \big) \, (\beta_2 - \beta_4) (\delta_2 - \delta_3) \big) \big/ \nonumber\\ &\phantom{=\;\:} \big( g^{\beta_1}(0) \, \big( {(\beta_1 \!- \beta_4) (\beta_2 \!- \beta_3) (\delta_1\! - \delta_3) (\delta_2\! - \delta_4) - (\beta_1 - \beta_3) (\beta_2 - \beta_4) (\delta_1 - \delta_4) (\delta_2 - \delta_3)} \big), \end{align}

(5.40) \begin{align} C_2(0) &= \big( \big( (W_y(g(0), 1) - A_y(g(0), 1)) \, g(0) - \beta_4 \, (W(g(0), 1) - A(g(0), 1)) \nonumber\\ &\phantom{= \big( \big( \;\:} + \delta_3 \, \big( A_y(g(0), 0) \, g(0) + \beta_4 \, (K - A(g(0), 0)) \big) \big) \, (\beta_1 - \beta_3) (\delta_1 - \delta_4) \nonumber\\ &\phantom{= \big( \;\:} - \big( (W_x(g(0), 1) - A_x(g(0), 1)) \, g(0) + \delta_4 \, A_y(g(0), 0) \, g(0) \nonumber\\ &\phantom{= \big( \big( \;\:} - \beta_3 \, \big( W(g(0), 1) - A(g(0), 1) - \delta_4 \, (K - A(g(0), 0)) \big) \big) \, (\beta_1 - \beta_4) (\delta_1 - \delta_3) \big) \big/ \nonumber\\ &\phantom{=\;\:} \big( g^{\beta_2}(0) \, \big( {(\beta_1\! - \beta_3) (\beta_2\! - \beta_4) (\delta_1 \!- \delta_4) (\delta_2 \!- \delta_3) - (\beta_1 - \beta_4) (\beta_2 - \beta_3) (\delta_1 - \delta_3) (\delta_2 - \delta_4)} \big), \end{align}

(5.41) \begin{align} C_3(0) &= \big( \big( (W_y(g(0), 1) - A_y(g(0), 1)) \, g(0) - \beta_1 \, (W(g(0), 1) - A(g(0), 1)) \nonumber\\ &\phantom{= \big( \big( \;\:} + \delta_2 \, \big( A_y(g(0), 0) \, g(0) + \beta_1 \, (K - A(g(0), 0)) \big) \big) \, (\beta_2 - \beta_4) (\delta_1 - \delta_4) \nonumber\\ &\phantom{= \big( \;\:}- \big( (W_y(g(0), 1) - A_y(g(0), 1)) \, g(0) + \delta_1 \, A_y(g(0), 0) \, g(0) \nonumber\\ &\phantom{= \big( \big( \;\:} - \beta_2 \, \big( W(g(0), 1) - A(g(0), 1) - \delta_1 \, (K - A(g(0), 0)) \big) \big) \, (\beta_1 - \beta_4) (\delta_2 - \delta_4) \big) \big/ \nonumber\\ &\phantom{=\;\:} \big( g^{\beta_3}(0) \, \big( {(\beta_1\! - \beta_3) (\beta_2\! - \beta_4) (\delta_1 \!- \delta_4) (\delta_2\! - \delta_3) - (\beta_1 - \beta_4) (\beta_2 - \beta_3) (\delta_1 - \delta_3) (\delta_2 - \delta_4)} \big), \end{align}

and

(5.42) \begin{align} C_4(0) &= \big( \big( (W_y(g(0), 1) - A_y(g(0), 1)) \, g(0) - \beta_1 \, (W(g(0), 1) - A(g(0), 1)) \nonumber\\ &\phantom{= \big( \big( \;\:} + \delta_2 \, \big( A_y(g(0), 0) \, g(0) + \beta_1 \, (K - A(g(0), 0)) \big) \big) \, (\beta_2 - \beta_3) (\delta_1 - \delta_3) \nonumber\\ &\phantom{= \big( \;\:}- \big( (W_y(g(0), 1) - A_y(g(0), 1)) \, g(0) + \delta_1 \, A_y(g(0), 0) \, g(0) \nonumber\\ &\phantom{= \big( \big( \;\:} - \beta_2 \, \big( W(g(0), 1) - A(g(0), 1) - \delta_1 \, (K - A(g(0), 0)) \big) \big) \, (\beta_1 - \beta_3) (\delta_2 - \delta_3) \big) \big/ \nonumber\\ &\phantom{=\;\:} \big( g^{\beta_4}(0) \, \big( {(\beta_1\! - \beta_4) (\beta_2\! - \beta_3) (\delta_1\! - \delta_3) (\delta_2\! - \delta_4) - (\beta_1 - \beta_3) (\beta_2 - \beta_4) (\delta_1 - \delta_4) (\delta_2 - \delta_3)} \big), \end{align}

with $\delta_k = P_0(\beta_k)/\lambda_1$ , for $k = 1, 2, 3, 4$ , as well as the relations

(5.43) \begin{align} &C_k(j; g(0), g(1), h(0), h(1)) \, P_j(\beta_k) = C_k(1-j; g(0), g(1), h(0), h(1)) \, \lambda_{1-j} \end{align}

are satisfied, for $j = 0, 1$ and $k = 1, 2, 3, 4$ (see also [Reference Guo44, Section 2] and [Reference Guo and Zhang45, Section 2] for similar calculations related to other optimal stopping problems in models with observable continuous-time Markov chains).

5.3. The result

Summarizing the facts proved above, let us formulate the following result which can be proved by means of the same arguments as the appropriate verification assertions from [Reference Gapeev36; Corollary A.1], [Reference Guo44, Theorem 2.1] and [Reference Guo and Zhang45, Theorem 2] related to the optimal stopping problems in the models with observable continuous-time Markov chains with two states.

Theorem 5.1. Suppose that the process Y is defined by (1.1) and (1.2), with some ${r} > 0$ , $\eta_0 > \eta_1 > 0$ , and $\sigma > 0$ , while $\boldsymbol{\Theta}$ is a continuous-time Markov chain with the state space $\{ 0, 1 \}$ and transition intensities $\lambda_j > 0$ , for $j = 0, 1$ , which is independent of standard Brownian motion B. Then the value functions $W^*(y, j) = W_*(y, j)$ of the optimal stopping zero-sum game in (5.1) and (5.2), with $L > K > 0$ fixed, are given by

(5.44) \begin{equation} W^*(y, 0) = W_*(y, 0) = \begin{cases} K, & \textit{if} \; \; y \le {\overline g}(0) \\ W(y, 0; {\overline g}(0), {\overline g}(1), {\underline h}(0), {\underline h}(1)), & \textit{if} \; \; {\overline g}(0) < y < {\underline h}(1) \\ W(y, 0; {\underline h}(0)), & \textit{if} \; \; {\underline h}(1) \le y < {\underline h}(0) \\ L, & \textit{if} \; \; y \ge {\underline h}(0) \end{cases} \end{equation}

and

(5.45) \begin{equation} W^*(y, 1) = W_*(y, 1) = \begin{cases} K, & \textit{if} \; \; y \le {\overline g}(1) \\ W(y, 1; {\overline g}(1)), & \textit{if} \; \; {\overline g}(1) < y \le {\overline g}(0) \\ W(y, 1; {\overline g}(0), {\overline g}(1), {\underline h}(0), {\underline h}(1)), & \textit{if} \; \; {\overline g}(0) < y < {\underline h}(1) \\ L, & \textit{if} \; \; y \ge {\underline h}(1) \end{cases} \end{equation}

and the stopping times ${\overline \tau}$ and ${\underline \zeta}$ forming a Nash equilibrium have the form of (5.3), where the function $W(y, 1; p(1))$ is given by (5.32), the functions $W(y, j; g(0), g(1), h(0), h(1))$ , for $j = 0, 1$ , are given by (5.33) with (5.39)–(5.43), the function $W(y, 0; h(0))$ is given by (5.34), and the numbers $0 < {\overline g}(1) \le {\overline g}(0) \le {r} K < {r} L \le {\underline g}(1) \le {\underline g}((0)$ are uniquely determined by the system of arithmetic equations in (5.35)–(5.38).

5.4. Solutions to the auxiliary optimal stopping problems

By means of straightforward computations, we obtain that the function ${\underline W}(y, j)$ in (2.6) and the boundary ${\underline g}(j)$ in (2.8) solve the left-hand system in (5.4)+(5.5)–(5.9)+(5.10), while the function ${\overline W}(y, j)$ in (2.7) and the boundary ${\overline h}(j)$ in (2.8) solve the right-hand system in (5.4)+(5.5)–(5.9)+(5.11). In this case, the candidate value function for ${\underline W}(y, j)$ should be of the form of (5.12) for $g(1) < y \le g(0)$ , and of (5.13) for $y > g(0)$ , while the candidate value function for ${\overline W}(y, j)$ should be of the form of (5.13) for $y < h(1)$ , and of (5.14) for $h(1) \le y < h(0)$ , with (5.15). Moreover, on the one hand, we observe that ${C}_1(j) = {C}_2(j) = 0$ should hold in the representation related to (5.13) for ${\underline W}(y, j)$ , since otherwise, we would have $W(y, j) \rightarrow \pm \infty$ with more than a linear growth as $y \uparrow \infty$ , which must be excluded by virtue of the obvious fact that the value function in (2.6) is of a linear growth under $y \uparrow \infty$ , for any $j = 0, 1$ fixed. On the other hand, we observe that ${C}_3(j) = {C}_4(j) = 0$ should hold in the representation related to (5.13) for ${\overline W}(y, j)$ , since otherwise we would have $W(y, j) \rightarrow \pm \infty$ with less than linear growth as $y \downarrow 0$ , which must be excluded by virtue of the obvious fact that the value function in (2.7) is of a linear growth under $y \downarrow 0$ , for any $j = 0, 1$ fixed.

Hence, following arguments similar to those applied in Section 5.2 above (see also [Reference Gapeev36, Appendix]) and solving the systems of arithmetic equations in (5.19)–(5.23) and (5.24)–(5.25), under ${C}_1(j) = {C}_2(j) = 0$ , we obtain that the functions

(5.46) \begin{align} &W(y, 1; g(1)) = \sum_{j = 1}^2 \frac{B_y(g(1), 1) g(1) - \gamma_{3-j} B(g(1), 1)} {\gamma_{3-j} - \gamma_{j}} \, \bigg( \frac{y}{g(1)} \bigg)^{\gamma_{j}} + B(y, 1) \end{align}

for $g(1) < y \le g(0)$ , as well as

(5.47) \begin{equation} W(y, 0; g(0)) = \sum_{k = 3}^4 \frac{A_y(g(0), 0) g(0) - \beta_{7-k} A(g(0), 0)}{\beta_{7-k} - \beta_{k}} \, \bigg( \frac{y}{g(0)} \bigg)^{\beta_k} + A(y, 0) \end{equation}

and

(5.48) \begin{equation} W(y, 1; g(0)) = \sum_{k = 3}^4 \frac{A_y(g(0), 0) g(0) - \beta_{7-k} A(g(0), 0)}{\beta_{7-k} - \beta_{k}} \, \frac{P_0(\beta_k)}{\lambda_1} \, \bigg( \frac{y}{g(0)} \bigg)^{\beta_k} + A(y, 1) \end{equation}

for $y > g(0)$ , satisfy the left-hand system in (5.4)–(5.5) and (5.6), while the left-hand condition of (5.7) is also satisfied when the equalities

(5.49) \begin{align} &\sum_{k = 3}^4 \frac{A_y(g(0), 0) g(0) - \beta_{7-k} A(g(0), 0)}{\beta_{7-k} - \beta_{k}} \, \frac{P_0(\beta_k)}{\eta_1} + A(g(0), 1) \nonumber\\ &= \sum_{j = 1}^2 \frac{B_y(g(1), 1) g(1) - \gamma_{3-j} B(g(1), 1)}{\gamma_{3-j} - \gamma_{j}} \, \bigg( \frac{g(0)}{g(1)} \bigg)^{\gamma_{j}} + B(g(0), 1) \end{align}

and

(5.50) \begin{align} &\sum_{k = 3}^4 \frac{A_y(g(0), 0) g(0) - \beta_{7-k} A(g(0), 0)}{\beta_{7-k} - \beta_{k}} \, \frac{P_0(\beta_k)}{\eta_1} \, \beta_k + A_y(g(0), 1) \, g(0) \nonumber\\ &= \sum_{j = 1}^2 \frac{B_y(g(1), 1) g(1) - \gamma_{3-j} B(g(1), 1)} {\gamma_{3-j} - \gamma_{j}} \, \gamma_j \, \bigg( \frac{g(0)}{g(1)} \bigg)^{\gamma_{j}} + B_y(g(0), 1) \, g(0) \end{align}

hold. Thus, following arguments similar to those applied in Section 5.2 above and solving the systems of arithmetic equations in (5.19)–(5.23) and (5.24)–(5.25), under ${C}_3(j) = {C}_4(j) = 0$ , we obtain that the functions

(5.51) \begin{align} &W(y, 0; h(0)) = \sum_{j = 1}^2 \frac{B_y(h(0), 0) h(0) - \alpha_{3-j} B(h(0), 0)} {\alpha_{3-j} - \alpha_{j}} \, \bigg( \frac{y}{h(0)} \bigg)^{\alpha_{j}} + B(y, 0) \end{align}

for $h(1) \le y < h(0)$ , as well as

(5.52) \begin{equation} W(y, 1; h(1)) = \sum_{k = 1}^2 \frac{A_y(h(1), 1) h(1) - \beta_{3-k} A(h(1), 1)}{\beta_{3-k} - \beta_{k}} \, \bigg( \frac{y}{h(1)} \bigg)^{\beta_k} + A(y, 1) \end{equation}

and

(5.53) \begin{equation} W(y, 0; h(1)) = \sum_{k = 1}^2 \frac{A_y(h(1), 1) h(1) - \beta_{3-k} A(h(1), 1)}{\beta_{3-k} - \beta_{k}} \, \frac{P_1(\beta_k)}{\eta_0} \, \bigg( \frac{y}{h(1)} \bigg)^{\beta_k} + A(y, 0) \end{equation}

for $y < h(1)$ , satisfy the right-hand system in (5.4)–(5.5) and (5.6), while the right-hand condition of (5.7) is also satisfied when the equalities

(5.54) \begin{align} &\sum_{k = 1}^2 \frac{A_y(h(1), 1) h(1) - \beta_{3-k} A(h(1), 1)}{\beta_{3-k} - \beta_{k}} \, \frac{P_1(\beta_k)}{\lambda_0} + A(h(1), 0) \\ \notag &= \sum_{j = 1}^2 \frac{B_y(h(0), 0) h(0) - \alpha_{3-j} B(h(0), 0)}{\alpha_{3-j} - \alpha_{j}} \, \bigg( \frac{h(1)}{h(0)} \bigg)^{\alpha_{j}} + B(h(1), 0) \end{align}

and

(5.55) \begin{align} &\sum_{k = 1}^2 \frac{A_y(h(1), 1) h(1) - \beta_{3-k} A(h(1), 1)}{\beta_{3-k} - \beta_{k}} \, \frac{P_1(\beta_k)}{\lambda_0} \, \beta_k + A_y(h(1), 0) \, h(1) \\ \notag &= \sum_{j = 1}^2 \frac{B_y(h(0), 0) h(0) - \alpha_{3-j} B(h(0), 0)}{\alpha_{3-j} - \alpha_{j}} \, \alpha_j \, \bigg( \frac{h(1)}{h(0)} \bigg)^{\alpha_{j}} + B_y(h(1), 0) \, h(1) \end{align}

hold.

Summarizing the facts shown above, let us formulate the following result which can be proved by means of the same arguments as Theorem 5.1 above.

Corollary 5.1. Suppose that the process Y is defined by (1.1) and (1.2), with some ${r} > 0$ , $\eta_0 > \eta_1 > 0$ , and $\sigma > 0$ , while $\boldsymbol{\Theta}$ is a continuous-time Markov chain with state space $\{ 0, 1 \}$ and transition intensities $\lambda_j > 0$ , for $j = 0, 1$ , which is independent of standard Brownian motion B. Then the value function ${\underline W}(y, j)$ of the optimal stopping problem in (2.6), with some $K > 0$ fixed, admits the representations

(5.56) \begin{equation} {\underline W}(y, 0)= \begin{cases} W(y, 0; {\underline g}(0)), & \textit{if} \; \; y > {\underline g}(0), \\ K, & \textit{if} \; \; y \le {\underline g}(0), \end{cases} \end{equation}

and

(5.57) \begin{equation} {\underline W}(y, 1)= \begin{cases} W(y, 1; {\underline g}(0)), & \textit{if} \; \; y > {\underline g}(0), \\ W(y, 1; {\underline g}(1)), & \textit{if} \; \; {\underline g}(1) < y \le {\underline g}(0), \\ K, & \textit{if} \; \; y \le {\underline g}(1), \end{cases} \end{equation}

and the optimal stopping time ${\underline \tau}$ has the form of (2.8), where the functions $W(y, j; g(0))$ , for $j = 0, 1$ , are given by (5.47) and (5.48), the function $W(y, 1; g(1))$ is given by (5.46), and the numbers ${\underline g}(j)$ , for $j = 0, 1$ , are determined by the system of arithmetic equations in (5.49) and (5.50). Also, the value function ${\overline W}(y, j)$ of the optimal stopping problem in (2.7), with some $L > 0$ fixed, admits the representations

(5.58) \begin{equation} {\overline W}(y, 0)= \begin{cases} W(y, 0; {\overline h}(1)), & \textit{if} \; \; y < {\overline h}(1), \\ W(y, 0; {\overline h}(0)), & \textit{if} \; \; {\overline h}(1) \le y < {\overline h}(0), \\ L, & \textit{if} \; \; y \ge {\overline h}(0), \end{cases} \end{equation}

and

(5.59) \begin{equation} {\overline W}(y, 1)= \begin{cases} W(y, 1; {\overline h}(1)), & \textit{if} \; \; y < {\overline h}(1), \\ L, & \textit{if} \; \; y \ge {\overline h}(1), \end{cases} \end{equation}

and the optimal stopping time ${\overline \zeta}$ has the form of (2.8), where the functions $W(y, j; h(1))$ , for $j = 0, 1$ , are given by (5.52) and (5.53), the function $W(y, 0; h(0))$ is given by (5.51), and the numbers ${\overline h}(j)$ , for $j = 0, 1$ , are uniquely determined by the system of arithmetic equations in (5.54) and (5.55).