2.1. The three-dimensional optimal stopping zero-sum game

Suppose that an investor writes a perpetual lookback game option and sells the contract to another investor at time 0. The holder of the option can then exercise the contract at some random time $\tau$ , which they can choose, by collecting the amount of the running maximum S and paying the floating strike K X to the writer, for some $K > 0$ . At the same time, the writer of the option can either recall the contract at some random time $\zeta$ , which they choose, by paying the amount of the running minimum Q to the holder and collecting the floating strike L X, for some $L > 0$ , when $Q - L X > S - K X$ holds, or agree with the holder on the payment of S and the collection of K X. Consequently, the holder of the option looks for the exercise time $\tau_*$ maximising the expected total payoff received from the writer, while, at the same time, the writer of the contract looks for the recall time $\zeta_*$ minimising the expected total payoff sent to the holder. In other words, the perpetual lookback game option pricing problem seeks to determine the pair of stopping times $\tau_*$ and $\zeta_*$ of the process X that corresponds to a saddle point for the total expected reward functional given by

(2.1) \begin{align} {\mathbb J}(\tau, \zeta) &= {\mathbb E} \big[ e^{- r \tau} \, F(X_{\tau}, S_{\tau}) \, I(\tau < \zeta) + e^{- r \zeta} \, G(X_{\zeta}, S_{\zeta}, Q_{\zeta}) \, I(\zeta \le \tau) \big], \end{align}

with the functions F(x, s) and G(x, s, q) defined in (1.5), for some $0 < L < K < L + 1$ fixed, which means that the inequalities

(2.2) \begin{equation} {\mathbb J}(\tau, \zeta_*) \le {\mathbb J}(\tau_*, \zeta_*) \le {\mathbb J}(\tau_*, \zeta) \end{equation}

should hold, for any exercise and recall times $\tau$ and $\zeta$ . Such a pair $\tau_*$ and $\zeta_*$ satisfying the inequalities of (2.2) with the functional defined in (2.1) is called a Nash equilibrium in the optimal stopping zero-sum game of (1.4) (see e.g. [Reference Bensoussan and Friedman8, Reference Ekström and Peskir14, Reference Peskir55] for a precise definition of this notion).

It thus follows from the results of [Reference Kallsen and Kühn39, Reference Kifer41] that the rational (or no-arbitrage) price of the game contingent claim described above coincides with the value function $V_*(x, s, q)$ of the optimal stopping zero-sum game for the (time-homogeneous strong) Markov process $(X, S, Q) = (X_t, S_t, Q_t)_{t \ge 0}$ of the form

(2.3) \begin{equation} V_*(x, s, q) = \sup_{\tau} \inf_{\zeta} {\mathbb E}_{x, s, q} \big[ e^{- r \tau} \, F(X_{\tau}, S_{\tau}) \, I(\tau < \zeta) + e^{- r \zeta} \, G(X_{\zeta}, S_{\zeta}, Q_{\zeta}) \, I(\zeta \le \tau) \big], \end{equation}

where the supremum and infimum are taken over all stopping times $\tau$ and $\zeta$ with respect to the natural filtration $(\mathcal{F}_t)_{t \ge 0}$ of the process X, and the functions F(x, s) and G(x, s, q) are given by (1.5), for some $0 < L < K < L + 1$ fixed. Here, we denote by ${\mathbb E}_{x, s, q}$ the expectation with respect to the probability measure ${\mathbb P}$ under the assumption that the three-dimensional (strong Markov) process (X, S, Q) defined in (1.1)–(1.2) and (1.3) starts at $(x, s, q) \in E$ , and by $E = \{ (x, s, q) \in {\mathbb R}^3 \, | \, 0 < q \leq x \leq s \}$ the state space of the process (X, S, Q). It therefore follows from the results of [Reference Cvitanić and Karatzas11, Theorem 4.1], based on the solutions of the associated (doubly) reflected backward stochastic differential equations, that the game-type optimal stopping problem of (2.3) has a value. The existence of the associated Stackelberg equilibria in various optimal stopping games is proved in the results of [Reference Lepeltier and Mainguenau47, Reference Peskir55–Reference Peskir56, Reference Stettner69–Reference Stettner70], among others. We further establish the existence and describe the structure of the stopping times $\tau_*$ and $\zeta_*$ forming a Nash equilibrium of the optimal stopping zero-sum game of (2.3).

2.2. The structure of the optimal stopping times

Let us first determine the structure of the stopping times forming a Nash equilibrium in the optimal stopping game of (2.3).

(i) By means of the results of general optimal stopping theory for Markov processes (see e.g. [Reference Peskir and Shiryaev59, Chapter I, Section 2.2]) and the results of the general theory of optimal stopping games (see e.g. [Reference Bensoussan and Friedman7–Reference Bensoussan and Friedman8, Reference Cvitanić and Karatzas11, Reference Friedman16–Reference Friedman17, Reference Krylov43, Reference Lepeltier and Mainguenau47, Reference Stettner69], among others), we obtain from the structure of the reward functional that the stopping times forming a Nash equilibrium in the optimal stopping game of (2.3) exist and are given by

(2.4) \begin{align}&\tau_* = \inf \big\{ t \ge 0 \; \big| \; V_*(X_t, S_t, Q_t) = F(X_t, S_t) \big\}\end{align}

and

(2.5) \begin{align}&\zeta_* = \inf \big\{ t \ge 0 \; \big| \; V_*(X_t, S_t, Q_t) = G(X_t, S_t, Q_t) \big\},\end{align}

so that the associated continuation and stopping regions have the forms

(2.6) \begin{equation}C_* = \big\{ (x, s, q) \in E \; \big| \; F(x, s) < V_*(x, s, q) < G(x, s, q) \big\}\end{equation}

and

(2.7) \begin{equation}D_* = \big\{ (x, s, q) \in E \; \big| \;\; \text{either} \;\;V_*(x, s, q) = F(x, s) \;\; \text{or} \;\; V_*(x, s, q) = G(x, s, q) \big\}\end{equation}

respectively, where the functions F(x, s) and G(x, s, q) are given by (1.5), for some $0 < L < K < L + 1$ fixed. It can be seen from the results of Theorem 1 below that the value function $V_*(x, s, q)$ is continuous, so that the set $C_*$ is open and the set $D_*$ is closed.

It follows from the structure of the payoff functions F(x, s) and G(x, s, q) given by (1.5) that the inequality $F(X_t, S_t) < G(X_t, S_t, Q_t)$ holds when $X_t > (S_t - Q_t)/(K - L)$ , while the equality $F(X_t, S_t) = G(X_t, S_t, Q_t)$ holds when $X_t \le (S_t - Q_t)/(K - L)$ , for any $t \ge 0$ . Moreover, by virtue of the structure of the processes S and Q in (1.3), we may conclude that the inequalities $S_t - K X_t \ge (1 - K) X_t \ge Q_t - L X_t$ hold when $X_t \ge Q_t/(L - K + 1)$ , because of the assumption $0 < L < K < L + 1$ , so that the equality $F(X_t, S_t) = G(X_t, S_t, Q_t)$ is also satisfied in that case, for any $t \ge 0$ . Observe that the latter condition in particular holds in the case $0 < L < K \le 1$ , which is more restrictive but allows us to keep both the payoffs $F(X_t, S_t)$ and $G(X_t, S_t, Q_t)$ positive, for all $t \ge 0$ , which also represents an important feature from the point of view of callable financial contracts. Note that, in the case $L \ge K$ , the inequality $S_t - K X_t < Q_t - L X_t$ cannot be satisfied for any $t \ge 0$ , which implies that the solution of the optimal stopping zero-sum game in (2.3) is trivial in that case, for each $0 < q < s$ fixed.

Summarising the arguments above, we realise that we are considering the three-dimensional continuous strong Markov process (X, S, Q) defined in (1.1)–(1.2) and (1.3) within the discounted optimal stopping zero-sum game of (2.3) with the continuous payoff functions F(x, s) and G(x, s, q) from (1.5) such that the equality $F(x, s) = G(x, s, q)$ holds, when $x \le (s - q)/(K - L)$ as well as $x \ge q/(L - K + 1)$ , for all $(x, s, q) \in E$ , under the assumption $0 < L < K < L + 1$ . Observe that the arguments of the proofs from [Reference Ekström and Peskir14, Theorem 2.1] and [Reference Peskir55, Theorem 2.1] (also applicable for continuous strong Markov time-space processes with constant killing rates) can be naturally extended to the case of the discounted optimal stopping game of (2.3) for the process (X, S, Q) with the second and third components changing only at the diagonals $d_1 = \{ (x, s, q) \in {\mathbb R}^3 \, | \, 0 < q \le x = s \}$ and $d_2 = \{ (x, s, q) \in {\mathbb R}^3 \, | \, 0 < q = x \le s \}$ , respectively, of the state space E (see also [Reference Peskir56, Theorem 3.1] for the corresponding result in a model based on a standard Brownian motion in the interval [0, 1] absorbed at either 0 or 1). Note that the natural analogues of the conditions of [Reference Ekström and Peskir14, Formula (2.1)] and [Reference Peskir55, Formulae (2.9) and (2.12)] are clearly satisfied for the discounted payoffs $e^{- r t} F(X_t, S_t)$ and $e^{- r t} G(X_t, S_t, Q_t)$ from (1.5) (see e.g. [Reference Shepp and Shiryaev64, Formula (2.16)]). Hence, by applying the resulting extensions mentioned above, we may conclude that the continuation region $C_*$ in (2.6) should belong to the set

(2.8) \begin{align}&E^{\prime} = \big\{ (x, s, q) \in {\mathbb R}^3 \; \big| \; 0 < a^{\prime}(s, q) \vee q \le x \le s \wedge b^{\prime}(s, q) \big\}\end{align}

with

(2.9) \begin{equation}a^{\prime}(s, q) = (s - q)/(K - L) \quad \text{and} \quad b^{\prime}(s, q) = q/(L - K + 1)\end{equation}

for $0 < q < s$ , which represents all points (x, s, q) from the state space E of the process (X, S, Q) for which the solution of the original problem of (2.3) may be nontrivial, while the complement $E \setminus E^{\prime}$ surely belongs to the stopping region $D_*$ in (2.7), under the assumption $0 < L < K < L + 1$ . Therefore, the property $\tau_* \wedge \zeta_* \le \theta$ ( $\mathbb{P}_{x, s, q}$ -almost surely (a.s.)) holds, for $\tau_*$ and $\zeta_*$ from (2.4) and (2.5) and for any point $(x, s, q) \in E^{\prime}$ , under the assumption $0 < L < K < L + 1$ , where we put

(2.10) \begin{equation}\theta = \inf \big\{ t \ge 0 \; \big| \;\; \text{either} \;\;X_t \le a^{\prime}(S_t, Q_t) \;\; \text{or} \;\; X_t \ge b^{\prime}(S_t, Q_t) \big\},\end{equation}

which is a stopping time of the process (X, S, Q).

(ii) We now describe the structure of the continuation and stopping regions $C_*$ and $D_*$ from (2.6)–(2.7). For this purpose, by means of standard applications of Itô’s formula (see e.g. [Reference Liptser and Shiryaev48, Theorem 4.4] or [Reference Revuz and Yor62, Chapter IV, Theorem 3.3]) to the processes $e^{-r t} (S_t - K X_t)$ and $e^{-r t} (Q_t - L X_t)$ , we obtain the representations

(2.11) \begin{align}&e^{- r t} \, (S_t - K \, X_t) = s - K \, x+ \int_0^t e^{- r u} \, \big( K \, \delta(S_u, Q_u) \, X_u - r \, S_u \big) \, du + \int_0^t e^{- r u} \, dS_u + N^1_t\end{align}

and

(2.12) \begin{align}&e^{- r t} \, (Q_t - L \, X_t) = q - L \, x+ \int_0^t e^{- r u} \, \big( L \, \delta(S_u, Q_u) \, X_u - r \, Q_u \big) \, du + \int_0^t e^{- r u} \, dQ_u + N^2_t\end{align}

for all $t \ge 0$ . Here, the processes $N^i = (N^i_t)_{t \ge 0}$ , for $i = 1, 2$ , defined by

(2.13) \begin{align}&N^1_t = - K \int_0^t e^{- r u} \, \sigma(S_u, Q_u) \, X_u \, d{B}_u\quad \text{and} \quad N^2_t = - L \int_0^t e^{- r u} \, \sigma(S_u, Q_u) \, X_u \, d{B}_u\end{align}

are continuous uniformly integrable martingales under the probability measure ${\mathbb P}_{x, s, q}$ . Then, inserting $\tau \wedge \zeta$ in place of t and applying Doob’s optional sampling theorem (see e.g. [Reference Liptser and Shiryaev48, Chapter III, Theorem 3.6] and [Reference Revuz and Yor62, Chapter II, Theorem 3.2]) to the expressions in (2.11) and (2.12), we get that the equalities

(2.14) \begin{align}&{\mathbb E}_{x, s, q} \big[ e^{- r \tau} \, (S_{\tau} - K \, X_{\tau}) \, I(\tau < \zeta)+ e^{- r \zeta} \, (Q_{\zeta} - L \, X_{\zeta}) \, I(\zeta \le \tau) \big] \\\notag&= {\mathbb E}_{x, s, q} \big[ e^{- r (\tau \wedge \zeta)} \, (S_{\tau \wedge \zeta} - K \, X_{\tau \wedge \zeta})- e^{- r \zeta} \, (S_{\zeta} - K \, X_{\zeta} - Q_{\zeta} + L \, X_{\zeta}) \, I(\zeta \le \tau) \big] \\\notag&= s - K \, x - {\mathbb E}_{x, s, q} \big[ e^{- r \zeta} \, (S_{\zeta} - K \, X_{\zeta} - Q_{\zeta} + L \, X_{\zeta})\, I(\zeta \le \tau) \big] \\\notag&\phantom{=\;\:}+ {\mathbb E}_{x, s, q} \bigg[ \int_0^{\tau \wedge \zeta} e^{- r u} \, \big( K \, \delta(S_u, Q_u) \, X_u- r \, S_u \big) \, du + \int_0^{\tau \wedge \zeta} e^{- r u} \, dS_u \bigg]\end{align}

and

(2.15) \begin{align}&{\mathbb E}_{x, s, q} \big[ e^{- r \tau} \, (S_{\tau} - K \, X_{\tau}) \, I(\tau < \zeta)+ e^{- r \zeta} \, (Q_{\zeta} - L \, X_{\zeta}) \, I(\zeta \le \tau) \big] \\\notag&= {\mathbb E}_{x, s, q} \big[ e^{- r (\tau \wedge \zeta)} \, (Q_{\tau \wedge \zeta} - L \, X_{\tau \wedge \zeta})+ e^{- r \tau} \, (S_{\tau} - K \, X_{\tau} - Q_{\tau} + L \, X_{\tau}) \, I(\tau < \zeta) \big] \\\notag&= q - L \, x + {\mathbb E}_{x, s, q} \big[ e^{- r \tau} \, (S_{\tau} - K \, X_{\tau} - Q_{\tau} + L \, X_{\tau})\, I(\tau < \zeta) \big] \\\notag&\phantom{=\;\:}+ {\mathbb E}_{x, s, q} \bigg[ \int_0^{\tau \wedge \zeta} e^{- r u} \, \big( L \, \delta(S_u, Q_u)\, X_u - r \, Q_u \big) \, du + \int_0^{\tau \wedge \zeta} e^{- r u} \, dQ_u \bigg]\end{align}

hold, for any stopping times $\tau$ and $\zeta$ such that $\tau \wedge \zeta \le \theta$ ( $\mathbb{P}_{x, s, q}$ -a.s.) holds with $\theta$ defined in (2.10), and for any starting point $(x, s, q) \in E^{\prime}$ of (X, S, Q). Hence, it follows from the expressions in (2.14) and (2.15) and the structure of the optimal stopping times in (2.4) and (2.5) that the value function of the optimal stopping game in (2.3) admits the representations

(2.16) \begin{align}V_*(x, s, q) &= s - K \, x - {\mathbb E}_{x, s, q} \big[ e^{- r \zeta_*} \, (S_{\zeta_*}- K \, X_{\zeta_*} - Q_{\zeta_*} + L \, X_{\zeta_*}) \, I(\zeta_* \le \tau_*) \big] \\\notag&\phantom{=\;\:}+ {\mathbb E}_{x, s, q} \bigg[ \int_0^{\tau_* \wedge \zeta_*} e^{- r u} \,\big( K \, \delta(S_u, Q_u) \, X_u - r \, S_u \big) \, du + \int_0^{\tau_* \wedge \zeta_*}e^{- r u} \, dS_u \bigg]\end{align}

and

(2.17) \begin{align}V_*(x, s, q) &= q - L \, x + {\mathbb E}_{x, s, q} \big[ e^{- r \tau_*} \, (S_{\tau_*} - K \, X_{\tau_*}- Q_{\tau_*} + L \, X_{\tau_*}) \, I(\tau_* < \zeta_*) \big] \\\notag&\phantom{=\;\:}+ {\mathbb E}_{x, s, q} \bigg[ \int_0^{\tau_* \wedge \zeta_*} e^{- r u} \, \big( L \,\delta(S_u, Q_u) \, X_u - r \, Q_u \big) \, du + \int_0^{\tau_* \wedge \zeta_*} e^{- r u} \, dQ_u \bigg]\end{align}

for the optimal stopping times $\tau_*$ and $\zeta_*$ forming a Nash equilibrium in (2.3), because the property $\tau_* \wedge \zeta_* \le \theta$ ( $\mathbb{P}_{x, s, q}$ -a.s.) holds with $\theta$ as defined in (2.10), for any $(x, s, q) \in E^{\prime}$ .

Here and subsequently, we denote by $\tau_* = \tau_*(x, s, q)$ and $\zeta_* = \zeta_*(x, s, q)$ the optimal stopping times forming a Nash equilibrium in (2.3) for the starting point $(x, s, q) \in E^{\prime}$ of the process (X, S, Q), where E ^′ is defined in (2.8) above. Thus, on the one hand, it follows from the structure of the integrand in the first integral of (2.16) and the fact that the second integral there increases whenever the process (X, S, Q) is located at the diagonal $d_1 = \{ (x, s, q) \in {\mathbb R}^3 \, | \, 0 < q \leq x = s \}$ that it should not be optimal for the option holder (maximiser of the expected reward) to exercise the contract earlier than the option writer recalls it, when the inequalities $a^{\prime}(S_t, Q_t) \vee r S_t/(K \delta(S_t, Q_t)) < X_t \le S_t \wedge b^{\prime}(S_t, Q_t)$ hold, for any $t \ge 0$ . Moreover, it follows from the structure of the integrand in first integral of (2.17) and the fact that the second integral there decreases whenever the process (X, S, Q) is located at the diagonal $d_2 = \{ (x, s, q) \in {\mathbb R}^3 \, | \, 0 < q = x \le s \}$ that it should not be optimal for the option writer (minimiser of the expected reward) to recall the contract earlier than the option holder exercises it, when the inequalities $a^{\prime}(S_t, Q_t) \vee Q_t \le X_t < r Q_t/(L \delta(S_t, Q_t)) \wedge b^{\prime}(S_t, Q_t)$ hold, for any $t \ge 0$ . Since both participants of the contract are acting simultaneously, these facts yield that the set

(2.18) \begin{align}&C^{\prime} = \big\{ (x, s, q) \in E^{\prime} \; \big| \; a^{\prime}(s, q) \vee q \vee {\overline a}(s, q)< x < {\underline b}(s, q) \wedge s \wedge b^{\prime}(s, q) \big\}\end{align}

with

(2.19) \begin{equation}{\overline a}(s, q) = r s/(K \delta(s, q)) \quad \text{and} \quad {\underline b}(s, q) = r q/(L \delta(s, q))\end{equation}

for $0 < q < s$ , which may be nonempty because of the assumption that $0 < L < K < L + 1$ holds, represents a part of the continuation region $C_*$ in (2.6).

(iii) Let us finally prove the connectivity of the left-hand and right-hand parts of the stopping region $D_*$ from (2.7) which are located outside the region C ^′ from (2.18). For this purpose, we first recall from the arguments of Part (i) above that all the points (x ^′, s, q) and (x ^′′, s, q) of the set $E \setminus E^{\prime}$ , with E ^′ as defined in (2.8), such that $0 < x^{\prime} < a^{\prime}(s, q)$ and $x^{\prime\prime} > b^{\prime}(s, q)$ belong to the stopping region $D_*$ , for each $0 < q < s$ fixed, under the assumption $0 < L < K < L + 1$ . We also recall that the process X admits the explicit expression of (1.1) and provides a (pathwise) unique strong solution of the stochastic differential equation in (1.2) with the processes S and Q given by (1.3), so that the solutions starting from the different points $x > 0$ do not intersect each other over the entire infinite time interval. For ease of presentation, in the rest of this section we indicate by $(X^{(x)}, S^{(s, x)}, Q^{(q, x)})$ the dependence of the process (X, S, Q) defined in (1.1)–(1.2) and (1.3) on its starting point $(x, s, q) \in E^{\prime}$ .

Observe that, if we take some $(x, s, q) \in D_*$ such that $a^{\prime}(s, q) < x < {\overline a}(s, q) \wedge s \wedge b^{\prime}(s, q)$ holds, for each $0 < q < s$ fixed, then the arguments of Part (ii) above would imply that it is not optimal for the option writer to recall the contract earlier than the holder exercises it, so that the value in (2.3) would admit the representation

(2.20) \begin{align}&V_*(x^{\prime}, s, q) - (s - K \, x^{\prime}) \\\notag&= {\mathbb E} \bigg[ \int_0^{\tau^{\prime}_*} e^{- r u} \,\big( K \, \delta \big( S^{(s, x^{\prime})}_u, Q^{(q, x^{\prime})}_u \big) \, X^{(x^{\prime})}_u - r \, S^{(s, x^{\prime})}_u \big) \, du+ \int_0^{\tau^{\prime}_*} e^{- r u} \, dS^{(s, x^{\prime})}_u \bigg],\end{align}

where $\tau^{\prime}_* = \tau_*(x^{\prime}, s, q)$ denotes the optimal exercise time for the holder in the problem of (2.3) under the assumption that the process (X, S, Q) starts at (x ^′, s, q), for all $a^{\prime}(s, q) \le x^{\prime} < x$ and each $0 < q < s$ fixed. Hence, because of the assumption that the function $\delta(s, q)$ is increasing in both the variables s and q on $[0, \infty]^2$ and the fact that the process (X, S, Q) started at (x ^′, s, q) reaches the point (x, s, q ^′) for some $0 < q^{\prime} \le q$ before hitting the upper diagonal $d_1 = \{ (x, s, q) \in {\mathbb R}^3 \, | \, 0 < q \le x = s \}$ , we see that the integrand in the first integral in the right-hand side of (2.20) (and thus the resulting total expected reward functional there) is increasing in x ^′ for $a^{\prime}(s, q) \le x^{\prime} < x$ , for each $0 < q < s$ fixed. Thus, we may conclude that the inequalities

(2.21) \begin{align}&V_*(x^{\prime}, s, q) - (s - K \, x^{\prime}) \le V_*(x, s, q) - (s - K \, x) = 0\end{align}

hold, so that the point (x ^′, s, q) such that $a^{\prime}(s, q) \le x^{\prime} < x < {\overline a}(s, q) \wedge s \wedge b^{\prime}(s, q)$ also belongs to the left-hand part of the stopping region $D_*$ in (2.7).

Similarly, if we take some $(x, s, q) \in D_*$ such that $a^{\prime}(s, q) \vee q \vee {\underline b}(s, q) < x < b^{\prime}(s, q)$ holds, for each $0 < q < s$ fixed, then the arguments of Part (ii) above would imply that it is not optimal for the option holder to exercise the contract earlier than the writer recalls it, so that the value in (2.3) would admit the representation

(2.22) \begin{align}&V_*(x^{\prime\prime}, s, q) - (q - L \, x^{\prime\prime}) \\\notag&= {\mathbb E} \bigg[ \int_0^{\zeta^{\prime}_*} e^{- r u} \, \big( L \,\delta \big( S^{(s, x^{\prime\prime})}_u, Q^{(q, x^{\prime\prime})}_u \big) \, X^{(x^{\prime\prime})}_u - r \, Q^{(q, x^{\prime\prime})}_u \big) \, du+ \int_0^{\zeta^{\prime}_*} e^{- r u} \, dQ^{(q, x^{\prime\prime})}_u \bigg],\end{align}

where $\zeta^{\prime}_* = \zeta_*(x^{\prime\prime}, s, q)$ denotes the optimal exercise time for the writer in the problem of (2.3) under the assumption that the process (X, S, Q) starts at (x ^′′, s, q), for all $x < x^{\prime\prime} \le b^{\prime}(s, q)$ and each $0 < q < s$ fixed. Hence, by the assumption that the function $\delta(s, q)$ is increasing in both the variables s and q on $[0, \infty]^2$ and the fact that the process (X, S, Q) started at (x ^′′, s, q) reaches the point (x, s ^′, q) for some $0 < s \le s^{\prime}$ before hitting the lower diagonal $d_2 = \{ (x, s, q) \in {\mathbb R}^3 \, | \, 0 < q = x \le s \}$ , we see that the integrand in the first integral in the right-hand side of (2.22) (and thus the resulting total expected reward functional there) is increasing in x ^′′ for $x < x^{\prime\prime} \le b^{\prime}(s, q)$ , for each $0 < q < s$ fixed. Thus, we may conclude that the inequalities

(2.23) \begin{align}&V_*(x^{\prime\prime}, s, q) - (q - L \, x^{\prime\prime}) \ge V_*(x, s, q) - (q - L \, x) = 0\end{align}

hold, so that the point (x ^′′, s, q) such that $a^{\prime}(s, q) \vee q \vee {\underline b}(s, q) < x < x^{\prime\prime} \le b^{\prime}(s, q)$ also belongs to the right-hand part of the stopping region $D_*$ in (2.7).

Figure 1. A computer drawing of the optimal exercise boundaries $a_*(s, q)$ , $b_*(s, q)$ , and ${\underline a}(s, q)$ , for each $q > 0$ fixed.

Figure 2. A computer drawing of the optimal exercise boundaries $a_*(s, q)$ , $b_*(s, q)$ , and ${\overline b}(s, q)$ , for each $s > 0$ fixed.

Combining these arguments with the facts deduced in Parts (i)–(ii) above and noting the comments in [Reference Dubins, Shepp and Shiryaev13, Subsection 3.3] and [Reference Peskir52, Subsection 3.3], we conclude that there exist functions $a_*(s, q)$ and $b_*(s, q)$ satisfying the inequalities $a^{\prime}(s, q) \le a_*(s, q) \le {\overline a}(s, q) \wedge s \wedge b^{\prime}(s, q)$ and $a^{\prime}(s, q) \vee q \vee {\underline b}(s, q) \le b_*(s, q) \le b^{\prime}(s, q)$ , for $0 < q < s$ , such that the continuation region $C_*$ in (2.6) has the form

(2.24) \begin{align} &C_* = \big\{ (x, s, q) \in E^{\prime} \; \big| \; a_*(s, q) < x < b_*(s, q) \big\}, \end{align}

while the stopping region ${D}_*$ in (2.7) is given by

(2.25) \begin{align} &D_* = \big\{ (x, s, q) \in E \; \big| \;\; \text{either} \;\; x \le a_*(s, q) \;\; \text{or} \;\; x \ge b_*(s, q) \big\}. \end{align}

(iv) Finally, in order to determine upper and lower bounds for the value function in (2.3) and optimal stopping boundaries in (2.24) and (2.25), we consider the optimal stopping problems with the value functions ${\overline V}(x, s, q)$ and ${\underline V}(x, s, q)$ from (5.1). It is shown in Section 5 below that the functions ${\overline V}(x, s, q)$ and ${\underline V}(x, s, q)$ admit the explicit expressions in (5.11) and (5.12), while the associated optimal stopping times ${\overline \tau}$ and ${\underline \zeta}$ are given by (5.2), where the boundaries ${\underline a}(s, q)$ and ${\overline b}(s, q)$ are determined as the maximal and minimal solutions of the first-order nonlinear ordinary differential equations in (3.24) and (3.27) staying below or above the diagonals $d_1 = \{ (x, s, q) \in {\mathbb R}^3 \, | \, 0 < q \leq x = s \}$ and $d_2 = \{ (x, s, q) \in {\mathbb R}^3 \, | \, 0 < q = x \le s \}$ , respectively. If we suppose that either the inequality $a_*(s, q) < {\underline a}(s, q)$ or the inequality $b_*(s, q) > {\overline b}(s, q)$ holds, then, for each given and fixed $x > 0$ such that either $x \in (a_*(s, q), {\underline a}(s, q))$ or $x \in ({\overline b}(s, q), b_*(s, q))$ , we would have either $V_*(x, s, q) > s - K x = {\overline V}(x, s, q)$ or $V_*(x, s, q) < q - L x = {\underline V}(x, s, q)$ , respectively, contradicting the obvious fact that the inequalities $V_*(x, s, q) \le {\overline V}(x, s, q)$ and $V_*(x, s, q) \ge {\underline V}(x, s, q)$ hold for all $(x, s, q) \in E^{\prime}$ . Thus, we may conclude that the inequalities $a_*(s, q) \ge {\underline a}(s, q)$ and $b_*(s, q) \le {\overline b}(s, q)$ should be satisfied for all $0 < q < s$ (see Figures 1 and 2 for computer drawings of the optimal stopping boundaries $a_*(s, q)$ and $b_*(s, q)$ , as well as the estimates ${\underline a}(s, q)$ and ${\overline b}(s, q)$ ).

2.3. The three-dimensional coupled free-boundary problem

By means of standard arguments based on an application of Itô’s formula, it can be shown that the infinitesimal operator ${\mathbb L}$ of the process (X, S, Q) has the form

(2.26) \begin{align}&{\mathbb L} = \big( r - \delta(s, q) \big) \, x \, \partial_x+ \frac{\sigma^2(s, q) x^2}{2} \, \partial_{xx} \quad \text{in} \quad 0 < q < x < s,\end{align}

(2.27) \begin{align}&\partial_s = 0 \quad \text{at} \quad 0 < q \le x = s, \quad \text{and}\quad \partial_q = 0 \quad \text{at} \quad 0 < q = x \le s\end{align}

(see e.g. [Reference Peskir52, Subsection 3.1]). In order to find analytic expressions for the unknown value function $V_*(x, s, q)$ from (2.3) and the unknown boundaries $a_*(s, q)$ and $b_*(s, q)$ from (2.24)–(2.25), let us build on the results of the general theory of optimal stopping problems for Markov processes (see e.g. [Reference Peskir and Shiryaev59, Chapter IV, Section 8]). We can reduce the optimal stopping game of (2.3) to the equivalent coupled free-boundary problem for V(x, s, q) with a(s, q) and b(s, q) given by

(2.28) \begin{align} ({{\mathbb L}} V - r V)(x, s, q) = 0 \quad &\text{for} \quad q \vee a(s, q) < x < b(s, q) \wedge s, \end{align}

(2.29) \begin{align} V(x, s, q) \big|_{x = a(s, q)+} = s - K \, a(s, q), \quad &V(x, s, q) \big|_{x = b(s, q)-} = q - L \, b(s, q), \end{align}

(2.30) \begin{align} V(x, s, q) = s - K \, x \quad \text{for} \quad x < a(s, q), \quad &V(x, s, q) = q - L \, x \quad \text{for} \quad x > b(s, q), \end{align}

(2.31) \begin{align} s - K \, x < V(x, s, q) < q - L \, x \quad &\text{for} \quad a(s, q) < x < b(s, q), \end{align}

(2.32) \begin{align} ({{\mathbb L}} V - r V)(x, s, q) < 0 \quad \text{for} \quad x < a(s, q), \quad &({{\mathbb L}} V - r V)(x, s, q) > 0 \quad \text{for} \quad x > b(s, q), \end{align}

where the instantaneous-stopping conditions in (2.29) are respectively satisfied when either $a(s, q) \ge a^{\prime}(s, q) \vee q$ or $b(s, q) \le s \wedge b^{\prime}(s, q)$ holds, for each $0 < q < s$ . Moreover, we further assume that the smooth-fit conditions

(2.33) \begin{align} \partial_x V(x, s, q) \big|_{x = a(s, q)+} = - K, \quad &\partial_x V(x, s, q) \big|_{x = b(s, q)-} = - L \end{align}

are satisfied, when either $a(s, q) > a^{\prime}(s, q) \vee q$ or $b(s, q) < s \wedge b^{\prime}(s, q)$ holds, while the normal-reflection conditions

(2.34) \begin{align} \partial_s V(x, s, q) \big|_{x = s-} = 0, \quad &\partial_q V(x, s, q) \big|_{x = q+} = 0\end{align}

are satisfied, when either $b^{\prime}(s, q) \ge b(s, q) > s$ or $a^{\prime}(s, q) \le a(s, q) < q$ holds, for each $0 < q < s$ . On the one hand, when either the inequality $a(s, q) > a^{\prime}(s, q) \vee q$ or the inequality $b(s, q) < s \wedge b^{\prime}(s, q)$ holds, for some $0 < q < s$ , the continuous process X can cross the left-hand boundary a(S, Q) before hitting the lower diagonal $d_2$ or cross the right-hand boundary b(S, Q) before hitting the upper diagonal $d_1$ , so that we can assume that the left-hand or the right-hand smooth-fit conditions of (2.33) are satisfied for the candidate value function V(x, s, q) at a(s, q) or b(s, q), respectively. On the other hand, when either the inequalities $b^{\prime}(s, q) \ge b(s, q) > s$ or the inequalities $a^{\prime}(s, q) \le a(s, q) < q$ hold, for some $0 < q < s$ , the process X can hit the upper diagonal $d_1 = \{ (x, s, q) \in {\mathbb R}^3 \, | \, 0 < q \leq x = s \}$ before crossing the right-hand boundary b(S, Q) or hit the lower diagonal $d_2 = \{ (x, s, q) \in {\mathbb R}^3 \, | \, 0 < q = x \le s \}$ before crossing the left-hand boundary a(S, Q), so that we can assume that either the left-hand or the right-hand normal-reflection conditions of (2.34) are satisfied for V(x, s, q) at $d_1$ or $d_2$ , respectively. These properties are verified in the proof of Theorem 1 below, while the inequalities in (2.32) follow directly from the arguments of Parts (i)–(ii) of Subsection 2.2 above.

3.1. The candidate value function

We first observe that the general solution of the second-order ordinary differential equation in (2.28) has the form

(3.1) \begin{equation}V(x, s, q) = C_{1}(s, q) \, x^{\gamma_1(s, q)} + C_{2}(s, q) \, x^{\gamma_2(s, q)},\end{equation}

where $C_{i}(s, q)$ , for $i = 1, 2$ , are some continuously differentiable functions and $\gamma_i(s, q)$ , for $i = 1, 2$ , are given by

(3.2) \begin{equation}\gamma_i(s, q) = \frac{1}{2} - \frac{r - \delta(s, q)}{\sigma^2(s, q)}- ({-}1)^i \sqrt{\bigg( \frac{1}{2} - \frac{r - \delta(s, q)}{\sigma^2(s, q)} \bigg)^2 + \frac{2 r}{\sigma^2(s, q)}},\end{equation}

so that $\gamma_2(s, q) < 0 < 1 < \gamma_1(s, q)$ holds for all $0 < q < s$ . Then, by applying the instantaneous-stopping conditions from (2.29) to the function in (3.1), we get that the equalities

(3.3) \begin{align}&C_{1}(s, q) \, a^{\gamma_1(s, q)}(s, q) + C_{2}(s, q) \, a^{\gamma_2(s, q)}(s, q) = s - K \, a(s, q),\end{align}

(3.4) \begin{align}&C_{1}(s, q) \, b^{\gamma_1(s, q)}(s, q) + C_{2}(s, q) \, b^{\gamma_2(s, q)}(s, q) = q - L \, b(s, q)\end{align}

are satisfied when $a(s, q) \ge a^{\prime}(s, q) \vee q$ and $b(s, q) \le s \wedge b^{\prime}(s, q)$ respectively hold, for each $0 < q < s$ . Hence, by using the smooth-fit conditions from (2.33), we obtain that the equalities

(3.5) \begin{align}&C_{1}(s, q) \, \gamma_1(s, q) \, a^{\gamma_1(s, q)}(s, q) + C_{2}(s, q) \,\gamma_2(s, q) \, a^{\gamma_2(s, q)}(s, q) = - K \, a(s, q),\end{align}

(3.6) \begin{align}&C_{1}(s, q) \, \gamma_1(s, q) \, b^{\gamma_1(s, q)}(s, q) + C_{2}(s, q) \,\gamma_2(s, q) \, b^{\gamma_2(s, q)}(s, q) = - L \, b(s, q)\end{align}

are satisfied when $a(s, q) > a^{\prime}(s, q) \vee q$ and $b(s, q) < s \wedge b^{\prime}(s, q)$ respectively hold, for each $0 < q < s$ . Thus, by applying the normal-reflection conditions from (2.34) to the function in (3.1), we obtain that the equalities

(3.7) \begin{align}&\sum_{i=1}^{2} \Big( \partial_s C_{i}(s, q) \, s^{\gamma_i(s, q)}+ C_{i}(s, q) \, \partial_s \gamma_i(s, q) \, s^{\gamma_i(s, q)} \ln s \Big) = 0,\end{align}

(3.8) \begin{align}&\sum_{i=1}^{2} \Big( \partial_q C_{i}(s, q) \, q^{\gamma_i(s, q)}+ C_{i}(s, q) \, \partial_q \gamma_i(s, q) \, q^{\gamma_i(s, q)} \ln q \Big) = 0\end{align}

are satisfied when $b^{\prime}(s, q) \ge b(s, q) > s$ and $a^{\prime}(s, q) \le a(s, q) < q$ respectively hold, for each $0 < q < s$ . Here, the partial derivatives $\partial_s \gamma_i(s, q)$ and $\partial_q \gamma_i(s, q)$ take the form

(3.9) \begin{align}&\partial_s \gamma_i(s, q) = \varphi(s, q) - ({-}1)^i\frac{\varphi(s, q) (\gamma_1(s, q) + \gamma_2(s, q)) \sigma^3(s, q) - 4 r \partial_s \sigma(s, q)}{\sigma^2(s, q) \sqrt{(\gamma_1(s, q) + \gamma_2(s, q))^2 \sigma^2(s, q) + 8 r}},\end{align}

(3.10) \begin{align}&\partial_q \gamma_i(s, q) = \psi(s, q) - ({-}1)^i\frac{\psi(s, q) (\gamma_1(s, q) + \gamma_2(s, q)) \sigma^3(s, q) - 4 r \partial_q \sigma(s, q)}{\sigma^2(s, q) \sqrt{(\gamma_1(s, q) + \gamma_2(s, q))^2 \sigma^2(s, q) + 8 r}}\end{align}

for $i = 1, 2$ , and the functions $\varphi(s, q)$ and $\psi(s, q)$ are defined by

(3.11) \begin{align}&\varphi(s, q) = \frac{\sigma(s, q) \partial_s \delta(s, q)+ 2 (r - \delta(s, q)) \partial_s \sigma(s, q)}{\sigma^3(s, q)},\end{align}

(3.12) \begin{align}&\psi(s, q) = \frac{\sigma(s, q) \partial_q \delta(s, q)+ 2 (r - \delta(s, q)) \partial_q \sigma(s, q)}{\sigma^3(s, q)}\end{align}

for $0 < q < s$ .

Now, by solving the system of equations in (3.3)–(3.4), we obtain that the function in (3.1) admits the representation

(3.13) \begin{align}&V(x, s, q;\, a(s, q), b(s, q))= \sum_{i = 1}^2 C_{i}(s, q;\, a(s, q), b(s, q)) \, x^{\gamma_i(s, q)}\end{align}

for $a^{\prime}(s, q) \vee q \le a(s, q) < x < b(s, q) \le s \wedge b^{\prime}(s, q)$ , where

(3.14) \begin{equation}C_{i}(s, q;\, a(s, q), b(s, q)) =\frac{(s - K a(s, q)) b^{\gamma_{3-i}(s, q)}(s, q) - (q - L b(s, q)) a^{\gamma_{3-i}(s, q)}(s, q)}{a^{\gamma_i(s, q)}(s, q) b^{\gamma_{3-i}(s, q)}(s, q) - b^{\gamma_i(s, q)}(s, q) a^{\gamma_{3-i}(s, q)}(s, q)}\end{equation}

when $a^{\prime}(s, q) \vee q \le a(s, q) < b(s, q) \le s \wedge b^{\prime}(s, q)$ holds, for every $i = 1, 2$ . Then, by solving the system of equations in (3.3) and (3.5), we obtain that the function in (3.1) admits the representation

(3.15) \begin{align}&V(x, s, q;\, a(s, q)) = C_{1}(s, q;\, a(s, q)) \, x^{\gamma_1(s, q)} + C_{2}(s, q;\, a(s, q)) \, x^{\gamma_2(s, q)}\end{align}

for $a^{\prime}(s, q) \vee q \le a(s, q) < x \le s < b(s, q) \le b^{\prime}(s, q)$ , where

(3.16) \begin{equation}C_{i}(s, q;\, a(s, q)) = \frac{\gamma_{3-i}(s, q) (s - K a(s, q)) + K a(s, q)}{(\gamma_{3-i}(s, q) - \gamma_{i}(s, q)) a^{\gamma_i(s, q)}(s, q)}\end{equation}

when $a^{\prime}(s, q) \vee q \le a(s, q) < s < b(s, q) \le b^{\prime}(s, q)$ holds, for every $i = 1, 2$ . Also, by solving the system of equations in (3.4) and (3.6), we obtain that the function in (3.1) admits the representation

(3.17) \begin{align}&V(x, s, q;\, b(s, q)) = C_{1}(s, q;\, b(s, q)) \, x^{\gamma_1(s, q)}+ C_{2}(s, q;\, b(s, q)) \, x^{\gamma_2(s, q)}\end{align}

for $a^{\prime}(s, q) \le a(s, q) < q \le x < b(s, q) \le s \wedge b^{\prime}(s, q)$ , where

(3.18) \begin{equation}C_{i}(s, q;\, b(s, q)) = \frac{\gamma_{3-i}(s, q) (q - L b(s, q)) + L b(s, q)}{(\gamma_{3-i}(s, q) - \gamma_{i}(s, q)) b^{\gamma_i(s, q)}(s, q)}\end{equation}

when $a^{\prime}(s, q) \le a(s, q) < q < b(s, q) \le s \wedge b^{\prime}(s, q)$ holds, for every $i = 1, 2$ .

Finally, by means of straightforward computations, it can be deduced from the expression in (3.13) that the first-order and second-order partial derivatives $\partial_x V(x, s, q;\, a(s, q), b(s, q))$ and $\partial_{xx} V(x, s, q;\, a(s, q), b(s, q))$ of the function $V(x, s, q;\, a(s, q), b(s, q))$ take the forms

(3.19) \begin{align}&\partial_x V(x, s, q;\, a(s, q), b(s, q)) =\sum_{i = 1}^2 C_{i}(s, q;\, a(s, q), b(s, q)) \, \gamma_i(s, q) \, x^{\gamma_i(s, q)-1}\end{align}

and

(3.20) \begin{align}&\partial_{xx} V(x, s, q;\, a(s, q), b(s, q)) =\sum_{i = 1}^2 C_{i}(s, q;\, a(s, q), b(s, q)) \, \gamma_i(s, q) (\gamma_i(s, q) - 1) \, x^{\gamma_i(s, q)-2}\end{align}

for $a^{\prime}(s, q) \vee q \le a(s, q) < x < b(s, q) \le s \wedge b^{\prime}(s, q)$ . Note that the same first-order and second-order partial derivatives of the functions $V(x, s, q;\, a(s, q))$ , for $a^{\prime}(s, q) \vee q \le a(s, q) < s < b(s, q) \le b^{\prime}(s, q)$ , and $V(x, s, q;\, b(s, q))$ , for $a^{\prime}(s, q) \le a(s, q) < q < b(s, q) \le s \wedge b^{\prime}(s, q)$ , from (3.15) with (3.16) and (3.17) with (3.18) are computed similarly.

3.2. The candidate stopping boundaries

We now apply the conditions of (3.5)–(3.6) to the functions $C_{i}(s, q;\, a(s, q), b(s, q))$ , for $i = 1, 2$ , in (3.14) to obtain the equalities

(3.21) \begin{align}&\frac{\gamma_i(s, q) (s - K a(s, q)) + K a(s, q)}{\gamma_i(s, q) (q - L b(s, q)) + L b(s, q)}= \bigg( \frac{a(s, q)}{b(s, q)} \bigg)^{\gamma_{3-i}(s, q)}\end{align}

for $a^{\prime}(s, q) \vee q \le a(s, q) < b(s, q) \le s \wedge b^{\prime}(s, q)$ and $i = 1, 2$ . Then we set $b(s, q) = b^{\prime}(s, q)$ and apply the condition of (3.5) to the same functions to obtain the equality

(3.22) \begin{align}&\sum_{i = 1}^2\frac{(s - K a(s, q)) {b^{\prime}}^{\gamma_{3-i}(s, q)}(s, q) - (q - L b^{\prime}(s, q)) a^{\gamma_{3-i}(s, q)}(s, q)}{a^{\gamma_i(s, q)}(s, q) {b^{\prime}}^{\gamma_{3-i}(s, q)}(s, q) - {b^{\prime}}^{\gamma_i(s, q)}(s, q) a^{\gamma_{3-i}(s, q)}(s, q)} \, \gamma_i(s, q) \, a^{\gamma_i(s, q)}(s, q) \\\notag&= - K \, a(s, q)\end{align}

for $a^{\prime}(s, q) \vee q \le a(s, q) < s \wedge b^{\prime}(s, q) < b(s, q)$ . Also, we set $a(s, q) = a^{\prime}(s, q)$ and apply the condition of (3.6) to the same functions to obtain the equality

(3.23) \begin{align}&\sum_{i = 1}^2\frac{(s - K a^{\prime}(s, q)) b^{\gamma_{3-i}(s, q)}(s, q) - (q - L b(s, q)) {a^{\prime}}^{\gamma_{3-i}(s, q)}(s, q)}{{a^{\prime}}^{\gamma_i(s, q)}(s, q) b^{\gamma_{3-i}(s, q)}(s, q) - b^{\gamma_i(s, q)}(s, q) {a^{\prime}}^{\gamma_{3-i}(s, q)}(s, q)} \, \gamma_i(s, q) \, b^{\gamma_i(s, q)}(s, q) \\\notag&= - L \, b(s, q)\end{align}

for $a(s, q) < a^{\prime}(s, q) \vee q < b(s, q) \le s \wedge b^{\prime}(s, q)$ .

The existence and uniqueness of solutions of the system of arithmetic equations in (3.21) as well as of the equations in (3.22) and (3.23) on the admissible intervals follow from the arguments of Subsection 3.4 below. Observe that the system of arithmetic equations in (3.21) and the equation in (3.23) satisfy the conditions of the classical (two-dimensional) implicit function theorem, so that the resulting solutions $a_*(s, q)$ and $b_*(s, q)$ turn out to be continuously differentiable. Furthermore, assuming that the candidate boundary functions a(s, q) and b(s, q) are continuously differentiable, we apply the condition of (3.7) to the functions $C_{i}(s, q;\, a(s, q))$ , for $i = 1, 2$ , in (3.16) to conclude that the candidate boundary a(s, q) satisfies the ordinary differential equation

(3.24) \begin{align}&\partial_s a(s, q) = \sum_{j = 1}^2 \frac{C_{i}(s, q;\, a(s, q)) \partial_s \gamma_i(s, q)s^{\gamma_i(s, q)} \ln s + \Psi_{1, i}(a(s, q), s, q) (s/a(s, q))^{\gamma_i(s, q)}}{\Phi_{1}(a(s, q), s, q) ((s/a(s, q))^{\gamma_1(s, q)} - (s/a(s, q))^{\gamma_2(s, q)})}\end{align}

for $a^{\prime}(s, q) \vee q \le a(s, q) < s < b(s, q) \le b^{\prime}(s, q)$ , where we set

(3.25) \begin{align}&\Phi_{1}(x, s, q) = \frac{(\gamma_1(s, q) + \gamma_2(s, q)) K+ \gamma_1(s, q) \gamma_2(s, q) (s - K x)/x}{\gamma_2(s, q) - \gamma_1(s, q)}\end{align}

and

(3.26) \begin{align}&\Psi_{1, i}(x, s, q) = \frac{\partial_s \gamma_{3-i}(s, q) (s - K x) + \gamma_{3-i}(s, q)}{\gamma_{3-i}(s, q) - \gamma_{i}(s, q)} \\\notag&- \frac{(K x + \gamma_{3-i}(s, q) (s - K x)) (\partial_s \gamma_i(s, q) \, \ln x+ \partial_s \ln (\gamma_{3-i}(s, q) - \gamma_i(s, q)))}{\gamma_{3-i}(s, q) - \gamma_{i}(s, q)}\end{align}

for all $0 < q \le x \le s$ and every $i = 1, 2$ . We also apply the condition of (3.8) to the functions $C_{i}(s, q;\, b(s, q))$ , for $i = 1, 2$ , in (3.18) to conclude that the candidate boundary b(s, q) satisfies the ordinary differential equation

(3.27) \begin{align}&\partial_q b(s, q) =\sum_{i = 1}^2 \frac{C_{i}(s, q;\, b(s, q)) \partial_q \gamma_j(s, q) q^{\gamma_i(s, q)} \ln q +\Psi_{2, i}(b(s, q), s, q) (q/b(s, q))^{\gamma_i(s, q)}}{\Phi_{2}(b(s, q), s, q) ((q/b(s, q))^{\gamma_1(s, q)} - (q/b(s, q))^{\gamma_2(s, q)})}\end{align}

for $a^{\prime}(s, q) \le a(s, q) < q < b(s, q) \le s \wedge b^{\prime}(s, q)$ , where we set

(3.28) \begin{align}&\Phi_{2}(x, s, q) = \frac{(\gamma_1(s, q) + \gamma_2(s, q)) L+ \gamma_1(s, q) \gamma_2(s, q) (q - L x)/x}{\gamma_2(s, q) - \gamma_1(s, q)}\end{align}

and

(3.29) \begin{align}&\Psi_{2, i}(x, s, q) =\frac{\partial_q \gamma_{3-i}(s, q) (q - L x) + \gamma_{3-i}(s, q)}{\gamma_{3-i}(s, q) - \gamma_{i}(s, q)} \\\notag&- \frac{(L x + \gamma_{3-i}(s, q) (q - L x))(\partial_q \gamma_i(s, q) \, \ln x+ \partial_q \ln (\gamma_{3-i}(s, q) - \gamma_i(s, q)))}{\gamma_{3-i}(s, q) - \gamma_{i}(s, q)}\end{align}

for all $0 < q \le x \le s$ and every $i = 1, 2$ . Note that, by virtue of the assumptions on the coefficients $\delta(s, q) > 0$ and $\sigma(s, q) > 0$ of the diffusion-type process X from (1.1)–(1.2) and (1.3), the right-hand sides of the expressions in (3.24) with (3.25)–(3.26) and in (3.27) with (3.28)–(3.29) are (locally) continuous in (s, q, a(s, q)) and (s, q, b(s, q)) and (locally) Lipschitz in a(s, q) and b(s, q), for each $0 < q < s$ fixed. Thus, by the classical results on the existence and uniqueness of solutions for first-order nonlinear ordinary differential equations, the equations in (3.24) and (3.27) admit (locally) unique solutions, which can be constructed by means of Picard’s method of successive approximations (see Subsection 5.3 below for further constructions and references).

3.3. The structure of the continuation region

In order to specify the optimal exercise boundaries for the floating-strike lookback game options, let us consider the functions $a_*(s, q)$ and $b_*(s, q)$ , which provide a unique solution to the system of arithmetic equations in (3.21) such that $a^{\prime}(s, q) \le a_*(s, q) < b_*(s, q) \le b^{\prime}(s, q)$ , or the function $a_*(s, q) < b_*(s, q) = b^{\prime}(s, q)$ represents the largest root of the equation in (3.22), or the function $b_*(s, q) > a_*(s, q) = a^{\prime}(s, q)$ represents the largest root of the equation in (3.23), or otherwise the properties $a_*(s, q) = a^{\prime}(s, q)$ and $b_*(s, q) = b^{\prime}(s, q)$ hold, for each $0 < q < s$ fixed.

On the one hand, we can set $s^*_{0}(q) = \infty$ and define the functions $s^*_{2k - 1}(q) = \sup \{ s < s^*_{2k - 2}(q) \, | \, b_*(s, q) > s \}$ and $s^*_{2k}(q) = \sup \{ s < s^*_{2k - 1}(q) \, | \, b_*(s, q) < s \}$ , whenever they exist, and put $s^*_{2k - 1}(q) = s^*_{2k}(q) = 0$ otherwise, so that the inequalities $0 \le s^*_{2k - 1}(q) \le s^*_{2k - 2}(q) \le \infty$ hold, for all $k \in {\mathbb N}$ , and each $q > 0$ fixed. In other words, the boundary $b_*(s, q)$ exits the region E from the side of the diagonal $d_1 = \{ (x, s, q) \in {\mathbb R}^3 \, | \, 0 < q \le x = s \}$ passing through the points $(s^*_{2k - 1}(q), s^*_{2k - 1}(q), q)$ and comes back to E from the side of $d_1$ passing through the points $(s^*_{2k}(q), s^*_{2k}(q), q)$ , for $k \in {\mathbb N}$ , for each $q > 0$ fixed. Hence, the candidate value function $V(x, s, q;\, a_*(s, q), b_*(s, q))$ admits the representation of (3.13) with (3.14) and the candidate stopping boundaries $a_*(s, q)$ and $b_*(s, q)$ solve the system of arithmetic equations in (3.21) in the regions

(3.30) \begin{align}&{\widetilde R}_{2k - 1}(a_*, b_*) = \big\{ (x, s, q) \in E^{\prime} \, \big| \, s^*_{2k - 1}(q) \le s < s^*_{2k - 2}(q) \big\},\end{align}

while the candidate value function $V(x, s, q;\, a_*(s, q))$ admits the representation of (3.15) with (3.16) and the candidate stopping boundary $a_*(s, q)$ either solves the first-order nonlinear ordinary differential equation in the regions

(3.31) \begin{align}&{\widetilde R}_{2k}(a_*) = \big\{ (x, s, q) \in E^{\prime} \, \big| \, s^*_{2k}(q) \le s < s^*_{2k - 1}(q) \big\}\end{align}

both representing subsets of the continuation region $C_*$ in (2.24), or coincides with a ^′ (s, q), for each $q > 0$ fixed, and every $k \in {\mathbb N}$ . Furthermore, we observe that the process (X, S, Q) can enter the region ${\widetilde R}_{2k - 1}(a_*, b_*)$ in (3.31) from the region ${\widetilde R}_{2k}(a_*)$ in (3.30) only through the point $(s^*_{2k - 1}(q), s^*_{2k - 1}(q), q)$ , for any $k \in {\mathbb N}$ , by hitting the plane $d_1 = \{ (x, s, q) \in {\mathbb R}^3 \, | \, 0 < q \le x = s \}$ , that is, by increasing its second component S. Therefore, the candidate value function should satisfy the instantaneous-stopping and smooth-fit conditions at the points $(s^*_{2k - 1}(q), s^*_{2k - 1}(q), q)$ , which are expressed by the equalities

(3.32) \begin{align} & \sum_{i = 1}^2 C_{i} \big( s^*_{2k - 1}(q)-, q;\, a_*(s^*_{2k - 1}(q)-, q) \big) \, (s^*_{2k - 1}(q))^{\gamma_i(s^*_{2k - 1}(q), q)}\\&= V \big( s^*_{2k - 1}(q), s^*_{2k - 1}(q), q;\, a_*(s^*_{2k - 1}(q), q), b_*(s^*_{2k - 1}(q), q) \big), \notag \qquad\qquad\qquad\qquad\end{align}

(3.33) \begin{align} & \sum_{i = 1}^2 C_{i} \big( s^*_{2k - 1}(q)-, q;\, a_*(s^*_{2k - 1}(q)-, q) \big)\, \gamma_i(s^*_{2k - 1}(q)-, q) \, (s^*_{2k - 1}(q))^{\gamma_i(s^*_{2k - 1}(q), q)} \\&= s^*_{2k - 1}(q) \, \partial_x V \big( s^*_{2k - 1}(q), s^*_{2k - 1}(q), q;\, a_*(s^*_{2k - 1}(q), q), b_*(s^*_{2k - 1}(q), q) \big)\notag\end{align}

where the functions $C_{i}(s, q;\, a_*(s, q))$ , for $i = 1, 2$ , are given by (3.16) and the function $V(x, s, q;\, a_*(s, q), b_*(s, q))$ has the form of (3.13) with (3.14), while the boundary $a_*(s, q)$ provides a unique solution of the first-order nonlinear ordinary differential equation in (3.24) in the region ${\widetilde R}_{2k}(a_*)$ from (3.31) satisfying the (starting) conditions of (3.32)–(3.33) above, for each $q > 0$ fixed, and every $k \in {\mathbb N}$ , respectively.

On the other hand, we can set $q^*_{0}(s) = 0$ , define the functions $q^*_{2l - 1}(s) = \inf \{ q > q^*_{2l - 2}(s) \, | \, a_*(s, q) < q \}$ and $q^*_{2l}(s) = \inf \{ q > q^*_{2l - 1}(s) \, | \, a_*(s, q) > q \}$ , whenever they exist, and put $q^*_{2l - 1}(s) = q^*_{2l}(s) = \infty$ otherwise, so that the inequalities $0 \le q^*_{2l - 2}(s) \le q^*_{2l - 1}(s) \le \infty$ hold for all $l \in {\mathbb N}$ and each $s > 0$ fixed. In other words, the boundary $a_*(s, q)$ exits the region E from the side of the diagonal $d_2 = \{ (x, s, q) \in {\mathbb R}^3 \, | \, 0 < q = x \le s \}$ passing through the points $(q^*_{2l - 1}(s), s, q^*_{2l - 1}(q))$ and comes back to E from the side of the diagonal $d_2$ passing through the points $(q^*_{2l-1}(s), s, q^*_{2l-1}(s))$ , for $k \in {\mathbb N}$ , for each $s > 0$ fixed. Hence, the candidate value function $V(x, s, q;\, a_*(s, q), b_*(s, q))$ admits the representation of (3.13) with (3.14), and the candidate stopping boundaries $a_*(s, q)$ and $b_*(s, q)$ solve the system of arithmetic equations in (3.21) in the regions

(3.34) \begin{align}&{\widehat R}_{2l - 1}(a_*, b_*) = \big\{ (x, s, q) \in E^{\prime} \, \big| \, q^*_{2l - 2}(s) < q \le q^*_{2l - 1}(s) \big\},\end{align}

while the candidate value function $V(x, s, q;\, b_*(s, q))$ admits the representation of (3.17) with (3.18), and the candidate stopping boundary $b_*(s, q)$ solves the first-order nonlinear ordinary differential equation in the regions

(3.35) \begin{align}&{\widehat R}_{2l}(b_*) = \big\{ (x, s, q) \in E^{\prime} \, \big| \, q^*_{2l - 1}(s) < q \le q^*_{2l}(s) \big\},\end{align}

both representing subsets of the continuation region $C_*$ in (2.24), for each $s > 0$ fixed and every $l \in {\mathbb N}$ . Furthermore, we observe that the process (X, S, Q) can enter the region ${\widehat R}_{2l - 1}(a_*, b_*)$ in (3.35) from the region ${\widehat R}_{2l}(b_*)$ in (3.34) only through the point $(q^*_{2l - 1}(s), s, q^*_{2l - 1}(s))$ , for any $l \in {\mathbb N}$ , by hitting the plane $d_2 = \{ (x, s, q) \in {\mathbb R}^3 \, | \, 0 < q = x \le s \}$ , that is, by decreasing its third component Q. Therefore, the candidate value function should satisfy the instantaneous-stopping and smooth-fit conditions at the points $(q^*_{2l - 1}(s), s, q^*_{2l - 1}(s))$ , which are expressed by the equalities

(3.36) \begin{align}&\sum_{i = 1}^2 C_{i} \big( s, q^*_{2l - 1}(s)+;\, b_*(s, q^*_{2l - 1}(s)+) \big)\, (q^*_{2l - 1}(s))^{\gamma_j(s, q^*_{2l - 1}(s))} \\&= V \big( q^*_{2l - 1}(s), s, q^*_{2l - 1}(s);\, a_*(s, q^*_{2l - 1}(s)), b_*(s, q^*_{2l - 1}(s)) \big),\notag\qquad\qquad\qquad\qquad\end{align}

(3.37) \begin{align}&\sum_{i = 1}^2 C_{i} \big( s, q^*_{2l - 1}(s)+;\, b_*(s, q^*_{2l - 1}(s)+) \big)\, \gamma_i(s, q^*_{2l - 1}(s)) \, (q^*_{2l - 1}(s))^{\gamma_i(s, q^*_{2l - 1}(s))} \\\notag&= q^*_{2l - 1}(s) \, \partial_x V \big( q^*_{2l - 1}(s), s, q^*_{2l - 1}(s);\, a_*(s, q^*_{2l - 1}(s)), b_*(s, q^*_{2l - 1}(s)) \big),\end{align}

where the functions $C_{i}(s, q;\, b_*(s, q))$ , for $i = 1, 2$ , are given by (3.18) and the function $V(x, s, q;\, a_*(s, q), b_*(s, q))$ has the form of (3.13) with (3.14), while the boundary $b_*(s, q)$ provides a unique solution of the first-order nonlinear ordinary differential equation in (3.27) in the region ${\widehat R}_{2l}(b_*)$ from (3.35) satisfying the (starting) conditions of (3.36)–(3.37) above, for each $s > 0$ fixed and every $k \in {\mathbb N}$ . Note that the process (X, S, Q) cannot come from the region ${\widetilde R}_{2k}(a_*)$ in (3.31) directly to the region ${\widehat R}_{2l}(b_*)$ in (3.35), or vice versa, without crossing the region ${\widetilde R}_{2k}(a_*, b_*)$ in (3.30) or ${\widehat R}_{2l}(a_*, b_*)$ in (3.34), respectively, for every $k, l \in {\mathbb N}$ .

Finally we observe that if we have $\gamma_i(s, q) = \gamma_i(q)$ , for $i = 1, 2$ , in (3.2), then the appropriate left-hand exercise boundary for the floating-strike lookback game option in (2.24)–(2.25) takes the form $a_*(s, q) = a^{\prime}(s, q) \vee g_*(q) s$ with some function $0 < g_*(q) < 1$ , while, if we have $\gamma_i(s, q) = \gamma_i(s)$ , for $i = 1, 2$ , in (3.2), then the appropriate right-hand exercise boundary for that contract there takes the form $b_*(s, q) = b^{\prime}(s, q) \wedge h_*(s) q$ with some function $h_*(s) > 1$ , for all $0 < q < s$ . In these cases, we have solely the regions ${\widetilde R}_{1}(a_*, b_*)$ , ${\widetilde R}_{2}(a_*)$ in (3.30)–(3.31) with $k = 1$ and ${\widehat R}_{1}(a_*, b_*)$ , ${\widehat R}_{2}(b_*)$ in (3.34)–(3.35) with $l = 1$ , respectively. Moreover, if we have $\gamma_i(s, q) = \gamma_i$ , for $i = 1, 2$ , in (3.2), then we have $a_*(s, q) = a^{\prime}(s, q) \vee a_*(s) = a^{\prime}(s, q) \vee g_* s$ and $b_*(s, q) = b^{\prime}(s, q) \wedge b_*(q) = b^{\prime}(s, q) \wedge h_* q$ with some constants $0 < g_* < 1$ and $h_* > 1$ , for all $0 < q < s$ . The latter property can be explained by the fact that the original problem of (1.4), and thus the three-dimensional problem of (2.3), can be reduced to an optimal stopping zero-sum game for the process $(S/X, Q/X) = (S_t/X_t, Q_t/X_t)_{t \ge 0}$ representing a two-dimensional Markov diffusion process with reflection, by means of the change-of-measure arguments from [Reference Shepp and Shiryaev65] (see also [Reference Gapeev21]).

3.4. The system of arithmetic equations for the boundaries

Let us now extend the arguments from [Reference Gapeev and Lerche26, Example 4.2] (see also [Reference Gapeev18, Section 3] and [Reference Qiu60, Theorem 1]) to show that the system of arithmetic power equations in (3.21) admits a unique solution. For this purpose, by virtue of straightforward calculations, we first observe that the system of equations in (3.21) is equivalent to the following one:

(3.38) \begin{equation}\Xi_i(a(s, q);\, s, q) = \Upsilon_i(b(s, q);\, s, q)\end{equation}

for $a^{\prime}(s, q) \vee q \le a(s, q) < b(s, q) \le s \wedge b^{\prime}(s, q)$ , where we set

(3.39) \begin{equation}\Xi_i(a;\, s, q) = \frac{\gamma_i(s, q) (s - K a) + K a}{a^{\gamma_{3-i}(s, q)}}\quad \text{and} \quad \Upsilon_i(b;\, s, q) = \frac{\gamma_i(s, q) (q - L b) + L b}{b^{\gamma_{3-i}(s, q)}}\end{equation}

for all $0 < q < a < b < s$ and every $i = 1, 2$ .

In order to show the existence and uniqueness of a solution of the system of arithmetic power equations in (3.38) with (3.39), we develop the ideas used to prove the existence and uniqueness of solutions to the systems of arithmetic power equations in [Reference Gapeev and Lerche26, Example 4.2] (see also the systems (4.73)–(4.74) in [Reference Shiryaev67, Chapter IV, Section 2], the system (3.16)–(3.17) in [Reference Gapeev18, Section 3], and [Reference Qiu60, Theorem 1]). For this purpose, we observe that, for the derivatives of the functions $\Xi_i(a)$ and $\Upsilon_i(b)$ , for $i = 1, 2$ , defined in (3.38), the equations

(3.40) \begin{align} \Xi^{\prime}_i (a;\, s, q) &= \frac{(\gamma_1(s, q) - 1) (\gamma_2(s, q) - 1) K (a - {\overline a}(s, q))}{a^{\gamma_{3-i}(s, q) + 1}} \end{align}

and

(3.41) \begin{align} \Upsilon^{\prime}_i (b;\, s, q) &= \frac{(\gamma_1(s, q) - 1) (\gamma_2(s, q) - 1) L (b - {\underline b}(s, q))}{b^{\gamma_{3-i}(s, q) + 1}} \end{align}

hold, so that the inequalities

(3.42) \begin{align} &\Xi^{\prime}_i (a;\, s, q) > 0 \quad \text{for} \quad a < {\overline a}(s, q) \quad \text{and} \quad \Upsilon^{\prime}_i (b;\, s, q) < 0 \quad \text{for} \quad b > {\underline b}(s, q) \end{align}

are satisfied, for $a^{\prime}(s, q) \vee q < a < b < s \wedge b^{\prime}(s, q)$ and every $i = 1, 2$ , where the functions a ^′(s, q) and b ^′(s, q) are given by (2.9). Hence, we may conclude that the following properties hold, for each $0 < q < s$ fixed. The function $\Xi_1(a;\, s, q)$ is increasing on the interval $(0, {\overline a}(s, q))$ , with $\Xi_1(0+;\, s, q) = 0$ and $\Xi_1({\overline a}(s, q);\, s, q) > 0$ , so that the range of its values is given by the interval $(0, \Xi_1({\overline a}(s, q);\, s, q))$ . The function $\Upsilon_1(b;\, s, q)$ is decreasing on the interval $({\underline b}(s, q), \infty)$ , with $\Upsilon_1({\underline b}(s, q);\, s, q) > 0$ and $\Upsilon_1(\infty;\, s, q) = - \infty$ , so that the range of its values is given by the interval $({-} \infty, \Upsilon_1({\underline b}(s, q);\, s, q))$ . The function $\Xi_2(a;\, s, q)$ is increasing on the interval $(0, {\overline a}(s, q))$ with $\Xi_2(0+;\, s, q) = - \infty$ and $\Xi_2({\overline a}(s, q);\, s, q) > 0$ , so that the range of its values is given by the interval $({-} \infty, \Xi_2({\overline a}(s, q);\, s, q))$ . The function $\Upsilon_2(b;\, s, q)$ is decreasing on the interval $({\underline b}(s, q), \infty)$ with $\Upsilon_2({\underline b}(s, q);\, s, q) > 0$ and $\Upsilon_2(\infty;\, s, q) = 0$ , so that the range of its values is given by the interval $(0, \Upsilon_2({\underline b}(s, q);\, s, q))$ .

We now observe that, when $\Xi_i({\overline a}(s, q);\, s, q) \le \Upsilon_i({\underline b}(s, q);\, s, q)$ holds, one can determine some ${\widehat b}_i(s, q) \ge {\underline b}(s, q)$ from the equation $\Xi_i({\overline a}(s, q);\, s, q) = \Upsilon_i({\widehat b}_i(s, q);\, s, q)$ , while, when $\Xi_i({\overline a}(s, q);\, s, q) \ge \Upsilon_i({\underline b}(s, q);\, s, q)$ holds, one can determine some ${\widehat a}_i(s, q) \le {\overline a}(s, q)$ from the equation $\Xi_i({\widehat a}_i(s, q);\, s, q) = \Upsilon_i({\underline b}(s, q);\, s, q)$ , for each $0 < q < s$ fixed, and every $i = 1, 2$ . Hence, it follows from the equations in (3.38) with (3.39) that, for each $a \in ({\underline a}(s, q), {\widehat a}_1(s, q) \wedge {\overline a}(s, q))$ , there exists a unique number $b \in ({\underline b}(s, q) \vee {\widehat b}_1(s, q), {\overline b}(s, q))$ , while, for each $a \in ({\underline a}(s, q), {\widehat a}_2(s, q) \wedge {\overline a}(s, q))$ , there exists a unique number $b \in ({\underline b}(s, q) \vee {\widehat b}_2(s, q), {\overline b}(s, q))$ , for each $0 < q < s$ fixed. (Recall that the values ${\overline a}(s, q)$ and ${\underline b}(s, q)$ are specified in Part (ii) of Subsection 2.2 above and given by the expressions in (2.19), while the values ${\underline a}(s, q)$ and ${\overline b}(s, q)$ are determined in Theorem 2 below, for each $0 < q < s$ fixed.)

In other words, we may conclude that the first and second equations in (3.38), respectively, uniquely determine the function $b_1(a)$ on $({\underline a}(s, q), {\widehat a}_1(s, q) \wedge {\overline a}(s, q))$ with the range $({\underline b}(s, q) \vee {\widehat b}_1(s, q), {\overline b}(s, q))$ and the function $b_2(a)$ on $({\underline a}(s, q), {\widehat a}_2(s, q) \wedge {\overline a}(s, q))$ with the range $({\underline b}(s, q) \vee {\widehat b}_2(s, q), {\overline b}(s, q))$ , for each $0 < q < s$ fixed. These arguments also imply that the expression ${\underline b}(s, q) \vee {\widehat b}_1(s, q)< b_1({\underline a}(s, q)) < {\overline b}(s, q) < b_2({\underline a}(s, q))$ holds. Moreover, the same arguments and assumptions directly yield that there exists exactly one intersection point, with the coordinates $a_*(s, q)$ and $b_*(s, q)$ , of the curves associated with the functions $b_1(a)$ and $b_2(a)$ on the interval $a \in ({\underline a}(s, q), {\widehat a}_2(s, q) \wedge {\overline a}(s, q))$ such that ${\underline b}(s, q) \vee {\widehat b}_1(s, q) < b_1(a_*(s, q)) \equiv b_*(s, q) \equiv b_2(a_*(s, q)) < {\overline b}(s, q)$ holds, for each $0 < q < s$ fixed (see Figure 3 above).

Figure 3. A computer drawing of the boundary functions $b_1(a)$ and $b_2(a)$ in the case $a^{\prime}(s, q) \le a_*(s, q) < b_*(s, q) \le b^{\prime}(s, q)$ , for each $0 < q < s$ fixed.

More precisely, let us assume that there exist at least two intersection points, $(a_*(s, q), b_*(s, q))$ and $({\widetilde a}(s, q), {\widetilde b}(s, q))$ , of the curves $b_1(a)$ and $b_2(a)$ , such that ${\underline a}(s, q) < {\widetilde a}(s, q) < a_*(s, q) \le {\overline a}(s, q) \wedge {\widehat a}_2(s, q)$ and ${\underline b}(s, q) \vee {\widehat b}_1(s, q) \le {\widetilde b}(s, q) < b_*(s, q) < {\overline b}(s, q)$ (or ${\underline a}(s, q) < a_*(s, q) < {\widetilde a}(s, q) \le {\overline a}(s, q) \wedge {\widehat a}_2(s, q)$ and ${\underline b}(s, q) \vee {\widehat b}_1(s, q) \le b_*(s, q) < {\widetilde b}(s, q) < {\overline b}(s, q)$ ), as well as $b_2(a) > b_1(a)$ , for $a \in ({\widetilde a}(s, q), a_*(s, q))$ and any $0 < q < s$ fixed. Observe that, by virtue of the assumptions made above and according to the implicit function theorem, it follows from the representations in (3.40)–(3.41) that the expressions

(3.43) \begin{align} &b^{\prime}_i (a) = \frac{\Xi^{\prime}_i (a)}{\Upsilon^{\prime}_i (b)} = \frac{K (a - {\overline a}(s, q))} {L (b - {\underline b}(s, q))} \, \bigg( \frac{b}{a} \bigg)^{\gamma_{3-i}(s, q) + 1} < 0 \end{align}

hold, for every $i = 1, 2$ , for all $a \in ({\widetilde a}(s, q), a_*(s, q))$ and $b \in ({\widetilde b}(s, q),b_*(s, q))$ , from which it directly follows that the inequality

(3.44) \begin{align} &\frac{b^{\prime}_2 (a)}{b^{\prime}_1 (a)} = \bigg( \frac{b}{a} \bigg)^{\gamma_1(s, q) - \gamma_2(s, q)} > 1 \end{align}

is satisfied for all $a \in ({\widetilde a}(s, q), a_*(s, q))$ . Since the derivatives $b^{\prime}_i (a)$ , for $i = 1, 2$ , from (3.43) are continuous functions on $({\widetilde a}(s, q), a_*(s, q))$ , we may conclude that there exists an open interval $({\widetilde a}(s, q) - {\varepsilon}, {\widetilde a}(s, q) + {\varepsilon})$ , for some relatively small ${\varepsilon} > 0$ , such that the inequality $b^{\prime}_2 (a) > b^{\prime}_1 (a)$ holds, so that the inequality $b_2(a) > b_1(a)$ should hold for $a \in ({\widetilde a}(s, q) - {\varepsilon}, {\widetilde a}(s, q) + {\varepsilon})$ too. However, the latter fact contradicts the assumption that the equality $b_1({\widetilde a}(s, q)) = b_2({\widetilde a}(s, q))$ holds, which means that the curves $b_1(a)$ and $b_2(a)$ may have only one intersection point; this completes the proof of the claim.

Furthermore, we recall that the functions $a_*(s, q)$ and $b_*(s, q)$ determined by means of the arguments above provide the candidate optimal stopping boundaries whenever the inequalities $a^{\prime}(s, q) \le a_*(s, q) < b_*(s, q) \le b^{\prime}(s, q)$ hold for the solution of the system in (3.38) with (3.39), with a ^′ (s, q) and b ^′ (s, q) from (2.9), for each $0 < q < s$ fixed. Otherwise, on the one hand, it can be shown by means of arguments similar to the ones used above that, if the inequalities $a^{\prime}(s, q) \le a_*(s, q) < b^{\prime}(s, q) < b_*(s, q)$ hold for the solution of the system in (3.38) with (3.39), then the right-hand candidate stopping boundary $b_*(s, q)$ should coincide with b ^′ (s, q), while the the left-hand candidate stopping boundary $a_*(s, q) < b^{\prime}(s, q)$ represents the smallest root (or the minimal solution) of the arithmetic equation in (3.22), which takes the form

(3.45) \begin{align}&\sum_{i = 1}^2 ({-}1)^i \, \bigg( \frac{\gamma_{3-i}(s, q) s}{b^{\prime}(s, q)} +(1 - \gamma_{3-i}(s, q)) \, K \, \frac{a(s, q)}{b^{\prime}(s, q)} \bigg) \,\bigg( \frac{a(s, q)}{b^{\prime}(s, q)} \bigg)^{- \gamma_i(s, q)} \\\notag&= (\gamma_1(s, q) - \gamma_2(s, q)) \, ({q}/{b^{\prime}(s, q)} - L)\end{align}

for $0 < q < s$ . On the other hand, it can be shown by means of arguments similar to the ones used above that, if the inequalities $a_*(s, q) < a^{\prime}(s, q) < b_*(s, q) \le b^{\prime}(s, q)$ hold for the solution of the system in (3.38) with (3.39), then the left-hand candidate stopping boundary $a_*(s, q)$ should coincide with a ^′ (s, q), while the the right-hand candidate stopping boundary $b_*(s, q) > a^{\prime}(s, q)$ represents the largest root (or the maximal solution) of the arithmetic equation in (3.23), which takes the form

(3.46) \begin{align}&\sum_{i = 1}^2 ({-}1)^i \, \bigg( \frac{\gamma_{3-i}(s, q) q}{a^{\prime}(s, q)} +(1 - \gamma_{3-i}(s, q)) \, L \, \frac{b(s, q)}{a^{\prime}(s, q)} \bigg) \,\bigg( \frac{b(s, q)}{a^{\prime}(s, q)} \bigg)^{- \gamma_i(s, q)} \\\notag&= (\gamma_1(s, q) - \gamma_2(s, q)) \, ({s}/{a^{\prime}(s, q)} - K)\end{align}

for $0 < q < s$ . We finally note that, in the case in which neither the system of arithmetic equations in (3.38) with (3.39) nor the equations in (3.45) and (3.46) admit solutions for which either the inequalities $a_*(s, q) \ge a^{\prime}(s, q)$ or $b_*(s, q) \le b^{\prime}(s, q)$ hold, respectively, we may conclude that both the candidate stopping boundaries $a_*(s, q)$ and $b_*(s, q)$ should coincide with the boundaries a ^′ (s, q) and b ^′ (s, q), for $0 < q < s$ , respectively.

Figure 4. A computer drawing of the value function $V_*(x, s, q)$ and optimal exercise boundaries $a^{\prime}(s, q) < a_*(s, q) < b_*(s, q) < b^{\prime}(s, q)$ , for each $0 < q < s$ fixed.

5.1. The optimal stopping and free-boundary problems

In order to provide the upper and lower bounds for the value functions and optimal stopping boundaries in the optimal stopping game of (2.3) above, let us introduce the value functions ${\overline V}(x, s, q)$ and ${\underline V}(x, s, q)$ of the optimal stopping problems

(5.1) \begin{equation}{\overline V}(x, s, q) = \sup_{\tau} {\mathbb E}_{x, s, q} \big[ e^{- r \tau} \, (S_{\tau} - K \, X_{\tau}) \big]\quad \text{and} \quad {\underline V}(x, s, q) = \inf_{\zeta} {\mathbb E}_{x, s, q} \big[ e^{- r \zeta} \, (Q_{\zeta} - L \, X_{\zeta}) \big]\end{equation}

for some given constants $K, L > 0$ , where the supremum and infimum are taken with respect to all stopping times $\tau$ and $\zeta$ of the process X. It can be shown by means of arguments similar to the ones used in Part (ii) of Subsection 2.2, based on the representations (2.14) and (2.15) for the reward functional, under the assumptions $\zeta = \infty$ and $\tau = \infty$ , respectively, or by using the (easily proved) convexity and concavity of the value functions ${\overline V}(x, s, q)$ and ${\underline V}(x, s, q)$ , that the optimal stopping times in the problems of (5.1) have the form

(5.2) \begin{equation}{\overline \tau} = \inf \big\{ t \ge 0 \; \big| \; X_t \le {\underline a}(S_t, Q_t) \big\}\quad \text{and} \quad {\underline \zeta} = \inf \big\{ t \ge 0 \; \big| \; X_t \ge {\overline b}(S_t, Q_t) \big\}\end{equation}

with some functions ${\underline a}(s, q)$ and ${\overline b}(s, q)$ , for $0 < q < s$ , to be determined (see [Reference Guo and Shepp36, Reference Pedersen51] as well as [Reference Gapeev, Kort and Lavrutich23, Section 3] and [Reference Gapeev21]). By means of arguments similar to the ones applied in [Reference Dubins, Shepp and Shiryaev13, Subsection 3.2] and [Reference Peskir52, Proposition 2.1], the existence of such boundaries ${\underline a}(s, q)$ and ${\overline b}(s, q)$ can be explained by the facts that the costs for the holder (maximiser of the expected discounted payoff) of waiting until the process X from (1.1) coming from a small $x > 0$ increases to the current value of the running maximum process S and the costs for the writer (minimiser of the expected discounted payoff) of waiting until the process X coming from a large $x > 0$ decreases to the current value of the running minimum process Q may be too large, owing to the structure of the integrands in the reward functionals of (2.16) and (2.17), respectively.

Extending the arguments from [Reference Guo and Shepp36, Reference Pedersen51] (see also [Reference Gapeev, Kort and Lavrutich23, Section 3] and [Reference Gapeev21]) to the three-dimensional model under consideration here, we may conclude that the unknown value functions ${\overline V}(x, s, q)$ and ${\underline V}(x, s, q)$ from (5.1) and the unknown boundaries ${\underline a}(s, q)$ and ${\overline b}(s, q)$ from (5.2) should solve the equivalent free-boundary problems

(5.3) \begin{align} ({{\mathbb L}} V - r V)(x, s, q) = 0 \quad \text{for} \quad &q \vee a(s, q) < x < s \quad \text{or} \end{align}

(5.4) \begin{align} ({{\mathbb L}} V - r V)(x, s, q) = 0 \quad \text{for} \quad &q < x < b(s, q) \wedge s, \end{align}

(5.5) \begin{align} V(x, s, q) \big|_{x = a(s, q)+} = s - K \, a(s, q), \quad &V(x, s, q) \big|_{x = b(s, q)-} = q - L \, b(s, q), \end{align}

(5.6) \begin{align} \partial_x V(x, s, q) \big|_{x = a(s, q)+} = - K, \quad &\partial_x V(x, s, q) \big|_{x = b(s, q)-} = - L, \end{align}

(5.7) \begin{align} \partial_s V(x, s, q) \big|_{x = s-} = 0, \quad &\partial_q V(x, s, q) \big|_{x = q+} = 0, \end{align}

(5.8) \begin{align} V(x, s, q) = s - K \, x \quad \text{for} \quad x < a(s, q), \quad &V(x, s, q) = q - L \, x \quad \text{for} \quad x > b(s, q), \end{align}

(5.9) \begin{align} V(x, s, q) > s - K \, x \quad \text{for} \quad a(s, q) < x \le s, \quad &V(x, s, q) < q - L \, x \quad \text{for} \quad q \le x < b(s, q), \end{align}

(5.10) \begin{align} ({{\mathbb L}} V - r V)(x, s, q) < 0 \quad \text{for} \quad x < a(s, q), \quad &({{\mathbb L}} V - r V)(x, s, q) > 0 \quad \text{for} \quad x > b(s, q), \end{align}

where the conditions in (5.5)–(5.7) are satisfied, for each $0 < q < s$ .

5.2. Solutions to the free-boundary problems

It follows from the arguments of [Reference Guo and Shepp36, Reference Pedersen51] (see also [Reference Gapeev, Kort and Lavrutich23, Section 3] and [Reference Gapeev21]) that the solution to the left-hand system in (5.3) and (5.5)–(5.10) has the form of (3.15) with (3.16), for $0 < q \le a(s, q) < x \le s$ , while the boundary a(s, q) solves the first-order nonlinear ordinary differential equation in (3.24), for any $q > 0$ fixed. We also see that the solution to the right-hand system in (5.4) and (5.5)–(5.10) has the form of (3.17) with (3.18), for $0 < q \le x < b(s, q) \le s$ , while the boundary b(s, q) solves the first-order nonlinear ordinary differential equation in (3.27), for any $s > 0$ fixed.

We further define the maximal and minimal admissible solutions of the first-order nonlinear ordinary differential equations as the largest and smallest possible solutions ${\underline a}(s, q)$ and ${\overline b}(s, q)$ of the equations in (3.24) and (3.27) which satisfy the inequalities ${\underline a}(s, q) < s$ and ${\overline b}(s, q) > q$ for all $0 < q < s$ . By virtue of the classical results on the existence and uniqueness of solutions for first-order nonlinear ordinary differential equations, we may conclude that these equations admit (locally) unique solutions, in view of the fact that the right-hand sides in (3.24) and (3.27) are (locally) continuous in (s, q, a(s, q)) and (s, q, b(s, q)) and (locally) Lipschitz in a(s, q) and b(s, q), for each (s, q) fixed (see [Reference Peskir52, Subsection 3.9] for similar arguments based on the analysis of other first-order nonlinear ordinary differential equations). It can then be shown by means of technical arguments based on Picard’s method of successive approximations that there exist unique solutions a(s, q) and b(s, q) to the equations in (3.24) and (3.27), started at some points $(s_0, s_0, q)$ and $(s, q_0, q_0)$ such that $s_0 > 0$ and $q_0 > 0$ , for each $0 < q < s$ fixed (see also [Reference Graversen and Peskir34, Subsection 3.2] and [Reference Peskir52, Example 4.4] for similar arguments based on the analysis of other first-order nonlinear ordinary differential equations).

Hence, in order to construct the appropriate functions ${\underline a}(s, q)$ and ${\overline b}(s, q)$ which satisfy the equations in (3.24) and (3.27) and stay strictly above or below the appropriate diagonal for $0 < q < s$ , we can follow the arguments from [Reference Peskir58, Subsection 3.5] (among others), which are based on the construction of sequences of so-called bad–good solutions which intersect the diagonals. For this purpose, for any positive sequences $(s_{k}, q_k)_{k \in {\mathbb N}}$ and $(s_l, q_{l})_{l \in {\mathbb N}}$ such that $s_{k} \uparrow \infty$ as $k \to \infty$ and $q_{l} \downarrow 0$ as $l \to \infty$ , we can construct the sequence of solutions $a_{k}(s, q)$ , for $k \in {\mathbb N}$ , and $b_{l}(s, q)$ , for $l \in {\mathbb N}$ , to the equations (3.24) and (3.27) such that $a_{k}(s_k, q_k) = s_k$ and $b_{l}(s_l, q_l) = q_l$ holds, for each $k, l \in {\mathbb N}$ . It follows from the structure of the equations in (3.24) and (3.27) that the properties $\partial_s a_{k}(s_{k}, q_k) < 1$ and $\partial_q b_{l}(s_l, q_{l}) > 1$ hold, for each $k, l \in {\mathbb N}$ (see also [Reference Pedersen51, pp. 979--982] for the analysis of solutions of the non-parametrised version of the first-order nonlinear differential equation of (3.24)). Observe that, by virtue of the uniqueness of solutions mentioned above, we know that the two curves $s \mapsto a_{k}(s, q)$ and $s \mapsto a_{m}(s, q)$ cannot intersect, and similarly $q \mapsto b_{l}(s, q)$ and $q \mapsto b_{n}(s, q)$ cannot intersect, for $l, k, m, n \in {\mathbb N}$ such that $k \neq m$ and $l \neq n$ ; thus, the sequence $(a_{k}(s, q))_{k \in {\mathbb N}}$ is increasing and the sequence $(b_{l}(s, q))_{l \in {\mathbb N}}$ is decreasing, so that the limits ${\underline a}(s, q) = \lim_{k \to \infty} a_{k}(s, q)$ and ${\overline b}(s, q) = \lim_{l \to \infty} b_{l}(s, q)$ exist, for each $0 < q < s$ . We may therefore conclude that ${\underline a}(s, q)$ and ${\overline b}(s, q)$ provide the maximal and minimal solutions to the equations in (3.24) and (3.27) such that ${\underline a}_{k}(s, q) < s$ holds for each $k \in {\mathbb N}$ and ${\overline b}_{l}(s, q) > q$ holds for each $l \in {\mathbb N}$ , for all $0 < q < s$ .

Moreover, since the right-hand sides of the first-order nonlinear ordinary differential equations in (3.24) and (3.27) are (locally) Lipschitz in s and q, for each $0 < q < s$ , one can deduce by means of Gronwall’s inequality that the functions $a_{k}(s, q)$ and $b_{l}(s, q)$ , for each $k, l \in {\mathbb N}$ , are continuous, so that the functions ${\underline a}(s, q)$ and ${\overline b}(s, q)$ are continuous too. The appropriate maximal admissible solutions of first-order nonlinear ordinary differential equations, and the associated maximality principle for solutions of optimal stopping problems, which is equivalent to the superharmonic characterisation of the payoff functions, were established in [Reference Peskir52] and further developed in [Reference Baurdoux and Kyprianou5, Reference Gapeev20, Reference Gapeev, Kort and Lavrutich23, Reference Gapeev and Rodosthenous30, Reference Gapeev and Rodosthenous32–Reference Guo and Zervos37, Reference Kyprianou and Ott46, Reference Ott50–Reference Pedersen51, Reference Peskir57–Reference Peskir58, Reference Rodosthenous and Zervos63], as well as other subsequent papers (see [Reference Peskir and Shiryaev59, Chapter I; Chapter V, Section 17] for further references).

5.3. The results

Summarising the facts shown above, we state the following result, which can be proved by means of the same arguments as used for Theorem 1 above, in combination with the arguments from [Reference Gapeev21] (see [Reference Shepp and Shiryaev64]–[Reference Shepp and Shiryaev66] for the original derivation and [Reference Gapeev20]–[Reference Gapeev, Kort and Lavrutich23] for the related comparison arguments).