1. Introduction
The main aim of this paper is to present closed-form solutions to the discounted optimal stopping problems with the values
\begin{align}V^*_1 &= \sup_{\tau} {\mathbb E} \biggl[ {\mathrm{e}}^{- r \tau} (K_1 - X_{\tau}) \, I(\tau < \theta_1)\notag \\[5pt] &\phantom{= \sup_{\tau} {\mathbb E} \biggl[\,} + {\mathrm{e}}^{- r \theta_1} (\alpha_1 + \beta_1 \, X_{\theta_1}\kern-0.3pt)\, I(\theta_1 \le \tau) + \dfrac{\nu_1}{r} \bigl(1 - {\mathrm{e}}^{- r (\tau \wedge \theta_1)}\bigr) \biggr]\end{align}
and
\begin{align}V^*_2 &= \sup_{\tau} {\mathbb E} \biggl[ {\mathrm{e}}^{- r \tau} (X_{\tau} - K_2) \, I(\tau < \theta_2)\notag \\&\phantom{= \sup_{\tau} {\mathbb E} \biggl[\,} - {\mathrm{e}}^{- r \theta_2} (\alpha_2 + \beta_2 \, X_{\theta_2})\, I(\theta_2 \le \tau) + \dfrac{\nu_2}{r} \bigl(1 - {\mathrm{e}}^{- r (\tau \wedge \theta_2)}\bigr) \biggr]\end{align}
for some given constants
$K_i > 0$
,
$\alpha_i \ge 0$
,
$\beta_i \ge 0$
, and
$\nu_i \ge 0$
, for every
$i = 1, 2$
, where
$I(\cdot)$
denotes the indicator function. Here, for a precise formulation of the problem, we consider a probability space
$(\Omega, \mathcal{G}, {\mathbb P})$
with a standard Brownian motion
$B=(B_t)_{t \ge 0}$
and strictly positive integrable random variables
$\eta$
and
$\xi$
, which have strictly increasing continuously differentiable cumulative distribution functions
$F_i(x)$
such that
$F_i(0) \equiv 1 - F_i(\infty) = 0$
and
$0 < F_i(x) < 1$
as well as
$F'_{\!\!i}(x) > 0$
, for all
$x > 0$
and
$i = 1, 2$
(B and
$\eta$
or
$\xi$
are assumed to be independent under the probability measure
${\mathbb P}$
). We assume that the process
$X=(X_t)_{t \ge 0}$
is given by
\begin{equation} X_t = x \, \exp \bigl( \bigl( r - \delta - {\sigma^2}/{2} \bigr) t + \sigma \, B_t \bigr) \end{equation}
so that it solves the stochastic differential equation
\begin{equation} {\mathrm{d}} X_t = (r - \delta) \, X_{t} \,{\mathrm{d}} t + \sigma \, X_{t} \,{\mathrm{d}} B_t \quad (X_0=x), \end{equation}
where
$r > 0$
,
$\delta > 0$
, and
$\sigma > 0$
are some given constants and
$x > 0$
is fixed. Suppose that the process X describes the price of a risky asset in a financial market, where r is the riskless interest rate,
$\delta$
is the dividend rate paid to the asset holders, and
$\sigma$
is the volatility rate. Here
$K_i$
, for
$i = 1, 2$
, are the strike prices,
$\alpha_1 + \beta_1 X$
is a (linear) recovery (in the put option case), and
$\alpha_2 + \beta_2 X$
is a (linear) penalty (in the call option case), while
$\nu_i$
, for
$i = 1, 2$
, are the promised rates of continuously paid dividends of certain contingent claims. We also define the random times
$\theta_i$
, for
$i = 1, 2$
, by
\begin{equation} \theta_1 = \inf \{ t \ge 0 \mid X_t \geq \eta \} \quad \text{and} \quad \theta_2 = \inf \{ t \ge 0 \mid X_t \leq \xi \} \end{equation}
and assume that cancellations of these dividend-paying contingent claims are announced by the issuers of those products at these times, which are based on the market price of the underlying risky asset. In particular, these properties mean that the holders of such contingent claims may impose some prior (Bayesian) distribution on the unknown and, to them, unobservable (random) cancellation thresholds
$\eta$
and
$\xi$
chosen by the issuers. Note that European-type defaultable contingent claims with fixed finite-time horizon which have similar payoff and dividend structure were described in Bielecki and Rutkowski [Reference Bielecki and Rutkowski7, Section 2.1] and Linetsky [Reference Linetsky29], among others (see also the related references therein).
Suppose that the suprema in (1.1) and (1.2) are taken over all stopping times
$\tau$
with respect to the natural filtration
$(\mathcal{F}_t)_{t \ge 0}$
of the process X, and the expectations there are taken with respect to the risk-neutral probability measure
${\mathbb P}$
. In this view, the values
$V^*_1$
and
$V^*_2$
in (1.1) and (1.2) can be interpreted as the rational (or no-arbitrage) ex-dividend prices of the perpetual American cancellable dividend-paying put and call options in an extension of the Black–Merton–Scholes model (see e.g. [Reference Shiryaev43, Chapter VII, Section 3g]). Observe that the structure of the reward functionals in (1.1) and (1.2) allows us to describe the associated contracts as standard game (or Israeli) contingent claims introduced by Kifer [Reference Kifer25]. Such contacts enable their issuers to exercise their right to withdraw the contracts prematurely, by paying some penalties agreed in advance. Further developments of the Israeli options and the associated zero-sum optimal stopping (Dynkin) games were provided by Kyprianou [Reference Kyprianou27], Kühn and Kyprianou [Reference Kühn and Kyprianou26], Kallsen and Kühn [Reference Kallsen and Kühn24], Baurdoux and Kyprianou [Reference Baurdoux and Kyprianou3, Reference Baurdoux and Kyprianou4, Reference Baurdoux and Kyprianou5], Ekström and Villeneuve [Reference Ekström and Villeneuve12], and Baurdoux, Kyprianou, and Pardo [Reference Baurdoux, Kyprianou and Pardo6], among others. In contrast to the concept of game contingent claims mentioned above, in the present paper we study the cancellable perpetual American options in which the exogenous terminations of the contracts occur at the first times when the underlying risky asset price processes reach certain random thresholds, which are unknown and unobservable to the holders of the claims. We assume that these thresholds are independent of the geometric Brownian motion describing the underlying risky asset price. Some extensive overviews of the perpetual American options in diffusion models of financial markets and other related results in the area are provided in Shiryaev [Reference Shiryaev43, Chapter VIII, Section 2a], Peskir and Shiryaev [Reference Peskir and Shiryaev37, Chapter VII, Section 25], and Detemple [Reference Detemple10], among others. Note that other applications of the concept described above include the consideration of perpetual American dividend-paying options with credit risk which are defaulted at the times when the underlying risky asset price processes reach such random thresholds. Other perpetual American defaultable and withdrawable dividend-paying options were recently considered in [Reference Gapeev and Li14] and [Reference Gapeev and Al Motairi15] in some other diffusion-type models of financial markets with full and partial information.
We further study the problems of (1.1) and (1.2) as the associated optimal stopping problems of (2.4) and (2.5) for the two-dimensional continuous Markov processes having the underlying risky asset price X and its running maximum S or minimum Q as their state space components. The resulting problems turn out to be necessarily two-dimensional in the sense that they cannot be reduced to optimal stopping problems for one-dimensional Markov processes. Note that the reward functionals of the optimal stopping problems in (2.4) and (2.5) contain complicated stochastic integrals with respect to the running maximum and minimum processes. This feature initiates further developments of techniques to determine the structure of the associated continuation and stopping regions as well as appropriate modifications of the normal-reflection conditions in the equivalent free-boundary problems. Discounted optimal stopping problems for the running maxima and minima of the initial continuous (diffusion-type) processes were initiated by Shepp and Shiryaev [Reference Shepp and Shiryaev40, Reference Shepp and Shiryaev41, Reference Shepp and Shiryaev42] and further developed by Pedersen [Reference Pedersen32], Guo and Shepp [Reference Guo and Shepp22], Gapeev [Reference Gapeev13], Guo and Zervos [Reference Guo and Zervos23], Peskir [Reference Peskir35, Reference Peskir36], Glover, Hulley, and Peskir [Reference Glover, Hulley and Peskir20], Gapeev and Rodosthenous [Reference Gapeev and Rodosthenous16, Reference Gapeev and Rodosthenous17, Reference Gapeev and Rodosthenous18], Rodosthenous and Zervos [Reference Rodosthenous and Zervos39], and Gapeev, Kort, and Lavrutich [Reference Gapeev, Kort and Lavrutich19], among others. It was shown, by means of the maximality principle established by Peskir [Reference Peskir33] for solutions of optimal stopping problems, which is equivalent to the superharmonic characterisation of payoff functions, that the optimal stopping boundaries are given by the appropriate extremal solutions of certain (systems of) first-order nonlinear ordinary differential equations. More complicated optimal stopping problems in models with spectrally negative Lévy processes and their running maxima were studied by Asmussen, Avram, and Pistorius [Reference Asmussen, Avram and Pistorius1], Avram, Kyprianou, and Pistorius [Reference Avram, Kyprianou and Pistorius2], Ott [Reference Ott31], and Kyprianou and Ott [Reference Kyprianou and Ott28], among others.
The rest of the paper is organised as follows. In Section 2 we embed the original problems of (1.1) and (1.2) into the optimal stopping problems of (2.4) and (2.5) for the two-dimensional continuous Markov processes (X, S) and (X, Q) defined in (1.3) and (2.1). It is shown that the optimal stopping times
$\tau^*_1$
and
$\tau^*_2$
are the first times at which the process X reaches some lower or upper boundaries
$a^*(S)$
and
$b^*(Q)$
depending on the current values of the processes S and Q, respectively. In Section 3 we derive closed-form expressions for the associated value functions
$V^*_1(x, s)$
and
$V^*_2(x, q)$
as solutions to the equivalent free-boundary problems, and apply the modified normal-reflection conditions at the edges of the two-dimensional state spaces for (X, S) and (X, Q) to characterise the optimal stopping boundaries
$a^*(S)$
and
$b^*(Q)$
as the maximal and minimal solutions to the resulting first-order nonlinear ordinary differential equations, respectively. In Section 4, by using the change-of-variable formula with local time on surfaces from Peskir [Reference Peskir34], we verify that the solutions of the free-boundary problems provide the solutions of the original optimal stopping problems. The main results of the paper are stated in Lemma 2.1 and Theorem 4.1.
2. Preliminaries
In this section we introduce the setting and notation of two-dimensional optimal stopping problems, which are related to the pricing of perpetual American cancellable dividend-paying put and call options. We then formulate the equivalent free-boundary problems.
2.1. The optimal stopping problems
Let us now define the running maximum and minimum processes
$S=(S_t)_{t \ge 0}$
and
$Q=(Q_t)_{t \ge 0}$
, associated with X, by
\begin{equation}S_t = s \vee \Bigl( \max_{0 \le u \le t} X_u \Bigr) \quad \text{and} \quad Q_t = q \wedge \Bigl( \min_{0 \le u \le t} X_u \Bigr)\end{equation}
for some arbitrary
$s \ge x \ge q > 0$
. Then the conditional probabilities of the events that cancellation occurs before any time
$t \ge 0$
take the form
\begin{align*} &{\mathbb P}(\theta_1 \le t \mid \mathcal{F}_t) = {\mathbb P}(S_t \ge \eta \mid \mathcal{F}_t) = F_1(S_t)\end{align*}
and
\begin{align*} &{\mathbb P}(\theta_2 \le t \mid \mathcal{F}_t) = {\mathbb P}(Q_t \le \xi \mid \mathcal{F}_t) = G_2(Q_t),\end{align*}
where
$F_i(x)$
, for
$i = 1, 2$
, are the cumulative distribution functions of
$\eta$
and
$\xi$
, respectively, and we set
$G_i(x) = 1 - F_i(x)$
, for all
$x > 0$
and every
$i = 1, 2$
. Thus, by virtue of the assumptions made above, we have
$G_i(0) = 1 - G_i(\infty) = 1$
and
$0 < G_i(x) < 1$
as well as
$G'_{\!\!i}(x) < 0$
, for all
$x > 0$
and every
$i = 1, 2$
. In this case the values of (1.1) and (1.2) admit the representations
\begin{align}V^*_1 &= \sup_{\tau} {\mathbb E} \biggl[ {\mathrm{e}}^{- r \tau} (K_1 - X_{\tau}) G_1(S_{\tau}) \notag \\[5pt]&\phantom{= \sup_{\tau} {\mathbb E} \biggl[\,} + \int_0^{\tau} {\mathrm{e}}^{- r t} (\alpha_1 + \beta_1 \, X_{t}) \,{\mathrm{d}} F_1(S_t) +\int_0^{\tau} {\mathrm{e}}^{- r t} \, \nu_1 G_1(S_t) \,{\mathrm{d}} t \biggr]\end{align}
and
\begin{align}V^*_2 &= \sup_{\tau} {\mathbb E} \biggl[ {\mathrm{e}}^{- r \tau} (X_{\tau} - K_2) F_2(Q_{\tau})\notag \\[5pt]&\phantom{= \sup_{\tau} {\mathbb E} \biggl[\,}- \int_0^{\tau} {\mathrm{e}}^{- r t} (\alpha_2 + \beta_2 \, X_{t}) \,{\mathrm{d}} G_2(Q_t) +\int_0^{\tau} {\mathrm{e}}^{- r t} \, \nu_2 \, F_2(Q_t) \,{\mathrm{d}} t \biggr],\end{align}
where the suprema are taken over all stopping times of
$\tau$
with respect to
$(\mathcal{F}_t)_{t \ge 0}$
. In this case, taking into account the fact that the processes S and Q may change their values only when
$X_t = S_t$
and
$X_t = Q_t$
, for
$t \ge 0$
, respectively, we see that the problems in (2.2) and (2.3) can be naturally embedded into the optimal stopping problems for the (time-homogeneous strong) Markov processes
$(X, S)=(X_t, S_t)_{t \ge 0}$
and
$(X, Q)=(X_t, Q_t)_{t \ge 0}$
with the value functions
\begin{align}V^*_1(x, s) &= \sup_{\tau} {\mathbb E}_{x, s} \biggl[ {\mathrm{e}}^{- r \tau} (K_1 - X_{\tau}) G_1(S_{\tau}) \notag \\[5pt]&\phantom{= \sup_{\tau} {\mathbb E}_{x, s} \biggl[,}+ \int_0^{\tau} {\mathrm{e}}^{- r t} (\alpha_1 + \beta_1 \, S_{t}) \, F'_{\!\!1}(S_t) \,{\mathrm{d}} S_t+ \int_0^{\tau} {\mathrm{e}}^{- r t} \, \nu_1 \, G_1(S_t) \,{\mathrm{d}} t \biggr]\end{align}
and
\begin{align}V^*_2(x, q) &= \sup_{\tau} {\mathbb E}_{x, q} \biggl[ {\mathrm{e}}^{- r \tau} (X_{\tau} - K_2) F_2(Q_{\tau})\notag \\[5pt]&\phantom{= \sup_{\tau} {\mathbb E}_{x, q} \biggl[,}- \int_0^{\tau} {\mathrm{e}}^{- r t} (\alpha_2 + \beta_2 \, Q_{t}) G'_{\!\!2}(Q_t) \,{\mathrm{d}} Q_t+ \int_0^{\tau} {\mathrm{e}}^{- r t} \, \nu_2 \, F_2(Q_t) \,{\mathrm{d}} t \biggr],\end{align}
where
${\mathbb E}_{x, s}$
and
${\mathbb E}_{x, q}$
denote the expectations with respect to the probability measures
${\mathbb P}_{x, s}$
and
${\mathbb P}_{x, q}$
under which the two-dimensional Markov processes (X, S) and (X, Q) defined in (1.3) and (2.1) start at
$(x, s) \in E_1 = \bigl\{ (x, s) \in {\mathbb R}^2 \mid 0 < x \le s \bigr\}$
and
$(x, q) \in E_2 = \bigl\{ (x, q) \in {\mathbb R}^2 \mid 0 < q \le x \bigr\}$
, respectively. It follows from the results of [Reference Cvitanić and Karatzas9, Theorem 4.1] based on the solutions of the associated (doubly) reflected backward stochastic differential equations that the optimal stopping problems of (2.4) and (2.5) have values. We further obtain closed-form solutions to the optimal stopping problems in (2.4) and (2.5) and verify in Theorem 4.1 below that the value functions
$V^*_1(x, s)$
and
$V^*_2(x, q)$
are the solutions of the problems in (2.2) and (2.3), and thus of the original problems in (1.1) and (1.2) under
$s = x$
and
$q = x$
, respectively.
2.2. The structure of optimal stopping times
Let us now determine the structure of the optimal stopping times at which the holders should exercise the contracts.
(i) By means of standard applications of Itô’s formula (see e.g. [Reference Liptser and Shiryaev30, Theorem 4.4] or [Reference Revuz and Yor38, Chapter II, Theorem 3.2]) to the processes
${\mathrm{e}}^{-r t} (K_1 - X_t) G_1(S_t)$
and
${\mathrm{e}}^{-r t} (X_t - K_2) F_2(Q_t)$
, we obtain the representations
\begin{align*} &{\mathrm{e}}^{-r t} (K_1 - X_t) \, G_1(S_t) = (K_1 - x) \, G_1(s) + N^1_t \\[5pt]& \quad + \int_0^t {\mathrm{e}}^{-r u} \, (\delta \, X_u - r \, K_1) \, G_1(S_u) \,{\mathrm{d}} u+ \int_0^t {\mathrm{e}}^{-r u} (K_1 - X_u) \, I(X_u = S_u) \, G'_{\!\!1}(S_u) \,{\mathrm{d}} S_u\end{align*}
and
\begin{align*} &{\mathrm{e}}^{-r t} (X_t - K_2) \, F_2(Q_t) = (x - K_2) \, F_2(q) + N^2_t \\[5pt]& \quad + \int_0^t {\mathrm{e}}^{-r u} \, (r \, K_2 - \delta \, X_u) \, F_2(Q_u) \,{\mathrm{d}} u+ \int_0^t {\mathrm{e}}^{-r u} (X_u - K_2) \, I(X_u = Q_u) \, F'_{\!\!2}(Q_u) \,{\mathrm{d}} Q_u\end{align*}
for all
$t \ge 0$
. Here the processes
$N^i=(N^i_t)_{t \ge 0}$
,
$i = 1, 2$
, defined by
\begin{align*} N^1_t &= - \int_0^t {\mathrm{e}}^{-r u} \, \sigma \, X_u \, G_1(S_u) \,{\mathrm{d}} B_u\quad \text{and} \quad N^2_t = \int_0^t {\mathrm{e}}^{-r u} \, \sigma \, X_u \, F_2(Q_u) \,{\mathrm{d}} B_u,\end{align*}
are continuous uniformly integrable martingales under the probability measures
${\mathbb P}_{x, s}$
and
${\mathbb P}_{x, q}$
, for each
$(x, s) \in E_1$
and
$(x, q) \in E_2$
, respectively. Then, by applying Doob’s optional sampling theorem (see e.g. [Reference Liptser and Shiryaev30, Chapter III, Theorem 3.6] or [Reference Revuz and Yor38, Chapter II, Theorem 3.2]), we obtain that the expected rewards from (2.4) and (2.5) admit the representations
\begin{align}&{\mathbb E}_{x, s} \biggl[ {\mathrm{e}}^{- r \tau} (K_1 - X_{\tau}) \, G_1(S_{\tau})\notag \\[5pt]&\phantom{{\mathbb E}_{x, s} \biggl[\,}+ \int_0^{\tau} {\mathrm{e}}^{- r t} \, (\alpha_1 + \beta_1 \, S_{t}) \, F'_{\!\!1}(S_t) \,{\mathrm{d}} S_t+ \int_0^{\tau} {\mathrm{e}}^{- r t} \, \nu_1 \, G_1(S_t) \,{\mathrm{d}} t \biggr]\notag \\[5pt]&= (K_1 - x) \, G_1(s) + {\mathbb E}_{x, s} \biggl[ \int_0^{\tau} {\mathrm{e}}^{- r t} \,(\delta \, X_t - r \, K_1 + \nu_1) \, G_1(S_t) \,{\mathrm{d}} t \notag \\[5pt]&\phantom{= (K_1 - x) \, G_1(s) + {\mathbb E}_{x, s} \biggl[\,}+ \int_0^{\tau} {\mathrm{e}}^{- r t} ( K_1 - \alpha_1 - (1 + \beta_1) \, S_t )\, I(X_t = S_t) \, G'_{\!\!1}(S_t) \,{\mathrm{d}} S_t \biggr]\end{align}
and
\begin{align}&{\mathbb E}_{x, q} \biggl[ {\mathrm{e}}^{- r \tau} (X_{\tau} - K_2) \, F_2(Q_{\tau}) \notag \\[5pt]&\phantom{{\mathbb E}_{x, q} \biggl[\,} - \int_0^{\tau} {\mathrm{e}}^{- r t} \, (\alpha_2 + \beta_2 \, Q_{t}) \, G'_{\!\!2}(Q_t) \,{\mathrm{d}} Q_t+ \int_0^{\tau} {\mathrm{e}}^{- r t} \, \nu_2 \, F_2(Q_t) \,{\mathrm{d}} t \biggr] \notag \\[5pt]&= (x - K_2) \, F_2(q) + {\mathbb E}_{x, q}\biggl[ \int_0^{\tau} {\mathrm{e}}^{- r t} \, (r \, K_2 + \nu_2 - \delta \, X_t) \, F_2(Q_t) \,{\mathrm{d}} t \notag \\[5pt]&\phantom{= (x - K_2) \, F_2(q) + {\mathbb E}_{x, q} \biggl[\,}+ \int_0^{\tau} {\mathrm{e}}^{- r t} \, ( (1 + \beta_2) \, Q_t - K_2 + \alpha_2 ) \,I(X_t = Q_t) \, F'_{\!\!2}(Q_t) \,{\mathrm{d}} Q_t \biggr]\end{align}
for
$(x, s) \in E_1$
and
$(x, q) \in E_2$
, for any stopping time
$\tau$
of the process (X, S) or (X, Q), respectively. Observe from the structure of the integrands and the fact that
$0 < G_i(x) < 1$
and
$0 < F_i(x) < 1$
, for all
$x > 0$
and every
$i = 1, 2$
, that the expectations of the integrals in the second lines of the formulas in (2.6) and (2.7) are finite. Moreover, by virtue of the assumed integrability of the random variables
$\eta$
and
$\xi$
, it is seen that the expectations of the integrals in the third lines of the formulas in (2.6) and (2.7) are finite too.
We now recall the assumptions that
$0 < F_i(x) < 1$
and
$F'_{\!\!i}(x) > 0$
, so that
$0 < G_i(x) < 1$
and
$G'_{\!\!i}(x) < 0$
holds, for all
$x > 0$
and every
$i = 1, 2$
. Then, according to the properties that
$0 < G_i(S_t) < 1$
and
$G'_{\!\!i}(S_t) < 0$
, for any
$t \ge 0$
and every
$i = 1, 2$
, by virtue of the fact that the process S is positive and increasing, it is seen from the structure of the integrands in (2.6) that the optimal stopping time
$\tau^*_1$
is infinite, whenever
$K_1 \le \nu_1/r$
holds. Furthermore, by virtue of the properties that
$0 < G_i(S_t) < 1$
and
$0 < F_i(Q_t) < 1$
, for any
$t \ge 0$
and every
$i = 1, 2$
, it follows from the structure of the first integrands in (2.6) and (2.7) that it is not optimal to exercise the cancellable put option when
${\overline a} \le X_t < S_t$
with
${\overline a} = (r K_1 - \nu_1)/\delta$
under
$K_1 > \nu_1/r$
, while it is not optimal to exercise the cancellable call option when
$Q_t < X_t \le {\underline b}$
with
${\underline b} = (r K_2 + \nu_2)/\delta$
, for any
$t \ge 0$
, respectively. In other words, these facts mean that the set
$\{ (x, s) \in E_1 \mid {\overline a} \le x < s \}$
under
$K_1 > \nu_1/r$
belongs to the continuation region
$C^*_1$
that has the form
\begin{align}&C^*_1 = \{ (x, s) \in E_1 \mid V^*_1(x, s) > (K_1 - x) \, G_1(s) \},\end{align}
while the set
$\{ (x, q) \in E_2 \mid q < x \le {\underline b} \}$
belongs to the continuation region
$C^*_2$
given by
\begin{align}&C^*_2 = \{ (x, q) \in E_2 \mid V^*_2(x, q) > (x - K_2) \, F_2(q) \}\end{align}
(see e.g. [Reference Peskir and Shiryaev37, Chapter I, Section 2.2]).
(ii) Note that by virtue of properties of the running maximum S and minimum Q from (2.1) of the geometric Brownian motion X from (1.3)–(1.4) (see e.g. [Reference Dubins, Shepp and Shiryaev11, Section 3.3] for similar arguments applied to the running maxima of the Bessel processes), it is seen that, for any
$s > 0$
and
$q > 0$
fixed and an infinitesimally small deterministic time interval
$\Delta$
, we have
\begin{align*} &S_{\Delta} = s \vee \max_{0 \le u \le \Delta} X_u = s \vee (s + \Delta X) + {\mathrm{o}} (\Delta)\quad \text{as $ \Delta \downarrow 0$}\end{align*}
and
\begin{align*} &Q_{\Delta} = q \wedge \min_{0 \le u \le \Delta} X_u = q \wedge (q + \Delta X) + {\mathrm{o}} (\Delta)\quad \text{as $ \Delta \downarrow 0$},\end{align*}
where we set
$\Delta X = X_{\Delta} - s$
and
$\Delta X = X_{\Delta} - q$
, respectively. Observe that
$\Delta S = {\mathrm{o}} (\Delta)$
when
$\Delta X \le 0$
,
$\Delta S = \Delta X + {\mathrm{o}} (\Delta)$
when
$\Delta X > 0$
,
$\Delta Q = {\mathrm{o}} (\Delta)$
when
$\Delta X \ge 0$
, and
$\Delta Q = \Delta X + {\mathrm{o}} (\Delta)$
when
$\Delta X < 0$
, where we set
$\Delta S = S_{\Delta} - s$
and
$\Delta Q = Q_{\Delta} - q$
, and recall that
${\mathrm{o}} (\Delta)$
denotes a random function satisfying
${\mathrm{o}} (\Delta)/\Delta \to 0$
as
$\Delta \downarrow 0$
(
${\mathbb P}$
-a.s.). In this case, using the asymptotic formulas
\begin{align*} &{\mathbb E}_{s, s} [ \Delta X ; \; \Delta X > 0 ] \equiv{\mathbb E}_{s, s} [ \Delta X \, I(\Delta X > 0) ] \sim s \, \sigma \, \sqrt{\dfrac{\Delta}{2 \pi}}\quad \text{as $\Delta \downarrow 0$}\end{align*}
and
\begin{align*}&{\mathbb E}_{q, q} [ \Delta X ; \; \Delta X < 0 ] \equiv{\mathbb E}_{q, q} [ \Delta X \, I(\Delta X < 0) ] \sim - q \, \sigma \, \sqrt{\dfrac{\Delta}{2 \pi}}\quad \text{as $ \Delta \downarrow 0$},\end{align*}
as well as applying the representations in (2.6) and (2.7), we get
\begin{align}&{\mathbb E}_{s, s} \bigl[ {\mathrm{e}}^{- r \Delta} \, (\delta \, s - r \, K_1 + \nu_1) \, G_1(s) \, \Delta +{\mathrm{e}}^{- r \Delta} \, ( K_1 - \alpha_1 - (1 + \beta_1) \, s ) \, G'_{\!\!1}(s) \, \Delta S \bigr] \notag \\[5pt]\notag&\quad \sim {\mathrm{e}}^{- r \Delta} \, (\delta \, s - r \, K_1 + \nu_1) \, G_1(s) \, \Delta +{\mathrm{e}}^{- r \Delta} \, ( K_1 - \alpha_1 - (1 + \beta_1) \, s ) \, G'_{\!\!1}(s) \,s \, \sigma \, \sqrt{\dfrac{\Delta}{2 \pi}} \\[5pt]&\quad \ \text{as $ \Delta \downarrow 0$}\end{align}
and
\begin{align}&{\mathbb E}_{q, q} \bigl[ {\mathrm{e}}^{- r \Delta} \, (r \, K_2 + \nu_2 - \delta \, q) \, F_2(q) \, \Delta +{\mathrm{e}}^{- r \Delta} \, ( (1 + \beta_2) \, q - K_2 + \alpha_2 ) \, F'_{\!\!2}(q) \, \Delta Q \bigr] \notag \\[5pt]\notag&\quad \sim {\mathrm{e}}^{- r \Delta} \, (r \, K_2 + \nu_2 - \delta \, q) \, F_2(q) \, \Delta -{\mathrm{e}}^{- r \Delta} \, ( (1 + \beta_2) \, q - K_2 + \alpha_2 ) \, F'_{\!\!2}(q) \,q \, \sigma \, \sqrt{\dfrac{\Delta}{2 \pi}} \\[5pt]&\quad \ \text{as $ \Delta \downarrow 0$}\end{align}
for each
$s > 0$
and
$q > 0$
fixed. Since we have
$G'_{\!\!i}(s) < 0$
and
$F'_{\!\!i}(q) > 0$
, for all
$s > 0$
,
$q > 0$
, and every
$i = 1, 2$
, we see that the resulting coefficients by the terms of order
$\sqrt{\Delta}$
in (2.10) and (2.11) are strictly positive, when
$s > s'$
with
$s' = (K_1 - \alpha_1)/(1 + \beta_1)$
under
$K_1 > \alpha_1$
(or when
$s > 0$
under
$K_1 \le \alpha_1$
), and
$q < q'$
with
$q' = (K_2 - \alpha_2)/(1 + \beta_2)$
under
$K_2 > \alpha_2$
. Hence, taking into account the fact that the process S is positive and increasing and the process Q is positive and decreasing, by virtue of the properties that
$G'_{\!\!i}(S_t) < 0$
and
$F'_{\!\!i}(Q_t) > 0$
, for any
$t \ge 0$
and every
$i = 1, 2$
, we may therefore conclude from the structure of the second integrands in (2.6) and (2.7), as well as the heuristic arguments presented in (2.10) and (2.11) above, that it is not optimal to exercise the cancellable put option when
$s' < S_t = X_t$
with
$s' = (K_1 - \alpha_1)/(1 + \beta_1)$
under
$K_1 > \alpha_1$
(or when
$0 < S_t = X_t$
under
$K_1 \le \alpha_1$
), while it is not optimal to exercise the cancellable call option when
$X_t = Q_t < q'$
with
$q' = (K_2 - \alpha_2)/(1 + \beta_2)$
under
$K_2 > \alpha_2$
, for any
$t \ge 0$
, respectively. In other words these facts mean that the sets
$d'_{\!\!1} = \{ (x, s) \in E_1 \mid x = s > s' \}$
under
$K_1 > \alpha_1$
(which becomes the whole diagonal
$d_1 = \{ (x, s) \in E_1 \mid x = s \}$
under
$K_1 \le \alpha_1$
) and
$d'_{\!\!2} = \{ (x, q) \in E_2 \mid x = q < q' \}$
under
$K_2 > \alpha_2$
(which becomes an empty set under
$K_2 \le \alpha_2$
) surely belong to the continuation regions
$C^*_1$
and
$C^*_2$
in (2.8) and (2.9) above. For simplicity of presentation, we further assume that the inequalities
$K_1 > \alpha_1 \vee (\nu_1/r)$
and
$K_2 > \alpha_2$
hold.
(iii) On the other hand, it follows from the definition of the processes (X, S) and (X, Q) in (1.3) and (2.1) and the structure of the rewards in (2.4) and (2.5) with the representations in (2.6) and (2.7) that for each
$s > 0$
fixed there exists a sufficiently small
$x > 0$
such that the point (x, s) belongs to the stopping region
$D^*_1$
, which has the form
\begin{align}&D^*_1 = \{ (x, s) \in E_1 \mid V^*_1(x, s) = (K_1 - x) \, G_1(s) \},\end{align}
while for each
$q > 0$
fixed there exists a sufficiently large
$x > 0$
such that the point (x, q) belongs to the stopping region
$D^*_2$
, which is given by
\begin{align}&D^*_2 = \{ (x, q) \in E_2 \mid V^*_2(x, q) = (x - K_2) \, F_2(q) \}\end{align}
(see e.g. [Reference Peskir and Shiryaev37, Chapter I, Section 2.2]). According to arguments similar to those applied in [Reference Dubins, Shepp and Shiryaev11, Section 3.3] and [Reference Peskir33, Section 3.3], the latter properties can be explained by the fact that the costs of waiting until the process X coming from such a small
$x > 0$
increases the current value of the running maximum process S, and that the costs of waiting until the process X coming from such a large
$x > 0$
decreases the current value of the running minimum process Q may be too large, due to the presence of the discounting factor in the reward functionals of (2.4) and (2.5). It is seen from the results of Theorem 4.1 proved below that the value functions
$V^*_1(x, s)$
and
$V^*_2(x, q)$
are continuous, so that the sets
$C^*_1$
and
$C^*_2$
in (2.8) and (2.9) are open while the sets
$D^*_1$
and
$D^*_2$
in (2.12) and (2.13) are closed.
Observe that if we take some
$(x, s) \in D^*_1$
from (2.12) and use the fact that the process (X, S) started at some
$(x_1, s)$
such that
$x_1 < x$
passes through the point (x, s) before hitting the diagonal
$d_1 = \{ (x, s) \in E_1 \mid x = s \}$
, then (2.4) and (2.6) imply
\begin{equation*} V^*_1(x_1, s) - (K_1 - x_1) G_1(s) \le V^*_1(x, s) - (K_1 - x) G_1(s) = 0, \end{equation*}
so that
$(x_1, s) \in D^*_1$
. Moreover, if we take some
$(x, q) \in D^*_2$
from (2.13) and use the fact that the process (X, Q) started at some
$(x_2, q)$
such that
$x_2 > x$
passes through the point (x, q) before hitting the diagonal
$d_2 = \{ (x, q) \in E_2 \mid x = q \}$
, then (2.5) and (2.7) imply
\begin{equation*} V^*_2(x_2, q) - (x_2 - K_2) F_2(q) \le V^*_2(x, q) - (x - K_2) F_2(q) = 0, \end{equation*}
so that
$(x_2, q) \in D^*_2$
. On the other hand, if we take some
$(x, s) \in C^*_1$
from (2.8) and use the fact that the process (X, S) started at (x, s) passes through some point
$(x'_{\!\!1}, s)$
such that
$x'_{\!\!1} > x$
before hitting the diagonal
$d_1$
, then (2.4) and (2.6) yield
\begin{equation*} V^*_1(x'_{\!\!1}, s) - (K_1 - x'_{\!\!1}) G_1(s) \ge V^*_1(x, s) - (K_1 - x) G_1(s) > 0, \end{equation*}
so that
$(x'_{\!\!1}, s) \in C^*_1$
. Moreover, if we take some
$(x, q) \in C^*_2$
from (2.9) and use the fact that the process (X, Q) started at (x, q) passes through some point
$(x'_{\!\!2}, q)$
such that
$x'_{\!\!2} < x$
before hitting the diagonal
$d_2$
, then (2.5) and (2.7) yield
\begin{equation*} V^*_2(x'_{\!\!2}, q) - (x'_{\!\!2} - K_2) F_2(q) \ge V^*_2(x, q) - (x - K_2) F_2(q) > 0, \end{equation*}
so that
$(x'_{\!\!2}, q) \in C^*_2$
. Hence, combining these arguments with the comments in [Reference Dubins, Shepp and Shiryaev11, Section 3.3] and [Reference Peskir33, Section 3.3] and recalling the fact that the sets
$d'_{\!\!1} = \{ (x, s) \in E_1 \mid x = s > s' \}$
and
$d'_{\!\!2} = \{ (x, q) \in E_2 \mid x = q < q' \}$
surely belong to the continuation regions
$C^*_1$
and
$C^*_2$
in (2.8) and (2.9), respectively, we may conclude that there exist functions
$a^*(s)$
and
$b^*(q)$
satisfying the inequalities
$a^*(s) < s \wedge {\overline a}$
with
${\overline a} = (r K_1 - \nu_1)/\delta$
and
$b^*(q) > q \vee {\underline b}$
with
${\underline b} = (r K_2 + \nu_2)/\delta$
, for all
$s > {\underline s}$
and
$q < {\overline q}$
and some
$0 \le {\underline s} \le s' \wedge {\overline a}$
and
${\overline q} \ge q' \vee {\underline b}$
fixed, as well as the equalities
$a^*(s) = s$
and
$b^*(q) = q$
, for all
$s \le {\underline s}$
and
$q \ge {\overline q}$
, such that the continuation regions
$C^*_1$
and
$C^*_2$
in (2.8) and (2.9) have the form
\begin{align}&C^*_1 = \{ (x, s) \in E_1 \mid a^*(s) < x \le s \}\quad \text{and} \quad C^*_2 = \{ (x, q) \in E_2 \mid q \le x < b^*(q) \},\end{align}
while the stopping regions
$D^*_1$
and
$D^*_2$
in (2.12)–(2.13) are given by
\begin{align}&D^*_1 = \{ (x, s) \in E_1 \mid x \le a^*(s) \}\quad \text{and} \quad D^*_2 = \{ (x, q) \in E_2 \mid x \ge b^*(q) \}\end{align}
under
$K_1 > \alpha_1 \vee (\nu_1/r)$
and
$K_2 > \alpha_2$
, respectively (see Figures 1 and 2 for depictions of the optimal exercise boundaries
$a^*(s)$
and
$b^*(q)$
).
The optimal exercise boundary
$a^*(s)$
.
The optimal exercise boundary
$b^*(q)$
.
We summarise the arguments shown above in the following assertion.
Lemma 2.1. Let the processes
$(X, S)$
and
$(X, Q)$
be given by (1.3) and (2.1), with some
$r > 0$
,
$\delta > 0$
, and
$\sigma > 0$
fixed, and suppose the inequalities
$K_1 > \alpha_1 \vee (\nu_1/r)$
and
$K_2 > \alpha_2$
hold, for some
$\alpha_i \ge 0$
,
$\beta_i \ge 0$
, and
$\nu_i \ge 0$
, for
$i = 1, 2$
fixed. Suppose that the random times
$\theta_i$
, for
$i = 1, 2$
, are defined in (1.5) for strictly positive continuous integrable random variables
$\eta$
and
$\xi$
with a strictly increasing continuously differentiable cumulative distribution function
$F_i(x) \equiv 1 - G_i(x)$
such that
$F_i(0) = 1 - F_i(\infty) = 0$
and
$0 < F_i(x) < 1$
as well as
$F'_{\!\!i}(x) > 0$
, for all
$x > 0$
and every
$i = 1, 2$
. Then the optimal stopping times in the problems of (2.4) and (2.5) have the structure
\begin{equation}\tau^*_1 = \inf \{t \ge 0 \mid X_t \le a^*(S_t) \}\quad \textit{and} \quad\tau^*_2 = \inf \{t \ge 0 \mid X_t \ge b^*(Q_t) \}\end{equation}
for some functions
$a^*(s)$
and
$b^*(q)$
satisfying the inequalities
$a^*(s) < s \wedge {\overline a}$
with
${\overline a} = (r K_1 - \nu_1)/\delta$
and
$b^*(q) > q \vee {\underline b}$
with
${\underline b} = (r K_2 + \nu_2)/\delta$
, for all
$s > {\underline s}$
and
$q < {\overline q}$
and some
$0 \le {\underline s} \le s' \wedge {\overline a}$
and
${\overline q} \ge q' \vee {\underline b}$
fixed, where
$s' = (K_1 - \alpha_1)/(1 + \beta_1)$
and
$q' = (K_2 - \alpha_2)/(1 + \beta_2)$
, as well as the equalities
$a^*(s) = s$
and
$b^*(q) = q$
, for all
$s \le {\underline s}$
and
$q \ge {\overline q}$
, respectively.
2.3. The free-boundary problems
By means of standard arguments based on the application of Itô’s formula, it is shown that the infinitesimal operator
${\mathbb L}$
of the process (X, S) or (X, Q) from (1.4) and (2.1) takes the form
\begin{align*} {\mathbb L} & = (r - \delta) \, x \, \partial_x + \dfrac{\sigma^2 x^2}{2} \, \partial_{xx} \quad \text{in $ 0 < x < s $ or $ 0 < q < x$} \\[5pt]\partial_s & = 0 \quad \text{at $ 0 < x = s$} \quad \text{or} \quad \partial_q = 0 \quad \text{at $ 0 < x = q$} \end{align*}
(see e.g. [Reference Peskir33, Section 3.1]). In order to find analytic expressions for the unknown value functions
$V^*_1(x, s)$
and
$V^*_2(x, q)$
in (2.4) and (2.5) and the unknown boundaries
$a^*(s)$
and
$b^*(q)$
from (2.16), we apply the results of general theory for solving optimal stopping problems for Markov processes presented in [Reference Peskir and Shiryaev37, Chapter IV, Section 8], among others (see also [Reference Peskir and Shiryaev37, Chapter V, Sections 15–20] for optimal stopping problems for maxima processes and other related references). More precisely, for the original optimal stopping problems in (2.4) and (2.5), we formulate the associated free-boundary problems (see e.g. [Reference Peskir and Shiryaev37, Chapter IV, Section 8]) and then verify in Theorem 4.1 below that the appropriate candidate solutions of the latter problems coincide with the solutions of the original problems. In other words we reduce the optimal stopping problems of (2.4) and (2.5) to the following equivalent free-boundary problems:
\begin{align} ({{\mathbb L}} V_1 - r V_1)(x, s) = - \nu_1 \, G_1(s) \quad &\text{for} \;\; (x, s) \in C_1 \setminus \{ (x, s) \in E_1 \mid x = s \}, \end{align}
\begin{align} ({{\mathbb L}} V_2 - r V_2)(x, q) = - \nu_2 \, F_2(q) \quad &\text{for} \;\; (x, q) \in C_2 \setminus \{ (x, q) \in E_2 \mid x = q \}, \end{align}
\begin{align} V_1(x, s) \big|_{x = a(s)+} = (K_1 - a(s)) \, G_1(s), \quad &V_2(x, q) \big|_{x = b(q)-} = (b(q) - K_2) \, F_2(q), \end{align}
\begin{align} \partial_x V_1(x, s) \big|_{x = a(s)+} = - G_1(s), \quad &\partial_x V_2(x, q) \big|_{x = b(q)-} = F_2(q), \end{align}
\begin{align} \partial_s V_1(x, s) \big|_{x = s-} = - (\alpha_1 + \beta_1 \, s) \, F'_{\!\!1}(s), \quad &\partial_q V_2(x, q) \big|_{x = q+} = (\alpha_2 + \beta_2 \, q) \, G'_{\!\!2}(q), \end{align}
\begin{align} V_1(x, s) = (K_1 - x) \, G_1(s) \quad &\text{for} \;\; (x, s) \in D_1, \end{align}
\begin{align} V_2(x, q) = (x - K_2) \, F_2(q) \quad &\text{for} \;\; (x, q) \in D_2, \end{align}
\begin{align} V_1(x, s) > (K_1 - x) \, G_1(s) \quad &\text{for} \;\; (x, s) \in C_1, \end{align}
\begin{align} V_2(x, q) > (x - K_2) \, F_2(q) \quad &\text{for} \;\; (x, q) \in C_2, \end{align}
\begin{align} ({{\mathbb L}} V_1 - r V_1)(x, s) < - \nu_1 \, G_1(s) \quad &\text{for} \;\; (x, s) \in D_1, \end{align}
\begin{align} ({{\mathbb L}} V_2 - r V_2)(x, q) < - \nu_2 \, F_2(q) \quad &\text{for} \;\; (x, q) \in D_2, \end{align}
where
$C_i$
and
$D_i$
,
$i = 1, 2$
, are defined as
$C^*_i$
and
$D^*_i$
,
$i = 1, 2$
, in (2.14) and (2.15) with a(s) and b(q) instead of
$a^*(s)$
and
$b^*(q)$
, respectively. Here the instantaneous stopping as well as the smooth-fit and modified normal-reflection conditions of (2.19)–(2.21) are satisfied, for all
$s > {\underline s}$
and
$q < {\overline q}$
, respectively. Observe that the superharmonic characterisation of the value function (see e.g. [Reference Peskir and Shiryaev37, Chapter IV, Section 9]) implies that
$V^*_1(x, s)$
and
$V^*_2(x, q)$
are the smallest functions satisfying (2.17)–(2.19) and (2.22)–(2.25) with the boundaries
$a^*(s)$
and
$b^*(q)$
, respectively. Note that (2.26)–(2.27) follow directly from the assertion of Lemma 2.1 proved in part (i) of Section 2.2 above.
3. Solutions to the free-boundary problems
In this section we obtain solutions to the free-boundary problems in (2.17)–(2.27) and derive first-order nonlinear ordinary differential equations for the candidate optimal stopping boundaries.
3.1. The candidate value functions
It is shown that the second-order ordinary differential equations in (2.17)–(2.18) have the general solutions
$${{V_1}(x,s) = {C_{1,1}}(s){x^{{\gamma _1}}} + {C_{1,2}}(s){x^{{\gamma _2}}} + {\nu _1}{G_1}(s)/r}$$
and
$${{V_2}(x,q) = {C_{2,1}}(q){x^{{\gamma _1}}} + {C_{2,2}}(q){x^{{\gamma _2}}} + {\nu _2}{F_2}(q)/r,}$$
where
$C_{1,j}(s)$
and
$C_{2,j}(q)$
,
$j = 1, 2$
, are some arbitrary (continuously differentiable) functions, and
$\gamma_j$
,
$j = 1, 2$
, are given by
\begin{equation*} \gamma_j = \dfrac{1}{2}-\dfrac{r-\delta}{\sigma^2} - (-1)^j \sqrt{\biggl( \dfrac{1}{2} - \dfrac{r-\delta}{\sigma^2} \biggr)^2 + \dfrac{2r}{\sigma^2}}, \end{equation*}
so that
$\gamma_2 < 0 < 1 < \gamma_1$
holds. Then, by applying the conditions of (2.19)–(2.21) to the functions in (3.1), we obtain the equalities
\begin{align} C_{1,1}(s) \, a^{\gamma_1}(s) + C_{1,2}(s) \, a^{\gamma_2}(s) + \nu_1 \, G_1(s)/r &= (K_1 - a(s)) \, G_1(s), \end{align}
\begin{align} \gamma_1 \, C_{1,1}(s) \, a^{\gamma_1}(s) + \gamma_2 \, C_{1,2}(s) \, a^{\gamma_2}(s) &= - a(s) \, G_1(s), \end{align}
\begin{align} C'_{\!\!1,1}(s) \, s^{\gamma_1} + C'_{\!\!1,2}(s) \, s^{\gamma_2} + \nu_1 \, G'_{\!\!1}(s)/r &= - (\alpha_1 + \beta_1 \, s) \, F'_{\!\!1}(s) \end{align}
for all
$s > {\underline s}$
, and
\begin{align} C_{2,1}(q) \, b^{\gamma_1}(q) + C_{2,2}(q) \, b^{\gamma_2}(q) + \nu_2 \, F_2(q)/r &= (b(q) - K_2) \, F_2(q), \end{align}
\begin{align} \gamma_1 \, C_{2,1}(q) \, b^{\gamma_1}(q) + \gamma_2 \, C_{2,2}(q) \, b^{\gamma_2}(q) &= b(q) \, F_2(q), \end{align}
\begin{align} C'_{\!\!2,1}(q) \, q^{\gamma_1} + C'_{\!\!2,2}(q) \, q^{\gamma_2} + \nu_2 \, F'_{\!\!2}(q)/r &= (\alpha_2 + \beta_2 \, q) \, G'_{\!\!2}(q) \end{align}
for all
$q < {\overline q}$
, respectively. Hence, by solving the systems of equations in (3.2)–(3.3) and (3.5)–(3.6), we obtain that the candidate value functions admit the representations
\begin{equation} V_1(x, s;\; a(s)) = C_{1,1}(s;\; a(s)) \, x^{\gamma_1} + C_{1,2}(s;\; a(s)) \, x^{\gamma_2} + \nu_1 \, G_1(s)/r \end{equation}
for
$a(s) < x \le s$
, with
\begin{align} &C_{1,j}(s;\; a(s)) = \dfrac{(\gamma_{3-j} (K_1 - \nu_1/r) - (\gamma_{3-j} - 1) a(s)) G_1(s)} {(\gamma_{3-j}-\gamma_j) a^{\gamma_j}(s)} \end{align}
for
$j = 1, 2$
, and
\begin{equation} V_2(x, q;\; b(q)) = C_{2,1}(q;\; b(q)) \, x^{\gamma_1} + C_{2,2}(q;\; b(q)) \, x^{\gamma_2} + \nu_2 \, F_2(q)/r \end{equation}
for
$q \le x < b(q)$
, with
\begin{align} &C_{2,j}(q;\; b(q)) = \dfrac{((\gamma_{3-j} - 1) b(q) - \gamma_{3-j} (K_2 + \nu_2/r)) F_2(q)} {(\gamma_{3-j}-\gamma_j) b^{\gamma_j}(q)} \end{align}
for
$j = 1, 2$
, respectively. Moreover, by means of straightforward computations, it can be deduced from (3.8) and (3.10) that the first-order and second-order partial derivatives
$\partial_x V_1(x, s;\; a(s))$
and
$\partial_{xx} V_1(x, s;\; a(s))$
of the function
$V_1(x, s;\; a(s))$
take the form
\begin{align*} \partial_x V_1(x, s;\; a(s)) &= C_{1,1}(s;\; a(s)) \, \gamma_1 \, x^{\gamma_1-1} + C_{1,2}(s;\; a(s)) \, \gamma_2 \, x^{\gamma_2-1}\end{align*}
and
\begin{align*} \partial_{xx} V_1(x, s;\; a(s)) &=C_{1,1}(s;\; a(s)) \, \gamma_1 (\gamma_1 - 1) \, x^{\gamma_1-2} + C_{1,2}(s;\; a(s)) \, \gamma_2 (\gamma_2 - 1) \, x^{\gamma_2-2}\end{align*}
on the interval
$a(s) < x \le s$
, for each
$s > {\underline s}$
, while the first-order and second-order partial derivatives
$\partial_x V_2(x, q;\; b(q))$
and
$\partial_{xx} V_2(x, q;\; b(q))$
of the function
$V_2(x, q;\; b(q))$
take the form
\begin{align}\partial_x V_2(x, q;\; b(q)) &= C_{2,1}(q;\; b(q)) \, \gamma_1 \, x^{\gamma_1-1} + C_{2,2}(q;\; b(q)) \, \gamma_2 \, x^{\gamma_2-1}\end{align}
and
\begin{align}\partial_{xx} V_2(x, q;\; b(q)) &=C_{2,1}(q;\; b(q)) \, \gamma_1 (\gamma_1 - 1) \, x^{\gamma_1-2} + C_{2,2}(q;\; b(q)) \, \gamma_2 (\gamma_2 - 1) \, x^{\gamma_2-2}\!\!\!\!\!\!\!\end{align}
on the interval
$q \le x < b(q)$
, for each
$q < {\overline q}$
fixed.
3.2. The candidate stopping boundaries
By applying the conditions of (3.4) and (3.7) to the functions in (3.9) and (3.11), we conclude that the candidate boundaries satisfy the first-order nonlinear ordinary differential equations
\begin{align} &a'(s) = \dfrac{\Psi_{1, 1}(s, a(s)) s^{\gamma_1} + \Psi_{1, 2}(s, a(s)) s^{\gamma_2} + (\alpha_1 + \beta_1 s - \nu_1/r) G'_{\!\!1}(s)} {\Phi_{1, 1}(s, a(s)) s^{\gamma_1} + \Phi_{1, 2}(s, a(s)) s^{\gamma_2}} \end{align}
for
$s > {\underline s}$
, and
\begin{align} &b'(q) = \dfrac{\Psi_{2, 1}(q, b(q)) q^{\gamma_1} + \Psi_{2, 2}(q, b(q)) q^{\gamma_2} + (\alpha_2 + \beta_2 q + \nu_2/r) F'_{\!\!2}(q)} {\Phi_{2, 1}(q, b(q)) q^{\gamma_1} + \Phi_{2, 2}(q, b(q)) q^{\gamma_2}} \end{align}
for
$q < {\overline q}$
, respectively. Here the functions
$\Phi_{1, j}(s, a(s))$
,
$\Psi_{1, j}(s, a(s))$
and
$\Phi_{2, j}(q, b(q))$
,
$\Psi_{2, j}(q, b(q))$
are defined by
\begin{align} &\Phi_{1, j}(s, a(s)) = \dfrac{((\gamma_1 - 1)(\gamma_2 - 1) - \gamma_1 \gamma_2 (K_1 - \nu_1/r)/a(s)) G_1(s)} {(\gamma_{3-j} - \gamma_{j}) a^{\gamma_j}(s)},\end{align}
\begin{align} &\Psi_{1, j}(s, a(s)) = \dfrac{((\gamma_{3-j} - 1) a(s) - \gamma_{3-j} (K_1 - \nu_1/r)) G'_{\!\!1}(s)} {(\gamma_{3-j} - \gamma_{j}) a^{\gamma_j}(s)} \end{align}
for
$s > 0$
, and
\begin{align} &\Phi_{2, j}(q, b(q)) = \dfrac{((\gamma_1 - 1)(\gamma_2 - 1) - \gamma_1 \gamma_2 (K_2 + \nu_2/r) / b(q)) F_2(q)}{(\gamma_{3-j} - \gamma_{j}) \, b^{\gamma_j}(q)}, \end{align}
\begin{align} &\Psi_{2, j}(q, b(q)) = \dfrac{((\gamma_{3-j} - 1) b(q) - \gamma_{3-j} (K_2 + \nu_2/r)) F'_{\!\!2}(q)} {(\gamma_{3-j} - \gamma_{j}) b^{\gamma_j}(q)} \end{align}
for
$q > 0$
and every
$j = 1, 2$
. We have also used the obvious facts that
$F'_{\!\!i}(s) = - G'_{\!\!i}(s)$
, for all
$s > 0$
, and
$G'_{\!\!i}(q) = - F'_{\!\!i}(q)$
, for all
$q > 0$
, by virtue of the definition of the function
$G_i(x) = 1 - F_i(x)$
, for all
$x > 0$
, and every
$i = 1, 2$
.
3.3. The maximal and minimal admissible solutions
$\boldsymbol{a}^*(\boldsymbol{s})$
and
$\boldsymbol{b}^*(\boldsymbol{q})$
We further consider the maximal and minimal admissible solutions of first-order nonlinear ordinary differential equations as the largest and smallest possible solutions
$a^*(s)$
and
$b^*(q)$
of the equations in (3.14) and (3.15) with (3.16)–(3.17) and (3.18)–(3.19) which satisfy the inequalities
$a^*(s) < s \wedge {\overline a}$
and
$b^*(q) > q \vee {\underline b}$
with
${\overline a} = (r K_1 - \nu_1)/\delta$
and
${\underline b} = (r K_2 + \nu_2)/\delta$
, for all
$s > {\underline s}$
and
$q < {\overline q}$
and some
$0 \le {\underline s} \le s'$
and
${\overline q} \ge q'$
fixed, with
$s' = (K_1 - \alpha_1)/(1 + \beta_1)$
under
$K_1 > \alpha_1$
and
$q' = (K_2 - \alpha_2)/(1 + \beta_2)$
under
$K_2 > \alpha_2$
, respectively. 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 (3.14) and (3.15) with (3.16)–(3.17) and (3.18)–(3.19) are (locally) continuous in (s, a(s)) and (q, b(q)) and (locally) Lipschitz in a(s) and b(q), for each
$s > {\underline s}$
and
$q < {\overline q}$
fixed (see also [Reference Peskir33, Section 3.9] for similar arguments based on the analysis of other first-order nonlinear ordinary differential equations). Then it is shown by means of technical arguments based on Picard’s method of successive approximations that there exist unique solutions a(s) and b(q) to the equations in (3.10) and (3.11) with (3.12)–(3.2) and (3.13)–(3.2), for
$s > {\underline s}$
and
$q < {\overline q}$
, started at some points
$(s_0, s_0)$
and
$(q_0, q_0)$
such that
$s_0 > {\underline s}$
and
$q_0 < {\overline q}$
(see also [Reference Graversen and Peskir21, Section 3.2] and [Reference Peskir33, 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
$a^*(s)$
and
$b^*(q)$
which satisfy (3.14) and (3.15) and which stay strictly above and below the appropriate diagonal, for
$s > {\underline s}$
and
$q < {\overline q}$
, respectively, we can follow the arguments from [Reference Peskir36, Section 3.5] (among others). These are based on the construction of sequences of the so-called bad–good solutions that intersect the diagonals. For this purpose, for any sequences
$(s_{l})_{l \in {\mathbb N}}$
and
$(q_{l})_{l \in {\mathbb N}}$
such that
$s_{l} > {\underline s}$
and
$q_{l} < {\overline q}$
as well as
$s_{l} \uparrow \infty$
and
$q_{l} \downarrow 0$
as
$l \to \infty$
, we can construct the sequence of solutions
$a_l(s)$
and
$b_l(q)$
,
$l \in {\mathbb N}$
, to the equations (3.14) and (3.15), for all
$s > {\underline s}$
and
$q < {\overline q}$
such that
$a_l(s_l) = s_l$
and
$b_l(q_l) = q_l$
hold, for each
$l \in {\mathbb N}$
. It follows from the structure of the equations in (3.14) and (3.15) as well as the functions in (3.16)–(3.17) and (3.18)–(3.19) that the properties
$a'_{\!\!l}(s_{l}) < 1$
and
$b'_{\!\!l}(q_{l}) < 1$
hold, for each
$l \in {\mathbb N}$
(see also [Reference Pedersen32, pp. 979–982] for the analysis of solutions of another first-order nonlinear differential equation). Observe that by virtue of the uniqueness of solutions mentioned above, we know that each pair of curves
$s \mapsto a_{l}(s)$
and
$s \mapsto a_{m}(s)$
as well as
$q \mapsto b_{l}(q)$
and
$q \mapsto b_{m}(q)$
cannot intersect, for
$l, m \in {\mathbb N}$
,
$l \neq m$
, and thus we see that the sequence
$(a_{l}(s))_{l \in {\mathbb N}}$
is increasing and the sequence
$(b_{l}(q))_{l \in {\mathbb N}}$
is decreasing, so that the limits
$a^*(s) = \lim_{l \to \infty} a_{l}(s)$
and
$b^*(q) = \lim_{l \to \infty} b_{l}(q)$
exist, for each
$s > {\underline s}$
and
$q < {\overline q}$
, respectively. We may therefore conclude that
$a^*(s)$
and
$b^*(q)$
provide the maximal and minimal solutions to the equations in (3.14) and (3.15) such that
$a^*(s) < s \wedge {\overline a}$
and
$b^*(q) > q \vee {\underline b}$
hold, for all
$s > {\underline s}$
and
$q < {\overline q}$
.
Moreover, since the right-hand sides of the first-order nonlinear ordinary differential equations in (3.14) and (3.15) with (3.16)–(3.17) and (3.18)–(3.19) are (locally) Lipschitz in s and q, respectively, one can deduce by means of Gronwall’s inequality that the functions
$a_{l}(s)$
and
$b_{l}(q)$
,
$l \in {\mathbb N}$
, are continuous, so that the functions
$a^*(s)$
and
$b^*(q)$
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 Peskir33] and further developed in [Reference Baurdoux and Kyprianou5], [Reference Gapeev13], [Reference Gapeev and Rodosthenous16, Reference Gapeev and Rodosthenous17, Reference Gapeev and Rodosthenous18], [Reference Gapeev, Kort and Lavrutich19], [Reference Glover, Hulley and Peskir20], [Reference Graversen and Peskir21], [Reference Guo and Shepp22], [Reference Guo and Zervos23], [Reference Kyprianou and Ott28], [Reference Ott31], [Reference Pedersen32], [Reference Peskir35, Reference Peskir36], and [Reference Rodosthenous and Zervos39], among other subsequent papers (see also [Reference Peskir and Shiryaev37, Chapter I; Chapter V, Section 17] for other references).
4. Main results and proofs
In this section, based on the expressions computed above, we formulate and prove the main results of the paper, which are based on the verification of the fact that the solution of the free-boundary problems in (2.17)–(2.27) provides the solutions of the optimal stopping problems of (2.4) and (2.5). Note that it also follows from the maximality principle established in [Reference Peskir33, Theorem 3.1] (see also [Reference Peskir33, Corollary 3.2] for the case of positive processes) that the solutions of the systems in (2.17)–(2.18) and (2.19)–(2.23) associated with the maximal and minimal admissible solutions of the first-order nonlinear ordinary differential equations in (3.14) and (3.15) satisfy (2.24)–(2.25) and (2.26)–(2.27). Recall that the existence of solutions of the optimal stopping problems in (2.4) and (2.5) follows from the results of [Reference Cvitanić and Karatzas9, Theorem 4.1], based on the solutions of the associated (doubly) reflected backward stochastic differential equations.
Theorem 4.1. Suppose that the assumptions of Lemma 2.1 are satisfied. Then the value functions of the perpetual American cancellable put and call option optimal stopping problems in (2.4) and (2.5) are given by
\begin{equation*} V^*_1(x, s)= \begin{cases} V_1(x, s;\; a^*(s)) \,\,\, {if} \, a^*(s) < x \le s \, {and} s > {\underline s}, \\[5pt] (K_1 - x) G_1(s) \,\,\, {if} \, 0 < x \le a^*(s) \, {and} \, s > {\underline s}, \\[5pt] (K_1 - x) G_1(s) \,\,\, {if} \, 0 < x \le s \le {\underline s}, \end{cases}\end{equation*}
and
\begin{equation} V^*_2(x, q)= \begin{cases} V_2(x, q;\; b^*(q)) \,\,\, {if} \, q \le x < b^*(q) \, {and} \, 0 < q < {\overline q}, \\[5pt] (x - K_2) F_2(q) \,\,\, {if} x \ge b^*(q) \, {and} \, 0 < q < {\overline q}, \\[5pt] (x - K_2) F_2(q) \,\,\, {if} \, x \ge q \ge {\overline q}, \end{cases}\end{equation}
and the optimal exercise times have the form (2.16). Here the functions
$V_1(x, s;\; a(s))$
and
$V_2(x, q;\; b(q))$
are given by (3.8) and (3.10) with (3.9) and (3.11), and the optimal exercise boundaries
$a^*(s)$
and
$b^*(q)$
provide the maximal and minimal solutions of the first-order nonlinear ordinary differential equations in (3.14) and (3.15) with (3.16)–(3.17) and (3.18)–(3.19), satisfying the inequalities
$a^*(s) < s \wedge {\overline a}$
with
${\overline a} = (r K_1 - \nu_1)/\delta$
and
$b^*(q) > q \vee {\underline b}$
with
${\underline b} = (r K_2 + \nu_2)/\delta$
, for all
$s > {\underline s}$
and
$q < {\overline q}$
and some
$0 \le {\underline s} \le s' \wedge {\overline a}$
and
${\overline q} \ge q' \vee {\underline b}$
fixed, with
$s' = (K_1 - \alpha_1)/(1 + \beta_1)$
and
$q' = (K_2 - \alpha_2)/(1 + \beta_2)$
. Further, the equalities
$a^*(s) = s$
and
$b^*(q) = q$
hold, for all
$s \le {\underline s}$
and
$q \ge {\overline q}$
, respectively.
Since the two assertions stated above are proved using similar arguments, we only give a proof for the case of the two-dimensional optimal stopping problem of (2.5) related to the dividend-paying perpetual American cancellable call option. Observe that we can put
$s = x$
and
$q = x$
to obtain the values of the original perpetual American cancellable option pricing problems of (2.2) and (2.3) from the values of the optimal stopping problems of (2.4) and (2.5).
Proof.In order to verify the assertion stated above, it remains for us to show that the function defined in (4.1) coincides with the value function in (2.5), and that the stopping time
$\tau^*_2$
in (2.16) is optimal with the boundary
$b^*(q)$
specified above. For this purpose, let b(q) be any solution of the ordinary differential equation in (3.15) satisfying the inequality
$b(q) > q \vee {\underline b}$
, for all
$q < {\overline q}$
and some
${\overline q} \ge q' \vee {\underline b}$
fixed. Also, let
$V^b_2(x, q)$
denote the right-hand side of (4.1) associated with b(q). Then it is shown by means of straightforward calculations from the previous section that the function
$V^b_2(x, q)$
solves the system of (2.18) with the right-hand sides of (2.22)–(2.25) and (2.27) and satisfies the right-hand conditions of (2.19)–(2.21). Recall that the function
$V^b_2(x, q)$
is
$C^{2,1}$
on the closure
${\overline C}_2$
of
$C_2$
and is equal to
$(x - K_2) F_2(q)$
on
$D_2$
, which are defined as
${\overline C}^*_2$
,
$C^*_2$
and
$D^*_2$
in (2.14) and (2.15) with b(q) instead of
$b^*(q)$
, respectively. Hence, taking into account the assumption that the boundary b(q) is continuously differentiable, for all
$q < {\overline q}$
, by applying the change-of-variable formula from [Reference Peskir34, Theorem 3.1] to the process
${\mathrm{e}}^{- r t} V^b_2(X_t, Q_t)$
(see also [Reference Peskir and Shiryaev37, Chapter II, Section 3.5] for a summary of the related results and further references), we obtain
\begin{align}&{\mathrm{e}}^{- r t} \, V^b_2(X_t, Q_t) = V^b_2(x, q) \notag \\[5pt]\notag& \quad + \int_0^t {\mathrm{e}}^{- r u} \, \bigl({\mathbb L} V^b_2 - r V^b_2\bigr)(X_{u}, Q_{u}) \, I(X_{u} \neq b(Q_u), X_u \neq Q_u) \,{\mathrm{d}} u \\[5pt]& \quad + \int_0^t {\mathrm{e}}^{- r u} \, \partial_q V^b_2(X_{u}, Q_{u}) \, I(X_u = Q_u) \,{\mathrm{d}} Q_u + M^2_t\end{align}
for all
$t \ge 0$
. Here the process
$M^2 = (M^2_t)_{t \ge 0}$
, defined by
\begin{equation}M^2_t = \int_0^t {\mathrm{e}}^{- r u} \, \partial_x V^b_2(X_u, Q_u) \, I(X_{u} \neq Q_{u}) \, \sigma \, X_u \,{\mathrm{d}} B_u,\end{equation}
is a continuous local martingale with respect to the probability measure
${\mathbb P}_{x, q}$
. Note that since the time spent by the process (X, Q) at the boundary surface
$\partial C_2 = \{ (x, q) \in E_2 \mid x = b(q) \}$
, as well as at the diagonal
$d_2 = \{ (x, q) \in E_2 \mid x = q \}$
, is of the Lebesgue measure zero (see e.g. [Reference Borodin and Salminen8, Chapter II, Section 1]), the indicators in the first line of the formula in (4.2) as well as in (4.3) can be ignored. Moreover, since the component Q decreases only when the process (X, Q) is located on the diagonal
$d_2 = \{ (x, q) \in E_2 \mid x = q \}$
, the indicator in the second line of (4.2) and the one in (4.3) can also be set equal to one. Observe that the integral in the second line of (4.2) will actually be compensated accordingly, due to the fact that the candidate value function
$V^b_2(x, q)$
satisfies the modified normal-reflection condition from the right-hand side of (2.21) at the diagonal
$d_2$
.
It follows from straightforward calculations and the arguments of the previous section that the function
$V^b_2(x, q)$
satisfies the second-order ordinary differential equation in (2.18) which, together with the conditions of (2.19)–(2.20) and (2.22) as well as the fact that inequality (2.27) holds, implies that the inequality
$({\mathbb L} V^b_2 - r V^b_2)(x, q) \le - \nu_2 F_2(q)$
is satisfied, for all
$0 < q < x$
such that
$q < {\overline q}$
and
$x \neq b(q)$
. Moreover, we observe directly from (3.10) with (3.11) as well as (3.12) and (3.13) that the function
$V^{b}_2(x, q) - (x - K_2) F_2(q)$
is convex and decreases to zero, because its first-order partial derivative
$\partial_x V^{b}_2(x, q) - F_2(q)$
is negative and increases to zero, while its second-order partial derivative
$\partial_{xx} V^{b}_2(x, q)$
is positive, on the interval
$q \le x < b(q)$
, for each
$q < {\overline q}$
fixed. Thus we may conclude that the right-hand inequality in (2.25) holds, which together with the right-hand conditions of (2.19)–(2.20) and (2.22) implies that the inequality
$V^{b}_2(x, q) \ge (x - K_2) F_2(q)$
is satisfied, for all
$(x, q) \in E_2$
. Let
$(\varkappa_n)_{n \in {\mathbb N}}$
be the localising sequence of stopping times for the process
$M^2$
from (4.3) such that
$\varkappa_n = \inf\{t \ge 0 \mid |M^2_t| \ge n \}$
, for each
$n \in {\mathbb N}$
. It therefore follows from (4.2) that we have the inequalities
\begin{align}&{\mathrm{e}}^{- r (\tau \wedge \varkappa_n)} \, (X_{\tau \wedge \varkappa_n} - K_2) \, F_2(Q_{\tau \wedge \varkappa_n})\notag \\[5pt]\notag& \quad - \int_0^{\tau \wedge \varkappa_n} {\mathrm{e}}^{- r u} \, (\alpha_2 + \beta_2 \, Q_u) \, G'_{\!\!2}(Q_u) \,{\mathrm{d}} Q_u+ \int_0^{\tau \wedge \varkappa_n} {\mathrm{e}}^{- r u} \, \nu_2 \, F_2(Q_u) \,{\mathrm{d}} u \\[5pt]\notag& \le {\mathrm{e}}^{- r (\tau \wedge \varkappa_n)} \, V_2(X_{\tau \wedge \varkappa_n}, Q_{\tau \wedge \varkappa_n}) \\[5pt]\notag& \quad - \int_0^{\tau \wedge \varkappa_n} {\mathrm{e}}^{- r u} \, (\alpha_2 + \beta_2 \, Q_u) \, G'_{\!\!2}(Q_u) \,{\mathrm{d}} Q_u+ \int_0^{\tau \wedge \varkappa_n} {\mathrm{e}}^{- r u} \, \nu_2 \, F_2(Q_u) \,{\mathrm{d}} u \\[5pt]& \le V^b_2(x, q) + M^2_{\tau \wedge \varkappa_n}\end{align}
for any stopping time
$\tau$
of the process X and each
$n \in {\mathbb N}$
fixed. Then, taking the expectation with respect to
${\mathbb P}_{x, q}$
in (4.4), by means of Doob’s optional sampling theorem, we get
\begin{align}&{\mathbb E}_{x, q} \biggl[ {\mathrm{e}}^{- r ({\tau \wedge \varkappa_n})} \, (X_{\tau \wedge \varkappa_n} - K_2) \, F_2(Q_{\tau \wedge \varkappa_n}) \notag \\[5pt]\notag&\phantom{{\mathbb E}_{x, q} \biggl[\,}- \int_0^{\tau \wedge \varkappa_n} {\mathrm{e}}^{- r u} \, (\alpha_2 + \beta_2 \, Q_u) \, G'_{\!\!2}(Q_u) \,{\mathrm{d}} Q_u+ \int_0^{\tau \wedge \varkappa_n} {\mathrm{e}}^{- r u} \, \nu_2 \, F_2(Q_u) \,{\mathrm{d}} u \biggr] \\[5pt]\notag& \le {\mathbb E}_{x, q}\biggl[ {\mathrm{e}}^{- r ({\tau \wedge \varkappa_n})} \, V^b_2(X_{\tau \wedge \varkappa_n}, Q_{\tau \wedge \varkappa_n}) \\[5pt]\notag&\phantom{\le {\mathbb E}_{x, q} \biggl[\,}- \int_0^{\tau \wedge \varkappa_n} {\mathrm{e}}^{- r u} \, (\alpha_2 + \beta_2 \, Q_u) \, G'_{\!\!2}(Q_u) \,{\mathrm{d}} Q_u+ \int_0^{\tau \wedge \varkappa_n} {\mathrm{e}}^{- r u} \, \nu_2 \, F_2(Q_u) \,{\mathrm{d}} u \biggr] \\[5pt]& \le V^b_2(x, q) + {\mathbb E}_{x, q} \big[ M^2_{\tau \wedge \varkappa_n} \big] \notag \\[5pt]&= V^b_2(x, q)\end{align}
for all
$0 < q \le x$
such that
$q < {\overline q}$
, and each
$n \in {\mathbb N}$
. Hence, letting n go to infinity and using Fatou’s lemma, we obtain from (4.5) the inequalities
\begin{align}&{\mathbb E}_{x, q} \biggl[ {\mathrm{e}}^{- r \tau} \, (X_{\tau} - K_2) \, F_2(Q_{\tau}) \notag \\[5pt]\notag&\phantom{{\mathbb E}_{x, q} \biggl[\,}- \int_0^{\tau} {\mathrm{e}}^{- r u} \, (\alpha_2 + \beta_2 \, Q_u) \, G'_{\!\!2}(Q_u) \,{\mathrm{d}} Q_u+ \int_0^{\tau} {\mathrm{e}}^{- r u} \, \nu_2 \, F_2(Q_u) \,{\mathrm{d}} u \biggr] \\[5pt]\notag& \le {\mathbb E}_{x, q} \biggl[ {\mathrm{e}}^{- r \tau} \, V^b_2(X_{\tau}, Q_{\tau}) \\[5pt]\notag&\phantom{\le {\mathbb E}_{x, q} \biggl[\,} - \int_0^{\tau} {\mathrm{e}}^{- r u} \, (\alpha_2 + \beta_2 \, Q_u) \, G'_{\!\!2}(Q_u) \,{\mathrm{d}} Q_u+ \int_0^{\tau} {\mathrm{e}}^{- r u} \, \nu_2 \, F_2(Q_u) \,{\mathrm{d}} u \biggr] \\[5pt]& \le V^b_2(x, q)\end{align}
for any stopping time
$\tau$
, and all
$0 < q \le x$
such that
$q < {\overline q}$
. Thus, taking the supremum over all stopping times
$\tau$
and then the infimum over all boundaries b in (4.6), we obtain the inequalities
\begin{align}&\sup_{\tau} {\mathbb E}_{x, q} \biggl[ {\mathrm{e}}^{- r \tau} \, (X_{\tau} - K_2) \, F_2(Q_{\tau})- \int_0^{\tau} {\mathrm{e}}^{- r u} \, (\alpha_2 + \beta_2 \, Q_u) \, G'_{\!\!2}(Q_u) \,{\mathrm{d}} Q_u \notag \\[5pt]&\phantom{\sup_{\tau} {\mathbb E}_{x, q} \biggl[\,}+ \int_0^{\tau} {\mathrm{e}}^{- r u} \, \nu_2 \, F_2(Q_u) \,{\mathrm{d}} u \biggr]\le \inf_b V^b_2(x, q) = V^{b^*}_2(x, q)\end{align}
for all
$0 < q \le x$
such that
$q < {\overline q}$
, where
$b^*(q)$
is the minimal solution of the ordinary differential equation in (3.15) and satisfies the inequality
$b^*(q) > q \vee {\underline b}$
, for all
$q < {\overline q}$
. By using the fact that the function
$V^b_2(x, q)$
is (strictly) increasing in the value b(q), for each
$q < {\overline q}$
fixed, we see that the infimum in (4.7) is attained over any sequence of solutions
$(b_m(q))_{m \in {\mathbb N}}$
to (3.15) satisfying the inequality
$b_m(q) > q \vee {\underline b}$
, for all
$q < {\overline q}$
, for each
$m \in {\mathbb N}$
, and such that
$b_m(q) \downarrow b^*(q)$
as
$m \to \infty$
, for each
$q < {\overline q}$
fixed. It follows from the (local) uniqueness of the solutions to the first-order (nonlinear) ordinary differential equation in (3.15) that no distinct solutions intersect, so that the sequence
$(b_m(q))_{m \in {\mathbb N}}$
is decreasing and the limit
$b^*(q) = \lim_{m \to \infty} b_m(q)$
exists, for each
$q < {\overline q}$
fixed. Since the inequalities in (4.6) hold for
$b^*(q)$
too, we see that (4.7) holds for
$b^*(q)$
and
$(x, q) \in E_2$
as well. We also note from inequality (4.5) that the function
$V^b_2(x, q)$
is superharmonic for the Markov process (X, Q) on
$E_2$
. Hence, taking into account the fact that
$V^b_2(x, q)$
is increasing in
$b(q) > q \vee {\underline b}$
, for all
$q < {\overline q}$
, and the inequality
$V^b_2(x, q) \ge (x - K_2) F_2(q)$
holds, for all
$(x, q) \in E_2$
, we observe that the selection of the minimal solution
$b^*(q)$
that stays strictly above the diagonal
$d_2 = \{ (x, q) \in E_2 \mid x = q \}$
and the level
$x = {\underline b}$
is equivalent to the implementation of the superharmonic characterisation of the value function as the smallest superharmonic function dominating the payoff function (see [Reference Peskir33] or [Reference Peskir and Shiryaev37, Chapter I; Chapter V, Section 17]).
In order to prove the fact that the boundary
$b^*(q)$
is optimal, we consider the sequence of stopping times
$\tau_m$
,
$m \in {\mathbb N}$
, defined as in the right-hand part of (2.16) with
$b_m(q)$
instead of
$b^*(q)$
, where
$b_m(q)$
is a solution to the first-order ordinary differential equation in (3.15), and such that
$b_m(q) \downarrow b^*(q)$
as
$m \to \infty$
, for each
$q < {\overline q}$
fixed. Then, by virtue of the fact that the function
$V^{b_m}_2(x, q)$
from the right-hand side of (4.1) associated with the boundary
$b_m(q)$
satisfies the equation of (2.18) and the condition of (2.19), and taking into account the structure of
$\tau^*_2$
in (2.16), it follows from an expression equivalent to (4.2) that we have
\begin{align}&{\mathrm{e}}^{- r (\tau_m \wedge \varkappa_n)} \, (X_{\tau_m \wedge \varkappa_n} - K_2) \, F_2(Q_{\tau_m \wedge \varkappa_n}) \notag \\[5pt]\notag& \quad- \int_0^{\tau_m \wedge \varkappa_n} {\mathrm{e}}^{- r u} \, (\alpha_2 + \beta_2 \, Q_u) \, G'_{\!\!2}(Q_u) \,{\mathrm{d}} Q_u+ \int_0^{\tau_m \wedge \varkappa_n} {\mathrm{e}}^{- r u} \, \nu_2 \, F_2(Q_u) \,{\mathrm{d}} u \\[5pt]\notag& = {\mathrm{e}}^{- r (\tau_m \wedge \varkappa_n)} \, V^{b_m}_2(X_{\tau_m \wedge \varkappa_n}, Q_{\tau_m \wedge \varkappa_n}) \\[5pt]\notag& =- \int_0^{\tau_m \wedge \varkappa_n} {\mathrm{e}}^{- r u} \, (\alpha_2 + \beta_2 \, Q_u) \, G'_{\!\!2}(Q_u) \,{\mathrm{d}} Q_u+ \int_0^{\tau_m \wedge \varkappa_n} {\mathrm{e}}^{- r u} \, \nu_2 \, F_2(Q_u) \,{\mathrm{d}} u \\[5pt]& = V^{b_m}_2(x, q) + M^2_{\tau_m \wedge \varkappa_n}\end{align}
for all
$0 < q \le x$
such that
$q < {\overline q}$
and each
$n, m \in {\mathbb N}$
. Observe that by virtue of the arguments from [Reference Shiryaev43, Chapter VIII, Section 2a], we have the property
\begin{align*} &{\mathbb E}_{x, q} \biggl[ \sup_{t \ge 0} \biggl( {\mathrm{e}}^{- r (\tau^*_2 \wedge t)} \, \bigl(X_{\tau^*_2 \wedge t} - K_2\bigr) \,F_2\bigl(Q_{\tau^*_2 \wedge t}\bigr) \\[5pt]&\phantom{{\mathbb E}_{x, q} \biggl[\,}- \int_0^{\tau^*_2 \wedge t} {\mathrm{e}}^{- r u} \, (\alpha_2 + \beta_2 \, Q_u) \, G'_{\!\!2}(Q_u) \,{\mathrm{d}} Q_u +\int_0^{\tau^*_2 \wedge t} {\mathrm{e}}^{- r u} \, \nu_2 \, F_2(Q_u) \,{\mathrm{d}} u \biggr) \biggr] < \infty\end{align*}
for all
$(x, q) \in E_2$
, and the variable
${\mathrm{e}}^{- r \tau^*_2} (X_{\tau^*_2} - K_2) F_2(Q_{\tau^*_2})$
is equal to zero on the event
$\{ \tau^*_2 = \infty \}$
(
${\mathbb P}_{x, q}$
-a.s.), because the value
$b^*(0+\!)$
is finite. Hence, letting m and n go to infinity and using the condition of (2.19) together with the property
$\tau_m \downarrow \tau^*_2$
(
${\mathbb P}_{x, q}$
-a.s.) as
$m \to \infty$
, we can apply the Lebesgue-dominated convergence theorem to the appropriate (diagonal) subsequence in (4.8) to obtain the equality
\begin{align*} &{\mathbb E}_{x, q} \biggl[ {\mathrm{e}}^{- r \tau^*_2} \, \bigl(X_{\tau^*_2} - K_2\bigr) \, F_2\bigl(Q_{\tau^*_2}\bigr) \\[5pt]&\phantom{{\mathbb E}_{x, q} \biggl[\,} - \int_0^{\tau^*_2} {\mathrm{e}}^{- r u} \, (\alpha_2 + \beta_2 \, Q_u)\, G'_{\!\!2}(Q_u) \,{\mathrm{d}} Q_u + \int_0^{\tau^*_2} {\mathrm{e}}^{- r u} \, \nu_2 \, F_2(Q_u) \,{\mathrm{d}} u \biggr]= V^{b^*}_2(x, q)\end{align*}
for all
$0 < x \le q$
such that
$q < {\overline q}$
, which together with (4.7) directly implies the desired assertion.
Acknowledgements
The authors are grateful to the Editors for their encouragement to prepare a revised version, and two anonymous referees for their valuable suggestions which helped to improve the motivation and presentation of the paper. The paper was essentially written during the time when the second author was visiting the Department of Mathematics at the London School of Economics and Political Science, and she is grateful for the hospitality.
Funding information
There are no funding bodies to thank relating to this creation of this article.
Competing interests
There were no competing interests to declare which arose during the preparation or publication process of this article.
