Discretionary stopping of stochastic differential equations with generalised drift

We consider the problem of optimally stopping a general one-dimensional stochastic differential equation (SDE) with generalised drift over an infinite time horizon. First, we derive a complete characterisation of the solution to this problem in terms of variational inequalities. In particular, we prove that the problem's value function is the difference of two convex functions and satisfies an appropriate variational inequality in the sense of distributions. We also establish a verification theorem that is the strongest one possible because it involves only the optimal stopping problem's data. Next, we derive the complete explicit solution to the problem that arises when the state process is a skew geometric Brownian motion and the reward function is the one of a financial call option. In this case, we show that the optimal stopping strategy can take several qualitatively different forms, depending on parameter values. Furthermore, the explicit solution to this special case reveals that the so-called"principle of smooth fit"does not hold in general for this type of optimal stopping problems in standard senses that this can be formulated.


Introduction
We consider the optimal stopping of the one-dimensional SDE with generalised drift where L z is the symmetric local time of X at level z, W is a standard one-dimensional Brownian motion andI = ]ι, ι[ is the interior of a given interval I ⊆ [−∞, ∞]. We assume that the signed Radon measure ν and the Borel-measurable function σ :I → R satisfy suitable conditions ensuring that the SDE (1) has a weak solution Ω, F , (F t ), P x , W, X that is unique in the sense of probability law up to a possible explosion time at which X hits the boundary {ι, ι} of I (see Assumption 1 in Section 2). If the boundary point ι (resp., ι) is inaccessible, then the interval I is open from the left (resp., open from the right).
On the other hand, if the boundary point ι (resp., ι) is not inaccessible, then we assume that it is absorbing and the interval I is closed from the left (resp., closed from the right).
Comprehensive studies of these SDEs as well as relevant literature surveys can be found in Engelbert and Schmidt [13], and Lejay [21]. The objective of the optimal stopping problem that we study aims at maximising the performance criterion over all stopping times τ , where the positive Borel-measurable discounting rate function r satisfies Assumption 2 in Section 2, while the positive and possibly unbounded reward function f satisfies Assumption 3 in Section 3. To the best of our knowledge, there exist no results in the literature addressing the solvability of such a problem by means of variational inequalities when ν is not absolutely continuous, even in special cases. We derive a complete characterisation of the solution to this problem in terms of variational inequalities. In particular, we prove that the value function v of the optimal stopping problem associated with (1) and (2) is the difference of two convex functions and satisfies the following variational inequality in the sense of distributions (see Definition 1 and Theorem 3.(I)-(II) in Section 3): where p is the scale function of the diffusion associated with the SDE (1). We also establish a verification theorem that is the strongest one possible because it involves only the optimal stopping problem's data. In particular, we derive a simple necessary and sufficient condition for a solution to (3) to identify with the problem's value function (see Theorem 3.(III)). The second main contribution of the paper is to derive the complete explicit solution to the special case that arises if f (x) = (x − K) + , for some constant K > 0, and X is a skew geometric Brownian motion, which is characterised by the choices ν(dx) = b σ 2 x dx + β δ z (dx), σ(x) = σx and I = ]0, ∞[, where b ∈ R, β ∈ ]−1, 1[ \ {0}, σ = 0 are constants and δ z (dx) is the Dirac probability measure that assigns mass 1 at {z}, for some z > 0. In this case, the SDE dX t = bX t dt + β dL z t + σX t dW t , X 0 = x > 0, (4) has a unique non-explosive strong solution. Given such a solution X, which exists on any given filtered probability space Ω, F , (F t ), P satisfying the usual conditions and supporting a standard one-dimensional (F t )-Brownian motion W , the value function of the discretionary stopping problem we solve is defined by where T is the family of all (F t )-stopping times and r, K > 0 are constants (we write E in place of E x because we consider strong rather than weak solutions here). We prove that the optimal stopping strategy can take several qualitatively different forms, depending on parameter values (see . In contrast to the optimal stopping of an SDE with absolutely continuous drift and reward function such as the one of a financial call option, the optimal stopping region may involve two distinct components, one of which may be an isolated point. Furthermore, the analysis of this problem reveals that the so-called "principle of smooth fit" does not hold even in a case of a "right-sided" optimal stopping strategy in the sense that none of the functions is continuous, where p (resp., ψ) is the scale function (resp., the increasing minimal excessive function) of the diffusion associated with the SDE (4) (see Remarks 3 and 5).
To derive the solution to the optimal stopping problem defined by (4) and (5), we need to consider a partition of the set R × (R \ {0}) × ]0, ∞[ × ]−1, 1[\{0} in which the parameter vector (b, σ, z, β) takes values into four sets (see Cases (I)-(IV) of Lemma 4). In Cases (I) and (II), β ∈ ]−1, 0[ and ψ is convex. In Cases (III) and (IV), β ∈ ]−1, 0[ and β ∈ ]0, 1[, respectively, and ψ fails to be convex. The best part of our analysis' complexity (including the whole of Section 5) is due to the facts that (a) the optimal strategy takes several qualitatively different forms, and (b) we establish necessary and sufficient conditions on the problem's data that differentiate between the different possible cases without leaving any "gap" in the parameter space. The solution to the problem in the easier Cases (I), (II) and (IV) has been presented in the PhD thesis Lon [22]. Although the analysis of Case (III) is not in itself harder, linking it with Cases (I) and (II) with necessary and sufficient conditions on the problem's data requires rather tedious analysis, due to the complex structure of ψ (see Figure 2). The solution to all cases was announced without proofs in the conference paper Lon, Rodosthenous and Zervos [23].
Variational inequalities take center stage in the continuous time optimal stopping theory because they are efficient for the investigation of specific problems. In the context of this paper, they can be used to easily identify critical parts of the state space I that are subsets of the so-called waiting region in a systematic way that involves no guesswork. For instance, the regularity of v implies that all points at which the reward function f is discontinuous as well as all "minimal" intervals in which f cannot be expressed as the difference of two convex functions (e.g., intervals in which f has the regularity of a Brownian sample path) should be parts of the closure of the waiting region. For further analysis and discussion in this direction, see Remark 2 at the end of Section 3. Beyond its usefulness in identifying optimal stopping strategies, the variational inequality characterisation is also very effective in verifying whether a candidate function identifies with the value function of a specific problem because, in the context of diffusions, it has a local character in the sense that it involves only derivatives.
The solution to optimal stopping problems using classical solutions to variational inequalities has been extensively studied. Results in Friedman [ [29], listed in chronological order, typically make strong regularity assumptions on the problem data (e.g., the problem data are assumed to be Lipschitz continuous). To relax such assumptions, Øksendal and Reikvam [27] and Bassan and Ceci [2] have considered viscosity solutions to the variational inequalities associated with the optimal stopping problems that they study. Results with minimal assumptions on the problem data, such as the ones that we derive here for the optimal stopping problem associated with (1) and (2), have been obtained by Lamberton [18] and Lamberton and Zervos [20] who consider the optimal stopping of the one-dimensional SDE which is the special case of (1) arising when ν(dx) = b(x)σ −2 (x) dx, over a finite and an infinite time horizon, respectively. Recently, the solution to suitable optimal stopping problems by means of variational inequalities has been used to characterise the boundary of Root's solution to the Skorokhod embedding problem (see Cox and Wang [8], Cox, Ob lój and Touzi [7], and references therein). Furthermore, variational inequalities arise most naturally in the study of optimal stopping problems involving controlled stochastic processes (e.g., see Bensoussan and Lions [4,Chapter 4], Krylov [17,Chapters 3,6], Beneš [3], Davis and Zervos [10], Karatzas and Sudderth [16], listed in chronological order, as well as many more recent contributions).
There exist few references in the literature addressing special cases of a general problem such as the one we study. Peskir [28] (see also Peskir and Shiryaev [29,Section IV.9.3]) considered the validity of the "principle of smooth fit" in terms of derivatives such as the first two ones in (6). In particular, Peskir [28] considered the optimal stopping of the process X = F (B), where B is a standard Brownian motion, for three examples of a function F that is C ∞ except at 0, at which it is only continuous. An application of Itô's formula reveals that the dynamics of such a state process X, which was termed "diffusion with angles", involves the local time of B. Therefore, such a process is not the solution of an SDE with generalised drift because the local times in (1) and (4) are the local times of the state process X and not the local time of its driving Brownian motion W . Nevertheless, the observations made by Peskir [28] are consistent with the results we derive here.
Recent references have considered the optimal stopping of a skew Brownian motion, which is given by the strong solution to the SDE with generalised drift with the objective to maximise the performance index given by (2) for r > 0 being a constant and for f being an increasing function associated with "right-sided" optimal stopping strategies. Crocce and Mordecki [9] studied the validity of the "principle of smooth fit" in terms of derivatives such as the ones given by (6), and presented two examples with optimal stopping strategies such as the ones in Theorem 7 that are illustrated by Figures 5 and 10 below. Alvarez and Salminen [1] derived sufficient conditions on f that are associated with optimal stopping strategies of the same qualitative nature as the ones in Theorems 7 and 9 that are illustrated by Figures 10, 12 and 13 below. Using a state space transformation of the form χ = ln x, we can see that the problem given by (4) and (5) corresponds to the problem analysed by Alvarez and Salminen [1] for the choices (the parameter β that we use here corresponds to 2β − 1 in their parametrisation). The analysis in these references is based on Dynkin's characterisation of an optimal stopping problem's value function as the minimal excessive majorant of its reward function and the Martin representation theory of excessive functions. In contrast to variational inequalities, which involve only the problem's primary data, this approach has a non-local character or it requires conditions involving the elements of the set As a consequence, its applicability has largely been limited to problems with "one-sided" optimal stopping strategies because, with notable exceptions such as the ones associated with a Brownian motion or a geometric Brownian motion, the minimal excessive functions are not in general expressible in terms of elementary functions. Beyond its contributions to the optimal stopping theory, the present paper has been motivated by applications to the optimal timing of investment decisions involving an underlying asset price or economic indicator. In mathematical finance and the theory of real options, such time series are typically modelled by SDEs driven by a standard Brownian motion or, more generally, a Lévy process. A skew geometric Brownian motion or, more generally, SDEs such as the ones considered in Example 1 can be used to model asset prices and economic indicators that exhibit support and resistance levels 1 (see Hämäläinen [15] for a recent survey of studies focusing on such directional predictability). Indeed, the skew geometric Brownian motion (4) behaves like a standard geometric Brownian motion, except that the sign of each excursion from z is chosen using an independent Bernoulli random variable of parameter p = 1 2 (β + 1), namely, P(X t > z) = p for any t > T z . Recently, several authors have studied financial models involving SDEs with generalised drift (e.g., see Corns and Satchell [6], Decamps, Goovaerts and Schoutens [11,12], Rosello [31], and references therein). Alternatively, a skew geometric Brownian motion can be used to capture phenomena of bounces and sinks that are exhibited by financial firms in distress (see Nilsen and Sayit [25]).
The paper is organised as follows. In Section 2, we derive an analytic characterisation of the minimal excessive functions of the one-dimensional diffusion associated with the SDE (1). In Section 3, we establish a complete characterisation of the solution to the general optimal stopping problem given by (1) and (2) in terms of variational inequalities. Section 4 presents a study of a skew geometric Brownian motion's minimal excessive functions. In Section 5, we prove a couple of technical results that will facilitate the streamlining of the presentation of the solution to the optimal stopping problem defined by (4) and (5). We present the complete solution to this problem in Section 6. Finally, the proofs of all results stated in Section 6 are collected in Section 7.

The SDE (1) and its associated minimal excessive functions
We start with the following assumption. 2 Also, ν is a signed Radon measure on I , B(I) such that ν {z} ∈ ]−1, 1[.
In the presence of this assumption, the SDE (1) has a weak solution that is unique in the sense of probability law (see Engelbert and Schmidt [13, Theorems 4.35 and 4.37]). In particular, given any initial point x ∈I, there is a collection S x = Ω, F , (F t ), P x , W, X such that (Ω, F , (F t ), P x ) is a filtered probability space satisfying the usual conditions, W is a standard (F t )-Brownian motion and X is a continuous (F t )-adapted stochastic process such that (1) holds true in the stochastic interval [0, with the usual assumption that inf ∅ = ∞. We assume that either of the endpoints ι, ι is either inaccessible or absorbing. Accordingly, if ι (resp., ι) is absorbing, then X t = ι (resp., X t = ι) for all t ≥ T ι (resp., t ≥ T ι ). The scale function of the diffusion associated with the SDE (1) is the unique, up to a strictly increasing affine transformation, continuous strictly increasing function p : I → R that satisfies for all points x < x < x in I. The restriction of p inI is the difference of two convex functions 3 and satisfies the ordinary differential equation (ODE) 2 We denote by B(I) the Borel σ-algebra onI. 3 A function g :I → R is the difference of two convex functions if and only if it is absolutely continuous in the sense that In particular, it is given by and p ′ where x 1 ∈I is an arbitrary fixed point. All these claims can be found, e.g., in Engelbert and Schmidt [13,Section 4.3]. For future reference, we also note that these expressions imply that We will need the following real analysis result. 4 Lemma 1 Let u : p(I) → R be a difference of two convex functions and define u(x) = u p(x) , for x ∈I. The following statements hold true: (I) u is the difference of two convex functions.
(II) The function u ′ − /p ′ − is of finite variation. Furthermore, the measure on I , B(I) that identifies with the function u ′ − /p ′ − is the image of the measure u ′′ under the function p −1 , namely, [ for all q < q in p(I).
(III) If u has absolutely continuous first derivative u ′ (= u ′ − = u ′ + ), then u ′ − /p ′ − is absolutely continuous and 5 with left-hand derivative that is a function of finite variation. Given such a function, we denote by g ′ ± its right-hand and left-hand side first derivatives, which are defined by and by g ′′ (dx) the measure that identifies with its second distributional derivative. 4 We denote by p(I) the interval ]p(ι), p(ι)[ and by B p(I) the Borel σ-algebra on p(I). 5 In this part of the lemma, we use the same notation for the signed Radon measure that identifies with the second distributional derivative of u as well as for the Radon-Nikodym derivative of this measure with respect to the Lebesgue measure, namely, we write u ′′ (dq) = u ′′ (q) dq. We refer to this footnote whenever we make such an abuse of notation; confusion is unlikely to occur.
Proof. We first note that u = u • p is absolutely continuous because it is the composition of absolutely continuous functions and p is increasing. Given any x ∈I, Combining this observation with the fact that the limit p ′ − (x) = lim ε↓0 exists for all x ∈I. Given any points x < x inI, we use the change of variables formula (e.g., see Revuz and Yor [30, Proposition 0.4.10]) to calculate and (II) follows. Furthermore, (I) follows from the absolute continuity of u and the fact that u ′ − is the product of the finite variation functions p ′ − and u ′ − /p ′ − . Finally, if u ′ is absolutely continuous (see also footnote 5), then and (III) follows.

Given a weak solution
for W = W . Conversely, given a weak solution S x = Ω, F, ( F t ), P x , W , X to the SDE (14), the collection Ω, F, ( F t ), P x , W , p −1 ( X) is a weak solution to the SDE (1) for W = W . These results, which are established in Engelbert and Schmidt [13, Proposition 4.29]), will play a fundamental role in our analysis.
To proceed further, we consider the discounting rate function r appearing in (2) and we make the following assumption.
Assumption 2 The function r :I → R + is Borel-measurable, uniformly bounded away from 0, namely, r(x) ≥ r 0 for all x ∈I, for some r 0 > 0, and such that The minimal r-excessive functions ϕ, ψ :I → R + of the diffusion associated with the SDE (1) are the unique, modulo multiplicative constants, functions that satisfy and where T x , T x are as in (8) Both of the functions ϕ and ψ are absolutely continuous. Furthermore, the functions ϕ ′ − /p ′ − and ψ ′ − /p ′ − are absolutely continuous and the homogeneous ODE 6 is satisfied Lebesgue-a.e. inI for g standing for either ϕ or ψ.
Proof. In view of the results connecting the solvability of (1) with the solvability of (14) that we have discussed above, we can see that, given any x < x inI, and ϕ, ψ : p(I) → R + are the minimal (r •p −1 )-excessive functions of the diffusion associated with the SDE (14), given by and It follows that In view of the general theory reviewed, e.g., in Borodin and Salminen [5, Section II.1], the functions ϕ, ψ are unique modulo multiplicative constants, C 1 with absolutely continuous first derivatives, and such that ϕ (resp., ψ) is strictly decreasing (resp., increasing). Also, 6 As in Lemma 1, we use here (ϕ ′ − /p ′ − ) ′ and (ψ ′ − /p ′ − ) ′ to denote the Radon-Nikodym derivatives of the measures identified with the functions ϕ ′ − /p ′ − and ψ ′ − /p ′ − with respect to the Lebesgue measure (see also footnote 5).
since ι (resp., ι) is an absorbing (resp., inaccessible) boundary point for X if and only if p(ι) (resp., p(ι)) is an absorbing (resp., inaccessible) boundary point for X ≡ p(X), if ι is absorbing for X, then ϕ p(ι) := lim Furthermore, ϕ and ψ satisfy the homogeneous ODE in g Lebesgue-a.e. in p(I). This fact and the absolute continuity of p imply that the ODE is satisfied Lebesgue-a.e. inI for g standing for either ϕ or ψ. Combining these observations with (19) and Lemma 1.(III), we obtain all of the required results.
Example 1 Suppose that the measure ν is of the form some constants β 1 , . . . , β k ∈ ]−1, 1[ and some distinct points z 1 , . . . , z k ∈I, where δ z j (dz) is the Dirac probability measure that assigns unit mass on {z j }. Using the occupation times formula, we can see that, in this case, X satisfies the SDE In view of (9), the restriction of the scale function p inI \ {z 1 , . . . , z k } has absolutely continuous first derivative Lebesgue-a.e. inI \ {z 1 , . . . , z k } (see also footnote 5 about p ′′ ). Furthermore, (13) implies that Using these observations, we derive the expression as well as a similar one for x < x 1 , where x 1 ∈I is an arbitrary fixed point. If we denote by g either of the excessive functions ϕ or ψ given by (15) and (16), then (21) and the (absolute) Furthermore, Lemma 2 and (20) imply that g satisfies the ODE Lebesgue-a.e. inI \ {z 1 , . . . , z k } (see also footnotes 5, 6 about g ′′ ).
3 The solution to the general optimal stopping problem The value function of the optimal stopping problem that aims at maximising the performance criterion appearing in (2) is defined by where the set of all stopping strategies T x consists of all pairs (S x , τ ) such that S x = Ω, F , (F t ), P x , W, X is a weak solution to (1) and τ is an associated (F t )-stopping time.
We assume that the discounting rate function r satisfies Assumption 2, while the reward function f satisfies the following assumption.

Assumption 3
The positive function f : I → R + is Borel-measurable and its restriction in I is upper semicontinuous, namely, Our main result in this section establishes a complete characterisation of the general optimal stopping problem defined by (1), (24) in terms of solutions to the variational inequality (3) in the sense of distributions, which are introduced by the following definition.
Theorem 3 Consider the optimal stopping problem defined by (1), (24), and recall the minimal excessive functions ϕ and ψ given by (15) and (16). The following statements hold true: then v(x) < ∞ for all x ∈ I. If any of the inequalities in (25) and and where S x is a weak solution to (1) and is optimal.
Proof. In view of the results connecting the solvability of (1) with the solvability of (14) that we have already used in the proof of Lemma 2, we can see that where where ϕ = ϕ • p −1 and ψ = ψ • p −1 are the minimal (r • p −1 )-excessive functions of the diffusion associated with the SDE (14), given by (17) and (18). The identities in (29) and Theorem 6.3.(I) in Lamberton and Zervos [20] imply (I). If the inequalities in (25) all hold true, then Theorem 6.3 in Lamberton and Zervos [20] asserts that the restriction of the value function v in p(I) is the difference of two convex functions and satisfies the variational inequality in the sense that the Radon measure on p(I), B(p(I)) defined by In view of Lemma 1.(I)-(II), v = v • p is the difference of two convex functions and Combining these observations with (28)-(29) and Theorems 6.3, 6.4 in Lamberton and Zervos [20], we obtain all of the required results in (II)-(IV).
Remark 1 It is worth stressing the precise nature of the boundary conditions appearing in (26) and (27). The existence of the limits on the left-hand side of (26) is a result, while, the existence of the limits on the left-hand side of (27) is an assumption. Also, the limits on the left-hand sides of (26), (27) are taken from inside the interiorI of I. On the other hand, the limsups on the right-hand sides of (26), (27) are taken from inside I itself. In view of these observations, we can see that, e.g., if ι is absorbing, then we are faced in (26) with either the possibility that where we have used the fact that, in this case, ϕ(ι) := lim x↓ι ϕ(x) < ∞ (see Lemma 2). Example 2 Suppose that the measure ν is as in Example 1. Given x < x such that [x, x[ ⊆I \ {z 1 , . . . , z k }, we use the integration by parts formula and the fact that the scale function p has absolutely continuous derivative satisfying (20) Also, we note that the measure µ v defined in Definition 1.(II) is such that It follows that, in this case, the variational inequality (3) takes the form Furthermore, if v has absolutely continuous first derivative, namely, if v ′′ (dx) is equal to v ′′ (x) dx (see also footnote 5), then v should satisfy Lebesgue-a.e. inI \ {z 1 , . . . , z k } as well as the conditions (31).

Remark 2
To appreciate how variational inequalities can be used to systematically identify critical parts of the state space I that belong to the waiting region, consider the previous example. In this context, we can make the following observations: (a) Given j = 1, . . . , k, if either f − (z j ) or f + (z j ) do not exist, then z j belongs to the waiting region. On the other hand, if both f ± (z j ) exist, then (31) implies that z j belongs to the waiting region if In particular, if f is C 1 at z j , then z j belongs to the waiting region if β j ∈ ]0, 1[. (b) Suppose that the restriction of f in an interval ]ι, ι[ ⊆ I is C 2 . The validity of (30) implies that x ∈ ]ι, ι[ is a subset of the waiting region. On the other hand, the intersection of the stopping region with ]ι, ι[ is a (usually strict) subset of the complement of this set.

Remark 3
In the context of Example 2, we can make the following observations relative to the so-called "principle of smooth fit": (a) If f is C 1 then (30) implies that the restriction of the value function v inI \ {z 1 , . . . , z k } is C 1 (see Lamberton and Zervos [20,Corollary 7.5]). (b) Given j = 1, . . . , k, if z j belongs to the stopping region, namely, f (z j ) = v(z j ), then (21) and (31) (c) Given j = 1, . . . , k, if z j belongs to the waiting region, namely, f (z j ) < v(z j ), then (21) and (31) The inequality in (33) can be strict: see Remark 5 in Section 6. It follows, that, in optimal stopping problems such as the ones we consider in this paper, the "principle of smooth fit" does not in general hold in the sense that none of the functions

The minimal excessive functions of a skew geometric Brownian motion
From this point onwards, we fix a filtered probability space Ω, F , (F t ), P satisfying the usual conditions and supporting a standard one-dimensional (F t )-Brownian motion W . In such a setting, we denote by X the unique non-explosive strong solution to the SDE (4). The conditions (22) in Example 1 reduce to while, given a constant r > 0, the ODE (23) in Example 1 reduces to the Euler ODE It is well-known that every solution to (35) is given by for some constants A, B ∈ R, where m < 0 < n are the solutions to the quadratic equation given by It is straightforward to verify that The excessive functions ψ = ψ(·; z) and ϕ = ϕ(·; z) that satisfy the ODE (35) inside ]0, z[ ∪ ]z, ∞[ as well as the condition (34) are given by for It is straightforward to verify that Here, as well as in the rest of the paper, we adopt the notation ψ ′ (x; z) = ∂ψ ∂x (x; z) and ψ ′′ (x; z) = ∂ 2 ψ ∂x 2 (x; z). In the rest of the paper, we make the following assumption, which is sufficient for the value function of the optimal stopping problem defined by (4) and (5) to be real-valued.
Indeed, if r < b ⇔ n < 1, then Theorem 3.(I) implies that the value function given by (5) is identically equal to ∞.
In the presence of Assumption 4, we can verify that and β c : Here, deriving the possible values of β c involves the observation that, if n + 2m − 1 < 0, then Combining the range of values of the point β c given by (43), with the observation that we can see that Furthermore, In the following result, we concentrate on the increasing function ψ because only this is involved in the solution to the optimal stopping problem we consider in this main section. In particular, the critical points defined by play a critical role in differentiating the different qualitative forms of the optimal strategy.
5 Preliminary analytic results for the solution to the optimal stopping problem defined by (4) and (5) We now establish a pair of technical analytic results that we will need for the solution of the optimal stopping problem defined by (4) and (5), which we derive in the next section.
(This section can easily be skipped at a first reading.) Given any z > 0 fixed, we consider the equation for x > z, where F is the function defined by which admits the expression The following result involves the critical points Z c , Z β , Z 0 defined by (47) and is structured based on the four cases of Lemma 4.

Lemma 5
In the presence of Assumption 4, the following statements hold true: (i) If the problem's parameters are as in Cases (I) or (II) of Lemma 4, then equation (52) defines uniquely a strictly decreasing C 1 function α : ]0, (ii) If the problem's parameters are as in Case (III) of Lemma 4, then K < CZ c , where C ∈ ]0, 1[ is given by (48), and equation (52) defines uniquely a strictly decreasing C 1 function Furthermore, Proof. Throughout the proof, we use repeatedly the expressions and signs of A, B, given by (39), (40), as well as the results in (36), (37) and (42) without special mention. We first note that We also calculate and F (z; z) The calculation (65) implies that Combining this observation with the limits which follow from (64), we can see that or there exist strictly positive constants or there exists a constant Keeping in mind that n > 1, and Z β > 0 if and only if n > 1+β 1−β , we can see that < 0, if (β < 0 and z < Z β ) or (β > 0 and z > Z β ) , > 0, if (β < 0 and z > Z β ) or (β > 0 and z ∈ ]0, Z β [) .
Differentiating the identity F α(z); z = 0 with respect to z, and using (66), the inequalities mK m−1 < Z c ≤ Z β < α(z) (see also (49)) and (56), we obtain which proves that α is strictly decreasing. Furthermore, the first limit in (55) follows from the calculation while, the second limit in (55) follows from (56) and (71). Proof of (ii). We first note that, in this case, where x † is given by (68) (see (42), (44) and the statement of Lemma 4.(III)). Combining this observation with (69) and the identities F (K; z) = ∂F ∂x (K; z) = 0, which follow from (72), we can see that Using the definition (53) of F and (64), we calculate

It follows that
where C is defined by (48). These inequalities and (74) imply that Combining this result with (75) and the fact that ∂F ∂z (x † ; z) > 0, which follows from (66) and (74), we can see that z < CZ c , which implies that K < CZ c .
The next result addresses the inequality which will play an important role in our analysis in the next section.
Proof. We first calculate for all z in the domain of α. (Note that ∂g ∂x (z, z) does not exist, the corresponding left and right partial derivatives of g(·, z) are discontinuous at z.) Proof of (i). This case follows immediately from (49), (56), (81) and the fact that α(z) > K.
Proof of (ii). We first recall that, in this case, K < CZ c (see Lemma 5.(ii)). In view of (54) and Lemma 4.(III), we can see that (in the inequalities here, we list only the cases we will use). Also, given any z ∈ ]0, CZ c [, we use the identities in (81) to calculate Defining α(CZ c ) := lim z→CZc α(z) = Z c , we note that (57), (58) imply that F (Z c ; CZ c ) = 0. This observation, the fact that K < CZ c and the second pair of inequalities in (82) imply that

This inequality and (83) imply that
Combining this result with the observation that g(z, z) > 0 for all z ≤ K, which follows from the definition (77) of g and the fact that α(z) > Z c > K for all z < CZ c , we can see that We will establish (78) if we show that g(z, z) < 0 for all z ∈ ]z ⊖ , CZ c [. To this end, we differentiate the expression of the function ]0, CZ c [ ∋ z → g(z) := g(z, z) given by (78) and we use the identities In view of the fact that α : ]0, CZ c [ → ]Z c , Z 0 [ is strictly decreasing and the inequalities mK m−1 < K < z ⊖ < CZ c < Z c < Z 0 (see (50) in Lemma 4 and Lemma 5.(ii)), the latter expression implies that the function ]0, CZ c [ ∋ z → g(z) is strictly convex. Combining this observation with the inequalities g(z ⊖ ) = 0 and g(CZ c ) < 0 (see (84) for the last one), we can see that g(z) ≡ g(z, z) < 0 for all z ∈ ]z ⊖ , CZ c ], as required.
To proceed further, we note that, if z < K, then the expression (54) of F implies that F (x, z) < 0 for all x ∈ [z, K]. Combining this observation with the identity F α(z); z = 0 and the first pair of inequalities in (82), we can see that given any z ∈ ]0, K[, F (x; z) < 0 for all x ∈ [z, α(z)[. On the other hand, the second pair of inequalities in (82) and the identity F α(z); z = 0 imply that given any z ∈ [K, CZ c [, either F (x; z) < 0 for all x ∈ ]z, α(z)[, These observations and (81) imply that either Given any z ∈ ]0, z ⊖ ], the inequality g(z, z) > 0 (see (78)), the identity g α(z), z = 0 and (85) imply that (77) holds true for all x ∈ [0, α(z)]. On the other hand, given any z ∈ [z ⊖ , CZ c [, the inequality g(z, z) < 0 (see (78)), the identity g α(z), z = 0 and (85) imply that there exists a unique z(z) ∈ [z ⊖ , Z c [ such that (79) holds true (note that g is as in the second case of (85) here).
Proof of (iii). In this case, (51) and (60) imply that It follows that On the other hand, (61) and (81) imply that Combining these observations with the calculation we can see that there exists a unique z ⊕ ∈ ]Z 0 , ∞[ such that (80) holds true. Finally, we fix any z ∈ ]0, z ⊕ ]. In view of the inequality g(Z 0 , z) ≥ 0, the identity g α(z), z) = 0 and the observation that which follows from (61) and (81), we can see that the inequality (77) holds true for all x ∈ ]0, α(z)[.

Remark 4
Our analysis in the next sections will make use of the following observation. Suppose that the problem's parameters are as in Case (III) of Lemma 4 and fix any z ∈ [z ⊖ , CZ c [, where z ⊖ is as in Lemma 6.(ii). The function u(·; z) : [z, ∞[ → R defined by where Γ(z) = α(z) − K /ψ α(z); z , is such that u z(z); z = 0 and u α(z); z = ∂u ∂x α(z); z = 0.
The first of these identities follows immediately from (79) and the fact that u(x; z) = g(x, z)ψ(x; z) for all x ≥ z. On the other hand, the identities for x = α(z) hold true because they are equivalent to the identity F α(z); z = 0.
6 The solution to the optimal stopping problem defined by (4) and (5) We expect that the value function v of the discretionary problem defined by (4) and (5) should be strictly positive. Combining this observation with the fact that the restriction of the function x → (x − K) + in R + \ {K} is C ∞ and the so-called "principle of smooth fit" (see also Remark 3), we expect that the restriction of v in ]0, ∞[ \ {z} should be C 1 with absolutely continuous first derivative. In view of (31), (32) in Example 2, we therefore expect that v should identify with a function w satisfying and max (1 + β)w ′ Furthermore, the strict positivity of v and Remark 2 imply that the waiting region includes the interval 0, K ∨ rK/(r − b) and, if β ∈ ]0, 1[, then z also belongs to the waiting region. We now solve the optimal stopping problem we consider in this section by constructing an appropriate solution to the variational inequality (86)-(87). In its simplest form, we expect that the required solution has the same qualitative form as the solution to the optimal stopping problem associated with the usual perpetual American call option, which involves a standard geometric Brownian motion (β = 0). Accordingly, we expect that the value function v should identify with the function for some constants a = a(z) > 0 and Γ(z) > 0, while should identify an optimal stopping time. It turns out that this is indeed the case for a wide range of parameter values (see . To determine the constant Γ(z) and the free-boundary point a, we first appeal to the continuity of the value function, which yields the expression With the exception of the possibilities depicted by Figures 5 and 8, we expect that the value function should be C 1 at a, which gives rise to the equation Γ(z)ψ ′ (a; z) = 1. =: Z 0 > 0 if a < z (see Figures 6 and 9).
The following result, which we prove in Section 7, involves the parameters Z c , Z β , Z 0 and z ⊖ , z ⊕ that are as in (47) and Lemma 6.(ii)-(iii), respectively.
Theorem 7 Consider the optimal stopping problem defined by (4), (5) and suppose that Assumption 4 holds true. If the problem parameters are as in Cases (I) or (II) of Lemma 4, define where the function α is as in Lemma 5.(i). If the problem parameters are as in Case (III) of Lemma 4, suppose that z ∈ ]0, where the function α is as in Lemma 5.(ii) and z ⊖ is as in Lemma 6.(ii). If β ∈ ]0, 1[ (Case (IV) of Lemma 4), suppose that z ∈ ]0, z ⊕ ] and define where the function α is as in Lemma 5.(iii) and z ⊕ is as in Lemma 6.(iii). For such choices of a and for Γ(z) > 0 given by (90), the function w defined by (88) identifies with the value function v of the discretionary stopping problem and the stopping time given by (89) is optimal.
In the context of (91), we can see that Figure 4 transforms "continuously" into Figure 5 and then into Figure 6 as z increases from 0 to ∞, thanks to the second limit in (55).      In view of the identity in (33) and Theorem 7, we can see that, given any We are thus faced with an example of "right-sided" optimal stopping of a skew geometric Brownian motion in which the "principle of smooth fit" does not hold in the sense that none If the problem parameters are as in Case (III) of Lemma 4, then the function w = w(·; z) given by (88), (90) is such that (see Lemma 6.(ii)). This observation and the "singularity" associated with z give rise to the following possibility. For z ≥ z ⊖ , the stopping time where ξ = ξ(z) > z is a constant, may be optimal. In such a context, we expect that the value function v should identify with the function for some C(z), D(z) ∈ R (see Figure 11). To determine the constants C(z), D(z) and the free-boundary point ξ = ξ(z), we require that w should be C 1 at ξ, which is suggested by the "principle of smooth fit", as well as continuous at z. The system of equations arising from these requirements is equivalent to the expressions and the algebraic equation where To establish the second identity here, we have used the definitions (39), (40) of A, B, as well as the definition (53) of F . We prove the following result in Section 7.
Theorem 8 Consider the optimal stopping problem defined by (4), (5) and suppose that Assumption 4 holds true. Also, suppose that the problem parameters are as in Case (III) of Lemma 4. Equation (98) defines uniquely a strictly decreasing function ξ : ]0, where α is as in Lemma 5.(ii) and z ⊖ is as in Lemma 6.(ii). Given any z ∈ ]z ⊖ , Z c [, the function w defined by (96)-(97) for ξ = ξ(z) identifies with the value function v of the discretionary stopping problem and the (F t )-stopping time τ ⋆ defined by (95) is optimal.
In the context of the previous result and the part of Theorem 7 addressing the case when the problem parameters are as in Case (III) of Lemma 4 (see (92) in particular), we can see that Figure 7 transforms "continuously" into Figure 11, then into Figure 8 and then into Figure 9 as z increases from 0 to ∞, thanks to the identities w(·; z ⊖ ) = w(·; z ⊖ ), ξ(z ⊖ ) = α(z ⊖ ) and lim (see (94) and (100)). If the problem parameters are as in Case (IV) of Lemma 4, then the function w = w(·; z) given by (88), (90) is such that (see Lemma 6.(iii) and Figure 12). This observation suggests the possibility for the stopping time where γ = γ(z) < ζ = ζ(z) are constants, to be optimal. In such a context, we expect that the value function v should identify with the function for some C l , D l , C r , D r ∈ R (see Figure 13). We suppress the actual dependence of C l , D l , C r , D r on z because we will not use variational arguments in the analysis of this case. To determine the constants C l , D l , C r , D r and the free-boundary points γ, ξ, we first require that w should be C 1 at γ and ζ, which is suggested by the "principle of smooth fit". This requirement yields the expressions We next require that w should be continuous at z and satisfy the identity that is associated with (87). These requirements yield the identities Using the expressions in (105), (106) to substitute for C l , D l , C r , D r , we obtain the system of equations By (a) subtracting (107) from (108) and (b) solving (108) for (m − 1)γ − mK z n γ −n and substituting for the resulting expression in (107), we obtain the system of equations G(γ, ζ; z) = 0 and H(γ, ζ; z) = 0, which is equivalent to (107) and (108), where and H(x, y; z) = y −n F (y; z) in which expressions, F is the function defined by (53). We prove the final result of the paper in Section 7.
(b) Given any z > z ⊕ and the associated solution (γ, ζ) to the system of equations (109), the function w defined by (104), for C l , D l , C r , D r > 0 given by (105)-(106) identifies with the value function v of the discretionary stopping problem and the (F t )-stopping time τ ⋆ defined by (103) is optimal.
In the context of the previous result and the part of Theorem 7 addressing the case when the problem parameters are as in Case (IV) of Lemma 4 (see (93) in particular), we can see that Figure 10 transforms "continuously" into Figure 12 and then into Figure 13 as z increases from 0 to ∞, thanks to the identity w(·; z ⊖ ) = w(·; z ⊖ ), which follows from (102). In view of this observation and Theorem 3.(III)-(IV), we can see that we will prove Theorems 7, 8 and 9 if we establish the claims made on the solvability of their associated free-boundary problems as well as show that the corresponding functions w, w and w satisfy the variational inequality (86)-(87).
Proof of Theorem 7. By construction, w is C 2 inside ]0, ∞[ \ {a, z} and C 1 at a if a = z.
It is straightforward to verify that w satisfies (87). In view of its structure, we will verify that w satisfies (86) if we prove that x − K ≤ w(x) for all x < a, and 1 2 σ 2 x 2 w ′′ (x) + bxw ′ (x) − rw(x) ≤ 0 for all x > a.
In view of (88) and (90), we can see that (112) is equivalent to for all x < a.
In the context of (91) with z < Z β or (92) with z ≤ z ⊖ or (93) with z ≤ z ⊕ , this inequality is equivalent to (77), which is true thanks to Lemma 6. In the context of (91) with z ≥ Z β or (92) with z ≥ Z c , this inequality follows immediately from the fact that the function x → (x − K)x −n is strictly increasing in ]0, Z 0 [. On the other hand, (113) is equivalent to bx − r(x − K) ≤ 0 for all x > a, which is true because, in all cases, a > Z c = rK r−b .
In view of (98) and the first expression in (99), we can see that this inequality is equivalent to (m − 1)ξ(z) − mK z n−m ξ −n (z) < (m − 1)z − mK z −m .
It follows that (128) is equivalent to z > z ⊕ , as required, thanks to Lemma 6.(iii). By construction, the function w defined by (104) is continuous, C 1 inside ]0, ∞[ \ {z} and C 2 inside ]0, ∞[\{z, ξ}. In view of its structure, we will verify that w satisfies the variational inequality (86)-(87) if we prove that and To establish (129), we first note that