Expected Supremum Representation and Optimal Stopping
aa r X i v : . [ m a t h . P R ] M a y EXPECTED SUPREMUM REPRESENTATION AND OPTIMAL STOPPING
LUIS H. R. ALVAREZ E. AND PEKKA MATOM ¨AKI
Abstract.
We consider the representation of the value of an optimal stopping problem of a lineardiffusion as an expected supremum of a known function. We establish an explicit integral represen-tation of this function by utilizing the explicitly known joint probability distribution of the extremalprocesses. We also delineate circumstances under which the value of a stopping problem inducesdirectly this representation and show how it is connected with the monotonicity of the generator.We compare our findings with existing literature and show, for example, how our representation islinked to the smooth fit principle and how it coincides with the optimal stopping signal represen-tation. The intricacies of the developed integral representation are explicitly illustrated in variousexamples arising in financial applications of optimal stopping.
Date : May 7, 2015.2010
Mathematics Subject Classification.
XPECTED SUPREMUM REPRESENTATION AND OPTIMAL STOPPING 1 Introduction
It is well-known from the literature on stochastic processes that the probability distributionsof first hitting times are closely related to the probability distributions of the running supremumand running infimum of the underlying diffusion. Consequently, the question of whether a lineardiffusion has exited from an open interval prior to a given date or not can be answered by studyingthe behavior of the extremal processes up to the date in question. If the extremal processes haveremained in the open interval up to the particular date, then the process has not yet hit theboundaries and vice versa. In this study we utilize this connection and develop a representation ofthe value function of an optimal stopping problem as the expected supremum of a function withknown properties in the spirit of the pioneering work by [19, 20] and the subsequent extension tooptimal stopping by [14].The relatively recent literature on stochastic control theory indicates that the connection be-tween, among others, the value functions and extremal processes in optimal stopping and singularstochastic control problems goes far beyond the standard connection between first hitting times andthe running supremum and infimum of the underlying process (see, for example, [7, 8, 9, 10, 18, 16]).Essentially, in these studies the determination of the optimal policy and its value is shown to beequivalent with the existence of an appropriate optional projection involving the running supremumof a progressively measurable process. The advantage of the representation utilized in these stud-ies is that it is very general and applies also outside the standard Markovian and infinite horizonsetting. Moreover, it can be utilized for studying and solving other than just optimal stopping andsingular stochastic control problems as well. For example, as was shown in [8, 9], the approach isapplicable in the analysis of the Gittins-index familiar from the literature on multi-armed bandits(cf. [17, 21, 22, 23, 27]).Instead of establishing directly how the value of an optimal stopping problem can be expressedas an expected supremum, we take an alternative route and compute first the joint probabilitydistribution of the running supremum and running infimum of the underlying diffusion at an inde-pendent exponentially distributed random time. We then compute explicitly the expected value ofthe supremum of an unknown function subject to a set of monotonicity and regularity conditions.Setting this expected value equal with the value of an optimal stopping problem then results intoa functional identity from which the unknown function can be explicitly determined. In the singleboundary setting the function admits a relatively simple characterization in terms of the minimalexcessive mappings for the underlying diffusion (cf. [7]). We find that the required monotonicity ofthe function needed for the representation is closely related with the monotonicity of the generatoron the state space of the underlying diffusion. However, since only the sign of the generator typicallyaffects the determination of the optimal strategy and its value, our results demonstrate that not allsingle boundary problems can be represented as the expected supremum of a monotonic function.We also investigate the regularity properties of the function needed for the representation and show
LUIS H. R. ALVAREZ E. AND PEKKA MATOM ¨AKI that it needs not be continuous. More precisely, we find that if the optimal boundary is attained at apoint where the exercise payoff is not differentiable, then the function needed for the representationis only upper semicontinuous. This is a result which is in line with the findings by [14].In the two boundary setting the representation becomes more involved and takes an integralform where the integration bounds are interdependent due to the dependence of the two extremalprocesses. However, since the representation is based on the minimal r -excessive functions and thescale of the underlying diffusion, our approach results into a representation which can be efficientlyutilized in numerical computations. We also compare our representation with previous represen-tations. Given that our approach is based on the study by [14] it naturally coincides with theirrepresentation the main difference being that we compute the expected supremum explicitly and inthat way state an explicit representation of the unknown function needed for the representation. Wealso establish that our representation coincides with the stopping signal representation originallydeveloped in [7]. Hence, our findings provide an explicit connection between these two seeminglydifferent approaches. Furthermore, we also demonstrate that the continuity requirement of the func-tional form needed for the representation is equivalent with the standard smooth fit principle. Inthis way, our study provides a link between the usual (e.g. free boundary/variational inequalities)approach and the more recent approaches based on the running supremum. In line with our find-ings in the single boundary case, our results indicate that the function needed for the representationdoes not need to be continuous. In this way, our numerical results appear to show that the stoppingsignal representation developed in [7] applies also in a nonsmooth environment.The contents of this study is as follows. In section two we formulate the considered problem,characterize the underlying stochastic dynamics, and state a set of auxiliary results needed in thesubsequent analysis of the problem. Section three focuses on a single boundary setting where theoptimal rule is to exercise as soon as a given exercise threshold is exceeded. The general two-boundary case is then investigated in detail in section four. Finally, section five concludes ourstudy. 2. Problem Formulation
Underlying stochastic dynamics.
We consider a linear, time homogeneous and regulardiffusion process X = { X ( t ); t ∈ [0 , ξ ) } , where ξ denotes the possible infinite life time of the diffusion.We assume that the diffusion is defined on a complete filtered probability space (Ω , P , {F t } t ≥ , F ),and that the state space of the diffusion is I = ( a, b ) ⊂ R . Moreover, we assume that the diffusiondoes not die inside I , implying that the boundaries a and b are either natural, entrance, exit orregular (see [12], pp. 18-20 for a characterization of the boundary behaviour of diffusions). Ifthe boundary is regular, we assume that the process is either killed or reflected at that boundary.Furthermore we will denote by I t = inf ≤ s ≤ t X s the running infimum and by M t = sup ≤ s ≤ t X s therunning supremum process of the considered diffusion X t . XPECTED SUPREMUM REPRESENTATION AND OPTIMAL STOPPING 3
As usually, we denote by A the differential operator representing the infinitesimal generator of X . For a given smooth mapping f : I 7→ R this operator is given by( A f )( x ) = 12 σ ( x ) d dx f ( x ) + µ ( x ) ddx f ( x ) , where µ : I 7→ R and σ : I 7→ R + are given continuous mappings. As is known from the classicaltheory on linear diffusions, there are two linearly independent fundamental solutions ψ ( x ) and ϕ ( x )satisfying a set of appropriate boundary conditions based on the boundary behavior of the process X and spanning the set of solutions of the ordinary differential equation ( G r u )( x ) = 0, where G r = A − r denotes the differential operator associated with the diffusion X killed at the constantrate r . Moreover, ψ ′ ( x ) ϕ ( x ) − ϕ ′ ( x ) ψ ( x ) = BS ′ ( x ) , where B > S ′ ( x ) = exp (cid:18) − Z x µ ( t ) σ ( t ) dt (cid:19) denotes the density of the scale function of X (for a comprehensive characterization of the funda-mental solutions, see [12], pp. 18-19). The functions ψ and ϕ are minimal in the sense that anynon-trivial r -excessive mapping for X can be expressed as a combination of these two (cf. [12], pp.32–35). Given the fundamental solutions, let u ( x ) = c ψ ( x ) + c ϕ ( x ) , c , c ∈ R be an arbitrarytwice continuously differentiable r -harmonic function and define for sufficiently smooth mappings g : I 7→ R the functional( L u g )( x ) = g ( x ) u ′ ( x ) S ′ ( x ) − g ′ ( x ) S ′ ( x ) u ( x ) = c ( L ψ g )( x ) + c ( L ϕ g )( x )associated with the representing measure for r -excessive functions (cf. [33]). Noticing that if g istwice continuously differentiable, then( L u g ) ′ ( x ) = − ( G r g )( x ) u ( x ) m ′ ( x )(1)where m ′ ( x ) = 2 / ( σ ( x ) S ′ ( x )) denotes the density of the speed measure m of X . Hence, we findthat ( L u g )( z ) − ( L u g )( y ) = Z yz ( G r g )( t ) u ( t ) m ′ ( t ) dt (2)for any a < z < y < b . Especially, if g is twice continuously differentiable, = I ( x ), and a < z < y < b , then the (symmetric) function R ( z, y ) = ( L u g )( z ) − ( L u g )( y )( L u )( z ) − ( L u )( y )(3)satisfies the limiting condition lim z ↑ y R ( z, y ) = − r ( G r g )( y )(4) LUIS H. R. ALVAREZ E. AND PEKKA MATOM ¨AKI which is independent of the harmonic function u . Finally, we denote by L r ( I ) the class of measurablefunctions f : I 7→ R + satisfying the integrability condition E x Z ∞ e − rs | f ( X s ) | ds < ∞ for all x ∈ I . As is known from the literature on linear diffusions, the expected cumulative presentvalue of a continuous function f ∈ L r ( I ), that is,( R r f )( x ) = E x Z ∞ e − rs f ( X s ) ds can be expressed as( R r f )( x ) = B − ϕ ( x ) Z xa ψ ( y ) f ( y ) m ′ ( y ) dy + B − ψ ( x ) Z bx ϕ ( y ) f ( y ) m ′ ( y ) dy. (5)2.2. The Optimal Stopping Problem and Auxiliary Results.
In this paper our objective isto examine an optimal stopping problem V ( x ) = sup τ E x (cid:2) e − rτ g ( X τ ) (cid:3) (6)for exercise payoff functions g satisfying a set of sufficient regularity conditions and establish arepresentation of the value V ( x ) as the expected supremum of an appropriately chosen functionalong the lines of the pioneering studies [8],[9],[14], [16], [18], [19], [20]. Our main results are basedon the following two representation theorems originally established in [14]. The first theorem focuseson the case of a one-sided stopping boundary. Theorem 2.1. ( [14] , Theorem 2.5) Let X t be a Hunt process on I and T ∼ Exp( r ) ⊥ X t . As-sume that the exercise payoff g is non-negative, lower semicontinuous, and satisfies the condition E x (cid:2) sup t ≥ e − rt g ( X t ) (cid:3) < ∞ for all x ∈ I . Assume also that there exists an upper semicontinuous ˆ f and a point y ∗ ∈ I such that(a) ˆ f ( x ) ≤ for x < y ∗ , ˆ f ( x ) is non-decreasing and positive for x ≥ y ∗ ,(b) E x h sup ≤ t ≤ T ˆ f ( X t ) i = g ( x ) for x ≥ y ∗ , and(c) E x h sup ≤ t ≤ T ˆ f ( X t ) i ≥ g ( x ) for x ≤ y ∗ .Then V ( x ) = E x " sup ≤ t ≤ T ˆ f ( X t ) [ y ∗ ,b ) ( X t ) = E x h ˆ f ( M T ) [ y ∗ ,b ) ( M T ) i (7) and τ ∗ = inf { t ≥ | X t > y ∗ } is an optimal stopping time. This theorem essentially says that if we can find a function satisfying the required conditions(a)-(c), then the optimal stopping policy for (6) constitutes an one-sided threshold rule. Moreover,in that case we also notice that the value can be expressed as an expected supremum attained at anindependent exponential random time. As we will prove later in this paper, the reverse argument
XPECTED SUPREMUM REPRESENTATION AND OPTIMAL STOPPING 5 is also sometimes true: under certain circumstances based on the behavior of the infinitesimalgenerator of the underlying diffusion the value of the optimal policy generates a continuous andmonotone function ˆ f for which the representation (7) is valid. However, as we will point out inExample 1, all single boundary stopping problems cannot be represented as proposed in Theorem2.1. The second representation theorem established in [14] focusing on two-sided stopping rules issummarized in the following . Theorem 2.2. ( [14] , Theorem 2.7) Let X t be a Hunt process on I and T ∼ Exp( r ) ⊥ X t . As-sume that the exercise payoff g is non-negative, lower semicontinuous, and satisfies the condition E x (cid:2) sup t ≥ e − rt g ( X t ) (cid:3) < ∞ for all x ∈ I . Assume also that there exists an upper semicontinuous ˆ f and a pair of points ( z ∗ , y ∗ ) such that(a) ˆ f ( x ) ≤ for x ∈ ( z ∗ , y ∗ ) , ˆ f ( x ) is non-increasing on ( a, z ∗ ) , nondecreasing on ( y ∗ , b ) , andpositive on ( a, z ∗ ) ∪ ( y ∗ , b ) ,(b) E x h sup ≤ t ≤ T ˆ f ( X t ) i = g ( x ) for x / ∈ [ z ∗ , y ∗ ] , and(c) E x h sup ≤ t ≤ T ˆ f ( X t ) i ≥ g ( x ) for x ∈ [ z ∗ , y ∗ ] .Then V ( x ) = E x " sup ≤ t ≤ T ˆ f ( X t ) ( a,z ∗ ) ∪ ( y ∗ ,b ) ( X t ) = E x hh ˆ f ( I T ) ( a,z ∗ ] ( I T ) i ∨ h ˆ f ( M T ) [ y ∗ ,b ) ( M T ) ii and τ ∗ = inf { t ≥ | X t / ∈ [ z ∗ , y ∗ ] } is an optimal stopping time. Theorem 2.2 essentially states a set of conditions extending the one sided representation con-sidered in Theorem 2.1 to the two-sided setting. It is, however, worth noticing that these theoremsdo not tell us how to come up with such functions ˆ f . Our objective is to identify these functionsin the ordinary linear diffusion setting and in this way establish a link between the supremumrepresentation and the standard solution techniques.Before proceeding in our analysis, we first establish two auxiliary lemmata characterizing thejoint probability distribution of the extreme processes and the underlying diffusion at an independentexponentially distributed random time. Our first findings on the joint probability distribution of M T and I T are summarized in the following. Lemma 2.3.
The joint probability distribution of the extreme processes M t and I t at an independentexponentially distributed random time T reads for all x ∈ ( i, m ) as P x ( I T ≤ i, M T ≤ m ) = − ψ ( x ) ψ ( m ) + ˆ ϕ m ( x )ˆ ϕ m ( i ) + ˆ ψ i ( x )ˆ ψ i ( m ) , (8) Both Theorem 2.1 and Theorem 2.2 are slightly modified versions of the original ones. Three minor misprints havebeen corrected based on a personal communication with P. Salminen
LUIS H. R. ALVAREZ E. AND PEKKA MATOM ¨AKI where ˆ ϕ m ( x ) = ψ ( m ) ϕ ( x ) − ϕ ( m ) ψ ( x ) and ˆ ψ i ( x ) = ϕ ( i ) ψ ( x ) − ψ ( i ) ϕ ( x ) . The marginal distributionsread as P x ( M T ≤ m ) = 1 − ψ ( x ) ψ ( m )(9) for x ∈ ( a, m ) and as P x ( I T ≤ i ) = ϕ ( x ) ϕ ( i ) for x ∈ ( i, b ) .Proof. It is known that (see [12], pp. 25–26) P x ( M T ≤ m ) = 1 − P x ( τ m < T ) = 1 − E x (cid:2) e − rτ m (cid:3) = 1 − ψ ( x ) ψ ( m )for all x ∈ ( a, m ). In a completely analogous fashion, we find that for x ∈ ( i, b ) it holds (see [12],pp. pp. 25–26) P x ( I T ≤ i ) = P x ( τ i < T ) = ϕ ( x ) ϕ ( i ) . For determining the joint probability distribution, we first notice that for all x ∈ ( i, m ) we have P x ( I T ≥ i, M T ≤ m ) = 1 − P x ( τ i,m ≤ T ) = 1 − E x (cid:2) e − rτ i,m (cid:3) = 1 − ˆ ϕ m ( x )ˆ ϕ m ( i ) − ˆ ψ i ( x )ˆ ψ i ( m )and that P x ( I T ≤ i, M T ≤ m ) = P x ( M T ≤ m ) − P x ( I T ≥ i, M T ≤ m ) = − ψ ( x ) ψ ( m ) + ˆ ϕ m ( x )ˆ ϕ m ( i ) + ˆ ψ i ( x )ˆ ψ i ( m ) , where P x ( M T ≤ m ) was calculated already in (9). (cid:3) Let ˜ X t = { X t ; t < τ v } , τ v = inf { t ≥ X t ≥ v } , denote the diffusion X killed at v ∈ I and ˆ X t = { X t ; t < τ i } , τ i = inf { t ≥ X t ≤ i } , denote the diffusion X killed at i ∈ I . Giventhese diffusions, we define ˆ M t = sup { ˆ X s , s ≤ t } and ˜ I t = inf { ˜ X s , s ≤ t } . We can now establish thefollowing useful result needed later in the characterization of the value of a stopping problem as anexpected supremum in the two-boundary setting. Lemma 2.4.
Assume that a < i < v < b . Then, P x h ˆ X T ∈ dy | ˆ M T = v i = r ˆ ψ i ( y ) m ′ ( y ) dy ˆ ψ ′ i ( v ) S ′ ( v ) − B P x h ˜ X T ∈ dy | ˜ I T = i i = r ˆ ϕ v ( y ) m ′ ( y ) dy − B − ˆ ϕ ′ v ( i ) S ′ ( i ) . XPECTED SUPREMUM REPRESENTATION AND OPTIMAL STOPPING 7 for all x ∈ ( i, v ) . Consequently, if h : ( i, v ) R is integrable, we have E x [ h ( ˆ X T ) | ˆ M T = v ] = R vi h ( y ) ˆ ψ i ( y ) m ′ ( y ) dy R vi ˆ ψ i ( y ) m ′ ( y ) dy E x [ h ( ˜ X T ) | ˜ I T = i ] = R vi h ( y ) ˆ ϕ v ( y ) m ′ ( y ) dy R vi ˆ ϕ v ( y ) m ′ ( y ) dy . Proof.
Assume that a < i < v < b and let ¯ X t = { X t , t < τ i ∧ τ v } denote the diffusion X killed atthe boundaries i and v . It is then clear by definition of the processes ˆ M t and ˜ I t that P x h ˆ M T ≤ v i = P x h ˜ I T ≥ i i = P x [ T < τ v ∧ τ i ] = 1 − ˆ ψ i ( x )ˆ ψ i ( v ) − ˆ ϕ v ( x )ˆ ϕ v ( i )implying that P x h ˆ M T ∈ dv i = ˆ ψ i ( x )ˆ ψ i ( v ) (cid:16) ˆ ψ ′ i ( v ) − BS ′ ( v ) (cid:17) dv P x h ˜ I T ∈ di i = ˆ ϕ v ( x )ˆ ϕ v ( i ) (cid:0) − BS ′ ( i ) − ˆ ϕ ′ v ( i ) (cid:1) di. On the other hand, P x h ˆ X T ∈ dy ; ˆ M T ≤ v i = P x h ˜ X T ∈ dy ; ˜ I T ≥ i i = P x (cid:2) ¯ X T ∈ dy (cid:3) = r ¯ G r ( x, y ) dy, where ¯ G r ( x, y ) = B − ˆ ϕ v ( x ) ˆ ψ i ( y )ˆ ψ i ( v ) x ≥ yB − ϕ v ( y )ˆ ϕ v ( i ) ˆ ψ i ( x ) x ≤ y. is the Green kernel associated with the killed diffusion ¯ X . Standard differentiation yields P x h ˆ X T ∈ dy ; ˆ M T ∈ dv i = r ˆ ψ i ( x )ˆ ψ i ( v ) ˆ ψ i ( y ) S ′ ( v ) m ′ ( y ) dydv P x h ˜ X T ∈ dy ; ˜ I T ∈ di i = r ˆ ϕ v ( x )ˆ ϕ v ( i ) ˆ ϕ v ( y ) S ′ ( i ) m ′ ( y ) dydi. The proposed conditional probability distributions follow from the definition of conditional proba-bility. The alleged conditional expectations are finally obtained by ordinary integration. (cid:3) One-boundary, increasing case
Problem Setting.
Our objective in this section is to delineate the circumstances under whichthe value of a one-sided threshold policy can be expressed as the expected supremum of a monotonicfunction and to identify that function explicitly. In what follows, we will focus on the case where theconsidered stopping policy can be characterized as a rule where the underlying process is stopped assoon as it exceeds a given constant threshold. The case where the single boundary stopping rule isto exercise as soon as the underlying falls below a given constant threshold is completely analogousand, therefore, left untreated.
LUIS H. R. ALVAREZ E. AND PEKKA MATOM ¨AKI
Let g : I 7→ R be a continuous payoff function for which g − ( R + ) = ∅ and satisfying E x (cid:20) sup t ≥ e − rt g ( X t ) (cid:21) < ∞ (10)for all x ∈ I . Assume also that g ∈ C ( I \ P ) ∩ C ( I \ P ), where
P ∈ I is a finite set of points in I and that | g ′ ( x ± ) | < ∞ and | g ′′ ( x ± ) | < ∞ for all x ∈ P .Given the assumed regularity conditions, let τ y = inf { t ≥ X t ≥ y } denote the first exit timeof the underlying diffusion from the set ( a, y ), where y ∈ g − ( R + ). Define now the nonnegativefunction V y : I 7→ R + as V y ( x ) = E x (cid:2) e − rτ y g ( X τ y ); τ y < ∞ (cid:3) = g ( x ) x ≥ yψ ( x ) g ( y ) ψ ( y ) x < y. (11)Given representation (11), we can now state our identification problem as follows. Problem 3.1.
For a given y ∈ g − ( R + ) , does there exist a nonnegative function ˆ f : I R + suchthat for all x ∈ I we would have J y ( x ) := E x h ˆ f ( M T ) [ y,b ) ( M T ) i = V y ( x ) , where T ∼ Exp( r ) ⊥ X t (cf. Theorem 2.1). Under which conditions on the function ˆ f we have ˆ f ( M T ) [ y,b ) ( M T ) sup t ∈ [0 ,T ] ˆ f ( X t ) [ y,b ) ( X t ) . It’s worth emphasizing that Problem 3.1 is twofold. The first representation problem essentiallyasks if the expected value of the exercise payoff accrued at the first hitting time to a constantboundary can be expressed as the expected value of an yet unknown function ˆ f at the runningmaximum of the underlying diffusion at an independent exponentially distributed date. The secondquestion essentially asks when the function ˆ f is such that the representation agrees with the generalfunctional form utilized in Theorem 2.1. As we will later establish in this paper, the class of functionssatisfying the first representation is strictly larger than the latter.Before proceeding in the derivation of the representation as an expected supremum, we firstestablish the following result characterizing the optimal policy. We apply this result later for theidentification of circumstances under which the value of the considered one-sided problem can beexpressed as the expected supremum of a monotonic function. Lemma 3.2.
Assume that the following conditions are satisfied: (i) there exists a y ∗ = argmax { g ( x ) /ψ ( x ) } ∈ I , (ii) ( G r g )( x ) ≤ for all x ∈ [ y ∗ , b ) \ P (iii) g ′ ( x +) ≤ g ′ ( x − ) for all x ∈ [ y ∗ , b ) ∩ P Then V ( x ) = V y ∗ ( x ) and τ y ∗ = inf { t ≥ X t ≥ y ∗ } is an optimal stopping time. XPECTED SUPREMUM REPRESENTATION AND OPTIMAL STOPPING 9
Proof.
It is clear that under our assumptions V y ∗ ( x ) is nonnegative, continuous, and dominates theexercise payoff g ( x ) for all x ∈ I . Let x ∈ ( y ∗ , b ) \ P be a fixed reference point and define theratio h x ( x ) = V y ∗ ( x ) /V y ∗ ( x ) = V y ∗ ( x ) /g ( x ). It is clear that our assumptions combined with (1)guarantee that σ h x x (( x, b ]) = ψ ( x ) Bg ( x ) (cid:20) g ′ ( x +) S ′ ( x ) ϕ ( x ) − g ( x ) ϕ ′ ( x ) S ′ ( x ) (cid:21) = − ψ ( x ) Bg ( x ) ( L ϕ g )( x +)is nonnegative and nonincreasing for all x ≥ x . Analogously, σ h x x ([ a, x )) = ϕ ( x ) Bg ( x ) (cid:20) g ( x ) ψ ′ ( x ) S ′ ( x ) − g ′ ( x − ) S ′ ( x ) ψ ( x ) (cid:21) ( y ∗ ,x ] ( x ) = ϕ ( x ) Bg ( x ) ( L ψ g )( x − ) ( y ∗ ,x ] ( x )is nonnegative and nondecreasing for all x ≤ x . Moreover, noticing that σ h x x ([ a, x ))+ σ h x x (( x , b ]) =1 shows, by imposing the condition σ h x x ( { x } ) = 0, that σ h x x constitutes a probability measure.Therefore, it induces an r -excessive function h x ( x ) via its Martin representation (cf. Proposition3.3 in [33]). However, since increasing linear transformations of excessive functions are excessiveand h x ( x ) g ( x ) = V y ∗ ( x ), we observe that V y ∗ ( x ) constitutes an r -excessive majorant of g for X .Invoking now (11) shows that V ( x ) = V y ∗ ( x ) and consequently, that τ y ∗ = inf { t ≥ X t ≥ y ∗ } isan optimal stopping time. (cid:3) Remark . It is at this point worth emphasizing that under the following slightly stricter assump-tions there always exists a unique maximizing threshold y ∗ = argmax { g ( x ) /ψ ( x ) } ∈ (˜ x, b ) and theconditions of Lemma 3.2 are satisfied (cf. Lemma 3.4 in [4]):(A) g − ( R + ) = ( x , b ), where a < x < b , and b is unattainable for X ,(B) there exists a ˜ x ∈ I so that ( G r g )( x ) > x ∈ ( a, ˜ x ) \ P and ( G r g )( x ) < x ∈ (˜ x, b ) \ P ,(C) g ′ ( x +) ≥ g ′ ( x − ) for all x ∈ ( a, ˜ x ) ∩ P and g ′ ( x +) ≤ g ′ ( x − ) for all x ∈ [˜ x, b ) ∩ P These assumptions are typically met in financial applications of optimal stopping. Note that theseconditions do not impose monotonicity requirements on the behavior of the generator ( G r g )( x ) on I \ P and only the sign of ( G r g )( x ) essentially counts. As we will later establish, it is precisely thisobservation which explains why not all single boundary stopping problems can be represented asexpected suprema.3.2. Characterization of f . Let y ∈ g − ( R + ) be given. Utilizing the distribution function char-acterized in (9) yields J y ( x ) = E x h ˆ f ( M T ) [ y,b ) ( M T ) i = ψ ( x ) Z bx ∨ y ˆ f ( z ) ψ ′ ( z ) ψ ( z ) dz. Given this expression, it is now sufficient to find a function ˆ f for which the identity V y ( x ) = J y ( x )holds. This identity holds for x ≥ y provided that the Volterra integral equation of the the first kind g ( x ) ψ ( x ) = Z bx ˆ f ( z ) ψ ′ ( z ) ψ ( z ) dz (12)is satisfied. Identity (12) has several important implications both on the regularity of g as well as onthe limiting behavior of the ratio g ( x ) /ψ ( x ) at b . First, we immediately notice that representation(12) implies that we necessarily need to have lim x → b − g ( x ) /ψ ( x ) = 0. Second, since the integral ofan integrable function is continuous, identity (12) implies that the exercise payoff g ( x ) has to becontinuous on [ y, b ). Moreover, if the unknown function ˆ f is continuous outside a finite set of points P ∈ [ y, b ), then identity (12) actually implies that the exercise payoff g ( x ) has to be continuouslydifferentiable on x ∈ [ y, b ) \ P and possesses both right and left derivatives on x ∈ P . Thus, (12)demonstrates that the proposed representation cannot hold unless the exercise payoff g satisfies aset of regularity conditions.Standard differentiation of identity (12) now shows that for all x ∈ [ y, b ) \ P we haveˆ f ( x ) = g ( x ) − ψ ( x ) g ′ ( x ) ψ ′ ( x ) = S ′ ( x ) ψ ′ ( x ) ( L ψ g )( x ) , (13)coinciding with the function ρ derived in [7] by relying on functional concavity arguments. Thus, withˆ f ( x ) defined in this way we have, by invoking identity (12) and condition lim x → b − g ( x ) /ψ ( x ) = 0,that J y ( x ) = ψ ( x ) Z bx ∨ y ψ ′ ( z ) g ( z ) − g ′ ( z ) ψ ( z ) ψ ( z ) dz = − ψ ( x ) Z bx ∨ y d (cid:18) g ( z ) ψ ( z ) (cid:19) = ψ ( x ) g ( x ∨ y ) ψ ( x ∨ y )(14)for all x ∈ [ y, b ) \ P . We summarize these findings in the following theorem. Theorem 3.4.
Fix y ∈ g − ( R + ) and let ˆ f be as in (13) . Then, if lim x → b − g ( x ) /ψ ( x ) = 0 , we have J y ( x ) = V y ( x ) . Moreover, if ˆ f ( x ) is also nonnegative and nondecreasing for all x ∈ [ y, b ) , then V y ( x ) is r -excessive for X .Proof. The first claim follows directly from the derivation of (14). If ˆ f is also nonnegative andnondecreasing for all x ∈ [ y, b ), then ˆ f ( x ) [ y,b ) ( x ) is nondecreasing, nonnegative, and upper semi-continuous on I . In that case ˆ f ( M T ) [ y,b ) ( M T ) = sup t ∈ [0 ,T ] ˆ f ( X t ) [ y,b ) ( X t ). Proposition 2.1 in [19](see also Lemma 2.2 in [14]) then guarantees that J y ( x ) is r -excessive for X . Since J y ( x ) = V y ( x )the alleged result follows. (cid:3) Theorem 3.4 shows that when ˆ f is chosen according to the rule (13) representation J y ( x ) = V y ( x )is valid provided that the limiting condition lim x → b − g ( x ) /ψ ( x ) = 0 is met. Moreover, Theorem3.4 also shows that if ˆ f ( x ) [ y,b ) ( x ) is also nondecreasing and nonnegative, then the representationis r -excessive for the underlying diffusion X . Note, however, that the representation needs notto majorize the exercise payoff and, therefore, it does not necessarily coincide with the value ofthe considered stopping problem. Moreover, the monotonicity and nonnegativity of ˆ f ( x ) [ y,b ) ( x ) is XPECTED SUPREMUM REPRESENTATION AND OPTIMAL STOPPING 11 sufficient but not necessary for the r -excessivity of J y ( x ). As we will later see, there are circumstanceswhere J y ( x ) is r -excessive even when ˆ f ( x ) [ y,b ) ( x ) is not monotonic.We are now in position to establish the following. Theorem 3.5.
Assume that the conditions of Lemma 3.2 are satisfied and that lim x → b g ( x ) /ψ ( x ) =0 . Then, V ( x ) = V y ∗ ( x ) = J y ∗ ( x ) = E x h ˆ f ( M T ) [ y ∗ ,b ) ( M T ) i . Proof.
It is clear that the conditions of the first claim of Theorem 3.4 are satisfied. Consequently, J y ∗ ( x ) = V y ∗ ( x ). The alleged result now follows from Lemma 3.2. (cid:3) Theorem 3.5 proves that the value of the optimal stopping strategy can be expressed as theexpected value of a mapping ˆ f at the running maximum of the underlying diffusion. This doesnot yet guarantee that the value of the stopping could be expressed as an expected supremum. Inwhat follows, our objective is to first determine a set of conditions under which we also have that J y ( x ) = E x h sup t ∈ [0 ,T ] ˆ f ( X t ) [ y,b ) ( X t ) i . In order to accomplish that objective, we first present anauxiliary result characterizing the circumstances under which the function ˆ f is indeed monotonic. Lemma 3.6.
Let y ∈ g − ( R + ) be given. Assume that either (A) g ( x ) is concave and ψ ( x ) is convex on [ y, b ) , or (B) there is a z ∈ ( a, y ) so that g ( x ) /ψ ( x ) is locally increasing at z , g ′ ( x +) ≤ g ′ ( x − ) for all x ∈ ( z, b ) ∩ P , and ( G r g )( x ) is non-increasing and non-positive for all x ∈ ( z, b ) .Then, the function ˆ f ( x ) characterized by (13) is non-decreasing on [ y, b ) .Proof. It is clear from (13) that the required monotonicity of ˆ f is met provided that inequality ddx (cid:18) g ′ ( x ) ψ ′ ( x ) (cid:19) < x ∈ [ y, b ) \ P andˆ f ( x +) − ˆ f ( x − ) = g ′ ( x − ) − g ′ ( x +) ψ ′ ( x ) > x ∈ [ y, b ) ∩ P . First, if g is concave and ψ is convex on [ y, b ), then the inequalities (15)and (16) are satisfied and g ′ ( x ) /ψ ′ ( x ) is non-increasing on [ y, b ) as claimed. Assume now insteadthat the conditions of part (B) are satisfied. It is clear that since [ y, b ) ⊂ ( z, b ) (16) is satisfied byassumption for all x ∈ [ y, b ) ∩ P . On the other hand, standard differentiation shows that for all x ∈ ( z, b ) \ P ddx (cid:18) g ′ ( x ) ψ ′ ( x ) (cid:19) = S ′ ( x ) ψ ′ ( x ) (cid:20) g ′′ ( x ) S ′ ( x ) ψ ′ ( x ) − ψ ′′ ( x ) S ′ ( x ) g ′ ( x ) (cid:21) = 2 S ′ ( x ) D ( x ) σ ( x ) ψ ′ ( x ) . where D ( x ) = ( G r g )( x ) ψ ′ ( x ) S ′ ( x ) + r ( L ψ g )( x ) . The assumed monotonicity and non-positivity of ( G r g )( x ) on ( z, b ) \ P now implies that D ( x ) = ( G r g )( x ) ψ ′ ( x ) S ′ ( x ) − r Z xz ψ ( t )( G r g )( t ) m ′ ( t ) dt + r ( L ψ g )( z +) ≤ ( G r g )( x ) ψ ′ ( z ) S ′ ( z ) + r ( L ψ g )( z +) ≤ r ( L ψ g )( z +)for all x ∈ ( z, b ) \ P . However, the assumed monotonicity of g ( x ) /ψ ( x ) in a neighborhood of z thenguarantees that ( L ψ g )( z +) ≤
0, proving that D ( x ) ≤ x ∈ ( z, b ) \ P . (cid:3) Lemma 3.6 states a set of conditions under which the function ˆ f ( x ) characterized by (13) isnon-decreasing on the set [ y, b ) and, therefore, the function ˆ f ( x ) [ y,b ) ( x ) is nondecreasing on I .Interestingly, the first of these conditions is based solely on the concavity of the exercise payoffand the convexity of the increasing fundamental solution without imposing further requirements.A sufficient condition for the convexity of the fundamental solution ψ ( x ) is that µ ( x ) − rx is non-increasing on I and a is unattainable for the underlying diffusion (see [1]). Consequently, part (A)of Lemma 3.6 essentially delineates circumstances under which the monotonicity of function ˆ f ( x )could be, in principle, characterized solely based on the infinitesimal characteristics of the underlyingdiffusion and the concavity of the exercise payoff. Part (B) of Lemma 3.6 shows, in turn, how themonotonicity of the function ˆ f ( x ) is associated with the monotonicity of the generator ( G r g )( x ). Theconditions of part (B) of Lemma 3.6 are satisfied, for example, under the assumptions of Remark3.3 provided that ( G r g )( x ) is non-increasing on (˜ x, b ) and z ∈ (˜ x, y ∧ y ∗ ).Moreover, it is clear that under the conditions of Lemma 3.6 we have J y ( x ) = V y ( x ) for all y ∈ I . However, without imposing further restrictions on the behavior of the payoff we do not knowwhether ˆ f ( x ) [ y,b ) ( x ) generates the smallest r -excessive majorant of the exercise payoff g or not, nordo we know how ˆ f ( x ) [ y,b ) ( x ) behaves in the neighborhood of the optimal stopping boundary. Ournext theorem summarizes a set of conditions under which these questions can be unambiguouslyanswered. Theorem 3.7.
Define y ∗ = inf { y : ˆ f ( y ) ≥ } and assume that the conditions (A) or (B) of Lemma3.6 are satisfied on [ y ∗ , b ) . Then, y ∗ = argmax { g ( x ) /ψ ( x ) } ∈ I . Especially, ˆ f ( y ∗ ) = 0 if y ∗ ∈ I \ P and ˆ f ( y ∗ ) = g ( y ∗ ) − ψ ( y ∗ ) ψ ′ ( y ∗ ) g ′ ( y ∗ +) > if y ∗ ∈ P . Moreover, ˆ f ( x ) = S ′ ( x ) ψ ′ ( x ) ( L ψ g )( x +) = ( L ψ g )( y ∗ +) − R xy ∗ ( G r g )( z ) ψ ( z ) m ′ ( z ) dzr R xa ψ ( z ) m ′ ( z ) dz + ψ ′ ( a +) /S ′ ( a +)(17) for all x ∈ ( y ∗ , b ) \ P , and V ( x ) = V y ∗ ( x ) = J y ∗ ( x ) = ψ ( x ) sup y ≥ x (cid:20) g ( y ) ψ ( y ) (cid:21) = ψ ( x ) g ( x ∨ y ∗ ) ψ ( x ∨ y ∗ ) = E x " sup t ∈ [0 ,T ] ˆ f ( X t ) [ y ∗ ,b ) ( X t ) . XPECTED SUPREMUM REPRESENTATION AND OPTIMAL STOPPING 13
Proof.
We first observe that condition (A) or (B) of Lemma 3.6 guarantee that ˆ f ( x ) is nondecreasingon [ y ∗ , b ). However, since ddx (cid:18) g ( x ) ψ ( x ) (cid:19) = − ψ ′ ( x ) ψ ( x ) ˆ f ( x ) , and the ratio g ( x ) /ψ ( x ) is continuous, we notice that g ( x ) /ψ ( x ) is increasing on ( a, y ∗ ) and decreas-ing on ( y ∗ , b ). Consequently, y ∗ = argmax { g ( x ) /ψ ( x ) } . As is clear, if y ∗ ∈ I \ P , then we necessarilyhave g ′ ( y ∗ ) ψ ( y ∗ ) = g ( y ∗ ) ψ ′ ( y ∗ ) showing that ˆ f ( y ∗ ) = 0 in that case. If the optimum is, however,attained on P , then we necessarily have that g ′ ( y ∗ − ) ψ ( y ∗ ) ≥ g ( y ∗ ) ψ ′ ( y ∗ ) ≥ g ′ ( y ∗ +) ψ ( y ∗ ), where atleast one of the inequalities is strict, proving that ˆ f ( y ∗ +) > f ( x ) = S ′ ( x ) ψ ′ ( x ) ( L ψ g )( x ) by invoking the canonical form ψ ′ ( x ) S ′ ( x ) − ψ ′ ( a +) S ′ ( a +) = r Z xa ψ ( z ) m ′ ( z ) dz and noticing that ( L ψ g ) ′ ( x ) = − ( G r g )( x ) ψ ( x ) m ′ ( x )for all x ∈ I \ P . Finally, identity V y ∗ ( x ) = J y ∗ ( x ) = V ( x ) follows from Theorem 3.4 after noticingthat identity y ∗ = argmax { g ( x ) /ψ ( x ) } guarantees that the proposed value dominates the exercisepayoff. (cid:3) Theorem 3.7 shows that the continuity of the function ˆ f at the optimal boundary y ∗ coincideswith the standard smooth fit principle requiring that the value should be continuously differentiableacross the optimal boundary. However, as is clear from Theorem 3.7, if the optimal boundary isattained at a threshold where the exercise payoff is not differentiable, then ˆ f is discontinuous at theoptimal boundary y ∗ . Furthermore, since the nonnegativity and monotonicity of ˆ f ( x ) [ y ∗ ,b ) ( x ) on[ y ∗ , b ) are sufficient for the validity of Theorem 3.7, we observe in accordance with the results by[14] that ˆ f ( x ) [ y ∗ ,b ) ( x ) is only upper semicontinuous on I .Theorem 3.7 also shows that ˆ f ( x ) has a neat integral representation (17) capturing the size ofthe potential discontinuity of ˆ f ( x ) at y ∗ . In the case where a is unattainable and the smooth fitprinciple is satisfied at y ∗ (17) can be re-expressed as (cf. Proposition 2.13 in [14])ˆ f ( x ) = R xy ∗ ( G r g )( z ) ψ ( z ) m ′ ( z ) dzr R xa ψ ( z ) m ′ ( z ) dz (18)and, hence, V ( x ) = E x " R M T y ∗ ( G r g )( z ) ψ ( z ) m ′ ( z ) dzr R M T a ψ ( z ) m ′ ( z ) dz [ y ∗ ,b ) ( M T ) (19)Finally, it is clear that if the sufficient conditions stated in Remark 3.3 are satisfied, and in addition( G r g )( x ) is non-increasing on ( y ∗ , b ), and a is unattainable for the underlying diffusion, then the conditions of Theorem 3.7 are met and V ( x ) = E x " sup t ∈ [0 ,T ] R X t y ∗ ( G r g )( z ) ψ ( z ) m ′ ( z ) dzr R X t a ψ ( z ) m ′ ( z ) dz [ y ∗ ,b ) ( X t ) ! . Examples.
We now illustrate our general findings in two separate examples. The first examplefocuses on a case where the payoff is smooth and the stopping strategy is of the single boundarytype. Despite these favorable properties, we will show that it does not always result into a valuecharacterizable as an expected supremum. The second example, in turn, focuses on a less smoothcase resulting into a representation where the function f ( x ) [ y,b ) ( x ) is monotone but not everywherecontinuous.3.3.1. Example 1: Smooth Payoff.
In order to illustrate our findings we now assume that the upperboundary b is unattainable for X and that the exercise payoff can be expressed as an expectedcumulative present value g ( x ) = ( R r π )( x ) for some continuous revenue flow π ∈ L ( I ) satisfyingthe conditions π ( x ) T x T x , where x ∈ ( a, b ), lim x ↓ a π ( x ) < − ε and lim x ↑ b π ( x ) > ε forsome ε > g ∈ C ( I )and ( G r g )( x ) = − π ( x ) S x T x . Moreover, utilizing representation (5) shows that under ourassumptions ( L ψ g )( x ) = Z xa ψ ( t ) π ( t ) m ′ ( t ) dt It is clear from our assumption that ( L ψ g )( x ) < x ≤ x and ( L ψ g )( x ) is monotonicallyincreasing on ( x , b ). Fix x > x . Then a standard application of the mean value theorem yields( L ψ g )( x ) = ( L ψ g )( x ) + Z xx ψ ( t ) π ( t ) m ′ ( t ) dt = ( L ψ g )( x ) + π ( ξ ) r (cid:20) ψ ′ ( x ) S ′ ( x ) − ψ ′ ( x ) S ′ ( x ) (cid:21) , where ξ ∈ ( x , x ). Letting x → b and noticing that ψ ′ ( x ) /S ′ ( x ) → ∞ as x → b (since b wasassumed to be unattainable for X , cf. p. 19 in [12]) then shows that lim x ↑ b ( L ψ g )( x ) = ∞ provingthat equation ( L ψ g )( x ) = 0 has a unique root y ∗ ∈ ( x , b ) and that y ∗ = argmax { ( R r π )( x ) /ψ ( x ) } .Moreover, the value (6) can be expressed as V ( x ) = ψ ( x ) sup y ≥ x (cid:20) ( R r π )( x ) ψ ( x ) (cid:21) = ( R r π )( x ) x ≥ y ∗ ( R r π )( y ∗ ) ψ ( y ∗ ) ψ ( x ) x < y ∗ . It is clear that under our assumptions the function f ( x ) characterized in Theorem 3.4 can beexpressed as f ( x ) = S ′ ( x ) ψ ′ ( x ) Z xa ψ ( y ) π ( y ) m ′ ( y ) dy. XPECTED SUPREMUM REPRESENTATION AND OPTIMAL STOPPING 15
As was established in Theorem 3.7, we have that f ( y ∗ ) = 0 and, therefore, V ( x ) = E x (cid:20) S ′ ( M T ) ψ ′ ( M T ) Z M T y ∗ ψ ( y ) π ( y ) m ′ ( y ) dy [ y ∗ ,b ) ( M T ) (cid:21) = ψ ( x ) ( R r π )( y ∗ ∨ x ) ψ ( y ∗ ∨ x ) . Moreover, standard differentiation now shows that for all x ∈ ( y ∗ , b ) we have f ′ ( x ) = 2 S ′ ( x ) ψ ( x ) ψ ′ ( x ) σ ( x ) (cid:20) π ( x ) ψ ′ ( x ) S ′ ( x ) − r Z xy ∗ ψ ( t ) π ( t ) m ′ ( t ) dt (cid:21) demonstrating that f is nondecreasing for x ∈ ( y ∗ , b ) only if π ( x ) ψ ′ ( x ) S ′ ( x ) ≥ r Z xy ∗ ψ ( t ) π ( t ) m ′ ( t ) dt for all x ≥ y ∗ . Otherwise it is clear from our results that the value of the considered optimal stop-ping problem cannot be expressed as an expected supremum (see Figure 1(A)). A simple sufficientcondition guaranteeing the required monotonicity is to assume that π ( x ) is nondecreasing on ( x , b )since in that case we have f ′ ( x ) ≥ S ′ ( x ) ψ ( x ) ψ ′ ( x ) σ ( x ) π ( x ) ψ ′ ( y ∗ ) S ′ ( y ∗ ) ≥ . If this is indeed the case, then sup t ∈ [0 ,T ] f ( X t ) [ y ∗ ,b ) ( X t ) = f ( M T ) [ y ∗ ,b ) ( M T ) and V ( x ) = E x " sup t ∈ [0 ,T ] f ( X t ) [ y ∗ ,b ) ( X t ) = E x (cid:2) f ( M T ) [ y ∗ ,b ) ( M T ) (cid:3) . Example 2: Capped Call Option.
In order to illustrate our findings in a nondifferentiablesetting, assume now that the upper boundary b is unattainable for X and that the exercise payoff g ( x ) = min(( x − K ) + , C ) (a capped call option ), with a < K < C < K + C < b , satisfies the limitinginequality lim x ↓ a | x − K | ϕ ( x ) < ∞ . (20)Assume also that the appreciation rate θ ( x ) = µ ( x ) − r ( x − K ) satisfies the conditions θ ∈ L r ( I ), θ ( x ) T x S x θ , where x θ ∈ I , and lim x → b θ ( x ) < − ε for ε > x ∗ = argmax { g ( x ) /ψ ( x ) } and V ( x ) = V x ∗ ( x ). Our objective isnow to prove that this threshold reads as x ∗ = min( C + K, y ∗ ), where y ∗ > x θ is the unique root ofequation Z y ∗ a ψ ( y ) θ ( y ) m ′ ( y ) dy = a − Kϕ ( a ) . To see that this is indeed the case, we first observe by applying part (A) of Corollary 3.2 in [2]combined with the limiting condition (20) that ψ ( x ) S ′ ( x ) ddx (cid:20) x − Kψ ( x ) (cid:21) = ψ ( x ) S ′ ( x ) − ( x − K ) ψ ′ ( x ) S ′ ( x ) = Z xa ψ ( t ) θ ( t ) m ′ ( t ) dt − a − Kϕ ( a ) . Applying analogous arguments with the ones in Example 1, we find that equation Z xa ψ ( t ) θ ( t ) m ′ ( t ) dt − a − Kϕ ( a ) = 0has a unique root y ∗ ∈ ( x θ , b ) so that y ∗ = argmax { ( x − K ) /ψ ( x ) } . Moreover, U ( x ) = sup τ E x (cid:2) e − rτ ( X τ − K ) + (cid:3) = x − K x ≥ y ∗ ( y ∗ − K ) ψ ( x ) ψ ( y ∗ ) x < y ∗ . In light of these observations, we find that if y ∗ ∈ ( K, K + C ), then it is sufficient to notice that V x ∗ ( x ) = min( C, U ( x )) is r -excessive since constants are r -excessive and U ( x ) is also r -excessive.Moreover, since both C and U ( x ) dominate the payoff, we notice that V x ∗ ( x ) = min( C, U ( x ))constitutes the smallest r -excessive majorant of g ( x ) and, therefore, V ( x ) = V x ∗ ( x ) = min( C, U ( x )).If instead y ∗ ≥ K + C , then x ∗ = K + C = argmax { g ( x ) /ψ ( x ) } and the optimal policy is to followthe stopping policy τ x ∗ = inf { t ≥ X t ≥ K + C } with a value˜ U ( x ) = C E x (cid:2) e − rτ x ∗ (cid:3) = C x ≥ C + KC ψ ( x ) ψ ( C + K ) x < C + K. Given these findings, we notice that if y ∗ ≥ K + C , then f ( x ) = C [ x ∗ ,b ) ( x ) ≥ V ( x ) = C E x (cid:2) [ x ∗ ,b ) ( M T ) (cid:3) = C P x [ M T ≥ K + C ] . However, since f ( x ∗ − ) = 0 and f ( x ∗ +) = C we notice that f is discontinuous at the optimalthreshold x ∗ (see Figure 1(B)). If y ∗ < K + C , then the nonnegative function f ( x ) = C x ≥ C + Kx − K − ψ ( x ) ψ ′ ( x ) x ∈ [ y ∗ , K + C )in nondecreasing only if the increasing fundamental solution is convex on ( y ∗ , K + C ) (it has to belocally convex at y ∗ ). If the convexity requirement is met, then V ( x ) = E x (cid:20)(cid:18) M T − K − ψ ( M T ) ψ ′ ( M T ) (cid:19) [ y ∗ ,C + K ) ( M T ) (cid:21) + C P x [ M T ≥ C + K ] . Moreover, since f ( C + K +) = C > C − ψ ( C + K − ) ψ ′ ( C + K − ) = f ( C + K − ), we notice that f is discontinuousat C + K . 4. Two-boundary case
Having considered the one-sided stopping policies our objective is to now extend our analysisto a two-boundary setting and determine a representation of the value in terms of a supremum of
XPECTED SUPREMUM REPRESENTATION AND OPTIMAL STOPPING 17 (a)
Example 1: Smooth payoff with π ( x ) = ( x − e − x + 1 leads to a non-increasing ˆ f . In this case therepresentation as an expected supremum fails to exist. (b) Example 2: Capped call option with g ( x ) =min { ( x − + , } leads to a discontinuous ˆ f . Figure 1.
Numerical examples based on geometric Brownian motion. Parameters havebeen chosen such that ψ = x and ϕ = x − a given function satisfying a set of regularity and monotonicity conditions. In order to accomplishthis task, we assume throughout this section that g : I 7→ R be a continuous payoff function forwhich g − ( R + ) = ∅ and satisfying condition E x (cid:20)(cid:18) sup t ≥ e − rt g ( X t ) (cid:19) ∨ (cid:18) − inf t ≥ e − rt g ( X t ) (cid:19)(cid:21) < ∞ (21)for all x ∈ I . Along the lines of the single boundary setting we also assume that g ∈ C ( I \ P ) ∩ C ( I \ P ), where
P ∈ I is a finite set of points in I and that | g ′ ( x ± ) | < ∞ and | g ′′ ( x ± ) | < ∞ forall x ∈ P .Let τ z,y = inf { t ≥ X t / ∈ ( z, y ) } denote the first exit time of X from the open set ( z, y ) ⊂ I with compact closure in I and denote by V z,y ( x ) := E x (cid:2) e − rτ z,y g ( X τ z,y ); τ z,y < ∞ (cid:3) the expected present value of the exercise payoff accrued from following that stopping strategy. Itis well known that in that case V can be rewritten as (cf. [31]) V z,y ( x ) = g ( x ) x ∈ ( a, z ] ∪ [ y, b ) ˆ ϕ y ( x )ˆ ϕ y ( z ) g ( z ) + ˆ ψ z ( x )ˆ ψ z ( y ) g ( y ) x ∈ ( z, y ) , (22)where ˆ ϕ y ( x ) = ϕ ( x ) ψ ( y ) − ϕ ( y ) ψ ( x ) denotes the decreasing and ˆ ψ z ( x ) = ψ ( x ) ϕ ( z ) − ψ ( z ) ϕ ( x )the increasing fundamental solution of the ordinary differential equation ( G r u )( x ) = 0 defined withrespect to the killed diffusion { X t ; t ∈ [0 , τ z,y ) } . Within this two-boundary setting our identificationproblem can be stated as follows: Problem 4.1.
For a given pair z, y ∈ g − ( R + ) satisfying the condition a < z < y < b , is therea function f ( x ) = f ( x ) ( a,z ] ( x ) + f ( x ) [ y,b ) ( x ) , where f ( x ) is nonincreasing and f ( x ) is nonde-creasing such that for all x ∈ I we would have J ( z,y ) ( x ) := E x (cid:2) f ( I T ) ( a,z ] ( I T ) ∨ f ( M T ) [ y,b ) ( M T ) (cid:3) = V z,y ( x ) , (23) where T ∼ Exp( r ) is independent of the underlying X . It is at this point worth pointing out that if f ( z − ) ∧ f ( y +) ≥
0, then we clearly havesup { f ( X t ); t ≤ T } = f ( I T ) ( a,z ] ( I T ) ∨ f ( M T ) [ y,b ) ( M T )Consequently, Problem 4.1 essentially asks if there exists a function such that the expected presentvalue of the payoff accrued at the first exit time from an open interval can be expressed as as anexpected supremum of that particular function or not. Especially, if the inequality f ( z − ) ∧ f ( y +) ≥ E x [sup { f ( X t ); t ≤ T } ] ≥ (cid:18) ψ ( x ) Z bx ∨ y f ( t ) ψ ′ ( t ) ψ ( t ) dt (cid:19) ∨ (cid:18) − ϕ ( x ) Z x ∧ za f ( t ) ϕ ′ ( t ) ϕ ( t ) dt (cid:19) . (24)Therefore, whenever V z,y ( x ) can be expressed as an expected supremum, it has to dominate thelower bound (24).On the other hand, the function f stated in Problem 4.1 has an additive form. One could, thus,be tempted to search for a similar additive representation of the supremum. Unfortunately, such anapproach is not possible since the assumed monotonicity of the functions f and f implies thatsup { f ( X t ); t ≤ T } ≤ sup { f ( X t ) ( a,z ] ( X t ); t ≤ T } + sup { f ( X t ) [ y,b ) ( X t ); t ≤ T }≤ sup { f +1 ( X t ) ( a,z ] ( X t ); t ≤ T } + sup { f +2 ( X t ) [ y,b ) ( X t ); t ≤ T } = f +1 ( I T ) ( a,z ] ( I T ) + f +2 ( M T ) [ y,b ) ( M T ) . Thus, if the inequality f ( z − ) ∧ f ( y +) ≥ { f ( X t ); t ≤ T } ≤ f ( I T ) ( a,z ] ( I T ) + f ( M T ) [ y,b ) ( M T )and, therefore, that E x [sup { f ( X t ); t ≤ T } ] ≤ ψ ( x ) Z bx ∨ y f ( t ) ψ ′ ( t ) ψ ( t ) dt − ϕ ( x ) Z x ∧ za f ( t ) ϕ ′ ( t ) ϕ ( t ) dt. (25)Based on these findings, we can establish the following. Lemma 4.2.
Assume that f ( z − ) ∨ f ( y +) ≥ . Then J ( z,y ) ( x ) is r -excessive for the underlyingdiffusion X . Moreover, if f ( z − ) ∧ f ( y +) ≥ , then J ( z,y ) ( x ) = E x [sup { f ( X t ); t ≤ T } ] satisfiesinequality (25) for all x ∈ I . XPECTED SUPREMUM REPRESENTATION AND OPTIMAL STOPPING 19
Proof.
We first observe that if f ( z − ) ∨ f ( y +) ≥
0, then f ( x ) ( a,z ] ( x ) ∨ f ( x ) [ y,b ) ( x ) is nonnegativeand upper semicontinuous for all x ∈ I . Proposition 2.1 in [19] then implies that J ( z,y ) ( x ) is r -excessive for X . The second claim was proven in the text. (cid:3) Lemma 4.2 states a set of easily verifiable conditions characterizing circumstances under whichthe proposed representation is r -excessive for the underlying X . It is, however, worth noticingthat Lemma 4.2 does not make statements on the relationship between the values J ( z,y ) ( x ) and V z,y ( x ). Thus, characterizing the expected value J ( z,y ) ( x ) without an explicit characterization ofthe functions f and f is not possible and more analysis is needed. It is also worth emphasizingthat Lemma 4.2 shows that if the auxiliary functions f and f are nonnegative on I , then theexpected supremum J ( z,y ) ( x ) is bounded from above by a functional form which, in principle, couldbe computed explicitly provided that the functions f and f were known.By reordering terms, the value (22) can also be expressed as(26) V z,y ( x ) = A ( z, y ) ϕ ( x ) + A ( z, y ) ψ ( x ) , where A ( z, y ) = ψ ( y ) g ( z ) − g ( y ) ψ ( z ) ψ ( y ) ϕ ( z ) − ψ ( z ) ϕ ( y )and A ( z, y ) = ϕ ( z ) g ( y ) − g ( z ) ϕ ( y ) ψ ( y ) ϕ ( z ) − ψ ( z ) ϕ ( y ) . Hence, if the exercise payoff is differentiable at the thresholds z and y , then1 ψ ( z ) ∂A ∂y ( z, y ) = − ϕ ( z ) ∂A ∂y ( z, y ) = S ′ ( y )ˆ ψ z ( y ) " g ( y ) ˆ ψ ′ z ( y ) S ′ ( y ) − ˆ ψ z ( y ) g ′ ( y ) S ′ ( y ) − Bg ( z ) (27)and 1 ψ ( y ) ∂A ∂z ( z, y ) = − ϕ ( y ) ∂A ∂z ( z, y ) = S ′ ( z )ˆ ψ z ( y ) (cid:20) ˆ ϕ y ( z ) g ′ ( z ) S ′ ( z ) − g ( z ) ˆ ϕ ′ y ( z ) S ′ ( z ) − Bg ( y ) (cid:21) . (28)We will apply these results later when deriving the auxiliary mappings needed for the representationof the value as an expected supremum. Before proceeding in our analysis, we first state the followingauxiliary lemma: Lemma 4.3.
Assume that the following conditions are satisfied: (i) there exists a unique pair ( z ∗ , y ∗ ) satisfying the inequality a < z ∗ < y ∗ < b such that V z ∗ ,y ∗ ( x ) = sup z,y ∈I V z,y ( x ) , (ii) lim x → a + ( g ′ ( x ) ψ ( x ) − g ( x ) ψ ′ ( x )) ≤ and lim x → b − ( g ′ ( x ) ϕ ( x ) − g ( x ) ϕ ′ ( x )) ≥ , (iii) ( G r g )( x ) ≤ for all x ∈ (( a, z ∗ ] ∪ [ y ∗ , b )) \ P , and (iv) g ′ ( x +) ≤ g ′ ( x − ) for all x ∈ (( a, z ∗ ] ∪ [ y ∗ , b )) ∩ P .Then, V ( x ) = V z ∗ ,y ∗ ( x ) and τ z ∗ ,y ∗ = inf { t ≥ X t ( z ∗ , y ∗ ) } is an optimal stopping time. Proof.
It is clear that under our assumptions V z ∗ ,y ∗ ( x ) is nonnegative, continuous, and dominatesthe exercise payoff g ( x ) for all x ∈ I . Consider now the behavior of the mappings ( L ψ V z ∗ ,y ∗ )( x )and ( L ϕ V z ∗ ,y ∗ )( x ). It is clear from (1) that ( L ψ V z ∗ ,y ∗ ) ′ ( x ) = ( L ϕ V z ∗ ,y ∗ ) ′ ( x ) = 0 for all x ∈ ( z ∗ , y ∗ )and ( L ψ V z ∗ ,y ∗ ) ′ ( x ) = − ψ ( x )( G r g )( x ) m ′ ( x ) ≥ , ( L ϕ V z ∗ ,y ∗ ) ′ ( x ) = − ϕ ( x )( G r g )( x ) m ′ ( x ) ≥ x ∈ (( a, z ∗ ) ∪ ( y ∗ , b )) \ P . However, since( L u V z ∗ ,y ∗ )( x − ) − ( L u V z ∗ ,y ∗ )( x +) = u ( x ) g ′ ( x +) − g ′ ( x − ) S ′ ( x ) ≤ x ∈ (( a, z ∗ ] ∪ [ y ∗ , b )) ∩ P when u = ψ or u = ϕ we find that ( L ψ V z ∗ ,y ∗ )( x ) and ( L ϕ V z ∗ ,y ∗ )( x )are nondecreasing on I . Combining these observations with assumption (ii) then proves that( L ψ V z ∗ ,y ∗ )( x ) ≥ L ϕ V z ∗ ,y ∗ )( x ) ≤ x ∈ I .Let x ∈ ( y ∗ , b ) \P be a fixed reference point and define the ratio h x ( x ) = V z ∗ ,y ∗ ( x ) /V z ∗ ,y ∗ ( x ) = V z ∗ ,y ∗ ( x ) /g ( x ). It is clear that our assumptions combined with identity (1) guarantee that σ h x x (( x, b ]) = − ψ ( x ) Bg ( x ) ( L ϕ g )( x +)is nonnegative and nonincreasing for all x ≥ x and σ h x x (( x , b ]) = − ψ ( x ) Bg ( x ) ( L ϕ g )( x ). Analogously, σ h x x ([ a, x )) = ϕ ( x ) Bg ( x ) (cid:2) ( L ψ g )( x − ) ( a,z ∗ ] ∪ [ y ∗ ,x ] ( x ) + ( L ψ V z ∗ ,y ∗ )( z ∗ − ) ( z ∗ ,y ∗ ) ( x ) (cid:3) is nonnegative and nondecreasing for all x ≤ x and satisfies σ h x x ([ a, x )) = ϕ ( x ) Bg ( x ) ( L ψ g )( x ). Theidentity V ( x ) = V z ∗ ,y ∗ ( x ) and optimality of the stopping time τ z ∗ ,y ∗ = inf { t ≥ X t ( z ∗ , y ∗ ) } results follow by utilizing analogous arguments with Lemma 3.2. (cid:3) Lemma 4.3 states a set of sufficient conditions under which the considered stopping problemconstitutes a two boundary problem where the underlying diffusion is stopped as soon as it exitsfrom the continuation region characterized by an open interval in the state space I . As in thecase of Lemma 3.2 no differentiability at the stopping boundaries is required nor do we imposeconditions on the monotonicity of the generator ( G r g )( x ) on I . An interesting implication of theresults of Lemma 4.3 is that at the optimal exercise boundaries we have V ′ z ∗ ,y ∗ ( z ∗ − ) ≥ V ′ z ∗ ,y ∗ ( z ∗ +)and V ′ z ∗ ,y ∗ ( y ∗ − ) ≥ V ′ z ∗ ,y ∗ ( y ∗ +) where the inequalities may be strict in case the smooth fit principleis not satisfied. As we will observe later in this section in our explicit numerical illustrations ofour principal findings, it is precisely the non-differentiability of the value at the exercise thresholdwhich may result in situations where the function needed for the representation of the value as anexpected supremum is discontinuous. Moreover, as in the single boundary setting, the potentialnon-monotonicity of the generator on the stopping set may result in situations where the value ofthe optimal policy cannot be represented as an expected supremum. Remark . Assume that the following conditions are met:(i) ( G r g )( x ) ≤ x ∈ (( a, ˜ x ) ∪ (˜ x , b )) \ P , where a < ˜ x < ˜ x < b . XPECTED SUPREMUM REPRESENTATION AND OPTIMAL STOPPING 21 (ii) the mappings ( L ψ g )( x ) and ( L ϕ g )( x ) are nondecreasing on ( a, ˜ x ] ∪ [˜ x , b ) and satisfy thelimiting conditions lim x ↓ a ( L ψ g )( x ) ≥
0, lim x ↑ b ( L ϕ g )( x ) ≤
0, lim x ↑ b ( L ψ g )( x ) = ∞ , andlim x ↓ a ( L ϕ g )( x ) = −∞ .Then, it can be shown by relying on the fixed point technique developed in [31] and [32] that thereexists a candidate pair z ∗ , y ∗ ∈ ( a, ˜ x ] ∪ [˜ x , b ) maximizing V z,y ( x ) and resulting in a r -excessivefunction V z ∗ ,y ∗ ( x ). Especially, if P ⊂ (˜ x , ˜ x ), then z ∗ , y ∗ ∈ ( a, ˜ x ] ∪ [˜ x , b ) constitutes the uniquepair maximizing V z,y ( x ) and V ( x ) = V z ∗ ,y ∗ ( x ).In order to characterize the functions f and f and determine J ( z,y ) ( x ) explicitly, we first needto make some further assumptions. Assumption 4.5.
We assume that either (a) f ( b ) = f ( a ), or (b) f ( b ) > f ( a ), g ( a ) < ∞ , andlim x ↑ b g ( x ) /ψ ( x ) = 0.It is at this point worthwhile to stress that a proof for the case ” f ( b ) < f ( a ), g ( b ) < ∞ ,and lim x ↓ a g ( x ) /ϕ ( x ) = 0” is completely analogous with the proof in case (b) of Assumption (4.5).Given these assumptions, define the state ζ := f − ( f ( a +)) and the functions α : [ y, b ) ( a, z ] and β : ( a, z ] [ y, ζ ) as (see Figure 2) α ( m ) := f − ( f ( m )) β ( i ) := f − ( f ( i )) . If these points do not exist, we interpret them by the generalized inverses: f − ( x ) = inf { m ∈ [ y, b ] | f ( m ) ≥ x } f − ( x ) = sup { i ∈ [ a, z ] | f ( i ) ≥ x } . Especially, we set α ( m ) = a for all m ≥ ζ and notice that β ( i ) ∈ [ y, b ) constitutes the point inthe domain of f for which the indifference condition f ( i ) = f ( β ( i )) holds, whenever f and f are continuous at the points i and β ( i ), respectively. Similarly, α ( m ) ∈ ( a, z ] constitutes a point inthe domain of f for which identity f ( α ( m )) = f ( m ) holds, whenever f and f are continuousat α ( m ) and m , respectively. In order to ease the notations in the sequel, we shall denote thesefunctions simply by α and β omitting the variables i and m from the notation.4.1. Calculating the expectation.
Utilizing the joint probability distribution (8) described inLemma 2.3 shows that P x ( I T ∈ di, M T ≤ β ( i )) = − BS ′ ( i ) − ˆ ϕ ′ β ( i )ˆ ϕ β ( i ) ˆ ϕ β ( x ) di P x ( I T ≥ α ( m ) , M T ∈ dm ) = − BS ′ ( m ) + ˆ ψ ′ α ( m )ˆ ψ α ( m ) ˆ ψ α ( x ) dm (29) Figure 2.
Illustrating f , f , α , β and ζ . Given these densities, we notice that J ( z,y ) ( x ) can be rewritten as J ( z,y ) ( x ) = Z xa f ( i ) P x ( I T ∈ di, M T < β ( i )) + Z bβ ( x ) f ( m ) P x ( I T > α ( m ) , M T ∈ dm ) x ≤ z Z za f ( i ) P x ( I T ∈ di, M T < β ( i )) + Z by f ( m ) P x ( I T > α ( m ) , M T ∈ dm ) x ∈ ( z, y ) Z α ( x ) a f ( i ) P x ( I T ∈ di, M T < β ( i )) + Z bx f ( m ) P x ( I T > α ( m ) , M T ∈ dm ) x ≥ y. Since our objective is to delineate circumstances under which J ( z,y ) ( x ) = V ( z,y ) ( x ) holds especiallyfor x ∈ ( z, y ), we can first determine for which f the equalitylim x z + ∂∂z J ( z,y ) ( x ) = lim x z + ∂∂z V ( z,y ) ( x )holds. We can then make an ansatz that the solution of this identity constitutes the required function f . In a completely analogous fashion, by differentiating V ( z,y ) with respect to y and setting x y − ,we can make a second ansatz that the solution of the resulting identity constitutes the required f .More precisely, we propose that the functions f and f should be of the form f ( i ) := − g ( β ( i )) BS ′ ( i ) + g ′ ( i ) ˆ ϕ β ( i ) ( i ) − g ( i ) ˆ ϕ ′ β ( i ) ( i ) − BS ′ ( i ) − ˆ ϕ ′ β ( i ) ( i ) f ( m ) := g ( α ( m )) BS ′ ( m ) + g ′ ( m ) ˆ ψ α ( m ) ( m ) − g ( m ) ˆ ψ ′ α ( m ) ( m ) BS ′ ( m ) − ˆ ψ ′ α ( m ) ( m ) . (30)4.2. Verifying our ansatz.
Our objective is now to delineate circumstances under which our ansatzcan be shown to be correct. To this end, at this point we assume that the problem specificationis such that f is non-increasing and f is non-decreasing, otherwise the functions α and β wouldnot be unambiguously defined. Later on, we shall state a set of sufficient conditions under whichthese monotonicity requirements indeed hold. In order to facilitate the explicit computation of the XPECTED SUPREMUM REPRESENTATION AND OPTIMAL STOPPING 23 functions f and f , we assume in what follows that the boundaries a and b are natural for theunderlying diffusion X .Let us now compute f ( m ) for m ∈ [ ζ, b ) \ P . We can rewrite f as f ( m ) = g ( α ) BS ′ ( m ) ϕ ( α ) + g ′ ( m ) ˜ ψ α ( m ) − g ( m ) ˜ ψ ′ α ( m ) BS ′ ( m ) ϕ ( α ) − ˜ ψ ′ α ( m ) , where ˜ ψ i ( m ) = ψ ( m ) − ψ ( i ) ϕ ( i ) ϕ ( m ). Clearly, lim i ↓ a ˜ ψ i ( m ) = ψ ( m ) and lim i ↓ a ˜ ψ ′ i ( m ) = ψ ′ ( m ). More-over, since a was assumed to be natural, and we interpreted α ( m ) = a for all m ≥ ζ , we get, for m ∈ [ ζ, b ) \ P , that f ( m ) = g ( m ) − g ′ ( m ) ψ ′ ( m ) ψ ( m ) . (31)Similarly, applying (29) shows that for all m ≥ ζ it holds P x ( I T ≥ α ( m ) , M T ∈ dm ) = P x ( I T ≥ a, M T ∈ dm ) = ψ ′ ( m ) ψ ( m ) ψ ( x ) . We observe that these are, in fact, the very same functionals we got in Section 3 with the increasingone-sided case. In order to verify our ansatz, let x ∈ ( z, y ), and substitute f and f from (30) and P x ’s from (29) to J ( z,y ) ( x ). After reordering terms, we get J ( z,y ) ( x ) = ϕ ( x ) Z za ψ ( β ) − g ( β ) BS ′ ( i ) + g ′ ( i ) ˆ ϕ β ( i ) − g ( i ) ˆ ϕ ′ β ( i )ˆ ϕ β ( i ) di + ϕ ( x ) Z ζy ψ ( α ) g ( α ) BS ′ ( m ) + g ′ ( m ) ˆ ψ α ( m ) − g ( m ) ˆ ψ ′ α ( m )ˆ ψ α ( m ) dm − ψ ( x ) Z za ϕ ( β ) − g ( β ) BS ′ ( i ) + g ′ ( i ) ˆ ϕ β ( i ) − g ( i ) ˆ ϕ ′ β ( i )ˆ ϕ β ( i ) di − ψ ( x ) Z ζy ϕ ( α ) g ( α ) BS ′ ( m ) + g ′ ( m ) ˆ ψ α ( m ) − g ( m ) ˆ ψ ′ α ( m )ˆ ψ α ( m ) dm + ψ ( x ) Z bζ g ( m ) ψ ′ ( m ) − g ′ ( m ) ψ ( m ) ψ ( m ) dm. Similar to one-sided case (Section 3), we notice that the last integral R bζ () dm equals g ( ζ ) /ψ ( ζ ).(Notice that it follows from our assumptions that if ζ = b , then g ( ζ ) /ψ ( ζ ) = 0.)Next let us make a change in variable in the integrals R ζy () dm : Substitute i := α ( m ) (or m = β ( i )), so that dm = β ′ ( i ) di and the boundaries change as y α ( y ) =: ˆ z ≤ z and ζ a . Wenotice that we can actually change the lower boundary as y z , since for all i ∈ (ˆ z, z ) we have β ′ ( i ) = 0, showing that the integrand between ˆ z and z equals zero. Doing this and reordering terms show that J ( z,y ) ( x ) can now be written as J ( z,y ) ( x ) = ϕ ( x ) Z za dA ( i, β ( i )) di di + ψ ( x ) Z za dA ( i, β ( i )) di di + ψ ( x ) g ( ζ ) ψ ( ζ )= ϕ ( x ) ( A ( z, y ) − A ( a, ζ )) + ψ ( x ) ( A ( z, y ) − A ( a, ζ )) + ψ ( x ) g ( ζ ) ψ ( ζ ) . Finally, since a was assumed to be a natural boundary for X , we obtain that A ( a, ζ ) = 0 and A ( a, ζ ) = g ( ζ ) /ψ ( ζ ). Consequently, J ( z,y ) ( x ) = A ( z, y ) ϕ ( x ) + A ( z, y ) ψ ( x ) = V ( z,y ) ( x ) for x ∈ ( z, y ) as claimed.Verifying the validity of our ansatz for x / ∈ ( z, y ) is entirely analogous. For x ≤ z we get J ( z,y ) ( x ) = ϕ ( x ) Z xa dA ( i, β ( i )) + ψ ( x ) Z xa dA ( i, β ( i )) + ψ ( x ) g ( ζ ) ψ ( ζ )= ϕ ( x ) A ( x, β ( x )) + ψ ( x ) A ( x, β ( x )) = ˆ ϕ β ( x )ˆ ϕ β ( x ) g ( x ) + ˆ ψ x ( x )ˆ ψ x ( β ) g ( β ) = g ( x ) . For x ∈ ( y, ζ ) we get J ( z,y ) ( x ) = ϕ ( x ) Z α ( x ) a dA ( i, β ( i )) + ψ ( x ) Z α ( x ) a dA ( i, β ( i )) + ψ ( x ) g ( ζ ) ψ ( ζ )= ˆ ϕ x ( x )ˆ ϕ x ( α ) g ( α ) + ˆ ψ α ( x )ˆ ψ α ( x ) g ( x ) = g ( x ) . Finally, for x ≥ ζ we get J ( z,y ) ( x ) = ψ ( x ) Z bx g ( m ) ψ ′ ( m ) − g ′ ( m ) ψ ( m ) ψ ( m ) dm = g ( x ) , where the equality follows from the derivation of the one-sided case (14). Let us now summarizethe analysis done so far into the following theorem. Theorem 4.6.
Assume that z, y ∈ g − ( R + ) satisfy the condition a < z < y < b , that a and b arenatural for X , and that Assumption 4.5 holds. Furthermore, assume that f and f are as in (30) .Then, if f is non-increasing and f is non-decreasing, J ( z,y ) ( x ) = V z,y ( x ) . Moreover, if inequality f ( z ) ∨ f ( y ) ≥ is satisfied as well, then V z,y ( x ) and J ( z,y ) ( x ) are r -excessive for X .Proof. The validity of identity J ( z,y ) ( x ) = V z,y ( x ) has been proven in the text. The alleged r -excessivity of J ( z,y ) ( x ) and, consequently, V z,y ( x ) now follows from Lemma 4.2. (cid:3) It is worth pointing out that we can replace assumption (B) of (4.5) with the condition ” f ( b ) Conditions under which f is as required. In the statement of our problem, we assumedthat f ( x ) = f ( x ) ( a,z ] ( x )+ f ( x ) [ y,b ) ( x ), where f ( x ) is non-increasing and f ( x ) is non-decreasing.In this section we state a set of sufficient conditions under which these requirements are unambigu-ously fulfilled. Before stating our principal characterization, we first make the following assumptions: Assumption 4.7. Assume that the exercise payoff g ∈ C ( I ) satisfies the conditions:(a) There is a threshold ˆ x = argmax { ( G r g )( x ) } ∈ I such that ( G r g )( x ) is nondecreasing on( a, ˆ x ), non-increasing on (ˆ x, b ), and ( G r g )(ˆ x ) > G r g )( b − ) ≤ ( G r g )( a +) < − ε, where ε > a < x 0. In that case thepoint ˆ ζ would be on the decreasing part f ( x ). Since the analysis is completely analogous, we leave itfor the interested reader. Moreover, as was shown in [31] and [32] our conditions are sufficient for theexistence of a unique extremal pair z ∗ ∈ ( a, x ) , y ∗ ∈ ( x , b ) s.t. τ z ∗ ,y ∗ = inf { t ≥ X t / ∈ ( z ∗ , y ∗ ) } constitutes the optimal stopping time, V z ∗ ,y ∗ ( x ) = V ( x ) constitutes the value of the optimal stoppingproblem, C = ( z ∗ , y ∗ ) is the continuation region, and Γ = ( a, z ∗ ] ∪ [ y ∗ , b ) is the stopping region.The existence of a pair of monotonic and nonnegative functions f and f is proven in thefollowing. Theorem 4.8. Let Assumption 4.7 hold. Then, f is non-increasing and f is non-decreasing.Moreover, f ( z ∗ ) = f ( y ∗ ) = 0 and f ( i ) = − R βi ˆ ϕ β ( t )( G r g )( t ) m ′ ( t ) dtr R βi ˆ ϕ β ( t ) m ′ ( t ) dt = − E x [( G r g )( ˜ X T ) | ˜ I T = i ] f ( m ) = − R mα ˆ ψ α ( t )( G r g )( t ) m ′ ( t ) dtr R mα ˆ ψ α ( t ) m ′ ( t ) dt = − E x [( G r g )( ˆ X T ) | ˆ M T = m ] , where ˜ X t = { X t ; t < τ β } , ˆ X t = { X t ; t < τ α } , ˜ I t = inf { X s ; s ≤ t ∧ τ β } , and ˆ M t = sup { X s ; s ≤ t ∧ τ α } .Proof. In order to establish the existence and monotonicity of the mappings f , f consider first thefunctions F y ( z ) = g ′ ( z ) S ′ ( z ) ˆ ϕ y ( z ) − g ( z ) ˆ ϕ ′ y ( z ) S ′ ( z ) − Bg ( y ) − B − ˆ ϕ ′ y ( z ) S ′ ( z ) F z ( y ) = g ′ ( y ) S ′ ( y ) ˆ ψ z ( y ) − g ( y ) ˆ ψ ′ z ( y ) S ′ ( y ) + Bg ( z ) B − ˆ ψ ′ z ( y ) S ′ ( y )6 LUIS H. R. ALVAREZ E. AND PEKKA MATOM ¨AKI derived in (30). Utilizing the identities (1) and (2) show that these mappings can be re-expressedin the simpler integral form F y ( z ) = ( L ˆ ϕ y g )( y ) − ( L ˆ ϕ y g )( z )( L ˆ ϕ y )( y ) − ( L ˆ ϕ y )( z ) = − R yz ( G r g )( t ) ˆ ϕ y ( t ) m ′ ( t ) dtr R yz ˆ ϕ y ( t ) m ′ ( t ) dt (32) F z ( y ) = ( L ˆ ψ z g )( y ) − ( L ˆ ψ z g )( z )( L ˆ ψ z )( y ) − ( L ˆ ψ z )( z ) = − R yz ( G r g )( t ) ˆ ψ z ( t ) m ′ ( t ) dtr R yz ˆ ψ z ( t ) m ′ ( t ) dt . (33)The alleged representation of the functions f and f follow directly from (32), (33), and Lemma2.4 provided that the existence of a root of equation F y ( z ) = F z ( y ) can be assured. Utilizingthe identities (32) and (33) show that the solutions have to satisfy identity H ( z, y ) = 0, where H : I R is defined by H ( z, y ) = Z yz ( G r g )( t ) u ( t ) m ′ ( t ) dt, (34)and u ( x ) = ϕ ( x ) (cid:18) ψ ′ ( y ) S ′ ( y ) − ψ ′ ( z ) S ′ ( z ) (cid:19) − ψ ( x ) (cid:18) ϕ ′ ( y ) S ′ ( y ) − ϕ ′ ( z ) S ′ ( z ) (cid:19) is monotonically decreasing and r -harmonic and satisfies the boundary conditions u ( z ) = ˆ ψ ′ z ( y ) /S ′ ( y ) − B > u ( y ) = B + ˆ ϕ ′ y ( z ) /S ′ ( z ) < 0, and u ′ ( z ) S ′ ( y ) = u ′ ( y ) S ′ ( z ) . We first notice that assumptions 4.6. (a) and (b) are sufficient for the existence of a unique pair z ∗ ∈ ( a, x ) , y ∗ ∈ ( x , b ) satisfying the optimality conditions ( L ψ g )( z ∗ ) = ( L ψ g )( y ∗ ) and ( L ϕ g )( z ∗ ) =( L ϕ g )( y ∗ ) implying that for any r -harmonic map u ( x ) = c ψ ( x ) + c ϕ ( x ) , c , c ∈ R we have Z y ∗ z ∗ u ( t )( G r g )( t ) m ′ ( t ) dt = 0 . Thus, H ( z ∗ , y ∗ ) = 0 showing that equation H ( z, y ) = 0 has at least one solution such that y z ∗ = y ∗ ∈ ( x , b ). Moreover, invoking (27), (28), and (30) shows that the necessary conditions for optimalityof the pair ( z ∗ , y ∗ ) coincide with the conditions f ( z ∗ ) = 0 = f ( y ∗ ).Given the results above, fix now z ∈ ( a, z ∗ ) and consider the function H ( z, y ). Standarddifferentiation yields that H y ( z, y ) = rm ′ ( y ) Z yz ˆ ϕ y ( t )( G r g )( t ) m ′ ( t ) dt − ( G r g )( y ) m ′ ( y ) (cid:18) ˆ ϕ ′ y ( y ) S ′ ( y ) − ˆ ϕ ′ y ( z ) S ′ ( z ) (cid:19) (35) H z ( z, y ) = rm ′ ( z ) Z yz ˆ ψ z ( t )( G r g )( t ) m ′ ( t ) dt − ( G r g )( z ) m ′ ( z ) ˆ ψ ′ z ( y ) S ′ ( y ) − ˆ ψ ′ z ( z ) S ′ ( z ) ! (36)demonstrating that H ( z, z ) = H y ( z, z ) = 0. Moreover, if y ∈ ( z, ˆ x ], then the monotonicity ofthe generator ( G r g )( x ) on ( a, ˆ x ) guarantees that H y ( z, y ) < y ∈ ( z, ˆ x ]. Hence, equation H y ( z, y ) = 0 does not have roots satisfying condition z = y when y ∈ ( z, ˆ x ). In a completelyanalogous fashion (36) shows that H ( y, y ) = H z ( y, y ) = 0 and H z ( z, y ) < x ≤ z < y .Hence, H z ( z, y ) = 0 does not have roots satisfying condition z = y when z ∈ (ˆ x, y ). Given these XPECTED SUPREMUM REPRESENTATION AND OPTIMAL STOPPING 27 observations, we notice that the existence of a root y z ∈ ( y ∗ , b ) would be guaranteed provided thatlim y → b H ( z, y ) > z ∈ ( a, z ∗ ). To see that this is indeed the case, we first consider thelimiting behavior of the function ˆ H : ( a, ˆ x ) × (ˆ x, b ) R defined asˆ H ( z, y ) = H ( z, y ) (cid:16) ψ ′ ( y ) S ′ ( y ) − ψ ′ ( z ) S ′ ( z ) (cid:17) (cid:16) ϕ ′ ( y ) S ′ ( y ) − ϕ ′ ( z ) S ′ ( z ) (cid:17) . It is clear that ˆ H ( z, y ) = R yz ( G r g )( t ) ϕ ( t ) m ′ ( t ) dtr R yz ϕ ( t ) m ′ ( t ) dt − R yz ( G r g )( t ) ψ ( t ) m ′ ( t ) dtr R yz ψ ( t ) m ′ ( t ) dt . Utilizing (4) now implies that for all z ∈ I we havelim y ↑ b R yz ( G r g )( t ) ψ ( t ) m ′ ( t ) dtr R yz ψ ( t ) m ′ ( t ) dt = 1 r lim y ↑ b ( G r g )( y ) = 1 r ( G r g )( b − ) < . Hence, for all z ∈ I it holds thatlim y → b − ˆ H ( z, y ) = R bz (( G r g )( t ) − ( G r g )( b − )) ϕ ( t ) m ′ ( t ) dtr R bz ϕ ( t ) m ′ ( t ) dt > b = argmin { ( G r g )( x ) } . The definition of ˆ H ( z, y ) now implies that lim y ↑ b H ( z, y ) = ∞ for all z ∈ I . Thus, for all z ∈ ( a, z ∗ ) equation H ( z, y ) = 0 has a root y z ∈ ( y ∗ , b ). Moreover, implicitdifferentiation shows that for all z ∈ ( a, z ∗ ) we have y ′ z = − H z ( z, y ) H y ( z, y ) = − m ′ ( z ) R yz (( G r g )( t ) − ( G r g )( z )) ˆ ψ z ( t ) m ′ ( t ) dtm ′ ( y ) R yz (( G r g )( t ) − ( G r g )( y )) ˆ ϕ y ( t ) m ′ ( t ) dt < (cid:3) Remark . Let u ( x ) = c ψ ( x ) + c ϕ ( x ) ≥ 0, where c , c ∈ R , and assume that ξ u is a randomvariable distributed on ( z, y ) according to the probability distribution P u with density p u ( t ) = u ( t ) m ′ ( t ) R yz u ( t ) m ′ ( t ) dt . Then, our results demonstrate that the functions f and f can be determined from the stationaryidentity E [( G r g )( ξ ϕ )] = E [( G r g )( ξ ψ )] . (37)By utilizing standard ergodic limit results, identity (37) can alternatively be expressed as (cf. SectionII.35 in [12])lim t →∞ R t ( G r g )( X s ) ϕ ( X s ) ( z,y ) ( X s ) ds R t ϕ ( X s ) ( z,y ) ( X s ) ds = lim t →∞ R t ( G r g )( X s ) ψ ( X s ) ( z,y ) ( X s ) ds R t ψ ( X s ) ( z,y ) ( X s ) ds . Theorem 4.8 characterizes the functions f and f in a smooth setting. According to Theorem4.8, the functions f and f vanish at the optimal boundaries z ∗ and y ∗ , respectively. Moreover, according to Theorem 4.8, the functions f and f can be expressed as conditional expectationsof the generator ( G r g )( x ). The decreasing mapping f ( i ) is associated with the diffusion X killedat the state β and its running infimum while the increasing mapping f ( m ) is associated with thediffusion X killed at the state α and its running supremum. Due to the interdependence of f and f it is not, however, clear beforehand whether the identities f ( z ∗ ) = 0 and f ( y ∗ ) = 0 continue tohold in a less smooth framework. As our subsequent examples indicate, there are cases under whichthese identities cease to hold as soon as the smooth pasting condition is not satisfied at one of theoptimal exercise boundaries.It is worth emphasizing that even though Theorem 4.8 assumes that the exercise payoff is smoothand that the boundaries of the state space of the underlying diffusion are natural, its results appearto be valid also under weaker regularity conditions and boundary classifications. More precisely, asis clear from the proof of Theorem 4.8 establishing the existence and monotonicity of the functions f and f can essentially be reduced to the analysis of the identity( L ϕ g )( y ) − ( L ϕ g )( z )( L ϕ )( y ) − ( L ϕ )( z ) = ( L ψ g )( y ) − ( L ψ g )( z )( L ψ )( y ) − ( L ψ )( z ) . (38)Since the monotonicity and limiting behavior of the functionals ( L ϕ g )( x ) and ( L ψ g )( x ) is principallydictated by the behavior of the generator ( G r g )( x ) (when defined), one could, in principle, attemptto delineate more general circumstances under which the uniqueness of a monotone solution for (38)could be guaranteed. A natural extension which could be utilized to accomplish this task would beto rely on the weak formulation of Dynkin’s theorem and, essentially, focus on those rewards whichadmit the representation (see, for example, [15],[25], and [30]) E x (cid:2) e − rτ g ( X τ ) τ< ∞ (cid:3) = g ( x ) + E x (cid:20)Z τ e − rs ˜ g ( X s ) ds ; τ < ∞ (cid:21) , where ˜ g ∈ L ( I ) coincides with the generator ( G r g )( x ) whenever the payoff is sufficiently smooth. Itis clear from the proof of Theorem 4.8 that if the function ˜ g satisfies parts (a) and (b) of Assumption4.7 with G r g replaced by ˜ g and ˜ g is continuous outside a finite set of points in I , then the identity R yz ψ ( t )˜ g ( t ) m ′ ( t ) dtr R yz ψ ( t ) m ′ ( t ) dt = R yz ϕ ( t )˜ g ( t ) m ′ ( t ) dt R yz ϕ ( t ) m ′ ( t ) dt generates a pair of functions f , f satisfying our monotonicity requirements and characterizing theoptimal exercise boundaries through the identities z ∗ = sup { x ∈ I : f ( x ) ≥ } and y ∗ = inf { x ∈I : f ( x ) ≥ } .It is also clear that the second integral expression stated in Theorem 4.8 resembles the expression(17) derived in the one-sided case. This is naturally not surprising in light of the fact that the one-sided cases can be derived from the two-sided case as limiting cases. Our main observation on thisis summarized in the following. Lemma 4.10. By setting z a , we retrieve the situation of Theorem 3.4. XPECTED SUPREMUM REPRESENTATION AND OPTIMAL STOPPING 29 Proof. Since f ( x ) = f ( x ) ( a,z ] ( x ) + f ( x ) [ y,b ) ( x ), we see that lim z a f ( x ) = f ( x ) [ y,b ) ( x ). More-over, now ζ = β ( a ) = β ( z ) = y , and thus, just as we derived (31), we get f ( m ) = g ( m ) − g ′ ( m ) ψ ′ ( m ) ψ ( m ) , for m ≥ y . (cid:3) Connection with the optimal stopping signal. As pointed out in the introduction, thereis a large variety of settings under which the values of stochastic control problems can be representedin terms of the expected value of the running supremum (see, for example, [7], [8], [9], [14], [19], and[20]). In what follows, our objective is to connect the developed approach to the optimal stoppingsignal approach developed in [7].Following [7], consider now the functionalˆ F ( x ; z, y ) := E x (cid:2) g ( x ) − e − rτ z,y g ( X τ z,y ) (cid:3) − E x [ e − rτ z,y ] , where τ z,y = inf { t ≥ X t ( z, y ) } denotes the first exit time from the open set ( z, y ) ⊂ I .Applying our previous computations yield that ˆ F can be re-expressed asˆ F ( x ; z, y ) = ˆ ψ z ( y ) g ( x ) − g ( z ) ˆ ϕ y ( x ) − g ( y ) ˆ ψ z ( x )ˆ ψ z ( y ) − ˆ ψ z ( x ) − ˆ ϕ y ( x ) . Letting first z ↑ x and then y ↓ x in this expression yields (by applying L’Hospital’s rule) h ( x, y ) := ˆ F ( x ; x − , y ) = g ( x ) ˆ ϕ ′ y ( x ) − g ′ ( x ) ˆ ϕ y ( x ) + BS ′ ( x ) g ( y )ˆ ϕ ′ y ( x ) + BS ′ ( x ) h ( x, z ) := ˆ F ( x ; z, x +) = ˆ ψ ′ z ( x ) g ( x ) − g ′ ( x ) ˆ ψ z ( x ) − BS ′ ( x ) g ( z )ˆ ψ ′ z ( x ) − BS ′ ( x ) . Utilizing the proof of Theorem 4.8 shows that the functions h , h can be re-expressed in the compactform h ( x, y ) = ( L ˆ ϕ g )( x ) − ( L ˆ ϕ y g )( y )( L ˆ ϕ y )( x ) − ( L ˆ ϕ y )( y ) h ( x, z ) = ( L ˆ ψ z g )( x ) − ( L ˆ ψ z g )( z )( L ˆ ψ z )( x ) − ( L ˆ ψ z )( z ) , proving that h ( x, y ) = F y ( x ) and h ( x, y ) = F z ( x ). Hence, we notice that the functions generat-ing f and f coincide with the functions characterizing the behavior of the functional ˆ F ( x ; z, y ).Theorem 13 in [7] tells us that the stopping set can in the present setting be represented in termsof the so called optimal stopping signal γ in the following way. Theorem 4.11. The stopping set Γ = { x ∈ I : g ( x ) = V ( x ) } = { x ∈ I : γ ( x ) ≥ } , where γ ( x ) = min y = x g ( x ) − g ′ ( x ) ψ ( x ) ψ ′ ( x ) , y = ah ( y, x ) , a < y < xh ( x, y ) , x < y < b.g ( x ) − g ′ ( x ) ϕ ( x ) ϕ ′ ( x ) , y = b If the smooth fit principle is met, then we know from Theorem 4.8 that the function f ( x ) = f ( x ) ( a,z ∗ ] ( x ) + f ( x ) [ y ∗ ,b ) ( x ) is positive on the same set as γ ( x ). In the next proposition we verifythe intuitively clear fact that our f ( x ) is indeed identical with γ on the stopping set Γ. Proposition 4.12. Let f ( x ) = f ( x ) ( a,z ∗ ] ( x ) + f ( x ) [ y ∗ ,b ) ( x ) . Then f ( x ) = γ ( x ) for x ∈ Γ .Proof. Let us redefine f on ( z ∗ , y ∗ ) to be negative. In this way, we can write the stopping setΓ = { x ∈ I : f ( x ) ≥ } . Consider now the auxiliary parameterized stopping problemsup τ E x (cid:2) e − rτ ( g ( X τ ) − k ) (cid:3) , (39)where k ≥ g is as in the initial problem (6). We know byTheorem 13 from [7] that for the problem (39) the stopping set can be written as Γ k = { x ∈ I : γ ( x ) ≥ k } . Thus, if we can also show that Γ k = { x ∈ I : f ( x ) ≥ k } , then we must necessarily have f ( x ) = γ ( x ) as k is arbitrary. In order to prove the desired result, let f k ( x ) = f k ( x ) ( a,z ∗ ] ( x ) + f k ( x ) [ y ∗ ,b ) ( x ) be the function f for the auxiliary problem (39). Using representation (30) now showsthat f k ( x ) = f ( x ) − k and f k ( x ) = f ( x ) − k . Hence, we have f k ( x ) = f ( x ) − k . Consequently, itfollows that Γ k = { x ∈ I : f k ( x ) ≥ } = { x ∈ I : f ( x ) ≥ k } and the claim follows. (cid:3) Unfortunately, neither function γ nor f can be expressed explicitly in a general setting despitethe fact that they both constitute alternative representations of the same value. The function γ istoo complex due to the minimization operator. Although f is more explicit than γ , it is neverthelessalso too complex for explicit expressions due to the implicit connection between f and f through α ( m ) and β ( i ). However, as our subsequent examples based on capped straddle options indicate, ourapproach applies even when the smooth pasting condition is not met. In this respect the approachdeveloped in our paper can generate the required representation in cases which do not appear inthe approach based on the stopping signal.4.5. Examples. Since the functions f and f depend on each other, it is very hard to expressthese functions explicitly. Fortunately, the derived integral representation is such that the functionscan be solved numerically in an efficient way. In what follows we shall illustrate these functions andtheir intricacies in several explicitly parameterized examples. XPECTED SUPREMUM REPRESENTATION AND OPTIMAL STOPPING 31 Example 3: Minimum guaranteed payment option. Set I = (0 , ∞ ) and consider the optimalstopping problem V ∗ ( x ) = sup τ E x (cid:2) e − rτ ( X τ ∨ c ) (cid:3) , (40)where c > V ∗ ( x ) = V ( z ∗ ,y ∗ ) ( x ) = x x ≥ y ∗ ˆ ϕ y ∗ ( x )ˆ ϕ y ∗ ( z ∗ ) c + ˆ ψ z ∗ ( x )ˆ ψ z ∗ ( y ∗ ) y ∗ z ∗ < x < y ∗ c x ≤ z ∗ (41)where the thresholds ( z ∗ , y ∗ ) constitutes the unique root of the first order optimality conditions ψ ′ ( y ∗ ) S ′ ( y ∗ ) y ∗ − ψ ( y ∗ ) S ′ ( y ∗ ) = ψ ′ ( z ∗ ) S ′ ( z ∗ ) cϕ ′ ( y ∗ ) S ′ ( y ∗ ) y ∗ − ϕ ( y ∗ ) S ′ ( y ∗ ) = ϕ ′ ( z ∗ ) S ′ ( z ∗ ) c. Geometric Brownian motion: Assume that X t constitutes a geometric Brownian motion char-acterized by the stochastic differential equation dX t = µX t dt + σX t dW t , where σ > µ < r . With these choices ψ ( x ) = x κ + , ϕ ( x ) = x κ − , where κ ± = 12 − µσ ± s(cid:18) − µσ (cid:19) + 2 rσ are the solutions of the characteristic equation σ κ ( κ − 1) + µκ − r = 0. Under these assumptions,problem (40) admits an explicit solution (cf. [24]) V ∗ ( x ) = V ( z ∗ ,y ∗ ) ( x ) = x x ≥ y ∗ (cid:0) κ + (cid:0) xz ∗ (cid:1) κ − − κ − (cid:0) xz ∗ (cid:1) κ + (cid:1) cκ + − κ − z ∗ < x < y ∗ c x ≤ z ∗ where z ∗ = (cid:18) κ + κ + − (cid:19) κ + − κ + − κ − (cid:18) κ − − κ − (cid:19) κ −− κ + − κ − c and y ∗ = (cid:18) κ + κ + − (cid:19) κ + κ + − κ − (cid:18) κ − − κ − (cid:19) κ − κ + − κ − c. Now the conditions of Theorem 4.8 are valid, so that we know that there exist a f and f suchthat f is non-increasing and f is non-decreasing, f ( z ∗ ) = 0 = f ( y ∗ ) and that E x (cid:2) sup ≤ t ≤ T f ( X t ) (cid:3) = V ( z ∗ ,y ∗ ) ( x ) for f ( x ) = f ( x ) ( a,z ] ( x ) + f ( x ) [ y,b ) ( x ). It can be calculated that lim i f ( i ) = c andthat lim m f ( m ) = ∞ , so that in this case ζ = b = ∞ . Unfortunately, the functions f and f cannot be expressed in analytically closed form. Logistic Diffusion: Assume that X t constitutes a logistic diffusion process characterized by thestochastic differential equation dX t = µX t (1 − γX t ) dt + σX t dW t , where σ > , γ ≥ µ > 0. In this case the fundamental solutions read as ψ ( x ) = x κ + M ( κ + , κ + − κ − , µγx/σ ) ϕ ( x ) = x κ − M ( κ − , − κ + + κ − , µγx/σ ) , where M denotes the confluent hypergeometric function. The functions f and f are now illustratednumerically in Figure 3. Figure 3. Illustrating f , f in logistic case. The parameters are µ = 0 . σ = 0 . γ = 0 . r = 0 . c = 1. With these choices ( z ∗ , y ∗ , ζ ) = (0 . , . , . Example 4: Capped straddle option. We now assume that the underlying follows a GBM andfocus on two straddle option variants. Namely, the symmetrically capped straddle with exercisepayoff g ( x ) = min( | X − K | , C ), where K > C > 0, and the asymmetrically capped straddle optionwith exercise payoff g ( x ) = min(( K − x ) + , C ) + min(( x − K ) + , C ) , where K > C > , C > 0. It is worth noticing that the asymmetrically capped straddle is relatedto minimum guaranteed payoff option treated in the previous example, since if C < C , then g ( x ) ≤ max( C , min(( x − K ) + , C )) and if C > C , then g ( x ) ≤ max( C , min(( K − x ) + , C )). Inthis way the value of the asymmetrically capped straddle is dominated by the value of a minimumguaranteed payoff option.It is now clear that the assumptions of our paper are met. Hence, the optimal exercise policyconstitutes a two-boundary stopping strategy. As in the capped call option case, the smooth fit XPECTED SUPREMUM REPRESENTATION AND OPTIMAL STOPPING 33 condition may, however, be violated depending on the precise parametrization of the model. Inthe present example the functions f and f are illustrated in Figure 4 under diffusion parameterspecifications resulting in ψ ( x ) = x and ϕ ( x ) = x − . Under these specifications, we observe from (a) Example 4: Capped straddle option with g ( x ) = min {| x − | , } . Smooth fit at z ∗ ≈ . y ∗ = 7. Both f and f arediscontinuous. (b) Example 5: Asymmetrically capped straddleoption with C = 1, K = 5, C = 3. Smooth fitat z ∗ ≈ . 78 and corner solution at y ∗ = 9. Now f is continuous. Figure 4. Numerical examples based on geometric Brownian motion. Figure 5(A) that the functions f and f may be discontinuous. In the case of Figure 5(A), thefirst discontinuity is based on the fact that the exercise payoff is not differentiable on the entirestopping region. The remaining discontinuity in Figure 5(A) is based on the fact that the valuedoes not satisfy the smooth fit principle at y ∗ . This observation illustrates the pronounced role ofthe interdependence between f and f and especially their sensitivity with respect to the potentialnonsmoothness of the problem.In both of these examples, ζ = y ∗ , which enables us to write down the functions f and f explicitly. Especially, in the case of Figure 4(B), they are f ( x ) = x − x + 256 x − x + 256 , i ∈ (0 , z ∗ ] f ( m ) = g ( m ) ≡ , m ∈ [ y ∗ , ∞ ) . Conclusions We considered the representation of the value of an optimal stopping problem of a linear dif-fusion as the expected supremum of a function with known regularity and monotonicity properties.We developed an integral representation for the above mentioned function by first computing thejoint probability distribution of the running supremum and infimum of the underlying diffusion andthen utilizing this distribution in determining the expected value explicitly in terms of the minimalexcessive mappings and the infinitesimal characteristics of the diffusion. There are at least two directions towards which our analysis could be potentially extended.First, given the close connection of optimal stopping with singular stochastic control it would natu-rally be of interest to analyze if our representation would function in that setting as well. It is clearthat this should be doable at least in some circumstances, since typically the marginal value of asingular stochastic control problem can be interpreted as a standard optimal stopping problem (see,for example, [5, 6, 11, 26, 28, 29]). Such an extension would be very interesting especially from thepoint of view of financial and economic applications, since a large class of control problems focusingon the rational management of a randomly fluctuating stock can be viewed as singular stochasticcontrol problems. Second, impulse control and switching problems can in most cases be interpretedas sequential stopping problems of the underlying process. Thus, extending our representation tothat setting would be interesting too (for a recent approach to this problem, see [13]). However,given the potential discreteness of the optimal policy in the impulse control policy setting seems tomake the explicit determination of the integral representation a very challenging problem which atthe moment is outside the scope of our study. Acknowledgements: The authors are grateful to Peter Bank and Paavo Salminen for suggestionsand helpful comments. References 1. L. H. R. Alvarez, On the properties of r -excessive mappings for a class of diffusions , Ann. Appl. Probab. (2003), no. 4, 1517–1533.2. , A class of solvable impulse control problems , Appl. Math. Optim. (2004), no. 3, 265–295.3. , Minimum guaranteed payments and costly cancellation rights: a stopping game perspective , Math. Finance (2010), no. 4, 733–751.4. L. H. R. Alvarez, P. Matom¨aki, and T. A. Rakkolainen, A class of solvable optimal stopping problems of spectrallynegative jump diffusions , SIAM J. Control Optim. (2014), no. 4, 2224–2249.5. F. M. Baldursson, Singular stochastic control and optimal stopping , Stochastics (1987), no. 1, 1–40.6. F. M. Baldursson and I. Karatzas, Irreversible investment and industry equilibrium , Finance and Stochastics (1997), 69 – 89.7. P. Bank and C. Baumgarten, Parameter-dependent optimal stopping problems for one-dimensional diffusions ,Electronic Journal of Probability (2010), 1971–1993.8. P. Bank and N. El Karoui, A stochastic representation theorem with applications to optimization and obstacleproblems , Annals of Probability (2004), 1030–1067.9. P. Bank and H. F¨ollmer, American options, multi-armed bandits, and optimal consumption plans: a unifying view ,Paris-Princeton Lectures on Mathematical Finance, 2002, Lecture Notes in Math., vol. 1814, Springer, Berlin,2003, pp. 1–42.10. P. Bank and F. Riedel, Optimal consumption choice with intertemporal substitution , The Annals of Applied Prob-ability (2001), 750–788.11. F. Boetius and M. Kohlmann, Connections between optimal stopping and singular stochastic control , StochasticProcesses and their Applications (1998), no. 2, 253–281. XPECTED SUPREMUM REPRESENTATION AND OPTIMAL STOPPING 35 12. A. Borodin and P. Salminen, Handbook of brownian motion - facts and formulae , Birkhauser, Basel, 2002.13. S. Christensen and P. Salminen, Impulse control and expected suprema , arXiv:1503.01253 (2015).14. S. Christensen, P. Salminen, and B. Q. Ta, Optimal stopping of strong markov processes , Stochastic Processes andtheir Applications (2013), 1138–1159.15. F. Crocce and E. Mordecki, Explicit solutions in one-sided optimal stopping problems for one-dimensional diffu-sions , Stochastics (2014), no. 3, 491–509.16. N. El Karoui and H. F¨ollmer, A non-linear riesz respresentation in probabilistic potential theory , Annales del’Institut Henri Poincare (B) Probability and Statistics (2005), 269–283.17. N. El Karoui and I. Karatzas, Dynamic allocation problems in continuous time , Ann. Appl. Probab. (1994),no. 2, 255–286.18. N. El Karoui and A. Meziou, Max–plus decomposition of supermartingales and convex order. application to amer-ican options and portfolio insurance , Annals of Probability (2008), 647–697.19. H. F¨ollmer and T. Knispel, A representation of excessive functions as expected suprema , Probabability and Math-ematical Statistics (2006), 379–394.20. , Potentials of a markov process are expected suprema , ESAIM: Probabability and Statistics (2007),89–101.21. J. C. Gittins, Bandit processes and dynamic allocation indices , J. Roy. Statist. Soc. Ser. B (1979), no. 2,148–177, With discussion.22. J. C. Gittins and K. D. Glazebrook, On Bayesian models in stochastic scheduling , J. Appl. Probability (1977),no. 3, 556–565.23. J. C. Gittins and D. M. Jones, A dynamic allocation index for the discounted multiarmed bandit problem , Biometrika (1979), no. 3, 561–565.24. X. Guo and L. Shepp, Some optimal stopping problems with nontrivial boundaries for pricing exotic options ,Journal of Applied Probability (2001), 647––658.25. K. Helmes and R. H. Stockbridge, Construction of the value function and optimal rules in optimal stopping ofone-dimensional diffusions , Adv. in Appl. Probab. (2010), no. 1, 158–182.26. I. Karatzas, A class of singular stochastic control problems , Advances in Applied Probability (1983), no. 2,225–254.27. , Gittins indices in the dynamic allocation problem for diffusion processes , Ann. Probab. (1984), no. 1,173–192.28. I. Karatzas and S. E. Shreve, Connections between optimal stopping and singular stochastic control I. Monotonefollower problems , SIAM Journal on Control and Optimization (1984), no. 6, 856–877.29. , Connections between optimal stopping and singular stochastic control II. Reflected follower problems ,SIAM Journal on Control and Optimization (1985), no. 3, 433–451.30. D. Lamberton and M. Zervos, On the optimal stopping of a one-dimensional diffusion , Electron. J. Probab. (2013), no. 34, 49.31. J. Lempa, A note on optimal stopping of diffusions with a two-sided optimal rule , Operations Research Letters (2010), 11–16.32. P. Matom¨aki, On solvability of a two-sided singular control problem , Math. Methods Oper. Res. (2012), no. 3,239–271.33. P. Salminen, Optimal stopping of one-dimensional diffusions , Mathematische Nachrichten124