Optimal Equilibria for Multi-dimensional Time-inconsistent Stopping Problems
OOptimal Equilibria for Multi-dimensional Time-inconsistentStopping Problems ∗ Yu-Jui Huang † Zhenhua Wang ‡ June 9, 2020
Abstract
We study an optimal stopping problem under non-exponential discounting, where the stateprocess is a multi-dimensional continuous strong Markov process. The discount function istaken to be log sub-additive, capturing decreasing impatience in behavioral economics. Onstrength of probabilistic potential theory, we establish the existence of an optimal equilibriumamong a sufficiently large collection of equilibria, consisting of finely closed equilibria satisfyinga boundary condition. This generalizes the existence of optimal equilibria for one-dimensionalstopping problems in prior literature.
MSC (2010):
Keywords: optimal stopping, time inconsistency, non-exponential discounting, probabilisticpotential theory, optimal equilibria.
Whereas consistent planning in Strotz [29] has been widely applied to time-inconsistent controland stopping problems, it gains little recognition as a two-phase procedure. First, an agent shouldfigure out strategies that he will actually follow over time, the so-called equilibria in the literature (Phase I) . Then, the agent needs to, according to Strotz [29, p.173], “find the best plan amongthose he will actually follow” (Phase II) .Thanks to the continuous-time formulation initiated in Ekeland and Lazrak [10], there hasbeen vibrant research on time-inconsistent problems in the communities of stochastic control andmathematical finance; see e.g. [12], [3], [13], [4], [11], [31], among many others. Note that allthe developments were focused on
Phase I of consistent planning. To the best of our knowledge,in-depth investigation of
Phase II was initiated fairly recently in Huang and Zhou [17, 19].The notion of an optimal equilibrium is proposed for the first time in [17]: an equilibrium isoptimal if it generates a larger value than any other equilibrium does, everywhere in the state space.This seems a fairly strong optimality criterion, as it requires a subgame perfect Nash equilibriumto be dominant on the entire state space. Nonetheless, [17] derives an optimal equilibrium for ∗ We would like to thank Tiziano De Angelis for introducing the fine topology to the first author. This led to theinitiation of this project. † University of Colorado, Department of Applied Mathematics, Boulder, CO 80309-0526, USA, email: [email protected] . Partially supported by National Science Foundation (DMS-1715439) and the Universityof Colorado (11003573). ‡ University of Colorado, Department of Mathematics, Boulder, CO 80309-0395, USA, email: [email protected] . a r X i v : . [ q -f i n . M F ] J un discrete-time stopping problem under non-exponential discounting. Corresponding results incontinuous time have been established in [19], relying on a detailed analysis of one-dimensionaldiffusions. Specifically, a key assumption in [19] is that for any initial state x ∈ R , the state process X satisfies P x [ X t > x ] = P x [ X t < x ] = 1 for all t > , (1.1)where X t := max s ∈ [0 ,t ] X s (resp. X t := min s ∈ [0 ,t ] X s ) is the running maximum (resp. minimum) of X . This condition ensures that X is “diffusive” enough, so that whenever X reaches the boundaryof a Borel subset R of R , it enters R immediately. This allows us to focus on only closed equilibrium,with no need to deal with equilibria of possibly pathological forms. Note that (1.1) is not restrictivein the one-dimensional case (i.e. d = 1): any regular diffusion (in the sense of [27, Definition V.45.2])readily fulfills (1.1), as recently observed in [16, Remark 3.1].There is, however, no natural extension of [19] to higher dimensions (i.e. d ≥ X exists (Assumption 2.1). This serves as a minimal condition to ensure Borelmeasurability of involved stopping policies; see Corollary 2.1. Since this assumption is satisfiedas long as a transition density of X exists (Lemma 2.1), Corollary 2.1 covers and generalizes allmeasurability results in [14], [15], and [19]. Next, we set out to devise a sufficiently large collection L of stopping policies, among which we will find an optimal equilibrium. The flexibility that L isnot required to contain all Borel stopping policies is important: focusing on more amenable andpractically relevant stopping policies will facilitate the search for an optimal equilibrium (optimalamong this class L ). In a sense, for the case d = 1, tractability comes from enhanced regularityof X , i.e. (1.1); for d ≥
2, it comes from additional structures of stopping policies. To constructthe collection L , we first restrict our attention to finely closed stopping policies, and then requirethe difference between them and their Euclidean closures to be sufficiently small, or more precisely, semipolar . The first restriction is without loss of any generality: as shown in Proposition 3.1, thefine closure of an equilibrium remains an equilibrium with the same values. The second restriction,on the other hand, is to exclude stopping policies with somewhat pathological and practically ir-relevant forms; see Remark 3.6. This class L already includes all closed stopping policies, and isclosed under finite unions and countable intersections; see Lemma 3.2.Our goal is to find an equilibrium optimal among (cid:101) E := E ∩ L , where E denotes the set of allequilibria. By invoking Hunt’s hypothesis (Assumption 4.1), which states that every semipolar setis polar , we obtain two important consequences. First, given a stopping policy R , if we performon R one round of the fixed-point iteration in [14] and get a smaller policy, this smaller one has tobe an equilibrium; see Proposition 4.1. Second, for any R ∈ (cid:101) E ⊆ L , since the difference between R and R , its Euclidean closure, is now polar (i.e. inaccessible to the process X ), the first hittingtime to R , denoted by ρ R , becomes much more tractable as it must coincide with ρ R . Based onall this, a machinery for improving equilibria in (cid:101) E is developed: for any R , T ∈ (cid:101) E , there existsanother equilibrium in (cid:101) E , contained in R ∩ T , which generates larger values than both R and T ; seeProposition 4.2. By carrying out this machinery recursively, we construct an equilibrium R that isoptimal among (cid:101) E , with R = (cid:84) R ∈ (cid:101) E R ; see Theorem 4.1, the main result of this paper.Given that ρ R = ρ R for all R ∈ (cid:101) E , it is tempting to believe that we could restrict our attentionfrom (cid:101) E to E := { R ∈ E : R is closed } , without loss of generality. This would make our results2ompletely in line with the one-dimensional analysis in [19], which is built upon E . This is however not the case, as the relation “ R ∈ (cid:101) E if and only if R ∈ E ” does not hold in general; see Remark 4.4for explanations, and Section 5 for an explicit example that demonstrates R ∈ (cid:101) E but R / ∈ E .Lately, marked progress has been made in applying Stortz’ equilibrium idea to time-inconsistentstopping problems in continuous-time diffusion models. It can be roughly categorized into twobranches. The first one is the fixed-point iterative approach introduced in Huang and Nguyen-Huu[14], and further developed in [15], [19], and [16]. The second branch adapts the stochastic controlformulation in [10] to the stopping context; this includes Ebert, Wei, and Zhou [9] and Christensenand Lindensj¨o [6, 7]. A large part of all these developments, notably, rely on the one-dimensionalstructure of the state process. The arguments in this paper can potentially shed new light onextending previous one-dimensional results to higher dimensions.The notion of optimal equilibria have recently been generalized in Bayraktar, Zhang, and Zhou[2], to reflect the subtle differences of how an equilibrium is formulated in Huang and Zhou [18],[14], and [6]. Also, the existence of optimal equilibria for a time-inconsistent dividend problem hasbeen analyzed in detail by Jin and Zhou [20]. These studies rely on more tractable continuous-timeMarkov chain or discrete-time models. It is of great interest as future research to extend theirdevelopments to a multi-dimensional diffusion model, like the one in this paper.The rest of this paper is organized as follows. Section 2 introduces the time-inconsistent stoppingproblem, a new measurability result, and the idea of optimal equilibria. Section 3 devises a suitableset of stopping policies for the rest of the paper to focus on. This set encodes desirable propertiesfrom probabilistic potential theory, yet remains general enough to cover practically relevant stoppingpolicies. Under Hunt’s hypothesis, Section 4 develops a machinery to improve equilibria, by whichan optimal equilibrium is constructed. Section 5 presents an example to demonstrate explicitlythat the Euclidean closure of a finely closed equilibrium need not be an equilibrium.
Let B be the Borel σ -algebra of R d , and P be the set of all probability measures on ( R d , B ).Consider an R d -valued time-homogeneous continuous strong Markov process ( X t ) t ≥ . Let thecollection ( P t : R d × B → [0 , t ≥ denote the transition function of X . If there exist a collection( p t : R d × R d → R + ) t ≥ of Borel measurable functions and a measure λ on ( R d , B ) such that P t ( x, A ) = (cid:90) A p t ( x, y ) λ ( dy ) , ∀ t ≥ , x ∈ R d , and A ∈ B , (2.1)we call ( p t ) t ≥ a transition density of X with respect to λ .On the path space Ω := C ([0 , ∞ ); R d ), the set of continuous functions mapping [0 , ∞ ) to R d ,let F X = {F Xt } t ≥ be the raw filtration generated by X . For each µ ∈ P , we denote by B µ thecompletion of B by µ , by X µ the process X with initial distribution µ , and by F µ = {F µt } t ≥ the µ -augmentation of F X . Moreover, the probability measure on (Ω , F X ∞ ) generated X µ (i.e. thelaw of ( X µt ) t ≥ ) is denoted by P µ , and the expectation taken under P µ is denoted by E µ . For any x ∈ R d , if µ = δ x (the Dirac measure concentrated at x ), we write X µ , P µ , and E µ simply as X x , P x and E x . We further consider the universal filtration F = {F t } t ≥ defined by F t := (cid:84) µ ∈P F µt forall 0 ≤ t < ∞ , and denote by T the set of F -stopping times.Consider a payoff function f : R d → R + , assumed to be continuous. Also consider a discountfunction δ : [0 , ∞ ) → [0 , δ (0) = 1 andlim t →∞ δ ( t ) = 0. A classical optimal stopping problem is formulated assup τ ∈T E x [ δ ( τ ) f ( X τ )] . (2.2)3t is well-known (see e.g. [22, Appendix D] and [28]) that under fairly general conditions, for anyinitial state x ∈ R d , there exists an optimal stopping time (cid:101) τ x ∈ T for (2.2). However, as long asthe discount function is not of the exponential form δ ( t ) := e − αt with α >
0, the problem (2.2) isin general time-inconsistent . That is, optimal stopping times obtained at different moments, suchas (cid:101) τ x at time 0 and (cid:101) τ X xt at time t >
0, may not be consistent with each other. This is problematic:even if a maximizer (cid:101) τ x of (2.2) can be found, our future self at any time t > (cid:101) τ X xt , optimal for him at time t , rather than stick with (cid:101) τ x ; see [14, Section 2] for a detaileddemonstration of such time inconsistency.Throughout this paper, we will assume that the discount function δ satisfies δ ( s ) δ ( t ) ≤ δ ( s + t ) , ∀ s, t > . (2.3)This covers a wide range of non-exponential discount functions from empirical studies; see thediscussion below [14, Assumption 3.12]. In economic terms, (2.3) captures decreasing impatience ,the fact that people discount more steeply over time intervals closer to the present. This feature ofempirical discounting is well-documented in behavioral economics; see e.g. [30], [24], and [23].Under (2.3), time inconsistency is a genuine problem. Strotz [29] proposes consistent planning as a solution: an agent should take into account his future selves’ disobedience, and find a strategythat once being enforced, none of his future selves would deviate from. Such strategies, called equilibiria in the literature, can be formulated using the game-theoretic framework initiated in [14]. Under our time-homogeneous Markovian setup, we will focus on first entry times to Borel subsetsof R d , i.e. τ R := inf { t ≥ X t ∈ R } for R ∈ B , instead of all general stopping times. The involvedregion R ∈ B will be called a stopping policy constantly.Suppose that an agent initially planned to take R ∈ B as his stopping policy. Now, at any state x ∈ R d , the agent carries out the game-theoretic reasoning: “assuming that all my future selveswill follow R ∈ B , what is the best stopping strategy today in response to that?” To this end, theagent compares the payoff of immediate stopping, i.e. f ( x ), and the payoff of continuation, i.e. J ( x, R ) := E x [ δ ( ρ R ) f ( X ρ R )] , (2.4)where ρ R is the first hitting time of X to R defined by ρ R := inf { t > X t ∈ R } . (2.5)As explained in detail in [19, Section 2.1] (see also [15, Section 2] or [14, Section 3.1]), the beststopping strategy for the agent at x ∈ R d is the first entry time to the regionΘ( R ) := S ( R ) ∪ ( I ( R ) ∩ R ) , (2.6)where S ( R ) := { x ∈ R d : J ( x, R ) < f ( x ) } ,I ( R ) := { x ∈ R d : J ( x, R ) = f ( x ) } ,C ( R ) := { x ∈ R d : J ( x, R ) > f ( x ) } . (2.7) Remark 2.1. In (2.4) , we need specifically the first hitting time ρ R , instead of the first entry time τ R = inf { t ≥ X t ∈ R } . This was explained in the discussion below [14, (3.5)]. x ∈ R d , E x (cid:20) sup t ∈ [0 , ∞ ] δ ( t ) f ( X t ) (cid:21) < ∞ , (2.8)where we take δ ( ∞ ) f ( X x ∞ ) := lim sup t →∞ δ ( t ) f ( X xt ), which is in line with [22, Appendix D].Second, to ensure that Θ( R ) in (2.6) is indeed a stopping policy, i.e. Θ( R ) ∈ B , certain regularityof X is required, which will be investigated closely below. Let us first recall the following concepts from probabilistic potential theory.
Definition 2.1 ([5], Definition II.3.1) . For any R ∈ B and x ∈ R d , the potential of R is defined by U ( x, R ) := (cid:90) ∞ P t ( x, R ) dt. We say R ∈ B is of zero potential if U ( x, R ) = 0 for all x ∈ R d . Definition 2.2 ([5], Definition V.1.1) . A measure λ on R d is called a reference measure for X ,provided that it is a countable sum of finite measures such that R ∈ B is of zero potential if andonly if λ ( R ) = 0 . Let us introduce the first main assumption of this paper.
Assumption 2.1.
A reference measure for X exists. A convenient sufficient condition of Assumption 2.1 is provided in the next result, whose proofis relegated to Appendix A.
Lemma 2.1.
Suppose that X has a transition density with respect to a measure λ on R d . If λ isa countable sum of finite measures, then for any α > , λ α : B → R + defined by λ α ( R ) := (cid:90) R d (cid:18)(cid:90) ∞ e − αt P t ( x, R ) dt (cid:19) λ ( dx ) is a reference measure for X . Lemma 2.1 indicates that Assumption 2.1 is not restrictive for a wide range of applications,as a large class of diffusion processes have a transition density with respect to the Lebesgue mea-sure in R d . For the d -dimensional Brownian motion B , such a transition density is p t ( x, y ) =(2 πt ) − d/ exp( − | x − y | t ); see [21, Exercise 5.6.17]. For an Itˆo diffusion given by dX t = b ( X t ) dt + σ ( X t ) dB t , (2.9)as long as there exists a weak solution unique in distribution, and the coefficients b and σ arecontinuous and grow at most linearly, the transition density of X with respect to the Lebesguemeasure exists and is characterized by the fundamental solution to a Cauchy problem; see [21,p.369]. For general Markov processes, we refer readers to [5, Proposition V.1.2] for a generalsufficient condition of the existence of a reference measure.When a reference measure exists, we have the following handy approximation for hitting times;see Proposition 10 in [8, Section 3.5] and [5, Exercise V.1.20].5 emma 2.2. Suppose Assumption 2.1 holds. For any R ∈ B , there exist a nondecreasing sequenceof compact sets K n ⊆ R such that ρ K n → ρ R P x -a.s. for all x ∈ R d . Borel measurability of the map x (cid:55)→ J ( x, R ) can then be established. Proposition 2.1. (i) For any closed R ∈ B , x (cid:55)→ J ( x, R ) is Borel measurable.(ii) Assume Assumption 2.1 and that (2.8) holds for all x ∈ R d . Then, for any R ∈ B , x (cid:55)→ J ( x, R ) is Borel measurable.Proof. (i) For any R ∈ B that is closed, ρ R is an F X -stopping time, thanks to [21, Problem I.2.7].Hence, the random variable H ( ω ) := δ ( ρ R ( ω )) f ( X ρ R ( ω )) is F X ∞ -measurable. By [5, Theorem I.3.6], x (cid:55)→ E x [ H ] = J ( x, R ) is Borel measurable.(ii) For any R ∈ B , Lemma 2.2 asserts the existence of compact sets K n ⊆ R such that ρ K n → ρ R P x -a.s. for all x ∈ R d . Thanks to this, the continuity of δ , f , and t (cid:55)→ X t , and (2.8), we concludefrom the dominated convergence theorem that J ( x, K n ) = E x [ δ ( ρ K n ) f ( X ρ Kn )] → E x [ δ ( ρ R ) f ( X ρ R )] = J ( x, R ) , ∀ x ∈ R d . (2.10)By part (a), x (cid:55)→ J ( x, K n ) is Borel measurable for all n ∈ N . Hence, J ( x, R ) is Borel measurablein view of (2.10). Corollary 2.1.
Assume Assumption 2.1 and that (2.8) holds for all x ∈ R d . For any R ∈ B , wehave Θ( R ) ∈ B .Proof. For any R ∈ B , by Proposition 2.1 (ii), x (cid:55)→ J ( x, R ) is Borel measurable. Hence, by definition S ( R ), I ( R ), and C ( R ) all belong to B . It follows that Θ( R ) = S ( R ) ∪ ( I ( R ) ∩ R ) ∈ B .By Corollary 2.1, Θ can be viewed as an operator acting on B , i.e. Θ : B → B . An equilibriumcan then be formulated as a fixed point of Θ.
Definition 2.3. R ∈ B is called an equilibrium if Θ( R ) = R . We denote by E the set of allequilibria. It can be checked directly that the entire space R = R d is an equilibrium. Moreover, a largenumber of (or even all) equilibria can be found, by the fixed-point iteration introduced in [14]: onestarts with an arbitrary R ∈ B , and apply Θ to it repetitively until an equilibrium is reached; seealso Remark 4.2. Finding equilibria, however, is only the first phase of consistent planning in Strotz [29]. In thesecond phase, the agent should choose the best one among all equilibria. This has not been studiedin the literature, except in [17] and [19]. Following [19, Section 2.2], for each R ∈ E , we define theassociated value function by V ( x, R ) := f ( x ) ∨ J ( x, R ) ∀ x ∈ R d . Definition 2.4.
Given any E (cid:48) ⊆ E , a set R ∈ E (cid:48) is called an optimal equilibrium among E (cid:48) if forany other T ∈ E (cid:48) , V ( x, R ) ≥ V ( x, T ) for all x ∈ R d . In the one-dimensional care (i.e. d = 1), the existence of an optimal equilibrium among theentire set E is established in [19]. 6 roposition 2.2 ([19], Theorem 4.1) . Suppose that for any x ∈ R , we have (1.1) , (2.8) , and δ ( t ) f ( X xt ) → as t → ∞ P x -a.s. (2.11) Then, the set R := (cid:92) R ∈E , R closed R (2.12) is an optimal equilibrium among E . As mentioned in the introduction, (1.1) ensures that whenever X reaches the boundary of aBorel subset R of R , it enters R immediately. This allows us to focus on only closed equilibrium,as indicated by (2.12). As shown in [15, Lemma 3.1], (1.1) is satisfied by a large class of one-dimensional Itˆo diffusions. Even more generally, any regular diffusion (in the sense of [27, DefinitionV.45.2]) fulfills (1.1), as recently observed in [16, Remark 3.1].Proposition 2.2 does not naturally extend to higher dimensions. First, due to the involved X and X processes, (1.1) is inherently one-dimensional, with no natural extension in higher dimensions.Moreover, the proof of Proposition 2.2 relies crucially on a consequence of (1.1): ρ { x } = 0 P x -a.s.for all x ∈ R , i.e. the process X re-visits its initial point immediately. This condition is mostlyviolated in higher dimensions. For instance, when X is a d -dimensional Brownian motion, ρ { x } = ∞ P x -a.s. for all x ∈ R d , whenever d ≥ E (cid:48) ⊆ E , which is allowed to be properly contained in E . Thisflexibility is important: focusing on more amenable and practically relevant stopping policies willfacilitate the search for an optimal equilibrium. In a sense, for d = 1, tractability comes fromdesirable regularity of X , i.e. (1.1); for d ≥
2, it will come from additional structures of stoppingpolicies. The search for an appropriate subset E (cid:48) ⊆ E , which needs to be amenable enough but stillsufficiently large, will be the focus of the next section. On strength of probabilistic potential theory, a suitable subset of E to focus on will be devised inthis section. We will first restrict our attention to finely closed stopping policies (see Definition 3.1below), and then further require the difference between them and their Euclidean closures to besufficiently small. As we will see, the first restriction is without loss of any generality, while thesecond is to exclude equilibria with possibly pathological and practically irrelevant forms. Let us recall several essential concepts from probabilistic potential theory.
Definition 3.1.
Given R ∈ B , a point x ∈ R d is said to be regular to R if ρ R = 0 P x -a.s. The setof all regular points to R (with respect to X ) is denoted by R r , and we call R ∗ := R ∪ R r . (3.1) the fine closure of R . In addition, R is said to be finely closed if R = R ∗ . Remark 3.1.
By Blumenthal’s zero-one law (Theorem 6 in [8, Section 2.3]), for any x ∈ R d and R ∈ B , P x ( ρ R = 0) is either 0 or . Hence, if x ∈ R d is not regular to R , then ρ R > P x -a.s. emark 3.2. Adding to a set all its regular points, as in (3.1) , is the closure operation under thefine topology (see e.g. [8, p.107]). Hence, for any R ∈ B , ( R ∗ ) ∗ = R ∗ , or ( R ∗ ) r ⊆ R ∗ . (3.2) Remark 3.3.
For any R ∈ B , X ρ R ∈ R ∗ P x -a.s. on { ρ R < ∞} , ∀ x ∈ R d . (3.3) Indeed, for P x -a.e. ω ∈ { ρ R < ∞} , if X ρ R ( ω ) / ∈ R , by the definition of ρ R , X ρ R ( ω ) must be regularto R , i.e. X ρ R ( ω ) ∈ R r . Hence, X ρ R ( ω ) ∈ R ∪ R r = R ∗ . Remark 3.4.
Fix R ∈ B . For any x ∈ R r , as ρ R = 0 P x -a.s., J ( x, R ) = f ( x ) . Hence, R r ⊆ I ( R ) . (3.4)Borel measurability of R r and R ∗ can be established under Assumption 2.1. Corollary 3.1.
Assume Assumption 2.1 and that (2.8) holds for all x ∈ R d . Then, for any R ∈ B , R r ∈ B and thus R ∗ ∈ B .Proof. For any R ∈ B , by the same arguments as in Proposition 2.1 (ii), with J ( x, R ) replaced by E x [ e − ρ R ], we can show that x (cid:55)→ E x [ e − ρ R ] is Borel measurable. Thus, R r = { x ∈ R d : E x [ e − ρ R ] =1 } ∈ B . It follows that R ∗ = R ∪ R r ∈ B .A key observation is that first hitting times to R and to R ∗ must coincide. Lemma 3.1.
For any R ∈ B , ρ R = ρ R ∗ P x -a.s. for all x ∈ R d . Hence, J ( x, R ) = J ( x, R ∗ ) ∀ x ∈ R d , (3.5) S ( R ) = S ( R ∗ ) , I ( R ) = I ( R ∗ ) , C ( R ) = C ( R ∗ ) . (3.6) Proof.
Fix R ∈ B and x ∈ R d . Since R ⊆ R ∗ , ρ R ∗ ≤ ρ R . Assume ρ R ∗ < ∞ , otherwise ρ R ∗ = ρ R trivially holds. By contradiction, assume that there exists ω ∈ Ω such that ρ R ∗ ( ω ) < ρ R ( ω ) . (3.7)By (3.3) and (3.2), X ρ R ∗ ( ω ) ∈ ( R ∗ ) ∗ = R ∗ . Then, (3.7) entails X ρ R ∗ ( ω ) ∈ R r \ R . This in turnimplies the existence of ( t n ( ω )) n ∈ N in R + such that t n > ρ R ∗ , X t n ∈ R , and t n ↓ ρ R ∗ . It followsthat ρ R = lim n →∞ t n = ρ R ∗ , a contradiction to (3.7). With ρ R = ρ R ∗ P x -a.s. for all x ∈ R d , (3.5)and (3.6) directly follow from (2.4) and (2.7). Proposition 3.1.
For any R ∈ B , R ∈ E if and only if R ∗ ∈ E . Moreover, if R ∈ E , then any T ∈ B with T ∗ = R ∗ belongs to E and satisfies J ( x, T ) = J ( x, R ) = J ( x, R ∗ ) for all x ∈ R d . Proof.
Fix R ∈ B . Suppose R ∈ E , i.e. R = Θ( R ) = S ( R ) ∪ ( I ( R ) ∩ R ) . (3.8)By (3.6), Θ( R ∗ ) = S ( R ∗ ) ∪ ( I ( R ∗ ) ∩ R ∗ ) = S ( R ) ∪ ( I ( R ) ∩ ( R ∪ R r ))= S ( R ) ∪ ( I ( R ) ∩ R ) ∪ ( I ( R ) ∩ R r )= R ∪ R r = R ∗ , (3.9)8here the third equality follows from (3.8) and (3.4). This shows that R ∗ also belongs to E .Conversely, suppose R ∗ ∈ E . Then (3.8) holds with R replaced by R ∗ , i.e. R ∗ = S ( R ∗ ) ∪ ( I ( R ∗ ) ∩ R ∗ ).This can be rewritten, using (3.6) and R ∗ = R ∪ R r = R ∪ ( R r \ R ), as R ∪ ( R r \ R ) = S ( R ) ∪ ( I ( R ) ∩ ( R ∪ ( R r \ R )))= S ( R ) ∪ ( I ( R ) ∩ R ) ∪ ( I ( R ) ∩ ( R r \ R ))= S ( R ) ∪ ( I ( R ) ∩ R ) ∪ ( R r \ R ) , (3.10)where the last equality follows from (3.4). Note that (3.4) also implies S ( R ) ∩ ( R r \ R ) ⊆ S ( R ) ∩ I ( R ) = ∅ . Hence, in (3.10), the left hand side is a disjoint union of R and R r \ R , and the righthand side is a disjoint union of S ( R ) ∪ ( I ( R ) ∩ R ) and R r \ R . We then conclude from (3.10) that R = S ( R ) ∪ ( I ( R ) ∩ R ) = Θ( R ), i.e. R ∈ E .Now, if R ∈ E , by part (i) R ∗ ∈ E . For any S ∈ B with S ∗ = R ∗ , as S ∗ = R ∗ ∈ E , we have S ∈ E (by part (a) again). Then, Lemma 3.1 directly gives J ( x, S ) = J ( x, S ∗ ) = J ( x, R ∗ ) = J ( x, R ), forall x ∈ R d .In view of Proposition 3.1, to find an optimal equilibrium, it suffices to restrict our attentionto finely closed stopping policies. After all, the fine closure of R ∈ E remains an equilibrium, withthe same values. In fact, as Lemma 3.1 indicates, R and R ∗ induce the same stopping behavior,with S ( R ), I ( R ), and C ( R ) in (2.7) staying intact after we take the fine closure of R . Definition 3.2 ([5], Definition II.3.1) . Given R ∈ B , we say that R is polar if ρ R = ∞ P x -a.s. forall x ∈ R d , that R is thin if R r = ∅ , and that R is semipolar if it is a countable union of thin sets. Instead of dealing with all stopping policies R ∈ B , we focus on those such that R \ R ∗ is semipolar , (3.11)where R denotes the (Euclidean) closure of R . Remark 3.5. (3.11) covers all closed subsets of R d . Indeed, if R ∈ B is closed, R \ R ∗ = R \ R = ∅ is trivially semipolar. Remark 3.6. (3.11) excludes some pathological sets that are so small that X will never reach, butso dense that their closures are immensely larger and will be hit by X with positive probability. Forinstance, if X is a d -dimensional Brownian motion with d ≥ , then Q := { x = ( x , x , ..., x d ) ∈ R d : x i ∈ Q , i = 1 , , ..., d } is polar, but Q = R d . Note that (3.11) excludes Q . Since Q is polar, Q r = ∅ and thus Q ∗ = Q ∪ Q r = Q is polar. Then, Q \ Q ∗ = R d \ Q will be hit by X continuously over time, and istherefore not semipolar (in view of [5, Proposition II.3.4]).In practice, one does not usually take into account a stopping policy like Q , but simply consider ∅ (giving the same effect “never stop” as Q ) or Q = R d (“stop immediately”). Remark 3.7.
In the one-dimensional case (i.e. d = 1 ), (1.1) ensures R ∗ = R for all R ∈ B , sothat (3.11) is trivially satisfied for all R ∈ B . Hence, (3.11) covers the one-dimensional setup in[19], and can be viewed as the multi-dimensional counterpart of (1.1) . L := { R ∈ B : R = R ∗ and R \ R is semipolar } . (3.12) Lemma 3.2. L contains all closed subsets of R d , and is closed under finite unions and countableintersections.Proof. The first assertion simply follows from Remark 3.5. For any
R, T ∈ L , using the fact that R ∪ T = R ∪ T , we get R ∪ T \ ( R ∪ T ) = ( R ∪ T ) \ ( R ∪ T ) ⊆ ( R \ R ) ∪ ( T \ T ) . As R \ R and T \ T are both semipolar, R ∪ T \ ( R ∪ T ) is semipolar, i.e. R ∪ T ∈ L . On the otherhand, given any nonincreasing sequence ( R n ) n ∈ N in L , set R := (cid:84) n R n . In view of Remark 3.2,since R n is finely closed for all n ∈ N , their intersection R is also finely closed. Moreover, since R ⊆ (cid:84) n R n , R \ R = R \ (cid:18) (cid:92) n R n (cid:19) ⊆ (cid:18) (cid:92) n R n (cid:19) \ (cid:18) (cid:92) n R n (cid:19) . Given any point x ∈ (cid:0) (cid:84) n R n (cid:1) \ ( (cid:84) n R n ), x is contained in every R n , and there exists n ∈ N suchthat x / ∈ R n ; hence, x ∈ R n \ R n . The above inclusion relation therefore implies R \ R ⊆ (cid:91) n ( R n \ R n ) . (3.13)Since R n \ R n is semipolar for all n ∈ N , the right hand side above, as a countable union of semipolarsets, is semipolar. Thus, R \ R is also semipolar, so that we can conclude R ∈ L .Based on the development in this section, the appropriate subset of E we will focus on is (cid:101) E := E ∩ L . (3.14) (cid:101) E In this section, we set out to find an optimal equilibrium among (cid:101) E defined in (3.14). We willfirst introduce a main assumption and its ramifications in Section 4.1, and develop a machinery toimprove equilibria in (cid:101) E in Section 4.2. The main result will be presented in Theorem 4.1. By Definition 3.2, a polar set is clearly semipolar. The converse is the celebrated Hunt hypothesis,which is the second main assumption of this paper.
Assumption 4.1 (Hunt’s hypothesis) . If R ∈ B is semipolar, then it is polar. Finding conditions which guarantee that a Markov process satisfies Assumption 4.1 is a classicaltopic in probabilistic potential theory. It is well-known that a d -dimensional Brownian motionsatisfies Assumption 4.1 for all d ∈ N . As a result, a large class of Itˆo diffusions, given by (2.9), alsosatisfies Assumption 4.1, as long as the Dol´eans-Dade exponential of t (cid:55)→ (cid:82) t b ( X s ) σ − ( X s ) ds is amartingale, thanks to Girsanov’s theorem; see e.g. [26, Section 9.2]. For general Markov processes,we refer readers to [8, Section 5.2] for a set of theoretic criteria that ensure Assumption 4.1.10 emark 4.1. For any R ∈ B , R \ R r is semipolar; see Theorem 6 in [8, Section 3.5]. Hence,under Assumption 4.1, R \ R r is polar. Assumption 4.1 leads to a very useful result in finding equilibria: if R ∈ B becomes smallerafter we apply Θ to R once, we immediately obtain an equilibrium. Proposition 4.1.
Suppose Assumptions 2.1 and 4.1 hold. Then, for any R ∈ B with Θ( R ) ⊆ R , R \ Θ( R ) is polar and Θ ( R ) = Θ( R ) , i.e. Θ( R ) ∈ E . In addition, if R is finely closed, so is Θ( R ) .Proof. For any R ∈ B , by (3.4), (2.6), and Θ( R ) ⊆ R , we have R r ∩ R ⊆ I ( R ) ∩ R ⊆ Θ( R ) ⊆ R. (4.1)As R \ R r is polar (Remark 4.1), this implies R \ Θ( R ) is also polar. It follows that ρ Θ( R ) = ρ R P x -a.s. for all x ∈ R d , which in turn implies J ( x, Θ( R )) = J ( x, R ) for all x ∈ R d . In view of (2.7),we obtain S ( R ) = S (Θ( R )) and I ( R ) = I (Θ( R )). Hence,Θ ( R ) = S (Θ( R )) ∪ ( I (Θ( R )) ∩ Θ( R ))= S ( R ) ∪ ( I ( R ) ∩ Θ( R ))= S ( R ) ∪ ( I ( R ) ∩ ( S ( R ) ∪ ( I ( R ) ∩ R )))= S ( R ) ∪ ( I ( R ) ∩ R ) = Θ( R ) , where the first, third, and fifth equalities follow from (2.6) and the fourth equality is due to S ( R ) ∩ I ( R ) = ∅ by definition. This shows that Θ( R ) ∈ E . Finally, as Θ( R ) ⊆ R , we have(Θ( R )) r ⊆ R r . If R is additionally finely closed, i.e. R r ⊆ R , then (4.1) yields R r ⊆ Θ( R ). Itfollows that (Θ( R )) r ⊆ R r ⊆ Θ( R ), i.e. Θ( R ) is finely closed. Remark 4.2.
Proposition 4.1 enhances the convergence of the fixed-point iteration introduced in[14]. When stated in the current context, [14, Proposition 3.3] asserts that whenever R ⊆ Θ( R ) , lim n →∞ Θ n ( R ) = (cid:91) n ∈ N Θ n ( R ) is well-defined and is an equilibrium. Proposition 4.1 complements the above result: for the oppositecase Θ( R ) ⊆ R , lim n →∞ Θ n ( R ) = Θ( R ) is an equilibrium. Remark 4.3.
Recall L in (3.12) . Under Assumption 4.1, R \ R is polar for all R ∈ L . Hence, forany R ∈ L , ρ R = ρ R P x -a.s. for all x ∈ R d . Remark 4.4.
It is tempting to conclude from Remark 4.3 that we can further restrict our attentionfrom (cid:101) E to E := { R ∈ E : R is closed } ; after all, the one-dimensional analysis in [19] is entirelybased on E . This is however not the case, as the relation “ R ∈ (cid:101) E if and only if R ∈ E ” does nothold in general. To illustrate, take any R ∈ (cid:101) E . For R to be in E , we need f ( x ) ≥ J ( x, R ) for x ∈ R .As R ∈ E , we must have f ( x ) ≤ J ( x, R ) = J ( x, R ) for x / ∈ R . Hence, “ R ∈ E ” boils down to thecondition “ f ( x ) = J ( x, R ) for x ∈ R \ R ”, which is not true in general. From this observation, weconstruct an example in Section 5, which explicitly demonstrates R ∈ (cid:101) E but R / ∈ E . .2 Improving an Equilibrium By repeating the arguments in the proof of [19, Lemma 3.1], we get the same result in the multi-dimensional case.
Lemma 4.1.
For any R , T ∈ B with R ⊆ T and R ∈ E , J ( x, R ) ≥ J ( x, T ) for all x ∈ R d . The next result is a multi-dimensional extension of [19, Proposition 4.8]
Proposition 4.2.
Assume Assumptions 2.1 and 4.1, and that (2.8) and (2.11) hold for all x ∈ R d .Then, for any R , T ∈ (cid:101) E , Θ( R ∩ T ) ⊆ R ∩ T belongs to (cid:101) E , and satisfies J ( x, Θ( R ∩ T )) ≥ J ( x, R ) ∨ J ( x, T ) , ∀ x ∈ R d . (4.2) Proof.
Fix R , T ∈ (cid:101) E = E ∩ L . By the same arguments in the proof of [19, Proposition 4.8], we get J ( x, R ∩ T ) ≥ J ( x, R ) ∨ J ( x, T ) , ∀ x ∈ ( R ∩ T ) c . (4.3)As R , T ∈ L , ρ R = ρ R and ρ T = ρ T P x -a.s. for all x ∈ R d (Remark 4.3). It follows that J ( x, R ) = J ( x, R ) and J ( x, T ) = J ( x, T ) , ∀ x ∈ R d . (4.4)Moreover, by the same argument above (3.13), we have ( R ∩ T ) \ ( R ∩ T ) ⊆ ( R \ R ) ∪ ( T \ T ). Since R \ R and T \ T are polar (Remark 4.3), so is ( R ∩ T ) \ ( R ∩ T ). It follows that J ( x, R ∩ T ) = J ( x, R ∩ T ) , ∀ x ∈ R d . (4.5)Now, by the fact R, T ∈ E , we obtain from (4.4), (4.3), and (4.5) that f ( x ) ≤ J ( x, R ) ∨ J ( x, T ) = J ( x, R ) ∨ J ( x, T ) ≤ J ( x, R ∩ T ) = J ( x, R ∩ T ) , ∀ x ∈ ( R ∩ T ) c . (4.6)This particularly implies S ( R ∩ T ) ⊆ R ∩ T , and thusΘ( R ∩ T ) = S ( R ∩ T ) ∪ ( I ( R ∩ T ) ∩ ( R ∩ T )) ⊆ R ∩ T. (4.7)By Proposition 4.1, this readily shows that Θ( R ∩ T ) ∈ E , Θ( R ∩ T ) is finely closed (as R ∩ T isfinely closed), and ( R ∩ T ) \ Θ( R ∩ T ) is polar . (4.8)Note that (4.7) also implies Θ( R ∩ T ) ⊆ Θ( R ∩ T ) ⊆ R ∩ T .
It follows that Θ( R ∩ T ) \ Θ( R ∩ T ) ⊆ (cid:0) R ∩ T (cid:1) \ Θ( R ∩ T ) ⊆ (cid:16) ( R ∩ T ) \ Θ( R ∩ T ) (cid:17) ∪ ( R \ R ) ∪ ( T \ T ) . As the three sets in the second line above are all polar (recall (4.8) and Remark 4.3), we concludethat Θ( R ∩ T ) \ Θ( R ∩ T ) is polar, and thus Θ( R ∩ T ) ∈ L . Hence, Θ( R ∩ T ) ∈ E ∩ L = (cid:101) E . Finally,thanks to Θ( R ∩ T ) ⊆ R (by (4.7)) and Θ( R ∩ T ) ∈ E , Lemma 4.1 asserts J ( x, Θ( R ∩ T )) ≥ J ( x, R )for all x ∈ R d . A similar argument shows that J ( x, Θ( R ∩ T )) ≥ J ( x, T ) for all x ∈ R d . We canthen conclude that (4.2) holds. 12 emark 4.5. In (4.3) , the inequality is guaranteed for only x ∈ ( R ∩ T ) c , although the correspondingone-dimensional result holds for all x ∈ R ; see [19, Proposition 4.8]. For instance, for d ≥ , if thereexist two closed equilibria R and T such that R ∩ T = { x } for some x ∈ R d , then ρ R ∩ T = ρ { x } = ∞ P x -a.s., for a wide range of Markov processes X . By (2.11) , this implies J ( x, R ∩ T ) = 0 , which isunlikely to be equal to J ( x, R ) ∨ J ( x, T ) . By contrast, for d = 1 , (1.1) ensures ρ R = ρ T = ρ R ∩ T = ρ { x } = 0 P x -a.s., so that J ( x, R ∩ T ) = f ( x ) = J ( x, R ) ∨ J ( x, T ) . Proposition 4.2 provides a partial order among elements in (cid:101) E . Corollary 4.1.
Assume Assumptions 2.1 and 4.1, and that (2.8) and (2.11) holds for all x ∈ R d .Then, for any R , T ∈ (cid:101) E with R ⊆ T , J ( x, R ) ≥ J ( x, T ) for all x ∈ R d .Proof. For any R , T ∈ (cid:101) E , note that R \ Θ( R ∩ T ) ⊆ ( R \ T ) ∪ (cid:0) ( R ∩ T ) \ Θ( R ∩ T ) (cid:1) . With R ⊆ T , R \ T ⊆ T \ T is polar (Remark 4.3). Recalling from (4.8) that ( R ∩ T ) \ Θ( R ∩ T ) is also polar,we conclude that R \ Θ( R ∩ T ) is polar. Hence, ρ R = ρ Θ( R ∩ T ) P x -a.s. for all x ∈ R d , and thus J ( x, R ) = J ( x, Θ( R ∩ T )) ≥ J ( x, T ) ∀ x ∈ R d , where the inequality follows from Proposition 4.2. Before we state the main result of this paper, we need a convergence result for first hitting times.
Lemma 4.2.
Suppose Assumption 4.1 holds. Let ( R n ) n ∈ N be a nonincreasing sequence in L , andset R := (cid:84) n ∈ N R n . Then lim n →∞ ρ R n = ρ R P x -a.s. ∀ x ∈ R c . (4.9) Proof.
Fix x ∈ R c . Set τ n := ρ R n , and define τ := lim n →∞ τ n . As R ⊆ R n , we must have τ ≤ ρ R .Hence, it suffices to prove τ ≥ ρ R P x -a.s. on { τ < ∞} . (4.10)For each m ∈ N , as ( R n ) n ∈ N is nonincreasing, ( R n ) n ∈ N is also nonincreasing. It follows that X xτ n ∈ R m ∀ n ≥ m, P x -a.s. on { τ < ∞} . As n → ∞ , by the continuity of t (cid:55)→ X xt , this implies X xτ ∈ R m P x -a.s. on { τ < ∞} . Since R m \ R m is polar (recall R m ∈ L and Remark 4.3), the above relation can be equivalently writtenas X xτ ∈ R m P x -a.s. on { τ < ∞} . By the arbitrariness of m ∈ N , we conclude X xτ ∈ (cid:92) m R m = R, P x -a.s. on { τ < ∞} . (4.11)As x ∈ R c , x / ∈ R n for some n ∈ N . Since R n is finely closed, x / ∈ R n implies that x is notregular to R n , i.e. τ n = ρ R n > P x -a.s. Consequently, τ > P x -a.s. We then deduce from τ > Remark 4.6.
We require “ x ∈ R c ” in (4.9) , as the convergence need not hold for x ∈ R . Forinstance, for any d ≥ , let X be a d -dimensional Brownian motion and R n ∈ L be the closed ballaround the origin O := (0 , , ..., ∈ R d with radius /n , for all n ∈ N . Clearly, R := (cid:84) n ∈ N R n = { O } . For x = O ∈ R , we have ρ R n = 0 for all n ∈ N , but ρ R = ∞ . Theorem 4.1.
Assume Assumptions 2.1 and 4.1, and that (2.8) and (2.11) hold for all x ∈ R d .Then, there exists R ∈ (cid:101) E that is optimal among (cid:101) E . Moreover, R = (cid:84) R ∈ (cid:101) E R. Proof.
Consider (cid:101) R := (cid:84) R ∈ (cid:101) E R . As an intersection of closed sets, (cid:101) R is closed. Since the indicatorfunction of a closed set is upper semi-continuous, [1, Proposition 4.1 ] implies that there exists acountable subset ( R n ) n ∈ N of (cid:101) E such that (cid:101) R = (cid:84) n R n . Define ( T n ) n ∈ N by T := R , T n := Θ( T n − ∩ R n ) for n ≥ . By applying Proposition 4.2 to ( T n ) n ∈ N recursively, we have T n ∈ (cid:101) E ∀ n ∈ N , (4.12)as well as T n +1 = Θ( T n ∩ R n +1 ) ⊆ T n ∩ R n +1 ⊆ T n = Θ( T n − ∩ R n ) ⊆ T n − ∩ R n ⊆ R n , ∀ n ≥ . (4.13)Consider R ◦ := (cid:84) n T n . We deduce from Lemma 3.2, (4.12), and (4.13) that R ◦ ∈ L and R ◦ ⊆ (cid:92) n R n = (cid:101) R. (4.14)Now, for any x ∈ ( R ◦ ) c , as ( T n ) n ∈ N in (cid:101) E = E ∩ L is nonincreasing (see (4.13)), Lemma 4.2 entails ρ T n → ρ R ◦ P x -a.s. Thanks to this, the continuity of δ , f , and t (cid:55)→ X t , and (2.8), we conclude fromthe dominated convergence theorem thatlim n →∞ J ( x, T n ) = lim n →∞ E x [ δ ( ρ T n ) f ( X ρ Tn )] = E x [ δ ( ρ R ◦ ) f ( X ρ R ◦ )] = J ( x, R ◦ ) . (4.15)On the other hand, the fact that x / ∈ R ◦ = (cid:84) n T n and ( T n ) n ∈ N is nonincreasing implies that thereexists n ∈ N such that x / ∈ T n for all n ≥ n . This, together with T n ∈ E (by (4.12)), indicates f ( x ) ≤ J ( x, T n ) for all n ≥ n . Combining this with (4.15), we obtain f ( x ) ≤ J ( x, R ◦ ) ∀ x ∈ ( R ◦ ) c . This shows that S ( R ◦ ) ⊆ R ◦ , so thatΘ( R ◦ ) = S ( R ◦ ) ∪ ( I ( R ◦ ) ∩ R ◦ ) ⊆ R ◦ . (4.16)By Proposition 4.1, this implies the following properties: R ◦ \ Θ( R ◦ ) is polar; (4.17) R := Θ( R ◦ ) belongs to E and is finely closed . (4.18)Note that (4.16) implies Θ( R ◦ ) ⊆ Θ( R ◦ ) ⊆ R ◦ , which givesΘ( R ◦ ) \ Θ( R ◦ ) ⊆ R ◦ \ Θ( R ◦ ) ⊆ ( R ◦ \ R ◦ ) ∪ ( R ◦ \ Θ( R ◦ )) . (4.19)As R ◦ ∈ L (by (4.14)), R ◦ \ R ◦ is polar (recall Remark 4.3). This, together with (4.17), shows thatthe right hand side of (4.19) is polar, and thus Θ( R ◦ ) \ Θ( R ◦ ) is polar. We then conclude from(4.18) that R = Θ( R ◦ ) ∈ (cid:101) E . By (4.16) and (4.14), R ⊆ (cid:101) R = (cid:92) R ∈ (cid:101) E R. (4.20)Hence, for any R ∈ (cid:101) E , we have R ⊆ R . With R ∈ E , Corollary 4.1 gives J ( x, R ) ≥ J ( x, R ) for all x ∈ R d . Therefore, R is optimal among (cid:101) E . Also, the fact R ∈ (cid:101) E implies (cid:101) R = (cid:84) R ∈ (cid:101) E R ⊆ R . This,together with (4.20), entails R = (cid:101) R = (cid:84) R ∈ (cid:101) E R . 14 An Example of R ∈ (cid:101) E but R / ∈ E
In this section, we take X to be a three-dimensional Brownian motion, and will construct an examplethat explicitly demonstrates R ∈ (cid:101) E but R / ∈ E . To this end, we need the following technical result,whose proof is relegated to Appendix B.
Lemma 5.1.
Let X be a three-dimensional Brownian motion. Given an open domain D ⊆ R ,suppose that f ≤ K on ∂D for some K > . Then,(i) for any x ∈ D and r > such that B ( x, r ) := { y ∈ R : (cid:107) y − x (cid:107) < r } ⊆ D , k ( r ) m ( B ( x, r )) (cid:90) B ( x,r ) J ( y, D c ) m ( dy ) ≤ J ( x, D c ) ≤ m ( B ( x, r )) (cid:90) B ( x,r ) J ( y, D c ) m ( dy ) , (5.1) where k ( r ) := E x [ δ ( ρ B ( x,r ) c )] is continuous and nonincreasing in r with lim r ↓ k ( r ) = 1 , and m ( · ) denotes the Lebesgue measure in R .(ii) x (cid:55)→ J ( x, D c ) is continuous on D . Furthermore, if z ∈ ∂D is regular to D c , then x (cid:55)→ J ( x, D c ) is also continuous at z , in the sense that lim x → z, x ∈ D J ( x, D c ) = J ( z, D c ) . (5.2)Now, let S be the collection of ( x , x , x ) ∈ R such that x = (cid:113) x + x , for x ≥ , ≤ x + x ≤ x = 2 − (cid:113) x + x , for x ≥ , ≤ x + x ≤ x + x = 4 , for x < . As shown in Figure 1,
S ⊂ R is the surface generated by rotating the curve l around the x -axis.It partitions R into two open sets G and G : G contains (1 , , G contains ( − , , ∂G = ∂G = S . Note that G = G ∪ S is a so-called Lebesgue thorn with the origin O := (0 , , O is not regular to either G or G , while all other points in S are regular to G . (5.3)Define h : R → R + by h ( x ) := (cid:40)(cid:112) x + x + x , for x + x + x ≤ , / (cid:112) x + x + x , otherwise . (5.4)Note that 0 < h ≤ S except at the origin O , where h ( O ) = 0. Then, we introduce h ( x ) := E x [ δ ( ρ G ) h ( X ρ G )] , ∀ x ∈ R . (5.5)Now, we define the payoff function f by f ( x ) := (cid:40) h ( x ) , x ∈ G , min { h ( x ) , h ( x ) } , x ∈ G . (5.6)15igure 1: Surface S is generated by rotating the curve (cid:96) around the x -axis Lemma 5.2. f in (5.6) is continuous on R , and satisfies (2.8) and (2.11) for all x ∈ R .Proof. As 0 ≤ h ≤ R , we have 0 ≤ f ≤ R by definition. Hence, (2.8) and (2.11) aretrivially satisfied. Clearly, h is continuous on R . By (5.3), we conclude from Lemma 5.1 that h is continuous on G \ { O } . Moreover, since every point in S \ { O } is regular to G , by definition h = h on S \ { O } . All this readily implies that f is continuous on R \ { O } . Finally, observethat 0 ≤ f ( x ) ≤ h ( x ) for all x ∈ R \ { O } . This, along with lim x → O h ( x ) = h ( O ) = 0, entailslim x → O f ( x ) = 0 = f ( O ). We therefore conclude that f is continuous on R .By (5.3), the fine closure of G is G ∗ = G ∪ ( S \ { O } ) . (5.7) Proposition 5.1. G ∗ ∈ (cid:101) E but G ∗ / ∈ E .Proof. By (5.3) and G being open, every point in G ∗ is regular to G ∗ . Hence, J ( x, G ∗ ) = E x [ δ ( ρ G ∗ ) f ( X ρ G ∗ )] = f ( x ) for all x ∈ G ∗ . For any x ∈ ( G ∗ ) c = G ∪ { O } , (5.3) implies X ρ G ∗ ∈ S \ { O } P x -a.s. Hence, by the fact that f = h on S , J ( x, G ∗ ) = E x [ δ ( ρ G ∗ ) f ( X ρ G ∗ )] = E x [ δ ( ρ G ∗ ) h ( X ρ G ∗ )] = h ( x ) ∀ x ∈ G ∪ { O } , (5.8)where the last equality follows from the fact that G ∗ and G only differ by the singleton { O } ,which is polar (with respect to X , a three-dimensional Brownian motion). By the definition of f ,this readily implies J ( x, G ∗ ) = h ( x ) ≥ f ( x ) for x ∈ G . Note that X ρ G ∗ ∈ S \ { O } also implies h ( X ρ G ∗ ) >
0. We then deduce from (5.8) that J ( x, G ∗ ) > x ∈ G ∪ { O } . In particular, J ( O, G ∗ ) > h ( O ) = f ( O ) . (5.9)Therefore, we conclude that G ∗ ⊆ I ( G ∗ ) and S ( G ∗ ) = ∅ . In view of (2.6), Θ( G ∗ ) = G ∗ , i.e. G ∗ ∈ E . In addition, since { O } is polar and G ∗ \ G ∗ = G \ G ∗ = { O } , we have G ∗ ∈ L . Hence, G ∗ ∈ E ∩ L = (cid:101) E . Finally, by (5.3) and (5.9), J ( O, G ∗ ) = J ( O, G ∗ ) > f ( O ), i.e. O ∈ C ( G ∗ ). As G ∗ = G ∗ ∪ { O } intersects C ( G ∗ ), we conclude G ∗ / ∈ E .16 Proof of Lemma 2.1
Fix R ∈ B . If R is of zero potential, then (cid:90) ∞ e − αt P t ( x, R ) dt ≤ (cid:90) ∞ P t ( x, R ) dt = U ( x, R ) = 0 ∀ x ∈ R d . (A.1)Hence, (cid:82) ∞ e − αt P t ( x, R ) dt = 0 for all x ∈ R d , and thus λ α ( R ) = 0. Conversely, if λ α ( R ) = 0, then λ ( E ) = 0 with E := { x ∈ R d : U α ( x, R ) > } , where we use the notation U α ( x, R ) := (cid:90) ∞ e − αt P t ( x, R ) dt. (A.2)Since there exists a transition density ( p t ) t ≥ for X , it follows from (2.1) and λ ( E ) = 0 that U ( x, E ) = (cid:90) ∞ P t ( x, E ) dt = (cid:90) ∞ (cid:90) E p t ( x, y ) λ ( dy ) dt = (cid:90) ∞ dt = 0 ∀ x ∈ R d . By the same argument as in (A.1), this implies U α ( x, E ) = 0 for all x ∈ R d . (A.3)Now, we claim that U α ( x, R ) = 0 for all x ∈ R d . Let us write R = R ∪ R , where R := { x ∈ R : U α ( x, R ) > } and R := { x ∈ R : U α ( x, R ) = 0 } . As R ⊆ E , we have U α ( x, R ) ≤ U α ( x, E ) for all x ∈ R d . Then, (A.3) entails U α ( x, R ) = 0 forall x ∈ R d , so that U α ( x, R ) = U α ( x, R ) ∀ x ∈ R d . (A.4)By contradiction, suppose that there exists x ∗ ∈ R d such that U α ( x ∗ , R ) >
0. Observe that U α ( x ∗ , R ) = E x ∗ (cid:20)(cid:90) ∞ e − αt R ( X t ) dt (cid:21) = E x ∗ (cid:20) (cid:90) ∞ ρ R e − αt R ( X t ) dt (cid:21) = E x ∗ (cid:104) e − αρ R U α ( X ρ R , R ) (cid:105) = E x ∗ (cid:104) e − αρ R U α ( X ρ R , R ) (cid:105) = 0 , where the first equality stems from the definition of U α in (A.2), the fourth equality follows from(A.4), and the last equality is due to U α ( X ρ R , R ) = 0 (by the definition of R ). This contradicts U α ( x ∗ , R ) = U α ( x ∗ , R ) >
0. Hence, we have U α ( x, R ) = 0 for all x ∈ R d . Thanks again toProposition 3 in [8, Section 3.5], this implies U ( x, R ) = 0 for all x ∈ R d , i.e. R is of zero potential. B Proof of Lemma 5.1 (i) Fix x ∈ D . As X is a three-dimensional Brownian motion and δ is continuous and nonincreasing, r (cid:55)→ k ( r ) := E x [ δ ( ρ B ( x,r ) )] is continuous and nonincreasing in r , and ρ B ( x,r ) ↓ r ↓ P x -a.s. Bythe dominated convergence theorem and δ (0) = 1, we get lim r ↓ k ( r ) = 1. Now, fix r > B ( x, r ) ⊆ D . For any 0 < s ≤ t , by (2.3) we get J ( x, D c ) = E x [ δ ( ρ D c ) f ( X ρ Dc )] ≥ E x [ δ ( ρ B ( x,s ) c ) E x [ δ ( ρ D c − ρ B ( x,s ) c ) f ( X ρ Dc ) | F ρ B ( x,s ) c ]]= E x [ δ ( ρ B ( x,s ) c ) J ( X ρ B ( x,s ) c , D c )] . (B.1)17s X is a three-dimensional Brownian motion, X xρ B ( x,s ) c and ρ B ( x,s ) c are independent, and X xρ B ( x,s ) c is uniformly distributed on ∂B ( x, s ). Thus, (B.1) implies J ( x, D c ) ≥ E x [ δ ( ρ B ( x,s ) c )] · E x [ J ( X ρ B ( x,s ) c , D c )] = k ( s ) · s ) (cid:90) ∂B ( x,s ) J ( y, D c )Σ( dy ) , (B.2)where Σ( s ) denotes the two-dimensional surface measure on ∂B ( x, s ) in R . On the other hand, as δ is nonincreasing, J ( x, D c ) ≤ E x [ δ ( ρ D c − ρ B ( x,s ) c ) f ( X ρ Dc )] = E x [ J ( X ρ B ( x,s ) c , D c )] = 1Σ( s ) (cid:90) ∂B ( x,s ) J ( y, D c )Σ( dy ) . (B.3)Combining (B.2) and (B.3), and using k ( r ) ≤ k ( s ) as s ≤ r , we obtain k ( r ) · (cid:90) ∂B ( x,s ) J ( y, D c )Σ( dy ) ≤ Σ( s ) J ( x, D c ) ≤ (cid:90) ∂B ( x,s ) J ( y, D c )Σ( dy ) , ∀ < s ≤ r. Integrating the above inequality with respect to s from 0 to r yields k ( r ) · (cid:90) B ( x,r ) J ( y, D c ) m ( dy ) ≤ m ( B ( x, r )) · J ( x, D c ) ≤ (cid:90) B ( x,r ) J ( y, D c ) m ( dy ) , which gives (5.1).(ii) First, we show that x (cid:55)→ J ( x, D c ) is continuous on D . As f ≤ K on ∂D , J ( x, D c ) ≤ K ∀ x ∈ D. (B.4)Fix an arbitrary ε >
0. For any x , x ∈ D , choose r > B ( x i , r ) ⊆ D for i = 1 , k ( r ) ≥ − ε . Then, by (5.1),1 − εm ( B ( x i , r )) (cid:90) B ( x i ,r ) J ( y, D c ) m ( dy ) ≤ J ( x i , D c ) ≤ m ( B ( x i , r )) (cid:90) B ( x i ,r ) J ( y, D c ) m ( dy ) , for i = 1 , . It follows that J ( x , D c ) − J ( x , D c ) ≤ m ( B ( x , r )) (cid:90) B ( x ,r ) J ( y, D c ) m ( dy ) − − εm ( B ( x , r )) (cid:90) B ( x ,r ) J ( y, D c ) m ( dy ) ≤ m ( B ( x , r )) (cid:90) B ( x ,r )∆ B ( x ,r ) J ( y, D c ) m ( dy ) + εm ( B ( x , r )) (cid:90) B ( x ,r ) J ( y, D c ) m ( dy ) ≤ K (cid:18) m ( B ( x , r )∆ B ( x , r )) m ( B ( x , r )) + ε (cid:19) , (B.5)where B ( x , r )∆ B ( x , r ) denotes the symmetric difference of B ( x , r ) and B ( x , r ), and the thirdequality is due to (B.4). By choosing x sufficiently close to x , we can make m ( B ( x ,r )∆ B ( x ,r )) m ( B ( x ,r )) ≤ ε ,so that J ( x , D c ) − J ( x , D c ) ≤ Kε . By switching x and x in (B.5), we obtain the similarresult that by choosing x close enough to x , we get J ( x , D c ) − J ( x , D c ) ≤ Kε . Hence, | J ( x , D c ) − J ( x , D c ) | ≤ Kε for x sufficiently close to x . That is, J ( x, D c ) is continuous at x .By the arbitrariness of x ∈ D , x (cid:55)→ J ( x, D c ) is continuous on D .It remains to prove (5.2). Fix z ∈ ∂D that is regular to D c . By Proposition 1 in [8, Section4.4], for any η > x (cid:55)→ P x ( ρ D c ≤ η ) is lower seimicontinuous. Hence,lim inf x → z P x ( ρ D c ≤ η ) ≥ P z ( ρ D c ≤ η ) = 1 , x → z P x ( ρ D c ≤ η ) = 1 . (B.6)Given r >
0, note that because X is a Brownian motion, x (cid:55)→ P x ( ρ B ( x,r ) c > η ) is constant . (B.7)Observe that it holds for all η > P x ( ρ D c < ρ B ( x,r ) c ) ≥ P x ( ρ D c ≤ η < ρ B ( x,r ) c ) ≥ P x ( ρ D c ≤ η ) + P x ( ρ B ( x,r ) c > η ) − . (B.8)By (B.6) and (B.7), this implieslim inf x → z P x ( ρ D c < ρ B ( x,r ) c ) ≥ P z ( ρ B ( z,r ) c > η ) , ∀ η > . As lim η ↓ P z ( ρ B ( z,r ) c > η ) = 1, thanks again to X being a Brownian motion, we conclude thatlim x → z P x ( ρ D c < ρ B ( x,r ) c ) = 1 . (B.9)On the set { ρ D c < ρ B ( x,r ) c } , we have (cid:107) X xρ Dc − x (cid:107) < r . Hence, for any ε >
0, by the continuity of f , δ , and t (cid:55)→ X t , we can choose r > x ∈ D with (cid:107) x − z (cid:107) ≤ r , | f ( x ) − f ( z ) | ≤ ε, | f ( X xρ Dc ) − f ( x ) | ≤ ε on { ρ D c < ρ B ( x,r ) c } , and E x [ δ ( ρ B ( x,r ) c )] ≥ − ε. (B.10)By (B.9), for this fixed r >
0, we can choose 0 < r (cid:48) < r such that for all x ∈ D with (cid:107) x − z (cid:107) < r (cid:48) , P x ( ρ D c ≥ ρ B ( x,r ) c ) ≤ ε. (B.11)As z is regular to D c , J ( z, D c ) = f ( z ). If follows that | J ( x, D c ) − J ( z, D c ) | ≤ E x [ | δ ( ρ D c ) f ( X ρ Dc ) − f ( z ) | ] . (B.12)Now, for any x ∈ D with (cid:107) x − z (cid:107) < r (cid:48) , observe that E x [ | δ ( ρ D c ) f ( X ρ Dc ) − f ( z ) | { ρ Dc <ρ B ( x,r ) c } ] ≤ E x [ | δ ( ρ D c ) f ( X ρ Dc ) − f ( x ) | { ρ Dc <ρ B ( x,r ) c } ] + | f ( x ) − f ( z ) |≤ E x [ | f ( X ρ Dc ) − f ( x ) | { ρ Dc ≤ ρ B ( x,r ) c } ] + E x [ | δ ( ρ D c ) − || f ( X ρ Dc ) | { ρ Dc ≤ ρ B ( x,r ) } ] + ε ≤ ( K + 2) ε, where the last inequality follows from (B.10) and 0 ≤ f ≤ K on ∂D . On the other hand, E x [ | δ ( ρ D c ) f ( X ρ Dc ) − f ( z ) | { ρ Dc ≥ ρ B ( x,r ) c } ] ≤ K P x ( ρ D c ≥ ρ B ( x,r ) c ) ≤ Kε, where the last inequality follows from (B.11). We then conclude from (B.12) that | J ( x, D c ) − J ( z, D c ) | ≤ (3 K + 2) ε , which completes the proof.19 eferences [1] Erhan Bayraktar and Mihai Sˆırbu. Stochastic Perron’s method and verification withoutsmoothness using viscosity comparison: the linear case. Proc. Amer. Math. Soc. , 140(10):3645–3654, 2012.[2] Erhan Bayraktar, Jingjie Zhang, and Zhou Zhou. On the notions of equilibria fortime-inconsistent stopping problems in continuous time. 2019. Preprint, available athttps://arxiv.org/abs/1909.01112.[3] Tomas Bj¨ork, Mariana Khapko, and Agatha Murgoci. On time-inconsistent stochastic controlin continuous time.
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