Failure of Smooth Pasting Principle and Nonexistence of Equilibrium Stopping Rules under Time-Inconsistency
aa r X i v : . [ q -f i n . M F ] S e p Failure of Smooth Pasting Principle and Nonexistence ofEquilibrium Stopping Rules under Time Inconsistency ∗ Ken Seng Tan Wei Wei Xun Yu Zhou † September 5, 2019
Abstract
This paper considers time-inconsistent stopping problems in which the inconsistencyarises from a class of non-exponential discount functions called the weighted discount func-tions. We show that the smooth pasting principle, the main approach that is used to con-struct explicit solutions for the classical time-consistent optimal stopping problems, mayfail under time-inconsistency. Specifically, we prove that the smooth pasting solves a time-inconsistent problem, within the intra-personal game theoretic framework with a generalnonlinear cost functional and a geometric Brownian motion, if and only if certain inequal-ities on the model primitives are satisfied. In the special case of a real option problem,we show that the violation of these inequalities can happen even for very simple non-exponential discount functions. Moreover, we show that the real option problem actuallydoes not admit any equilibrium whenever the smooth pasting approach fails. The negativeresults in this paper caution blindly extending the classical approach for time-consistentstopping problems to their time-inconsistent counterparts. ∗ Zhou gratefully acknowledges financial supports through a start-up grant at Columbia University and throughthe Nie Center for Intelligent Asset Management. The authors thank the associate editor and two anonymousreferees for their detailed and constructive comments that have led to a much improved version. † Tan: Department of Statistics and Actuarial Science University of Waterloo Mathematics 3, 200 UniversityAvenue West Waterloo, Ontario, Canada N2L 3G1. E-mail: [email protected]. Wei: Department of Statisticsand Actuarial Science, University of Waterloo, 200 University Avenue West, Waterloo, N2L 3G1, ON, Canada.E-mail: [email protected]. Zhou: Department of IEOR, Columbia University, New York, NY 10027, USA.Email: [email protected]. ey words : optimal stopping, weighted discount function, time inconsistency, equilibrium stop-ping, intra-personal game, smooth pasting, real option.2
Introduction
A crucial assumption imposed on classical optimal stopping models is that an agent has a con-stant time preference rate and hence discounts her future payoff exponentially. When this as-sumption is violated, an optimal stopping problem becomes generally time-inconsistent in thatany optimal stopping rule obtained today may no longer be optimal from the perspective ofa future date. The problem then becomes largely descriptive rather than normative becausethere is generally no dynamically optimal solution that can be used to guide the agent’s deci-sions. Different agents may react differently to a same time-inconsistent problem, and a goalof the study is to describe the different behaviors. Strotz (1955) is the first to observe thatnon-constant time preference rates result in time-inconsistency, and to categorize three types ofagents when facing such time-inconsistency. One of the types is called a “non-committed, sophis-ticated agent” who, at any given time, optimizes the underlying objective taking as constraintsthe stopping decisions chosen by her future selves. Such a problem has been formulated withinan intra-personal game theoretic framework and the corresponding equilibria are used to de-scribe the behaviors of this type of agents; see, for example, Phelps and Pollak (1968); Laibson(1997); O’Donoghue and Rabin (2001); Krussell and Smith (2003) and Luttmer and Mariotti(2003). An extended dynamic programming equation for continuous-time deterministic equi-librium controls is derived in Ekeland and Lazrak (2006), followed by a stochastic version inBjork and Murgoci (2010) and application to a mean–variance portfolio model in Bj¨ork et al.(2014).This paper studies a time-inconsistent stopping problem in continuous time within the intra-personal game framework, in which the source of time-inconsistency is the so-called weighteddiscount function (WDF), a very general class of non-exponential discount functions. We maketwo main contributions. First, we demonstrate that the smooth pasting (SP) principle, whichis almost the exclusive approach in solving classical optimal stopping problems, may fail when The WDF, proposed in Ebert et al. (2016), is a weighted average of a set of exponential discount functions.It has been shown in Ebert et al. (2016) that it can be used to model the time preference of a group of individualsas well as that of behavioral agents, and that most commonly used non-exponential discount functions are WDFs. C smooth pasting around the free boundary, and thenchecks that it solves the PDE under some standard regularity/convexity conditions on the modelprimitives. Finally it verifies that the first hitting time of the free boundary indeed solves the op-timal stopping problem using the standard verification technique. Recently, Grenadier and Wang(2007) and Hsiaw (2013), among others, extend the application of the SP principle to solvingtime-inconsistent stopping problems. While the SP happened to work in the specific settings ofthese papers, it is more an exception than a rule. Indeed, in the present paper we show that,for a geometric Brownian motion with a nonlinear cost functional, while the SP always yieldsa candidate solution, the latter actually gives rise to an equilibrium stopping rule if and only ifcertainty inequalities on the model primitives are satisfied. These inequalities hold trivially forthe time-consistent exponential discount case, but does not in general for its time-inconsistentnon-exponential counterpart, even if all the other parameters and assumptions (state dynam-ics, running cost, etc.) are identical . Indeed, the violation of such inequalities is not rare evenin very simple cases. For example, we show that in the special case of a real option problemwith some WDFs including the pseudo-exponential discount function (Ekeland and Lazrak 2006;Karp 2007; Harris and Laibson 2013), the inequalities do not hold for plausible sets of parametervalues of the chosen discount functions. The bottom line is that one cannot blindly apply theSP to any stopping model when time-inconsistency is present, even if the SP does work for itstime-consistent counterpart.The second contribution is on the nonexistence of an intra-personal equilibrium. For a time-2onsistent stopping problem, optimal stopping rules exist when the cost functional and theunderlying process satisfy some mild regularity conditions (see, e.g., Peskir and Shiryaev 2006).However, this is no longer the case for the time-inconsistent counterpart. To demonstrate this,we again take the real option problem with a WDF. For such a problem, we prove that theresimply does not exist any equilibrium stopping rule whenever the aforementioned inequalities areviolated and hence the SP principle fails. Our result therefore reveals that equilibrium stoppingrules within the intra-personal game theoretic framework may not exist no matter what regularityconditions are imposed on the underlying models.There are studies in the literature on time-inconsistent stopping including nonexistence re-sults, albeit in considerably different settings especially in terms of the source of time-inconsistencyand the definition of an equilibrium. Bayraktar et al. (2018) consider a stopping problem with adiscrete-time Markov chain, whereas the time-inconsistency comes from the mean-variance objec-tive functional. The Markov chain takes value in a set of finite numbers, which allows them to dis-cuss the nonexistence of equilibrium stopping rules by enumeration. Christensen and Lindensj¨o(2018a) and Christensen and Lindensjö (2018b) study continuous-time stopping problems wherethe time-inconsistency follows from the types of payoff functions (mean-variance or endogenoushabit formation). In particular, Christensen and Lindensj¨o (2018a) show that the candidate so-lution derived from the SP may not lead to an equilibrium stopping for some range of parameters.However, the definition of equilibria in these papers is entirely different from the one based onthe “first-order” spike variation; the latter seems to be widely adopted by many papers (see, e.g.,O’Donoghue and Rabin 2001; Bjork and Murgoci 2010; Ekeland et al. 2012; and Bj¨ork et al.2014) including the present one. Huang and Nguyen-Huu (2018) investigate a continuous-timestopping problem with non-exponential discount functions. They define an equilibrium via a fixedpoint of a mapping, which is essentially based on a zeroth-order condition and hence is differentfrom our definition. Under their setting, immediately stopping is always a (trivial) equilibrium Christensen and Lindensj¨o (2018a) also consider mixed strategies as opposed to the pure strategies studiedin our paper and many other papers. Time-inconsistent problems using mixed strategies are interesting, and it ispossible that no equilibrium may be found in the class of mixed strategies either. However, the main point of thispaper is to show that a change of discounting factor from exponential to non-exponential may cause a stoppingproblem that has an equilibrium to one that does not, even though both are using pure strategies . Throughout this paper we consider weighted discount functions defined as follows.
Definition 1 (Ebert et al. 2016)
Let h : [0 , ∞ ) → (0 , be strictly decreasing with h (0) = 1 . Wecall h a weighted discount function (WDF) if there exists a distribution function F concentratedon [0 , ∞ ) such that h ( t ) = Z ∞ e − rt dF ( r ) . (1) Moreover, we call F the weighting distribution of h. Many commonly used discount functions can be represented in weighted form. For example,exponential function h ( t ) = e − rt , r > (Samuelson 1937) and pseudo-exponential function h ( t ) = δe − rt +(1 − δ ) e − ( r + λ ) t , < δ < , r > , λ > (Ekeland and Lazrak 2006; Karp 2007) are WDFs withdegenerate and binary distributions respectively. A more complicated example is the generalized4yperbolic discount function (Loewenstein and Prelec 1992) with parameters γ > , β > , whichcan be represented as h ( t ) = 1(1 + γt ) βγ ≡ Z ∞ e − rt f (cid:18) r ; βγ , γ (cid:19) dr (2)where f ( r ; k, θ ) = r k − e − rθ θ k Γ( k ) is the density function of the Gamma distribution with parameters k and θ , and Γ( k ) = R ∞ x k − e − x dx the Gamma function evaluated at k . See Ebert et al. (2016) formore examples and discussions about the types of discount functions that are of weighted form.The following result is a restatement of the well-known Bernstein’s theorem in terms of WDFs,which actually provides a characterization of the latter. Theorem 1 (Bernstein 1928)
A discount function h is a WDF if and only if it is continuouson [0 , ∞ ) , infinitely differentiable on (0 , ∞ ) , and satisfies ( − n h ( n ) ( t ) ≥ , for all non-negativeintegers n and for all t > . Bernstein’s theorem can be used to examine if a given function is a WDF without necessarilyrepresenting it in the form of (1). For example, it follows from this theorem that the constantsensitivity discount function h ( t ) = e − at k , a, k > , and the constant absolute decreasing impatiencediscount function h ( t ) = e e − ct − , c > , are both WDFs. On a complete filtered probability space (Ω , F , F = {F t } t ≥ , P ) there lives a one dimensionalBrownian motion W , and a family of Markov diffusion processes X = X x parameterized by theinitial state X = x ∈ R and governed by the following stochastic differential equation (SDE) dX t = b ( X t ) dt + σ ( X t ) W t , X = x, (3)5here b, σ are Lipschitz continuous functions, i.e., there exists an L > such that for any x = y | b ( x ) − b ( y ) | + | σ ( x ) − σ ( y ) | ≤ L | x − y | . (4)We assume that F is the P -augmentation of the natural filtration generated by X . To avoid anuninteresting case we also assume that | σ ( x ) | ≥ c > ∀ x ∈ R so that X is non-degenerate. For any fixed x ∈ R , an agent monitors the process X = X x and aims to minimize the followingcost functional J ( x ; τ ) = E (cid:20)Z τ h ( s ) f ( X s ) ds + h ( τ ) g ( X τ ) (cid:12)(cid:12)(cid:12) X = x (cid:21) (5)by choosing τ ∈ T , the set of all F -stopping times. Here h is a WDF with a weighting distribution F , g is continuous and bounded, and f is continuous with polynomial growth, i.e., there exists m ≥ and C > such that | f ( x ) | ≤ C ( | x | m + 1) . (6)Moreover, we assume that there exists n ≥ , C ( r ) > satisfying R ∞ C ( r ) dF ( r ) + R ∞ rC ( r ) dF ( r ) < ∞ such that sup τ ∈T E (cid:20)Z τ e − rs | f ( X s ) | ds + e − rτ | g ( X τ ) | (cid:12)(cid:12)(cid:12) X = x (cid:21) ≤ C ( r )( | x | n + 1) , ∀ r ∈ supp ( F ) . (7)This is a (weak) assumption to ensure that the optimal value of the stopping problem is finite,and hence the problem is well-posed.We now define stopping rules which are essentially binary feedback controls. These stoppingrules induce Markovian stopping times for any given Markov process. Definition 2 (Stopping rule)
A stopping rule is a measurable function u : R → { , } where Here we assume that the Brownian motion is one dimensional just for notational simplicity. There is noessential difficulty with a multi-dimensional Brownian motion. indicates “continue” and indicates “stop”. For any given Markov process X = { X t } t ≥ , astopping rule u defines a Markovian stopping time τ u = inf { t ≥ , u ( X t ) = 1 } . (8)Given a stopping rule u , we can define the stopping region S u = { x ∈ (0 , ∞ ) : u ( x ) = 1 } . Forany x ∈ ¯ S u , since the underlying process X is non-degenerate, a standard result (e.g., Chapter of Ito and McKean Jr 1965) yields that P ( τ u = 0 | X = x ) = 1 , and hence J ( x ; τ u ) = g ( x ) . Thismeans that the agent stops immediately once the process reaches at any point in ¯ S u . As a result,in the setting of this paper, the continuation region is C u = ¯ S uc . As discussed earlier the non-exponential discount function h in the cost functional (5) rendersthe underlying optimal stopping problem generally time-inconsistent. In this paper we considera sophisticated and non-committed agent who is aware of the time-inconsistency but unable tocontrol her future actions. In this case, she seeks to find the so-called equilibrium strategieswithin the intra-personal game theoretic framework, in which the individual is represented bydifferent players at different dates. We now give the precise definition of an equilibrium stopping rule ˆ u , which essentially entailsa solution to a game in which no self at any time (or, equivalently in the current setting, at anystate) is willing to deviate from ˆ u . Definition 3 (Equilibrium stopping rule)
The stopping rule ˆ u is an equilibrium stoppingrule if lim sup ǫ → J ( x ; τ ˆ u ) − J ( x ; τ ǫ,a ) ǫ ≤ , ∀ x ∈ R , ∀ a ∈ { , } , (9) Given the infiniteness of the time horizon, the stationarity of the process X as well the time-homogeneity ofthe running objective function f , each self at any given time t faces exactly the same decision problem as theothers, which only depends on the current state X t = x , but not on time t directly. We can thus identify self t bythe current state X t = x . That is why we need to consider only stationary stopping rules u , which are functionsof the state variable x only. For details on this convention, see, e.g. Grenadier and Wang (2007); Ekeland et al.(2012); Harris and Laibson (2013) and, in particular, Section 3.2 of Ebert et al. (2016). here τ ǫ,a = inf { t ≥ ǫ, ˆ u ( X t ) = 1 } if a = 0 , if a = 1 (10) with { X t } t ≥ being the solution to (3). This definition of an equilibrium is consistent with the majority of definition for time-inconsistent control problems in the literature (see, e.g., Bjork and Murgoci 2010; Ekeland et al.2012; and Bj¨ork et al. 2014) when a stopping rule is interpreted as a binary control. Indeed, τ ǫ,a is a stopping time that might be different from τ ˆ u only in the very small initial time interval [0 , ǫ ) ; hence it is a “pertubation” of the latter. The following result, Theorem 2, formally establishes the Bellman system and provides theverification theorem for verifying equilibrium stoppings.
Theorem 2 (Equilibrium characterization)
Consider the cost functional (22) with WDF h ( t ) = Z ∞ e − rt dF ( r ) , a stopping rule ˆ u, an underlying process X defined by (3), functions w ( x ; r ) = E (cid:20)Z τ ˆ u e − rt f ( X t ) dt + e − rτ ˆ u g ( X τ ˆ u ) (cid:12)(cid:12)(cid:12) X = x (cid:21) and V ( x ) = Z ∞ w ( x ; r ) dF ( r ) . Suppose that w is contin-uous in x and V is continuously differentiable with its first-order derivative being absolutelycontinuous. If ( V, w, ˆ u ) solves min (cid:26) σ ( x ) V xx ( x ) + b ( x ) V x ( x ) + f ( x ) − Z ∞ rw ( x ; r ) dF ( r ) , g ( x ) − V ( x ) (cid:27) = 0 , x ∈ R , (11) ˆ u ( x ) = if V ( x ) = g ( x ) , otherwise , x ∈ R , (12) then ˆ u is an equilibrium stopping rule and V is the value function of the problem, i.e., V ( x ) = J ( x ; τ ˆ u ) ∀ x ∈ R . A proof to the above proposition is relegated to the appendix. A proof of this result in a different setting was provided in Ebert et al. (2016). Here we supply a proof forreader’s convenience. Failure of SP and Nonexistence of Equilibrium
In the classical literature on (time-consistent) stopping, optimal solutions are often obtained bythe SP, because the candidate solution obtained from the SP must solve the Bellman system(and hence the optimal stopping problem) under some mild conditions, such as the smoothnessand convexity/concavity of the cost functions. In economics terms, the SP principle amountsto the matching of the marginal cost at the stopped state; hence some economists apply theSP principle without even explicitly introducing the Bellman system. However, as we will showin this section, the SP approach in the presence of time-inconsistency may not yield a solutionto the Bellman system (and therefore not to the stopping problem within the game theoreticframework), no matter how smooth and convex/concave the cost functions might be.
Let us start with a time-consistent optimal stopping problem which we use as a benchmark forcomparison purpose and outline the way to use the SP principle in constructing explicit solutions.Consider the following classical optimal stopping problem inf τ ∈T E (cid:20)Z τ e − rs f ( X s ) ds + e − rτ K (cid:12)(cid:12)(cid:12) X = x (cid:21) , (13)where the underlying process X is a geometric Brownian motion dX t = bX t dt + σX t dW t , x > , (14)and T is the set of all stopping times with respect to F . In what follows we assume that the running cost f is continuously differentiable, increasingand concave. Moreover, to rule out the “trivial cases” where either immediately stopping ornever stopping is optimal for this time consistent benchmark, we assume that f (0) < rK, b < r, In this formulation the final cost is assumed to be a constant lump sum K without loss of generality. In fact,by properly modifying the running cost, we are able to reduce the stopping problem with a final cost function g to one with a final cost being any given constant K >
0. To see this, applying Ito’s formula to e − rt ( g ( X t ) − K ), lim x →∞ f x ( x ) x = ∞ .Define L ( x ; r ) = E [ R ∞ e − rs f ( X s ) ds | X = x ] . Noting that X is a geometric Brownian motion, wehave after straightforward manipulations L ( x ; r ) = Z ∞ Z ∞ f ( yx ) e − rs G ( y, s ) dyds, (15)where G ( y, s ) = √ π σy √ s e − (ln y − ( b − σ s )22 σ s . To ensure L and L x are well defined, we further assumethat f has linear growth and f x (0+) < ∞ .We now characterize the optimal stopping rule as follows. Proposition 1
There exists x B > such that the stopping rule u B ( x ) = x ≥ x B ( x ) solves optimalstopping problem (13). Moreover, x B is the unique solution of the following algebraic equationin y : α ( r )[ K − L ( y ; r )] + L x ( y ; r ) y = 0 (16) where α ( r ) = − ( b − σ ) + q ( b − σ ) + 2 σ rσ . (17)The key to proving this theorem is to make us of the SP; see Appendix A.2. we get E (cid:20)Z τ e − rs f ( X s ) ds + e − rτ g ( X τ ) (cid:12)(cid:12)(cid:12) X = x (cid:21) = E (cid:20)Z τ e − rs f ( X s ) ds + e − rτ K + e − rτ ( g ( X τ ) − K ) (cid:12)(cid:12)(cid:12) X = x (cid:21) = E (cid:26)Z τ e − rs f ( X s ) ds + e − rτ K + Z τ e − rs (cid:20) σ x g xx ( X s ) + bxg x ( X s ) − r ( g ( X s ) − K ) (cid:21) ds (cid:12)(cid:12)(cid:12) X = x (cid:27) . Letting ˜ f ( x ) := f ( x ) + σ x g xx ( x ) + bxg x ( x ) − r ( g ( x ) − K ), the cost functional now becomes the one in problem(13) with running cost ˜ f . .2 Equivalent conditions under time-inconsistency We now consider exactly the same stopping problem as the above time-consistent benchmarkexcept that the exponential discount function is replaced by a WDF, namely, the cost functionalis changed to J ( x ; τ ) = E (cid:20)Z τ h ( s ) f ( X s ) ds + h ( τ ) K (cid:12)(cid:12)(cid:12) X = x (cid:21) , (18)where h is a WDF with a weighting distribution F .As in the case of exponential discounting, we need to impose the following regularity condi-tions on the parameters of the problem: b < r, ∀ r ∈ supp ( F ); and max (cid:26)Z ∞ r − b dF ( r ) , Z ∞ r dF ( r ) , Z ∞ rdF ( r ) (cid:27) < ∞ . These conditions either hold automatically or reduce to the respective counterparts when thediscount function degenerates into the exponential one. On the other hand, they hold valid withmany genuine WDFs, including the generalized hyperbolic discount function (2) when γ < β andthe pseudo-exponential discount function.We now attempt to use the SP principle to solve the Bellman system in Theorem 2 with thecost functional (18). We start by conjecturing that the equilibrium stopping region is [ x ∗ , ∞ ) forsome x ∗ > . (As in the time-consistent case, x ∗ is called the triggering boundary or the stoppingthreshold.)It follows from the Feynman–Kac formula that w in the Bellman system is given by w ( x ; r ) = ( K − L ( x ∗ ; r )) (cid:16) xx ∗ (cid:17) α ( r ) + L ( x ; r ) , x < x ∗ ,K, x ≥ x ∗ , L ( x ; r ) is defined by (15) and α ( r ) by (17). Recall we have defined V and ˆ u by V ( x ) = Z ∞ w ( x ; r ) dF ( r ) = R ∞ (( K − L ( x ∗ ; r ))( xx ∗ ) α ( r ) + L ( x ; r )) dF ( r ) , x < x ∗ ,K, x ≥ x ∗ and ˆ u ( x ) = x < x ∗ , otherwise . The SP applied to V ( not to w ) yields V x ( x ∗ ) = 0 , implying that x ∗ is the solution to the followingalgebraic equation in y Z ∞ [ α ( r )( K − L ( y ; r )) + L x ( y ; r ) y ] dF ( r ) = 0 . (19)Clearly, this equation is a generalization of its time-consistent counterpart, (16). The followingproposition stipulates that it has a unique solution. Proposition 2
Equation (19) admits a unique solution in (0 , ∞ ) . Proof.
Following the same lines of proof of Proposition 1 (Appendix A.2), we have that Q ( x ) := R ∞ ( α ( r )( K − L ( x ; r )) + L x ( x ; r ) x ) dF ( r ) is strictly decreasing in x > , with Q (0) > and Q ( ∞ ) < . This completes the proof.Proposition 2 indicates that following the conventional SP line of argument does indeed giverise to a candidate solution to the Bellman system, even under time inconsistency. We may betempted to claim, as taken for granted in the time-consistent case, that this candidate solutionsolves the Bellman system in Theorem 2 and hence the corresponding stopping rule ˆ u solvesthe equilibrium stoppping problem. Unfortunately, this is not always the case, as shown in thefollowing result. Theorem 3
Assume that α ( r )[ α ( r ) − K − L ( x ∗ ; r )] is increasing in r ∈ supp ( F ) , and let x ∗ bethe unique solution to (19). Then the triplet ( V, w, ˆ u ) solves the Bellman system in Theorem 2 nd in particular ˆ u is an equilibrium stopping rule if and only if f ( x ∗ ) ≥ Z ∞ rdF ( r ) K, (20) and Z ∞ α ( r )[ α ( r ) − K − L ( x ∗ ; r )] dF ( r ) + Z ∞ x ∗ L xx ( x ∗ ; r ) dF ( r ) ≤ . (21)As a proof is lengthy, we defer it to Appendix A.3.The above theorem presents characterizing conditions (on the model primitives) for the SPto work for stopping problems with general WDFs. These conditions are satisfied automaticallyin the classical time-consistent case, but not in the time-inconsistent case in general. We willdemonstrate this with a classical real option problem in the next subsection. In this subsection we consider a special case of the model studied in the previous subsection,which is a time-inconsistent counterpart of the well-studied (time-consistent) problem of realoptions. Such a problem can be used to model, among others, when to start a new project orto abandon an ongoing project; see Dixit (1993) for a systematic account on the classical realoptions theory.The problem is to minimize E (cid:20)Z τ h ( s ) X s ds + h ( τ ) K (cid:12)(cid:12)(cid:12) X = x (cid:21) (22)by choosing τ ∈ T , where X = { X t } t ≥ is governed by Here we assume that the geometric Brownian motion is driftless without loss of generality. X t = σX t dW t . (23)We now apply Theorem 3 to this problem, and see what the equivalent conditions (20) and(21) boil down to.First of all, L ( x ; r ) = E [ Z ∞ e − rt X t dt (cid:12)(cid:12)(cid:12) X = x ] = xr . Hence w ( x ; r ) = (cid:0) K − x ∗ r (cid:1) (cid:16) xx ∗ (cid:17) α ( r ) + xr , x < x ∗ ,K, x ≥ x ∗ ,V ( x ) = R ∞ (cid:0) K − x ∗ r (cid:1) (cid:16) xx ∗ (cid:17) α ( r ) dF ( r ) + R ∞ xr dF ( r ) , x < x ∗ ,K, x ≥ x ∗ , and ˆ u ( x ) = x < x ∗ , otherwise , where α ( r ) = σ + q σ + 2 σ rσ . (24)Moreover, it follows from (19) that x ∗ is the solution to the following equation in y : Z ∞ (cid:18) K − yr (cid:19) α ( r ) dF ( r ) + Z ∞ yr dF ( r ) = 0 . x ∗ = Z ∞ α ( r ) dF ( r ) Z ∞ α ( r ) − r dF ( r ) K. (25)Next, it is easy to verify that α ( r )[ α ( r ) − K − L ( x ∗ ; r )] = σ ( Kr − x ∗ ); hence it is an increasingfunction in r ≥ . Moreover, substituting the explicit representation of x ∗ in (25) into (20) and(21) we find that the latter two inequalities are both identical to the following single inequality Z ∞ α ( r ) dF ( r ) ≥ Z ∞ rdF ( r ) Z ∞ α ( r ) − r dF ( r ) . (26)We have proved the following Proposition 3
The triplet ( V, w, ˆ u ) solves the Bellman system of the real option problem if andonly if (26) holds. Inequality (26) is a critical condition on the model primitives we must verify before we canbe sure that the solution constructed through the SP is indeed an equilibrium solution to thetime-inconsistent real option problem. It is immediate to see that the strict inequality of (26) issatisfied trivially when the distribution function F is degenerate corresponding to the classicaltime-consistent case with an exponential discount function. In this case, x ∗ defined by (25)coincides with the stopping threshold derived in Subsection 3.1. This reconciles with the time-consistent setting.The condition (26) may hold for some non-exponential discount functions. Consider a gener-alized hyperbolic discount function h ( t ) = 1(1 + γt ) βγ ≡ Z ∞ e − rt r βγ − e − rγ γ βγ Γ( βγ ) dr, γ > , β > . We assume that γ < β ≤ σ . Noting that α ( r ) − − + √ σ +2 σ rσ is a concave function in r , we15ave α ( r ) − ≤ ( α ( r ) − ′ | r =0 r + α (0) − σ r. Moreover, it is easy to see that Z ∞ rdF ( r ) = β and α ( r ) ≥ . Therefore, Z ∞ α ( r ) − r dF ( r ) Z ∞ rdF ( r ) ≤ β σ ≤ ≤ Z ∞ α ( r ) dF ( r ) which is (26). So, in this case the SP works and the stopping threshold x ∗ is given by x ∗ = Z ∞ α ( r ) r βγ − e − rγ γ βγ Γ( βγ ) dr Z ∞ α ( r ) − r r βγ − e − rγ γ βγ Γ( βγ ) dr K. However, it is also possible that (26) fails, which is the case even with the simplest classof non-exponential WDFs – the pseudo-exponential discount functions. To see this, let h ( t ) = δe − rt + (1 − δ ) e − ( r + λ ) t , < δ < , r > , λ > . It is straightforward to obtain that Z ∞ α ( r ) dF ( r ) = δα ( r ) + (1 − δ ) α ( r + λ ) and Z ∞ rdF ( r ) Z ∞ α ( r ) − r dF ( r ) > (1 − δ )( r + λ ) δ (cid:18) α ( r ) − r (cid:19) . Since (1 − δ )( r + λ ) δ ( α ( r ) − r ) grows faster than δα ( r ) + (1 − δ ) α ( r + λ ) when λ becomes large, weconclude that (26) is violated when r, δ are fixed and λ is sufficiently large.16hat we have discussed so far shows that the solution constructed through the SP does notsolve the time-inconsistent real option problem whenever inequality (26) fails. A natural questionin this case is whether there might exist equilibrium solutions that cannot be obtained by theSP or even by the Bellman system. The answer is resoundingly negative. Proposition 4
For the real option problem (22)–(23), if (26) does not hold, then no equilibriumstopping rule exists.
Proof.
We prove by contradiction. Suppose ˆ u is an equilibrium stopping rule. We first notethat C ˆ u ≡ { x > u ( x ) = 0 } 6 = (0 , ∞ ) ; otherwise ˆ u ≡ , leading to J ( x ; τ ˆ u ) = R ∞ L ( x ; r ) dF ( r ) = R ∞ xr dF ( r ) , and hence J ( x ; τ ˆ u ) → ∞ as x → ∞ contradicting Lemma 2 in Appendix A.2.Define x ∗ = inf { x : x ∈ ¯ S ˆ u } . It follows from Lemma 3 in Appendix A.2 that x ∗ ∈ (0 , ∞ ) . Astandard argument then leads to J ( x ; τ ˆ u ) = Z ∞ (cid:18) K − x ∗ r (cid:19) (cid:18) xx ∗ (cid:19) α ( r ) dF ( r ) + Z ∞ xr dF ( r ) , x ∈ (0 , x ∗ ] . Because J ( x ; τ ˆ u ) ≤ K and J ( x ∗ ; τ ˆ u ) = K, we have J x ( x ∗ − ; τ ˆ u ) ≥ , i.e., Z ∞ (cid:18) K − x ∗ r (cid:19) α ( r ) 1 x ∗ dF ( r ) + Z ∞ r dF ( r ) ≥ , which in turn gives x ∗ ≤ R ∞ α ( r ) dF ( r ) R ∞ α ( r ) − r dF ( r ) K. Combining with the failure of condition (26), we derive x ∗ < Z ∞ rdF ( r ) , which contradicts Lemma 3. This completes the proof.The above is a stronger result. It suggests that for the problem to have any equilibrium17topping rule at all (not necessarily the one obtainable by the SP principle), condition (26) must hold. So, when it comes to a time-inconsistent stopping problem with non-exponentialdiscounting, it is highly likely that no equilibrium stopping rule exists, even if the SP principledoes generate a “solution”, or even if the time-consistent counterpart (in which everything else isidentical except the discount function) is indeed solvable by the SP. Applying these conclusions tothe pseudo-exponential discount functions discussed above, we deduce that there is no equilibriumstopping when λ is sufficiently large.Having said this, a logical conclusion from Propositions 3 and 4 is that when equilibriado exist, one of them must be a solution generated by the SP. In general if there exists anequilibrium then there may be multiple ones; see, for example, Krussell and Smith (2003) andEkeland and Pirvu (2008) for multiple equilibria in time-inconsistent control problems. In thiscase, the SP can only generate one of them, but not necessarily all of them. (This statement istrue even for a classical time-consistent stopping problem.) So, after all, the SP is still a useful,proper method for time-inconsistent problems; we can simply apply it to generate a candidatesolution. If the solution is an equilibrium (which we must verify), then we have found one (butnot necessarily other equilibria); if it is not an equilibrium, then we know there is no equilibriumat all. While the SP principle has been widely used to study time-inconsistent stopping problems, ourresults indicate the risk of using this principle on such problems. We have shown that the SPprinciple solves the time-inconsistent problem if and only if certain inequalities are satisfied.By a simple model of the classical real option problem, we have found that these inequalitiesmay be violated even for simple and commonly used non-exponential discount functions. Whenthe SP principle fails, we have shown the intra-personal equilibrium does not exist. The nonex-istence result and the failure of the SP principle suggest that it is imperative that the techniquesfor conventional optimal stopping problems be used more carefully when extended to solving18ime-inconsistent stopping problems.
A Appendix: Proofs
A.1 Proof of Theorem 2
For the stopping time τ ǫ,a , if a = 1 , then J ( x ; τ ǫ,a ) = g ( x ) . The Bellman equation (11) impliesthat g ( x ) ≥ V ( x ) ≡ J ( x ; τ ˆ u ) . This yields (9).If a = 0 , then J ( x ; τ ǫ,a ) = E (cid:20)Z ǫ h ( s ) f ( X s ) ds (cid:12)(cid:12)(cid:12) X = x (cid:21) + E (cid:20)Z τ ǫ,a ǫ ( h ( s ) − h ( s − ǫ )) f ( X s ) ds (cid:12)(cid:12)(cid:12) X = x (cid:21) + E [( h ( τ ǫ,a ) − h ( τ ǫ,a − ǫ )) g ( X τ ǫ,a ) | X t = x ] + E [ V ( X ǫ ) | X = x ]= E (cid:20)Z ǫ h ( s ) f ( X s ) ds (cid:12)(cid:12)(cid:12) X = x (cid:21) + E (cid:20)Z τ ǫ,a ǫ Z ∞ e − r ( s − ǫ ) ( e − ǫr − dF ( r ) f ( X s ) ds (cid:12)(cid:12)(cid:12) X = x (cid:21) + E (cid:20)Z ∞ e − r ( τ ǫ,a − ǫ ) ( e − ǫr − dF ( r ) g ( X τ ǫ,a ) (cid:12)(cid:12)(cid:12) X = x (cid:21) + E [ V ( X ǫ ) | X = x ]= E (cid:20)Z ǫ h ( s ) f ( X s ) ds (cid:12)(cid:12)(cid:12) X = x (cid:21) + E (cid:20)Z ∞ ( e − ǫr − w ( X ǫ ; r ) dF ( r ) (cid:12)(cid:12)(cid:12) X = x (cid:21) + E [ V ( X ǫ ) | X = x ] . (27)Define τ n = inf { s ≥ σ ( X s ) V x ( X s ) > n } ∧ ǫ . Then it follows from Ito’s formula that E [ V ( X τ n ) | X = x ] = E (cid:20)Z τ n ( 12 σ ( X s ) V xx ( X s ) + b ( X s ) V x ( X s )) ds (cid:12)(cid:12)(cid:12) X = x (cid:21) + V ( x ) . By (11), we conclude E [ V ( X τ n ) | X = x ] = E (cid:20)Z τ n ( 12 σ ( X s ) V xx ( X s ) + b ( X s ) V x ( X s )) ds (cid:12)(cid:12)(cid:12) X = x (cid:21) + V ( x ) ≥ E (cid:20)Z τ n ( − f ( X s ) + Z ∞ rw ( X s ; r ) dF ( r )) ds (cid:12)(cid:12)(cid:12) X = x (cid:21) + V ( x ) . Note that conditions (6) and (7) ensure that − f ( x ) + R ∞ rw ( x ; r ) dF ( r ) has polynomial growth,19.e., there exist C > , m ≥ such that (cid:12)(cid:12)(cid:12) − f ( x ) + Z ∞ rw ( x ; r ) dF ( r ) (cid:12)(cid:12)(cid:12) ≤ C ( | x | m + 1) , which leads to sup ≤ t ≤ ǫ (cid:12)(cid:12)(cid:12) − f ( X s ) + Z ∞ rw ( X s ; r ) dF ( r ) (cid:12)(cid:12)(cid:12) ≤ C ( sup ≤ t ≤ ǫ | X t | m + 1) . Moreover, under condition (4), it follows from standard SDE theory (see, for example, Chapter of Yong and Zhou (1999)) that equation (3) admits a unique strong solution X satisfying E [ sup ≤ t ≤ ǫ | X t | m | X = x ] ≤ K ǫ ( | x | m + 1) with K ǫ > . Then letting n → ∞ , we conclude by the dominated convergence theorem that E [ V ( X ǫ ) | X = x ] ≥ E (cid:20)Z ǫ ( − f ( X s ) + Z ∞ rw ( X s ; r ) dF ( r )) ds (cid:12)(cid:12)(cid:12) X = x (cid:21) + V ( x ) . Consequently, lim inf ǫ → J ( x ; τ ǫ,a ) − J ( x ; τ ˆ u ) ǫ ≥ lim inf ǫ → E (cid:20)Z ǫ h ( s ) f ( X s ) ds (cid:12)(cid:12)(cid:12) X = x (cid:21) + E (cid:20)Z ∞ ( e − ǫr − w ( X ǫ ; r ) dF ( r ) (cid:12)(cid:12)(cid:12) X = x (cid:21) + lim inf ǫ → ǫ E (cid:20)Z ǫ Z ∞ ( rw ( X t ; r ) dF ( r ) − f ( X t )) dt (cid:12)(cid:12)(cid:12) X = x (cid:21) . The continuity of f and w along with the polynomial growth conditions (6) and (7) allow theuse of the dominated convergence theorem, which yields lim inf ǫ → J ( x ; τ ǫ,a ) − J ( x ; τ ˆ u ) ǫ ≥ . A.2 Proof of Proposition 1
Let V B be the value function of the optimal stopping problem. It follows from the standardargument (see, for example, Chapter of Krylov 2008) that V B is continuously differentiableand its first-order derivative is absolutely continuous. Moreover, V B solves the following Bellmanequation min (cid:26) σ x V Bxx ( x ) + bxV Bx ( x ) + f ( x ) − rV B ( x ) , K − V B ( x ) (cid:27) = 0 . (28)Define the continuation region C B = { x > V B ( x ) < K } and the stopping region S B = { x > V B ( x ) = K } .We claim that S B = (0 , ∞ ) . If not, then V B ≡ K. Thus σ x V Bxx ( x )+ bxV Bx ( x )+ f ( x ) − rV B ( x ) < whenever x ∈ { x > f ( x ) − rK < } . However, since f (0) < rK, the continuity of f implies { x > f ( x ) − rK < } 6 = ∅ . This contradicts the Bellman equation (28).We now show that C b = (0 , ∞ ) . If it is false, then we have V B ( x ) = L ( x ; r ) , with L defined by(15). Since f is increasing and bounded from below by , we have V B ( ∞ ) ≡ lim x →∞ V B ( x ) = Z ∞ Z ∞ lim x →∞ f ( yx ) e − rs G ( y, s ) dyds. The concavity of f yields f ( x ) ≥ xf x ( x ) + f (0) . It then follows from lim x →∞ xf x ( x ) = ∞ that lim x →∞ f ( x ) = ∞ , which yields that V B ( ∞ ) = ∞ . This contradicts the fact that V B ( x ) ≤ K. Next, since X is a geometric Brownian motion and f is increasing, it is clear that V isincreasing too. Now, we derive the value of the triggering boundary, x B , via the SP principle.Specifically, it follows from (28) that V b ( x ) = ( K − L ( x B ; r ))( xx B ) α ( r ) + L ( x ; r ) , x < x B V b ( x ) = K, x ≥ x B , α ( r ) is defined by (17). Then the SP implies that V Bx ( x B ) = 0 which after some calculationsyields that x B is the solution of the equation (16).To prove the unique existence of the solution of (16), define Q ( x ) := α ( r )( K − L ( x ; r ))+ L x ( x ; r ) x. Then Q x ( x ) = ( − α ( r ) + 1) L x ( x ; r ) + L xx ( x ; r ) x . As L is strictly increasing and concave and α ( r ) > ,we deduce that Q is strictly decreasing. It remains to show Q (0) > and Q ( ∞ ) < . It is easy tosee that Q (0) = α ( r )( K − L (0; r )) = α ( r )( K − f (0) r ) > and Q ( x ) = α ( r )( K − L (0; r ) − R x L x ( s ; r ) ds ) + L x ( x ; r ) x. Since L is concave, we have R x L x ( s ; r ) ds ≥ xL x ( x ; r ) . Thus Q ( x ) ≤ α ( r )( K − L (0; r )) +( − α ( r ) + 1) xL x ( x ; r ) . Recalling that lim x →∞ xL x ( x ; r ) = ∞ and α ( r ) > , we have Q ( ∞ ) = −∞ .This completes the proof. A.3 Proof of Theorem 3
We need to present a series of lemmas before giving a proof of Theorem 3.
Lemma 1
Given a stopping rule u and a discount rate r > , the function E ( x ; τ u , r ) := E [ R τ u e − rt f ( X t ) dt + e − rτ u K | X = x ] is continuous in x ∈ (0 , ∞ ) . Proof.
We prove the right continuity of E ( · ; τ u , r ) at a given x > ; the left continuity can bediscussed in the same way.If there exists δ > such that ( x , x + δ ) ∈ S u , then the right continuity of E ( · ; τ u , r ) at x isobtained immediately. If there exists δ > such that ( x , x + δ ) ∈ C u , then it follows from theFeynman-Kac formula that E ( · ; τ u , r ) is the solution to the differential equation σ x E xx + bxE x − rE + f = 0 on ( x , x + δ ) . This in particular implies that E ( · ; τ u , r ) ∈ C (( x , x + δ )) ∩ C ([ x , x + δ ]) due to the regularity of f and the coefficients of the differential equations; hence the rightcontinuity of E ( · ; τ u , r ) at x .Otherwise, we first assume that f ( x ) ≥ rK and consider the set C u ∩ ( x , ∞ ) . Since it is anopen set, we have C u ∩ ( x , ∞ ) = ∪ n ≥ ( a n , b n ) , where a n , b n ∈ ¯ S u , ∀ n ≥ . It is then easy to see that x is an accumulation point of { a n } n ≥ and hence x ∈ ¯ S u .Define I ( x ) := E ( x ; τ u , r ) − K for x ∈ ( a n , b n ) . It is easy to see that I solves the following22ifferential equation σ x I xx ( x ) + bxI x ( x ) − rI ( x ) + f ( x ) − rK = 0 . (29)with the boundary conditions I ( a n ) = I ( b n ) = 0 . Consider an auxiliary function H that solves the following differential equation σ x H xx ( x ) + bxH x ( x ) − rH ( x ) + f ( x ) − rK = 0 , with the boundary conditions H ( x ) = H ( b ) = 0 . Since f ( x ) > rK on ( x , ∞ ) , the comparison principle shows that H ( x ) ≥ , ∀ x ∈ [ x , b ] . Applyingthe comparison principle again on any ( a n , b n ) ∩ ( x , b ) , ∀ n ∈ N + , we have ≤ I ( x ) ≤ H ( x ) . Notingthat H ( x ) → H ( x ) = 0 as x → x + , we conclude that I ( · ) is right continuous at x and so is E ( · ; τ u , r ) .For the case f ( x ) < rK, a similar argument applies. Indeed, consider an auxiliary func-tion H satisfying the differential equation (29) on ( x , f − ( rK )) with the boundary condi-tion H ( x ) = H ( f − ( rK )) = 0 . The comparison principle yields that H ( x ) ≤ I ( x ) ≤ on ( a n , b n ) ∩ ( x , f − ( rK )) , ∀ n ∈ N + . The right continuity of I ( · ) and E ( · ; τ u , r ) then follows immedi-ately. Lemma 2 If ˆ u is an equilibrium stopping rule, then J ( x ; τ ˆ u ) ≤ K ∀ x ∈ (0 , ∞ ) . roof. If there exists x ∈ (0 , ∞ ) such that J ( x ; τ ˆ u ) > K, then we have lim sup ǫ → J ( x ; τ ˆ u ) − J ( x ; τ ǫ, ) ǫ = ∞ , where ˆ u ǫ, is given by (10). This contradicts the definition of an equilibrium stopping rule. Lemma 3 If ˆ u is an equilibrium stopping rule, then { x > f ( x ) < R ∞ rdF ( r ) K } ⊂ C ˆ u . Proof.
Suppose that there exists x ∈ { x > f ( x ) < R ∞ rdF ( r ) K } ∩ ¯ S ˆ u , then it follows fromLemma 2 that E [ J ( X t ; τ ˆ u ) | X = x ] ≤ K. Consider the stopping time τ ǫ, . Equation (27) and thefact that J ( x ; τ ˆ u ) = K give J ( x ; τ ǫ, ) − J ( x ; τ ˆ u ) ǫ ≤ ǫ E (cid:20)Z ǫ h ( s ) f ( X s ) ds (cid:12)(cid:12)(cid:12) X = x (cid:21) + E "Z ∞ e − ǫr − ǫ ! w ( X ǫ ; r ) dF ( r ) (cid:12)(cid:12)(cid:12) X = x . As w ( · , r ) is continuous (Lemma 1) and w ( x ; r ) = K , we have lim inf ǫ → J ( x ; τ ǫ, ) − J ( x ; τ ˆ u ) ǫ ≤ f ( x ) − Z ∞ rKdF ( r ) < . This contradicts the definition of an equilibrium stopping rule.We now turn to the proof of Theorem 3. We begin with the sufficiency. To this end it sufficesto show that V ( x ) ≤ K, x ∈ (0 , x ∗ ) and f ( x ) − R ∞ rdF ( r ) K ≥ , x ∈ ( x ∗ , ∞ ) . We first show that V xx ≤ , x ∈ (0 , x ∗ ) . By simple algebra, we have V xx ( x ) = Z ∞ α ( r )( α ( r ) − K − L ( x ; r ))( xx ∗ ) α ( r ) x dF ( r ) + Z ∞ L xx ( x ; r ) dF ( r ) . As L is concave, we only need to prove R ∞ α ( r )( α ( r ) − K − L ( x ; r ))( xx ∗ ) α ( r ) dF ( r ) ≤ . It is easyto see that ( xx ∗ ) α ( r ) is decreasing in r given that α ( r ) is increasing in r and x < x ∗ . Then the24earrangement inequality (e.g., Chapter of Hardy et al. 1952; Lehmann et al. 1966) yields Z ∞ α ( r )( α ( r ) − K − L ( x ; r ))( xx ∗ ) α ( r ) dF ( r ) ≤ Z ∞ α ( r )( α ( r ) − K − L ( x ; r )) dF ( r ) Z ∞ ( xx ∗ ) α ( r ) dF ( r ) . (30)Therefore it follows from (21) that V xx ( x ) ≤ , x ∈ (0 , x ∗ ) . Now, V x ( x ∗ ) = 0 . Thus V x ( x ) ≥ andconsequently V ( x ) ≤ K ∀ x ∈ (0 , x ∗ ) , due to V ( x ∗ ) = K. Next, the inequality f ( x ) − R ∞ rdF ( r ) K ∀ x ∈ ( x ∗ , ∞ ) follows from f being increasing alongwith inequality (20). This completes the proof of the sufficiency.We now turn to the necessity part. As (20) is an immediate corollary of Lemma 3, we onlyneed to prove (21). Suppose (21) does not hold. Then by a simple calculation, we have V xx ( x ∗ − ) = Z ∞ α ( r )( α ( r ) − K − L ( x ∗ ; r )) 1 x ∗ dF ( r ) + Z ∞ L xx ( x ∗ ; r ) dF ( r ) > . However, V x ( x ∗ ) = 0 , implying that there exists x ∈ (0 , x ∗ ) such that V x ( x ) < on x ∈ ( x , x ∗ ) . Then it follows from V ( x ∗ ) = K that V ( x ) > K when x ∈ ( x , x ∗ ) , which contradicts Lemma 2. References
Bayraktar, E., J. Zhang, and Z. Zhou (2018): “Time consistent stopping for the mean-standard deviation problem—The discrete time case,”
Available at SSRN 3128866 . Bernstein, S. (1928): “Sur les fonctions absolument monotones,”
Acta Mathematica , 52, 1–66.
Bjork, T. and A. Murgoci (2010): “A general theory of Markovian time inconsistent stochas-tic control problems,”
Available at SSRN 1694759 . Inequality (30) can be read as cov(
X, Y ) ≤ , with X = α ( R )( α ( R ) − K − L ( x ; R ))( xx ∗ ) α ( R ) and Y = ( xx ∗ ) α ( R ) , where R is a random variable with distributionfunction F . Because of the monotonicity of X, Y in R , X and Y are anti-comonotonic. Then inequality (30)follows from the fact that the covirance of two anti-comonotonic random variables is non-positive. j¨ork, T., A. Murgoci, and X. Zhou (2014): “Mean-variance portfolio optimization withstate dependent risk aversion,” Mathematical Finance , 24, 1–24.
Christensen, S. and K. Lindensjö (2018b): “On finding equilibrium stopping times for time-inconsistent Markovian problems,”
SIAM Journal on Control and Optimization , 56, 4228–4255.
Christensen, S. and K. Lindensj¨o (2018a): “On time-inconsistent stopping problems andmixed strategy stopping times,”
Available at arXiv 1804.07018 . Dixit, A. K. (1993):
The art of smooth pasting , vol. 2, Routledge, London and New York.
Ebert, S., W. Wei, and X. Zhou (2016): “Weighted discounting–On group diversity, time-inconsistency, and consequences for investment,”
Available at SSRN 2840240 . Ekeland, I. and A. Lazrak (2006): “Being serious about non-commitment: Subgame perfectequilibrium in continuous time,”
Available at arXiv 0604264 . Ekeland, I., O. Mbodji, and T. A. Pirvu (2012): “Time-consistent portfolio management,”
SIAM Journal on Financial Mathematics , 3, 1–32.
Ekeland, I. and T. Pirvu (2008): “Investment and consumption without commitment,”
Mathematics and Financial Economics , 2, 57–86.
Grenadier, S. and N. Wang (2007): “Investment under uncertainty and time inconsistentpreferences,”
Journal of Financial Economics , 84, 2–39.
Hardy, G. H., J. E. Littlewood, and G. P´olya (1952):
Inequalities , Cambridge universitypress, Cambridge.
Harris, C. and D. Laibson (2013): “Instantaneous gratification,”
Quarterly Journal of Eco-nomics , 128, 205–248.
Hsiaw, A. (2013): “Goal-setting and self-control,”
Journal of Economic Theory , 148, 601–626.26 uang, Y.-J. and A. Nguyen-Huu (2018): “Time-consistent stopping under decreasingimpatience,”
Finance and Stochastics , 22, 69–95.
Ito, K. and P. McKean Jr (1965):
Diffusion processes and their sample paths , Springer,Berlin Heidelberg.
Karp, L. (2007): “Non-constant discounting in continuous time,”
Journal of Economic Theory ,132, 557–568.
Krussell, P. and A. Smith (2003): “Consumption-savings decision with quasi-geometricdiscounting,”
Econometrica , 71, 365–375.
Krylov, N. V. (2008):
Controlled diffusion processes , vol. 14, Springer, Berlin Heidelberg.
Laibson, D. (1997): “Golden eggs and hyperbolic discounting,”
Quarterly Journal of Eco-nomics , 112, 443–378.
Lehmann, E. L. et al. (1966): “Some concepts of dependence,”
The Annals of MathematicalStatistics , 37, 1137–1153.
Loewenstein, G. and D. Prelec (1992): “Anomalies in intertemporal choice: Evidence andan interpretation,”
The Quarterly Journal of Economics , 573–597.
Luttmer, E. and T. Mariotti (2003): “Subjective discounting in an exchange economy,”
Journal of Political Economy , 11, 959–989.
O’Donoghue, T. and M. Rabin (2001): “Choice and procrastination,”
Quarterly Journal ofEconomics , 116, 112–160.
Peskir, G. and A. Shiryaev (2006):
Optimal stopping and free-boundary problems ,Birkh¨auser, Basel.
Phelps, E. and R. Pollak (1968): “On second-best national saving and game-equilibriumgrowth,”
Review of Economic Studies , 35, 185–199.27 amuelson, P. (1937): “A note on measurement of utility,”
Review of Economic Studies , 4,155–161.
Strotz, R. (1955): “Myopia and inconsistency in dynamic utility maximization,”
Review ofEconomic Studies , 23, 165–180.
Yong, J. and X. Y. Zhou (1999):