Mortality and Healthcare: a Stochastic Control Analysis under Epstein-Zin Preferences
MMortality and Healthcare: a Stochastic Control Analysis underEpstein-Zin Preferences
Joshua Aurand ∗ Yu-Jui Huang † April 22, 2020
Abstract
This paper studies optimal consumption, investment, and healthcare spending under Epstein-Zin preferences. Given consumption and healthcare spending plans, Epstein-Zin utilities aredefined over an agent’s random lifetime, partially controllable by the agent as healthcare re-duces mortality growth. To the best of our knowledge, this is the first time Epstein-Zin utilitiesare formulated on a controllable random horizon, via an infinite-horizon backward stochas-tic differential equation with superlinear growth. A new comparison result is established forthe uniqueness of associated utility value processes. In a Black-Scholes market, the stochasticcontrol problem is solved through the related Hamilton-Jacobi-Bellman (HJB) equation. Theverification argument features a delicate containment of the growth of the controlled moralityprocess, which is unique to our framework, relying on a combination of probabilistic argumentsand analysis of the HJB equation. In contrast to prior work under time-separable utilities,Epstein-Zin preferences largely facilitate calibration. In four countries we examined, the model-generated mortality closely approximates actual mortality data; moreover, the calibrated efficacyof healthcare is in broad agreement with empirical studies on healthcare across countries.
MSC (2010):
JEL:
G11, I12
Keywords:
Consumption-investment problem, Healthcare, Mortality, Gompertz’ law, Epstein-Zin utilities, Random horizons, Backward stochastic differential equations.
Mortality, the probability that someone alive today dies next year, exhibits an approximate ex-ponential growth with age, as observed by Gompertz [12] in 1825. Despite the steady decline ofmortality at all age groups across different generations, the exponential growth of mortality within each generation has remained remarkably stable, which is called the Gompertz law. Figure 1 dis-plays this clearly: in the US, mortality of the cohort born in 1900 and that of the cohort born in1940 grew exponentially at a similar rate; the latter is essentially shifted down from the former.At the intuitive level, this “shift down” of mortality across generations can be ascribed tocontinuous improvement of healthcare and accumulation of wealth. Understanding precisely how ∗ University of Colorado, Department of Applied Mathematics, Boulder, CO 80309-0526, USA, email: [email protected] . † 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). a r X i v : . [ q -f i n . M F ] A p r
900 cohort1940 cohort
40 45 50 55 60 65 700.10.51510 Age ( years ) M o r t a li t y ( % ) Figure 1:
Mortality rates (vertical axis, in logarithmic scale) at adults’ ages for the cohorts born in 1900and 1940 in the US. The dots are actual mortality data (Source: Berkeley Human Mortality Database), andthe lines are model-implied mortality curves. this “shift down” materializes demands careful modeling in which wealth evolution, healthcarechoices, and the resulting mortality are all endogenous . Standard models of consumption andinvestment do not seem to serve the purpose: the majority, e.g. [36], [24], [25], and [30], considerno more than exogenous mortality, leaving no room for healthcare. Recently, Guasoni and Huang [14] directly modeled the effect of healthcare on mortality: health-care reduces Gompertz’ natural growth rate of mortality, through an efficacy function that char-acterizes the effect of healthcare spending in a society. Healthcare, as a result, indirectly increasesutility from consumption accumulated over a longer lifetime. Under the constant relative riskaversion (CRRA) utility function U ( x ) = x − γ − γ , 0 < γ <
1, an optimal strategy of consumption,investment, and healthcare spending is derived in [14], where the constraint 0 < γ < /γ as an agent’s elasticity of intertemporal substitution (EIS). Specifically, to modelmortality endogenously, we need to be cautious of potential preference for death over life. To avoidthis, [14] assumes that an agent can leave a fraction ζ ∈ (0 , γ >
1. Indeed, with γ >
1, or EIS less than one,the income effect of future loss of wealth at death is so substantial that the agent reduces currentconsumption to zero, leading to the ill-posedness; see below [14, Proposition 3.2] for details.Despite the progress in [14], the artificial relation that EIS is the reciprocal of relative riskaversion , forced by CRRA utility functions, significantly restricts its applications. Although apreliminary calibration was carried out in [14, Section 5], it was not based on the full-fledged modelin [14], but a simplified version without any risky asset. Indeed, once a risky asset is considered, itis unclear whether γ should be calibrated to relative risk aversion or EIS. More crucially, empiricalstudies largely reject relative risk aversion and EIS being reciprocals to each other: it is widelyaccepted that EIS is larger than one (see e.g. [3], [2], [6], and [5]), while numerous estimates ofrelative risk aversion are also larger than one (see e.g. [33], [3], and [16]).In this paper, we investigate optimal consumption, investment, and healthcare spending under As an exception, the literature on health capital, initiated by [13], considers endogenous healthcare. Despite itsdevelopment towards more realistic models, e.g. [10], [9], [37], [17], [15], the Gompertz law remains largely absent. < γ (cid:54) = 1)and EIS (denoted by ψ > ψ > γ > /ψ, (1.1)which implies a preference for early resolution of uncertainty (as explained in [31]), and conformsto empirical estimations mentioned above.Our Epstein-Zin utility process has several distinctive features. First, it is defined on a randomhorizon τ , the death time of an agent. Prior studies on Epstein-Zin utilities focus on a fixed-timehorizon; see e.g. [8], [27], [21], [29], [20], and [35]. To the best our knowledge, random-horizonEpstein-Zin utilities are developed for the first time in Aurand and Huang [1], where the horizonis assumed to be a stopping time adapted to the market filtration. Our studies complement [1], byallowing for a stopping time (i.e. the death time) that need not depend on the financial market.Second, the random horizon τ is controllable : one slows the growth of mortality via healthcarespending, which in turn changes the distribution of τ . Note that a controllable random horizonis rarely discussed in stochastic control, even under time-separable utilities. Third, to formulateour Epstein-Zin utilities, we need not only a given consumption stream c (as in the literature),but also a specified healthcare spending process h . Given the pair ( c, h ), the Epstein-Zin utility isdefined as the right-continuous process (cid:101) V c,h that satisfies a random-horizon dynamics (i.e. (2.6)below), with a jump at time τ . Thanks to techniques of filtration expansion, we decompose (cid:101) V c,h asa function of τ and a process V c,h that solves an infinite-horizon backward stochastic differentialequation (BSDE) under solely the market filtration; see Proposition 2.1. That is, the randomnessfrom death and from the market can be dealt with separately. By deriving a comparison result forthis infinite-horizon BSDE (Proposition 2.2), we are able to uniquely determine the Epstein-Zinutility (cid:101) V c,h for any k -admissible strategy ( c, h ) (Definition 2.3); see Theorem 2.1.In a Black-Scholes financial market, we maximize the time-0 Epstein-Zin utility (cid:101) V c,h over per-missible strategies ( c, π, h ) of consumption, investment, and healthcare spending (Definition 4.2).First, we derive the associated Hamilton-Jacobi-Bellman (HJB) equation, from which a candidateoptimal strategy ( c ∗ , π ∗ , h ∗ ) is deduced. Taking advantage of a scaling property of the HJB equa-tion, we reduce it to a nonlinear ordinary differential equation (ODE), for which a unique classicalsolution exists on strength of the Perron method construction in [14]. This, together with a generalverification theorem (Theorem 3.1), yields the optimality of ( c ∗ , π ∗ , h ∗ ); see Theorem 4.1.Compared with classical Epstein-Zin utility maximization, the additional controlled mortalityprocess M h in our case adds nontrivial complexity. In deriving the comparison result Proposi-tion 2.2, standard Gronwall’s inequality cannot be applied due to the inclusion of M h . As shownin Appendix A.2, a transformation of processes, as well as the use of both forward and backwardGronwall’s inequalities, are required to circumvent this issue. On the other hand, in carrying outverification arguments, we need to contain the growth of M h to ensure that the Epstein-Zin utilityis well-defined. This is done through a combination of probabilistic arguments and analysis of theaforementioned nonlinear ODE; see Appendix A.4 for details.Our model is calibrated to mortality data in the US, the UK, the Netherlands, and Bulgaria.There are three intriguing findings. First, our model-implied mortality closely approximates actualmortality data. Under the simplifying assumptions that the cohort born in 1900 had no healthcareand the cohort born in 1940 had full access to healthcare, we generate an endogenous mortalitycurve for the 1940 cohort. Figure 1 shows that the model-implied mortality (red line) essentiallyreproduces actual data (red dots). Our model performs well for other countries as well; see Figure 3.Second, the calibrated efficacy of healthcare, shown in Figure 2, indicates a ranking among countriesin terms of the effectiveness of healthcare spending: across realistic levels of spending, healthcare is3ore effective in the Netherlands than in the UK, in the UK than in the US, and in the US than inBulgaria. This ranking is in broad agreement with empirical studies on healthcare across countries;see Section 5.3. Third, healthcare spendings in the four countries all increase steadily with age, butdiffer markedly in magnitude; see Figure 4. This, together with the ranking of efficacy in Figure 2,reveals that higher efficacy of healthcare induces lower healthcare spending.Figure 2: Calibrated efficacy of healthcare g ( h ), measured by the reduction in the growth of mortality,given proportions of wealth h spent on healthcare in different countries. The rest of the paper is organized as follows. Section 2 establishes Epstein-Zin utilities overone’s random lifetime, with healthcare spending incorporated. Section 3 introduces the problemof optimal consumption, investment, and healthcare spending under Epstein-Zin preferences, andderives the related HJB equation and a general verification theorem. Section 4 characterizes optimalconsumption, investment, and healthcare spending in three different settings of aging and accessto healthcare. Section 5 calibrates our model to mortality data in four countries, and discussesimportant implications. Most proofs are collected in Appendix A.
Let (Ω , F , P ) be a probability space equipped with a filtration F = ( F t ) t ≥ that satisfies the usualconditions. Consider another probability space (Ω (cid:48) , F (cid:48) , P (cid:48) ) supporting a random variable Z thathas an exponential law P (cid:48) ( Z > z ) = e − z , z ≥ . (2.1)We denote by ( ¯Ω , ¯ F , ¯ P ) the product probability space (Ω × Ω (cid:48) , F × F (cid:48) , P × P (cid:48) ). The expectationstaken under P , P (cid:48) , and ¯ P will be denoted by E , E (cid:48) , and ¯ E , respectively.Consider an agent who obtains utility from consumption, partially determines his lifespanthrough healthcare spending, and has bequest motives to leave his wealth at death to beneficiaries.Specifically, we assume that the mortality rate process M of the agent evolves as dM t = ( β − g ( h t )) M t dt, M = m > , (2.2)where h = ( h t ) t ≥ , a nonnegative F -progressively measurable process, represents the proportionof wealth spent on healthcare at each time t , while g : R + → R + is the efficacy function thatprescribes how much the natural growth rate of mortality β > t . For any ¯ ω = ( ω, ω (cid:48) ) ∈ ¯Ω, the random lifetime of the agent is formulated as τ (¯ ω ) := inf (cid:26) t ≥ (cid:90) t M hs ( ω ) ds ≥ Z ( ω (cid:48) ) (cid:27) . (2.3)The information available to the agent is then defined as G = ( G t ) t ≥ with G t := F t ∨ H t , where H t := σ (cid:0) { τ ≤ u } , u ∈ [0 , t ] (cid:1) . (2.4)That is, at any time t , the agent knows the information contained in F t and whether he is stillalive (i.e. whether τ > t holds); he has no further information of τ , as the random variable Z isinaccessible to him. Finally, we assume that the agent can leave a fraction ζ ∈ (0 , Remark 2.1.
The controlled mortality (2.2) , introduced by Guasoni and Huang [14], assumes thathealthcare expenses affect mortality growth relative to wealth rather than in absolute terms. Whilethis is a modeling simplification, there are empirical and theoretical justifications; see [14, p.319].
Now, let us define a non-standard Epstein-Zin utility process that incorporates healthcare spend-ing. First, recall the Epstein-Zin aggregator f : R + × R → R given by f ( c, v ) := δ (1 − γ ) v − ψ (cid:32)(cid:18) c ((1 − γ ) v ) − γ (cid:19) − ψ − (cid:33) = δ c − ψ − ψ (cid:0) (1 − γ ) v (cid:1) − θ − δθv, with θ := 1 − γ − ψ , (2.5)where γ and ψ represent the agent’s relative risk aversion and EIS, respectively, as stated inSection 1. Given a consumption stream c = ( c t ) t ≥ , assumed to be nonnegative F -progressivelymeasurable, and a healthcare spending process h = ( h t ) t ≥ introduced below (2.2), we define the Epstein-Zin utility on the random horizon τ to be a G -adapted semimartingale ( (cid:101) V c,ht ) t ≥ satisfying (cid:101) V c,ht = ¯ E t (cid:20)(cid:90) T ∧ τt ∧ τ f ( c s , (cid:101) V c,hs ) ds + ζ − γ (cid:101) V c,hτ − { τ ≤ T } + (cid:101) V c,hT { τ>T } (cid:21) , for all 0 ≤ t ≤ T < ∞ , (2.6)where we use the notation ¯ E t [ · ] = ¯ E [ ·|G t ]. In (2.6), we assert that the loss of wealth at death resultsin a decreased bequest utility, by a factor of ζ − γ . This assertion will be made clear and justifiedin Section 4, where a financial model is in place; see Remark 4.4 particularly.Before solving (2.6) for ( (cid:101) V c,ht ) t ≥ , we introduce a general definition of infinite-horizon BSDEs. Definition 2.1.
Let V be an F -progressively measurable process satisfying E [sup s ∈ [0 ,t ] | V s | ] < ∞ forall t ≥ . For any G : Ω × R + × R → R such that (cid:0) G ( · , t, V t ( · )) (cid:1) t ≥ is F -progressively measurable,we say V is a solution to the infinite-horizon BSDE dV t = − G ( ω, t, V t ) dt + d M t , (2.7) if for any T > there exists an F -martingale ( M t ) t ∈ [0 ,T ] such that (2.7) holds for ≤ t ≤ T . Remark 2.2.
Without a terminal condition, (2.7) can have infinitely many solutions. Indeed, aslong as G admits proper monotonicity, there are solutions to (2.7) that satisfy “ lim t →∞ V t = ξ for F -measurable random variable ξ ” or “ lim t →∞ E (cid:2) e ρt V t (cid:3) → for ρ > ”; see [7] and [11]. We willaddress this non-uniqueness issue by enforcing appropriate “terminal behavior”; see Remark 2.5. G -adapted (cid:101) V in (2.6) can be expressed as a function of τ andan F -adapted process V that satisfies an infinite-horizon BSDE. Proposition 2.1.
Let c, h be nonegative F -progressively measurable and (cid:101) V be a G -adapted semi-martingale, with ¯ E [sup s ∈ [0 ,t ] | (cid:101) V s | ] < ∞ for all t ≥ , that satisfies (2.6) . Then, (cid:101) V t = V t { t<τ } + ζ − γ V τ − { t ≥ τ } ∀ t ≥ , (2.8) where V is an F -adapted semimartingale, with E [sup s ∈ [0 ,t ] | V s | ] < ∞ for all t ≥ , that satisfies theinfinite-horizon BSDE dV t = − F ( c t , M ht , V t ) ds + d M t , (2.9) with F : R + × R + × R → R defined by F ( c, m, v ) := f ( c, v ) − (1 − ζ − γ ) mv. (2.10) Proof.
See Section A.1.In view of Proposition 2.1, to uniquely determine the Epstein-Zin utility process (cid:101) V , we need tofind a suitable class of stochastic processes among which there exists a unique solution to (2.9). Tothis end, we start with imposing appropriate integrability and transversality conditions. Definition 2.2.
For any k ∈ R , define Λ := δθ +(1 − θ ) k . Then, for any nonnegative F -progressivelymeasurable h , we denote by E hk the set of all F -adapted semimartingales Y that satisfy the followingintegrability and transversality conditions: E (cid:20) sup s ∈ [0 ,t ] | Y s | (cid:21) < ∞ ∀ t > and lim t →∞ e − Λ t E (cid:20) e − γ ( ψ − − ζ − γ − γ (cid:82) t M hs ds | Y t | (cid:21) = 0 . (2.11) Remark 2.3.
Condition (2.11) is similar to [22, (2.3)], but the controlled mortality M h in our casecomplicates the transversality condition: unlike [22, (2.3)], the exponential term no longer containsa constant rate, but a stochastic one involving M h . This adds nontrivial complexity to deriving acomparison result (Proposition 2.2) and the use of verification arguments (Theorem 4.1). Remark 2.4.
The constant
Λ := δθ + (1 − θ ) k in (2.11) can be negative, even when k > (as willbe assumed in Section 4). In such a case, (2.11) stipulates that M h must increase fast enough toneutralize the growth of e − Λ t , such that the transversality condition can be satisfied. We now introduce the appropriate collection of strategies ( c, h ) we will focus on.
Definition 2.3.
Let c, h be nonnegative F -progressively measurable. For any k ∈ R , we say ( c, h ) is k -admissible if there exists V ∈ E hk satisfying (2.9) and V s ≤ δ θ (cid:18) k + ( ψ −
1) 1 − ζ − γ − γ M hs (cid:19) − θ c − γs − γ , ∀ s ≥ . (2.12) Remark 2.5.
Condition (2.12) is the key to a comparison result for (2.9) , as shown in Proposi-tion 2.2 below. In a sense, (2.11) - (2.12) is the enforced “terminal behavior”, under which a solutionto (2.7) can be uniquely identified. Technically, (2.12) is similar to typical conditions imposed forinfinite-horizon BSDEs, such as [7, (H1’)] and the one in [11, Theorem 5.1]: all of them requirethe solution to be bounded from above by a tractable process. Moreover, for classical Epstein-Zinutilities (without healthcare), a similar condition was imposed in [22, (2.5)]. In fact, Definition 2.3is in line with [22, Definition 2.1], but adapted to include the controlled mortality M h . Proposition 2.2.
Let k ∈ R and c, h be nonnegative F -progressively measurable processes. Supposethat V ∈ E hk is a solution to (2.9) and V ∈ E hk is a solution to (2.7) . If V satisfies (2.12) and F ( c t , M t , V t ) ≤ G ( t, V t ) d P × dt -a.e., then V t ≤ V t for t ≥ P -a.s.Proof. See Section A.2.The next result is a direct consequence of Propositions 2.1 and 2.2.
Theorem 2.1.
Fix k ∈ R . For any k -adimissible ( c, h ) , there exists a unique solution V c,h ∈ E hk to (2.9) that satisfies (2.12) . Hence, the Epstein-Zin utility (cid:101) V c,h can be uniquely determined via (2.8) . Let B = ( B t ) t ≥ be an F -adapted standard Brownian motion. Consider a financial market with ariskfree rate r > S t given by dS t = ( µ + r ) S t dt + σS t dB t , (3.1)where µ ∈ R and σ > x >
0, at each time t ≥
0, anagent consumes a lump-sum c t of his wealth, invests a fraction π t of his wealth on the risky asset,and spends another fraction h t on healthcare. The resulting dynamics of the wealth process X is dX t = X t ( r + µπ t − h t ) dt − c t dt + X t σπ t dB t , X = x. (3.2) Definition 3.1.
For all k ∈ R , let H k be the set of strategies ( c, π, h ) such that ( c, h ) is k -admissible(Definition 2.3), π is F -progressively measurable, and a unique solution X c,π,h to (3.2) exists. The agent aims at maximizing his lifetime Epstein-Zin utility (cid:101) V c,h by choosing ( c, π, h ) in asuitable collection of strategies P , i.e.sup ( c,π,h ) ∈P (cid:101) V c,h = sup ( c,π,h ) ∈P V c,h , (3.3)where the equality follows from (2.8). In this section, we only require P to satisfy P ⊆ H k for some k ∈ R . (3.4)Our focus is to establish a versatile verification theorem under merely (3.4). A more precisedefinition of P , depending on specification of β , γ , and ζ , will be introduced in Definition 4.2. Under the current Markovian setting (i.e. (3.1) and (3.2)), we take v ( x, m ) := sup ( c,π,h ) ∈P V c,h , (3.5)i.e. the optimal value should be a function of the current wealth and mortality. The relation (A.10),derived from (2.6), suggests the following dynamic programming principle: With the shorthandnotation p = ( c, π, h ) and p s = ( c s , π s , h s ) for s ≥
0, for any
T > v ( x, m ) =sup p ∈P E (cid:20)(cid:90) T e − (cid:82) s M hr dr (cid:16) f ( c s , v ( X p s , M hs )) + ζ − γ M hs v ( X p s , M hs ) (cid:17) ds + e − (cid:82) T M s ds v ( X p T , M hT ) (cid:21) . (3.6)7y applying Itˆo’s formula to e − (cid:82) t M hs ds v ( X p t , M ht ), assuming enough regularity of v , we get e − (cid:82) T M hs ds v ( X p T , M hT ) − v ( x, m )= (cid:90) T (cid:16) L p s [ v ]( X p t , M ht ) dt − M ht v ( X p t , M ht ) (cid:17) dt + (cid:90) T e − (cid:82) t M hs ds σπX p t v x ( X p t , M ht ) dB t , where the operator L a,b,d [ · ] is defined by L a,b,d [ κ ]( x, m ) := (( r + µb − d ) x − a ) κ x ( x, m ) + ( β − g ( d )) mκ m ( x, m ) + 12 σ b x κ xx ( x, m ) , (3.7)for any κ ∈ C , ( R + × R + ). We can then rewrite (3.6) as0 = sup p ∈P E (cid:20)(cid:90) T e − (cid:82) s M ht dt (cid:16) f ( c s , v ( X p s , M hs )) ds + ( ζ − γ − M hs v ( X p s , M hs ) + L p s [ v ]( X p s , M hs ) (cid:17) ds (cid:21) . The HJB equation associated with v ( x, m ) is then0 = sup c ∈ R + { f ( c, w ( x, m )) − cw x ( x, m ) } + sup h ∈ R + {− g ( h ) mw m ( x, m ) − hxw x ( x, m ) } + sup π ∈ R (cid:26) µπxw x ( x, m ) + 12 σ π x w xx ( x, m ) (cid:27) (3.8)+ rxw x ( x, m ) + βmw m ( x, m ) + ( ζ − γ − mw ( x, m ) , ∀ ( x, m ) ∈ R . Equivalently, this can be written in the more compact formsup c,h ∈ R + ,π ∈ R (cid:110) L c,π,h [ w ]( x, m ) + f ( c, w ( x, m )) (cid:111) + ( ζ − γ − mw ( x, m ) = 0 , ∀ ( x, m ) ∈ R . (3.9) Theorem 3.1.
Let w ∈ C , ( R + × R + ) be a solution to (3.8) and P satisfy (3.4) . Suppose for any ( c, π, h ) ∈ P , the process w ( X c,π,ht , M ht ) , t ≥ , belongs to E hk (with k ∈ R specified by (3.4) ) and E (cid:20) sup s ∈ [0 ,t ] π s X c,π,hs w x ( X c,π,hs , M hs ) (cid:21) < ∞ , ∀ t > . (3.10) Then, the following holds.(i) w ( x, m ) ≥ v ( x, m ) on R + × R + .(ii) Suppose further that there exist Borel measurable functions ¯ c, ¯ π, ¯ h : R → R such that ¯ c ( x, m ) , ¯ π ( x, m ) , and ¯ h ( x, m ) are maximizers of sup c ∈ R + { f ( c, w ( x, m )) − cw x ( x, m ) } , sup π ∈ R (cid:26) µπxw x ( x, m ) + 12 σ π x w xx ( x, m ) (cid:27) , (3.11)sup h ∈ R + {− g ( h ) mw m ( x, m ) − hxw x ( x, m ) } , (3.12) respectively, for all ( x, m ) ∈ R . If ( c ∗ , π ∗ , h ∗ ) defined by c ∗ t := ¯ c ( X t , M t ) , π ∗ t := ¯ π ( X t , M t ) , h ∗ t := ¯ h ( X t , M t ) , t ≥ , (3.13) belongs to P and W ∗ t := w ( X c ∗ ,π ∗ ,h ∗ t , M h ∗ t ) satisfies (2.12) (with V , c , h replaced by W ∗ , c ∗ , h ∗ ), then ( c ∗ , π ∗ , h ∗ ) optimizes (3.5) and w ( x, m ) = v ( x, m ) on R + × R + . roof. (i) Fix ( x, m ) ∈ R . Consider an arbitrary p = ( c, π, h ) ∈ P . For any T ≥ t ∈ [0 , T ],by applying Itˆo’s formula to w ( X p s , M hs ), we get w ( X p T , M hT ) = w ( X p t , M ht ) + (cid:90) Tt L p s [ w ]( X p s , M hs ) ds + (cid:90) Tt σπ s X p s w x ( X p s , M hs ) dB s , where the operator L a,b,d [ · ] is defined in (3.7). Thanks to (3.10), u (cid:55)→ (cid:82) ut σπ s X p s w x ( X p s , M hs ) dB s is a true martingale. Hence, the above equality shows that W s := w ( X p s , M hs ) is a solution toBSDE (2.7), with G ( ω, s, v ) := − L p s ( ω ) [ w ]( X p s ( ω ) , M hs ( ω )). On the other hand, (3.4) implies that( c, h ) is k -admissible, so that there exists a unique solution V c,h ∈ E hk to (2.9) that satisfies (2.12)(Theorem 2.1). Since w is a solution to (3.8), and equivalently to (3.9), we have F ( c s , M hs , W s ) = f ( c s , W s ) + ( ζ − γ − M hs W s ≤ − L p s [ w ]( X p s , M hs ) . (3.14)We then conclude from Proposition 2.2 that W t ≥ V c,ht for all t ≥
0. In particular, w ( x, m ) = W ≥ V c,h . By the arbitrariness of ( c, π, h ) ∈ P , w ( x, m ) ≥ sup ( c,π,h ) ∈P V c,h = v ( x, m ), as desired.(ii) Fix ( x, m ) ∈ R . If ( c ∗ , π ∗ , h ∗ ) ∈ P , we can repeat the arguments in part (a), obtaining(3.14) with the inequality replaced by equality. This shows that W ∗ t = w ( X c ∗ ,π ∗ ,h ∗ t , M h ∗ t ) ∈ E h ∗ k is a solution to (2.9). Also, (3.4) implies that ( c ∗ , h ∗ ) is k -admissible, so that there is a uniquesolution V c ∗ ,h ∗ ∈ E h ∗ k to (2.9) satisfying (2.12) (Theorem 2.1). As W ∗ also satisfies (2.12), we have W ∗ t = V c ∗ ,h ∗ t for all t ≥
0; particularly, w ( x, m ) = W ∗ = V c ∗ ,h ∗ . With w ( x, m ) ≥ sup ( c,π,h ) ∈P V c,h = v ( x, m ) in part (a), we conclude w ( x, m ) = v ( x, m ) and ( c ∗ , π ∗ , h ∗ ) ∈ P is an optimal control. If we assume heuristically that w xx < w m < g is differentiable, and the inverse of g (cid:48) iswell-defined, then the optimizers stated in Theorem 3.1 (ii) can be uniquely determined as¯ c ( x, m ) = δ ψ [(1 − γ ) w ( x, m )] ψ (1 − θ ) w x ( x, m ) ψ , ¯ π ( x, m ) = − µσ w x ( x, m ) xw xx ( x, m ) , ¯ h ( x, m ) = ( g (cid:48) ) − (cid:18) − xw x ( x, m ) mw m ( x, m ) (cid:19) . (3.15)Plugging these into (3.8) yields0 = δ ψ ψ − − γ ) v ( x, m )] ψ (1 − θ ) v x ( x, m ) ψ − − δθv ( x, m ) − (cid:16) µσ (cid:17) v x ( x, m ) v xx ( x, m ) + rxv x ( x, m ) + βmv m ( x, m )+ ( ζ − γ − mv ( x, m ) − mv m ( x, m ) sup h ∈ R + (cid:26) g ( h ) + hxv x ( x, m ) mv m ( x, m ) (cid:27) . (3.16)Using the ansatz w ( x, m ) = δ θ x − γ − γ u ( m ) − θψ , the above equation reduces to0 = u ( m ) − ˜ c ( m ) u ( m ) − βmu (cid:48) ( m ) + mu (cid:48) ( m ) sup h ∈ R + (cid:26) g ( h ) − ( ψ − u ( m ) mu (cid:48) ( m ) h (cid:27) , m > , (3.17)where ˜ c ( m ) := ψδ + (1 − ψ ) (cid:18) ( ζ − γ − m − γ + r + 12 γ (cid:16) µσ (cid:17) (cid:19) . (3.18)9oreover, the maximizers in (3.15) now become¯ c ( x, m ) = xu ( m ) , ¯ π ≡ µγσ , ¯ h ( m ) = ( g (cid:48) ) − (cid:18) ( ψ − u ( m ) mu (cid:48) ( m ) (cid:19) . (3.19)These maximizers indeed characterize optimal consumption, investment, and healthcare spending,as will be shown in the next section. Let us now formulate the set P of permissible strategies ( c, π, h ) in the optimization problem (3.3).First, take k ∈ R in Definition 2.2 to be k ∗ := δψ + (1 − ψ ) (cid:18) r + 12 γ (cid:16) µσ (cid:17) (cid:19) , (4.1)so that Λ ∈ R in Definition 2.2 becomesΛ ∗ := δθ + (1 − θ ) k ∗ = δγψ + (1 − γψ ) (cid:18) r + 12 γ (cid:16) µσ (cid:17) (cid:19) . (4.2) Definition 4.1.
Let P the set of strategies ( c, π, h ) such that ( c, π, h ) ∈ H k ∗ , ( X c,π,h ) − γ satisfies (2.11) (with Λ ∈ R therein taken to be Λ ∗ ) as well as E (cid:2) sup s ∈ [0 ,t ] π s ( X c,π,hs ) − γ (cid:3) < ∞ for t ≥ . Let P be defined as P , except that the second part of (2.11) is replaced by lim t →∞ e − Λ ∗ t E (cid:20) e − ηγ ( ψ − − ζ − γ − γ (cid:82) t M hs ds ( X c,π,ht ) − γ (cid:21) = 0 , for some η ∈ (1 − γ , . (4.3) Definition 4.2.
The set of permissible strategies ( c, π, h ) , denoted by P , is defined as follows.(i) For the case β = 0 and g ≡ (i.e. with neither aging nor healthcare), P := P ;(ii) For the case β > (i.e. with aging), P := (cid:40) P , if γ ∈ (cid:0) ψ , (cid:1) or ζ = 1 , P , if γ > and ζ ∈ (0 , , Remark 4.1.
When there is aging ( β > ), for the case γ > and ζ ∈ (0 , , we need ( X c,π,h ) − γ to satisfy the slightly stronger condition (4.3) (than the transversality condition in (2.11) ), so thatthe general verification Theorem 3.1 can be applied; see Appendix A.4 for details. The rest of the section presents main results in three different settings of aging and access tohealthcare, in order of complexity.
When the natural growth rate of mortality is zero ( β = 0) and healthcare is unavailable ( g ≡ M t ≡ m . Consequently, in the HJB equation (3.8), all derivativesin m should vanish; also, as v ( x, m ) is nondecreasing in x by definition, the second supremum in(3.8) should be zero. Corresponding to this largely simplified HJB equation, (3.17) reduces to0 = u ( m ) − ˜ c ( m ) u ( m ) , which directly implies u ( m ) = ˜ c ( m ). The problem (3.5) can then be solved explicitly.10 roposition 4.1. Assume β = 0 and g ≡ . For any m ≥ , if ˜ c ( m ) > in (3.18) , then v ( x, m ) = δ θ x − γ − γ ˜ c ( m ) − θψ for x > .Furthermore, c ∗ t := ˜ c ( m ) X t , π ∗ t := µγσ , and h ∗ t := 0 , for t ≥ , form an optimal control for (3.5) .Proof. See Section A.3.Proposition 4.1 shows that without aging and healthcare, optimal investment follows classicalMerton’s proportion, while the optimal consumption rate is the constant ˜ c ( m ), dictated by thefixed mortality m . By (3.18), for the case ζ = 1, ˜ c ( m ) ≡ ψδ + (1 − ψ ) (cid:0) r + γ (cid:0) µσ (cid:1) (cid:1) no longerdepends on m . Indeed, with no loss of wealth (and thus utility) at death, dying sooner or laterdoes not make a difference to one who maximizes lifetime utility plus bequest utility.As ζ − γ − − γ < < γ (cid:54) = 1, we observe from (3.18) that a larger mortality rate m inducesa larger consumption rate due to EIS ψ >
1. This can be explained by the usual substitution effectin response to negative wealth shocks: a larger mortality rate means more pressing loss of wealthat death, encouraging the agent to consume more (i.e. consumption substitutes for saving).
When the natural growth of mortality is positive ( β >
0) but healthcare is unavailable ( g ≡ M t = me βt . As g ≡ v ( x, m ) is nondecreasing in x bydefinition, the second supremum in (3.8) vanishes. It follows that (3.17) reduces to0 = u ( m ) − ˜ c ( m ) u ( m ) − βmu (cid:48) ( m ) , m > . (4.4)This type of differential equations can be solved explicitly. Lemma 4.1.
Fix q > , and define the function u q : R + → R + by u q ( m ) := (cid:18) q (cid:90) ∞ e ψ − q (1 − γ ) ( ζ − γ − my ( y + 1) − (cid:16) k ∗ q (cid:17) dy (cid:19) − . (4.5) If k ∗ > in (4.1) , then u q is the unique solution to the ordinary differential equation u ( m ) − ˜ c ( m ) u ( m ) − qmu (cid:48) ( m ) , ∀ m > , (4.6) such that lim q → u q ( m ) = ˜ c ( m ) . Moreover, u q satisfies u q (0) = ˜ c (0) = k ∗ > , lim m →∞ [ u q ( m ) − (˜ c ( m ) + q )] = 0 , ˜ c ( m ) < u q ( m ) < ˜ c ( m ) + q, ∀ m > . (4.7) Proof.
Similarly to (A.8) in [14], (4.6) admits the general solution u ( m ) = qe ψθq ( ζ − γ − m (cid:18) Cβm kβ + (cid:90) ∞ e ψθq ( ζ − γ − mv v − (1+ kq ) dv (cid:19) − , with C ∈ R . To ensure lim q → u ( m ) = ˜ c ( m ), we need C = 0, which identifies the corresponding solution as u q ( m ) = qe ψθq ( ζ − γ − m (cid:18)(cid:90) ∞ e ψθq ( ζ − γ − mv v − (1+ kq ) dv (cid:19) − . A straightforward change of variable then gives the formula (4.5). Now, replacing the positiveconstants δ +( γ − rγ , β , and − ζ − γ γ in [14, Lemma A.1] by k ∗ , q , and − ψ − − γ ( ζ − γ −
1) in our setting,we immediately obtain the remaining assertions.11 roposition 4.2.
Assume β > and g ≡ . If k ∗ > in (4.1) , then v ( x, m ) = δ θ x − γ − γ u β ( m ) − θψ , ( x, m ) ∈ R , where u β : R + → R + is defined as in (4.5) , with q = β . Furthermore, c ∗ t := u β ( me βt ) X t , π ∗ t := µγσ ,and h ∗ t := 0 , for t ≥ , form an optimal control for (3.5) .Proof. See Section A.5.Observe from (3.18) and (4.1) that˜ c ( m ) = k ∗ + ( ψ −
1) (1 − ζ − γ ) m − γ . (4.8)As ψ > − ζ − γ − γ > < γ (cid:54) = 1, the condition k ∗ > c ( m ) > m > u β > ˜ c ((4.7) with q = β ), shows that k ∗ > u β ( me βt ) is strictly positivefor all t ≥
0. Moreover, with q = β , (4.7) stipulates that aging enlarges consumption rate, but theincrease does not exceed the growth of aging β >
0; note that the increase in consumption resultsfrom the same substitution effect as discussed below Proposition 4.1.
For the general case where the natural growth of mortality is positive ( β >
0) and healthcare isavailable ( g (cid:54)≡ Assumption 1.
Let g : R + → R + be twice differentiable with g (0) = 0 , g (cid:48) ( h ) > and g (cid:48)(cid:48) ( h ) < for h > , and satisfies the Inada condition g (cid:48) (0+) = ∞ and g (cid:48) ( ∞ ) = 0 , (4.9) as well as g ( I ( ψ − < β with I := ( g (cid:48) ) − . (4.10)Condition (4.10) was first introduced in [14]. Its purpose will be made clear after the optimalhealthcare spending strategy h ∗ is introduced in Theorem 4.1; see Remark 4.3. Lemma 4.2.
Suppose Assumption 1 holds. If k ∗ > in (4.1) , there exists a unique nonnegative,strictly increasing, strictly concave, classical solution u ∗ : R + → R + to (3.17) . Furthermore, define β := β − sup h ≥ { g ( h ) − ( ψ − h } ∈ (0 , β ) . Then, lim m →∞ (cid:2) u ∗ ( m ) − (˜ c ( m ) + β ) (cid:3) = 0 and u β ( m ) ≤ u ∗ ( m ) ≤ min { u β ( m ) , ˜ c ( m ) + β } ∀ m > . (4.11) Proof.
By replacing positive constants − γγ , δ +(1 − γ ) rγ , and − ζ − γ γ in [14, Appendix A.3] (particularlyTheorems 3.1 and 3.2) by ψ − k ∗ , and − ψ − − γ ( ζ − γ −
1) in our setting, we get the desired results.
Remark 4.2.
The tractable lower and upper bounds for u ∗ in (4.11) will play a crucial role inverification arguments in the proof of Theorem 4.1 below, as well as calibration in Section 5. heorem 4.1. Suppose Assumption 1 holds. If k ∗ > in (4.1) , then v ( x, m ) = δ θ x − γ − γ u ∗ ( m ) − θψ , ( x, m ) ∈ R , (4.12) where u ∗ : R + → R + is the unique nonnegative, strictly increasing, strictly concave, classicalsolution to (3.17) . Furthermore, ( c ∗ , π ∗ , h ∗ ) defined by c ∗ t := u ∗ ( M t ) X t , π ∗ t := µγσ , h ∗ t := ( g (cid:48) ) − (cid:18) ( ψ − u ∗ ( M t ) M t ( u ∗ ) (cid:48) ( M t ) (cid:19) , t ≥ is an optimal control for (3.5) .Proof. See Section A.4.Theorem 4.1 identifies the marginal efficacy of optimal healthcare spending, g (cid:48) ( h ∗ t ), to be in-versely proportional to m ( u ∗ ) (cid:48) ( m ) u ∗ ( m ) , the elasticity of consumption with respect to mortality, wherethe constant of proportionality depends on EIS ψ . Note that a larger EIS implies less healthcarespending, as ( g (cid:48) ) − is strictly decreasing. In a sense, healthcare spending is like saving: it crowdsout current consumption, but potentially enlarges future consumption by extending one’s lifetime.Since a larger EIS means a stronger substitution effect (as discussed below Proposition 4.1), onesubstitutes more consumption for saving-like healthcare spending with a larger ψ . Remark 4.3.
As the same argument in [14, Lemma A.2] implies u ∗ ( m ) m ( u ∗ ( m )) (cid:48) ≥ for m > , g ( h ∗ t ) = g (cid:18) I (cid:18) ( ψ − u ∗ ( M t ) M t ( u ∗ ) (cid:48) ( M t ) (cid:19)(cid:19) ≤ g ( I ( ψ − < β, (4.13) where the last inequality is due to (4.10) . In other words, (4.10) stipulates that optimizing healthcarespending can only reduce, but not reverse, the growth of mortality. Remark 4.4.
Since the transferred wealth at death is ζX c ∗ ,π ∗ ,h ∗ τ − , (4.12) indicates that δ θ ( ζX c ∗ ,π ∗ ,h ∗ τ − ) − γ − γ u ∗ ( M h ∗ τ − ) − θψ = ζ − γ v ( X c ∗ ,π ∗ ,h ∗ τ − , M h ∗ τ − ) , i.e. the loss of wealth at death reduces utility by a factor of ζ − γ , confirming the setup in (2.6) . Remark 4.5.
For the case ψ = 1 /γ > , Propositions 4.1, 4.2 and Theorem 4.1 reduce to resultsin [14] under time-separable utilities; see Propositions 3.1, 3.2, and Theorems 3.4, 4.1 therein. In this section, we calibrate the model in Section 4.3 to actual mortality data. We take as given r = 1%, δ = 3%, ψ = 1 . γ = 2, ζ = 50%, µ = 5 . σ = 15 . r = 1%approximates the long-term average real rate on Treasury bills in [4], and the time preference δ = 3% is also consistent with estimates therein; ψ = 1 . γ = 2 follows thespecification in [20] and [35]; µ = 5 .
2% and σ = 15 .
4% are taken from the long-term study [18]; ζ = 50% is a rough estimate of inheritance and estate taxes in developed countries. These valuesensure k ∗ > g : R + → R + to be g ( z ) = a · ( z q /q ) , with a > q ∈ (0 , . (5.1)13he equation (3.17) then becomes u ( m ) − ˜ c ( m ) u ( m ) − βmu (cid:48) ( m ) + ((1 − q ) /q ) a − q (( ψ − u ( m )) − q − q ( mu (cid:48) ( m )) − q = 0 , (5.2)and the optimal healthcare spending process is now h ∗ t = (cid:0) a − ( ψ − u ∗ ( M t ) M t ( u ∗ ) (cid:48) ( M t ) (cid:1) − − q , where u ∗ isthe unique solution to (5.2). The endogenous mortality is then dM t = M t (cid:32) β − q a − q (cid:18) ( ψ − u ∗ ( M t ) M t ( u ∗ ) (cid:48) ( M t ) (cid:19) − q − q (cid:33) dt, M = m > . (5.3)We calibrate β > a > q ∈ (0 , m > β > β >
0, healthcare parameters a > q ∈ (0 ,
1) in (5.1), as well as initial mortality m >
0, are calibrated by matching the endogenous mortality curve (5.3) with mortality data forthe cohort born in 1940, through minimizing the mean squared error (MSE). Essentially, we workunder the assumption that the 1900 cohort had no access to healthcare (whence its mortality grewexponentially with the Gompertz law) and the 1940 cohort had full access to healthcare. This is acrude simplification, but conforms to several realistic constraints; see [14, Section 5.2].
Table 1
Calibration ResultsCountry β (%) m × a q Model MSE × MSE × United States (US) 7.24069 1.34995 0.19 0.61 0.0436896 0.128984United Kingdom (UK) 7.79605 0.843827 0.19 0.60 0.0249924 0.12755Netherlands (NL) * ** * Mortality rates impacted during WWII were excluded when calculating β . ** Incomplete data for the 1900 cohort. β estimated from age range 47-77. Our calibration exploits the bounds in (4.11) to approximate the solution u ∗ to (5.2), insteadof solving (5.2) directly. Solving (5.2) is nontrivial: as the initial condition u (0) = 0 gives multiplesolutions, one needs Neumann boundary conditions u (cid:48) (0) = ∞ and u (cid:48) ( ∞ ) = 0, and solving (5.2)via sequential approximations. This is computationally taxing even for a fixed pair of ( a, q ). Asthe calibration needs to explore numerous possibilities of ( a, q ), we did not follow this approach. In Figure 1, the blue line is obtained by linearly regressing mortality data of the 1900 cohort (bluedots), while the red line is the model-implied mortality curve calibrated to mortality data of the1940 cohort (red dots). Clearly, our model reproduces declines in mortality that are very close toones observed historically. When compared with [14, Figure 5.2], Figure 1 provides a much betterfit. This improvement can be attributed to the use of Epstein-Zin utilities (so that γ and ψ can both take empirically relevant values), the inclusion of risky assets, and modifications of calibrationmethods. Figure 3 shows that our model performs well for other countries as well.We also compare our model performance with linear regression. Indeed, without any idea ofhealthcare, one can model mortality data of the 1940 cohort by linear regression (as we did for the1900 cohort). Our model outperforms linear regression: the sixth column of Table 1 reports MSEsunder our model, significantly smaller than those under linear regression in the seventh column.14
900 cohortwithouthealthcare1940 cohortwithhealthcare
40 45 50 55 60 65 700.10.51510 Age ( years ) M o r t a li t y ( % ) (a) UK
40 45 50 55 60 65 700.10.51510 Age ( years ) M o r t a li t y ( % ) (b) Netherlands
40 45 50 55 60 65 700.10.51510 Age ( years ) M o r t a li t y ( % ) (c) Bulgaria Figure 3:
Mortality rates (vertical axis, in logarithmic scale) at adults’ ages for the cohorts born in 1900 and1940 in three countries. The dots are actual mortality data (Source: Berkeley Human Mortality Database),and the lines are model-implied mortality curves.
Figure 4 displays the model-implied optimal healthcare spending in the four countries. The leftpanel reveals that the proportion of wealth spent on healthcare is negligible at age 40, but increasesquickly to 0.5-1% at age 80. The right panel further shows that healthcare spending increases withage much faster than consumption and investment combined: it accounts for less than 5% of totalspending at age 40, but increases continuously to 13-30% at age 80.For the US, UK, and Netherlands, healthcare-spending ratios reported above are in broad agree-ment with actual healthcare expenditure as a percentage of GDP, as shown in Figure 5. Bulgariais distinctively different: model-implied healthcare-spending ratios largely outsize its healthcareexpenditure as a percentage of GDP at 8.4%. This may indicate that Bulgaria’s healthcare expen-diture is less than optimal, while a detailed empirical investigation is certainly needed here. g Figure 2 presents calibrated efficacy functions g ( h ) = a h q q for the four countries. Intriguingly, itindicates a ranking among them in term of the effectiveness of healthcare spending: across realisticlevels of spending (0-30% of wealth), healthcare is more effective (in reducing mortality growth) inthe Netherlands than in the UK, in the UK than in the US, and in the US than in Bulgaria.Along with healthcare spending illustrated in Figure 4, this ranking of efficacy reveals that lowerefficacy of healthcare is compensated by larger healthcare spending, relative to total wealth and15igure 4: Optimal healthcare spending in the US, UK, Netherlands (NL), and Bulgaria (BG). Left panel:Healthcare-wealth ratio (vertical, log-scale) at adult ages (horizontal). Right panel: Healthcare as a fractionof total spending in consumption, investment, and healthcare (vertical) at adult ages (horizontal).
Bulgaria NetherlandsUK US (% GDP ) L i f e E x pe c t an cy ( Y ea r s ) Figure 5:
Life expectancy v.s. healthcare spending as a percentage of GDP (2017) for countries in OECDand European Union (Source: OECD Health Statistics Database and [23]). total spending. In other words, in the face of enhanced efficacy, our model stipulates less healthcarespending, instead of more to exploit the reduced marginal cost to curtail mortality growth.In addition, our model-implied ranking of efficacy is in broad agreement with empirical studies.Figure 5 displays life expectancy versus healthcare spending as a percentage of GDP for numerouscountries, and the black line represents average effectiveness of healthcare. The Netherlands isfurther away above average than the UK, while the US and Bulgaria are two outliers below average;this generally agrees with the ranking in Figure 2. Certainly, there are more comprehensive,multifaceted measures of healthcare. Tandon et al. [32], rated by [28] as the most reproducible andtransparent ranking of healthcare systems, studied 191 countries based on quality of care, access tocare, efficiency, equity, and healthiness of citizens. The Netherlands, the UK, the US, and Bulgariaranked number 17, 18, 37, and 102, respectively, again in line with the ranking in Figure 2.
A Proofs
A.1 Proof of Proposition 2.1
In view of (2.3) and (2.1), for any 0 ≤ t ≤ s , it holds for ¯ P -a.e. ¯ ω = ( ω, ω (cid:48) ) ∈ ¯Ω that¯ P ( τ > (cid:96) | F s ∨ H t )(¯ ω ) = e − (cid:82) (cid:96)t M hu ( ω ) du { τ>t } (¯ ω ) , ∀ t ≤ (cid:96) ≤ s. (A.1)16lso, since (cid:101) V is a G -adapted semimartingale, it follows from (2.4) that there exists an F -adaptedsemimartingale V such that (cid:101) V t = V t ¯ P -a.s. on { t < τ } , ∀ t ≥ . (A.2)Indeed, for any fixed ω ∈ Ω, consider A t ( ω ) := { ω (cid:48) ∈ Ω (cid:48) : t < τ ( ω, ω (cid:48) ) } for all t ≥
0. As (cid:101) V is G -adapted, (2.4) implies (cid:101) V t ( ω, ω (cid:48) ) is constant P (cid:48) -a.s. on A t ( ω ). By defining V t ( ω ) = (cid:101) V t ( ω, A t ( ω )) forall t ≥ V is an F -adapted semimartingale satisfying (A.2). Also note that E [sup s ∈ [0 ,t ] | V s | ] < ∞ ,as ¯ E [sup s ∈ [0 ,t ] | (cid:101) V s | ] < ∞ , for all t ≥
0. Now, observe that¯ E (cid:20)(cid:90) T ∧ τt ∧ τ f ( c s , (cid:101) V c,hs ) ds (cid:12)(cid:12)(cid:12)(cid:12) G t (cid:21) = ¯ E (cid:20)(cid:90) Tt { s<τ } f ( c s , (cid:101) V c,hs ) ds (cid:12)(cid:12)(cid:12)(cid:12) F t ∨ H t (cid:21) = (cid:90) Tt ¯ E (cid:104) { s<τ } f ( c s , (cid:101) V c,hs ) (cid:12)(cid:12)(cid:12) F t ∨ H t (cid:105) ds = (cid:90) Tt ¯ E (cid:104) ¯ E (cid:2) { s<τ } f ( c s , V c,hs ) | F s ∨ H t (cid:3) (cid:12)(cid:12)(cid:12) F t ∨ H t (cid:105) ds = (cid:90) Tt ¯ E (cid:104) f ( c s , V c,hs ) ¯ E (cid:2) { s<τ } | F s ∨ H t (cid:3) (cid:12)(cid:12)(cid:12) F t ∨ H t (cid:105) ds = (cid:90) Tt ¯ E (cid:104) f ( c s , V c,hs ) { t<τ } e − (cid:82) st M hu du (cid:12)(cid:12)(cid:12) F t ∨ H t (cid:105) ds = ¯ E (cid:20)(cid:90) Tt { t<τ } e − (cid:82) st M hu du f ( c s , V c,hs ) ds (cid:12)(cid:12)(cid:12)(cid:12) G t (cid:21) , (A.3)where the second and last equalities follow from Fubini’s theorem for conditional expectations (see[26, Theorem 27.17]), the third equality is due to the tower property of conditional expectations and(A.2), the fourth equality results from c s ∈ F s and V c,hs ∈ F s , and the fifth equality holds thanksto (A.1). Next, for ¯ P -a.e. fixed ¯ ω = ( ω, ω (cid:48) ) ∈ ¯Ω, consider the cumulative distribution function of τ given the information F T ∨ H t , i.e. F ( s ) := ¯ P ( τ ≤ s | F T ∨ H t )(¯ ω ) , s ≥ . Thanks to (A.1), F ( s ) = 1 − e − (cid:82) st M hu ( ω ) du { τ>t } (¯ ω ) for t ≤ s ≤ T . This implies η ( s ) := F (cid:48) ( s ) = M hs ( ω ) e − (cid:82) st M hu ( ω ) du { τ>t } (¯ ω ) , for t ≤ s ≤ T, (A.4)which is the density function of τ given the information F T ∨ H t . It follows that¯ E (cid:104) (cid:101) V c,hτ − { τ ≤ T } (cid:12)(cid:12)(cid:12) G t (cid:105) = ¯ E (cid:104) V c,hτ − { τ ≤ T } (cid:12)(cid:12)(cid:12) G t (cid:105) { τ ≤ t } + ¯ E (cid:104) V c,hτ − { τ ≤ T } (cid:12)(cid:12)(cid:12) G t (cid:105) { τ>t } = V c,hτ − { τ ≤ t } + ¯ E (cid:104) ¯ E (cid:2) V c,hτ − { t<τ ≤ T } | F T ∨ H t (cid:3) (cid:12)(cid:12)(cid:12) F t ∨ H t (cid:105) = V c,hτ − { τ ≤ t } + ¯ E (cid:20)(cid:90) Tt { t<τ } M hs e − (cid:82) st M hu du V c,hs ds (cid:12)(cid:12)(cid:12)(cid:12) G t (cid:21) , (A.5)where the first line results from (cid:101) V τ − = V τ − (by (A.2)), the second line follows from the towerproperty of conditional expectations, and the third line is due to the density formula (A.4). Since V is right-continuous, it has at most countably many jumps on [ t, T ], so that we may use V s (insteadof V s − ) in the last term of (A.5). Finally,¯ E (cid:104) (cid:101) V c,hT { τ>T } (cid:12)(cid:12)(cid:12) G t (cid:105) = ¯ E (cid:104) ¯ E (cid:2) V c,hT { τ>T } | F T ∨ H t (cid:3) (cid:12)(cid:12)(cid:12) F t ∨ H t (cid:105) = ¯ E (cid:104) V c,hT ¯ E (cid:2) { τ>T } | F T ∨ H t (cid:3) (cid:12)(cid:12)(cid:12) F t ∨ H t (cid:105) = ¯ E (cid:104) { t<τ } e − (cid:82) Tt M hu du V c,hT (cid:12)(cid:12)(cid:12) G t (cid:105) , (A.6)17here the first equality follows from the tower property of conditional expectations and (A.2),the second equality is due to V T ∈ F T , and the third equality is a consequence of (A.1). Now,combining (A.3), (A.5), and (A.6), we obtain from (2.6) and (cid:101) V τ − = V τ − that (cid:101) V c,ht = E t (cid:20) (cid:90) Tt e − (cid:82) st M hr dr (cid:16) f ( c s , V c,hs ) + ζ − γ M hs V c,hs (cid:17) ds + e − (cid:82) Tt M hs ds V c,hT (cid:21) { t<τ } + ζ − γ V c,hτ − { t ≥ τ } , for all 0 ≤ t ≤ T < ∞ , (A.7)where we use the notation E t [ · ] = E [ ·|F t ]. This, together with (A.2), particularly implies V t ( ω ) { t<τ } ( ω,ω (cid:48) ) = (cid:101) V t ( ω, ω (cid:48) ) { t<τ } ( ω,ω (cid:48) ) = E t,T ( ω ) { t<τ } ( ω,ω (cid:48) ) , (A.8)where E t,T ( ω ) := E t (cid:20) (cid:90) Tt e − (cid:82) st M hr dr (cid:16) f ( c s , V c,hs ) + ζ − γ M hs V c,hs (cid:17) ds + e − (cid:82) Tt M hs ds V c,hT (cid:21) ( ω ) . For any ω ∈ Ω, since there exists ω (cid:48) ∈ Ω (cid:48) such that { t<τ } ( ω,ω (cid:48) ) = 1 (in view of (2.3) and (2.1)), weconclude from (A.8) that V t ( ω ) = E t,T ( ω ). We can then simplify (A.7) as (cid:101) V t = V t { t<τ } + ζ − γ V τ − { t ≥ τ } , (A.9)where V satisfies V t = E t (cid:20) (cid:90) Tt e − (cid:82) st M hr dr (cid:16) f ( c s , V s ) + ζ − γ M hs V s (cid:17) ds + e − (cid:82) Tt M hs ds V T (cid:21) , ∀ ≤ t ≤ T < ∞ (A.10)Now, note that the above equation directly implies V (cid:48) t := e − (cid:82) t M hr dr V t = M (cid:48) t − (cid:90) t e − (cid:82) s M hr dr (cid:16) f ( c s , V s ) + ζ − γ M hs V s (cid:17) ds, where M (cid:48) t := E t (cid:20) (cid:90) T e − (cid:82) s M hr dr (cid:16) f ( c s , V s ) + ζ − γ M hs V s (cid:17) ds + e − (cid:82) T M hs ds V T (cid:21) is an F -martingale on [0 , T ], thanks to (A.10). Applying generalized Itˆo’s formula for semimartin-gales (see [19, Theorem I.4.57]) to V t = e (cid:82) t M hr dr V (cid:48) t gives dV t = − F ( c t , M ht , V t ) + e (cid:82) t M hr dr d M (cid:48) t . Since0 ≤ M ht ≤ me βt by definition (by (2.2)), M t := (cid:82) t e (cid:82) s M hr dr d M (cid:48) s is again an F -martingale. Hence, V is a solution to BSDE (2.9). This, together with (A.9), yields the desired result. A.2 Derivation of Proposition 2.2
Lemma A.1.
Let c, h, V and W be F -progressively measurable processes with W s ≤ V s for all s ≥ .If there exists k ∈ R such that V satisfies (2.12) , then F ( c s , M hs , V s ) − F ( c s , M hs , W s ) ≤ − Γ(Λ , M hs )( V s − W s ) , (A.11) where F is given in (2.10) , Λ := δθ + (1 − θ ) k (as in Definition 2.2), and Γ is defined by Γ( λ, m ) := λ + γ ( ψ − − γ (1 − ζ − γ ) m. (A.12)18 roof. As in the proof of [22, Lemma B.1], (A.11) holds by the mean value theorem provided that F v ( c s , M hs , u ) ≤ − Γ(Λ , M hs ) for all u ∈ [ W s , V s ]. To this end, note that F v ( c s , M hs , u ) = − (cid:18) δθ + (1 − ζ − γ ) M hs + δ (1 − θ ) (cid:18) c − γs (1 − γ ) u (cid:19) /θ (cid:19) . Thanks to (1.1), a direct calculation shows F vv ( c s , M hs , u ) >
0, i.e. F v ( c s , M hs , u ) is increasingin u . This, together with V satisfying (2.12), implies that for all u ∈ [ W s , V s ], F v ( c s , M hs , u ) ≤ F v ( c s , M hs , ˆ u ), where ˆ u := δ θ (cid:0) k − ψ − − γ ( ζ − γ − M hs (cid:1) − θ c − γs − γ . By direct calculation, F v ( c s , M hs , ˆ u ) = − (cid:18) δθ + (1 − ζ − γ ) M hs + (1 − θ ) (cid:18) k − ψ − − γ ( ζ − γ − M hs (cid:19) (cid:19) = − (cid:18) Λ + γ ( ψ − − γ (1 − ζ − γ ) M hs (cid:19) = − Γ(Λ , M hs ) , where the second equality follows from the definition of Λ and θ = − γ − /ψ .To prove Proposition 2.2, we intend to follow the idea in the proof of [22, Theorem 2.2]. Theinvolvement of the controlled mortality M h in (2.11), as well as the possibility that Λ therein canbe negative (Remark 2.4), result in additional technicalities. The proof below combines argumentsin [22, Theorem 2.2] and [11, Theorem 2.1], adapted to weaker regularity of processes. Proof of Proposition 2.2.
Recall the function Γ in (A.12). Fix 0 ≤ t < T , define∆ t := e − (cid:82) tt Γ(0 ,M hs ) ds (cid:0) V t − V t (cid:1) , t ∈ [ t , T ] , (A.13)and consider the stopping time θ := inf (cid:8) s ≥ t : V s ≤ V s (cid:9) . Applying generalized Itˆo’s formula(see [19, Theorem I.4.57]) to e − (cid:82) t Γ(0 ,M hs ) ds V it , i = 1 ,
2, yields d (cid:16) e − (cid:82) t Γ(0 ,M hs ) ds V t (cid:17) = − e − (cid:82) t Γ(0 ,M hs ) ds (cid:104) Γ(0 , M hs ) V t + F ( c t , M ht , V t ) (cid:105) dt + e − (cid:82) t Γ(0 ,M hs ) ds d M t ,d (cid:16) e − (cid:82) t Γ(0 ,M hs ) ds V t (cid:17) = − e − (cid:82) t Γ(0 ,M hs ) ds (cid:104) Γ(0 , M hs ) V t + G ( t, V t ) (cid:105) dt + e − (cid:82) t Γ(0 ,M hs ) ds d M t , where M , M are some F -martingales on [0 , T ]. As 0 ≤ Γ(0 , M ht ) ≤ γ ( ψ − − γ (1 − ζ − γ ) me βt by thedefinition of M h in (2.2), r (cid:55)→ (cid:82) rt e − (cid:82) t Γ(0 ,M hs ) ds d M it is a true martingale for i = 1 ,
2. Hence,∆ t = E t (cid:20)(cid:90) Tt { s<θ } (cid:104)(cid:16) F ( c s , M hs , V s ) − G ( s, V s ) (cid:17) + Γ(0 , M hs ) (cid:0) V s − V s (cid:1)(cid:105) e − (cid:82) st Γ(0 ,M hr ) dr ds + ∆ T ∧ θ (cid:21) . Observe that { s<θ } (cid:16) F ( c s , M hs , V s ) − G ( s, V s ) (cid:17) = { s<θ } (cid:16) F ( c s , M hs , V s ) − F ( c s , M hs , V s ) (cid:17) + { s<θ } (cid:16) F ( c s , M hs , V s ) − G ( s, V s ) (cid:17) ≤ { s<θ } (cid:16) F ( c s , M hs , V s ) − F ( c s , M hs , V s ) (cid:17) ≤ { s<θ } (cid:16) − Γ(Λ , M hs ) (cid:0) V s − V s (cid:1)(cid:17) , F ( c s , M hs , V s ) ≤ G ( s, V s ), and the second is due to LemmaA.1, which is applicable here as V s > V s for s ∈ [ t, θ ). Thanks to the above inequality,∆ t ≤ E t (cid:20)(cid:90) Tt { s<θ } (cid:104) − Γ(Λ , M hs ) + Γ(0 , M hs ) (cid:105) (cid:0) V s − V s (cid:1) e − (cid:82) st Γ(0 ,M hr ) dr ds + ∆ T ∧ θ (cid:21) = E t (cid:20) − (cid:90) Tt { s<θ } Λ∆ s ds + ∆ T ∧ θ (cid:21) , (A.14)where the second line follows from Γ(Λ , M hs ) = Λ + Γ(0 , M hs ) and (A.13). Multiplying both sidesby { t<θ } yields∆ t { t<θ } ≤ E t (cid:20) − (cid:90) Tt Λ∆ s { s<θ } ds + ∆ T ∧ θ { t<θ } (cid:21) ≤ E t (cid:20) − (cid:90) Tt Λ∆ s { s<θ } ds + ∆ T { T <θ } (cid:21) , where the second inequality follows from the right continuity of V and V . Indeed, the rightcontinuity implies V θ ≤ V θ , so that ∆ T ∧ θ = ∆ θ { θ ≤ T } + ∆ T { T <θ } ≤ ∆ T { T <θ } . Set ∆ + t :=∆ t { t<θ } , and write the previous inequality as ∆ + t ≤ E t (cid:2) − (cid:82) Tt Λ∆ + s ds + ∆ + T (cid:3) . Taking expectationson both sides and using Fubini’s theorem giveΘ t ≤ − (cid:90) Tt ΛΘ s ds + Θ T , (A.15)where Θ t := E (cid:2) ∆ + t (cid:3) ≥ , M s ) ≥ E (cid:2) sup t ∈ [0 ,T ] | V it | (cid:3) < ∞ , thanks to V i ∈ E hk (Definition 2.2), for i = 1 ,
2. Now, if Λ >
0, by writing Θ T ≥ Θ t + (cid:82) Tt ΛΘ s ds , we applystandard Gronwall’s inequality to get Θ T ≥ Θ t e (cid:82) Tt Λ ds , or equivalentlyΘ t ≤ Θ T e − (cid:82) Tt Λ ds , t ∈ [ t , T ] . (A.16)If Λ <
0, applying backward Gronwall’s inequality (see [34, Proposition 2]) to (A.15) also gives(A.16). By (A.16), (A.13), and (A.12), we obtainΘ t ≤ Θ T e − (cid:82) Tt Λ ds ≤ E (cid:20) e − (cid:82) Tt Γ(Λ ,M s ) ds (cid:0) | V T | + | V T | (cid:1)(cid:21) . (A.17)Since T > V t and V t immediately implies0 ≤ Θ t ≤ lim T →∞ E (cid:20) e − (cid:82) Tt Γ(Λ ,M s ) ds (cid:0) | V T | + | V T | (cid:1)(cid:21) = 0 . (A.18)That is, Θ t = E (cid:2)(cid:0) V t − V t (cid:1) { t <θ } (cid:3) = 0. This entails θ = t , and thus V t ≤ V t . Since t ≥ V t ≤ V t for all t ≥ A.3 Proof of Proposition 4.1
For any fixed m > c ( m ) >
0, define w ( x ) := δ θ x − γ − γ ˜ c ( m ) − θψ for x >
0. In order toapply Theorem 3.1, we need to verify all its conditions. It can be checked directly that w , as aone-variable function, solves (3.8) in a trivial way, with all derivatives in m being zero. For any( c, π, h ) ∈ P = P , since ( X c,π,h ) − γ satisfies (2.11) (with Λ ∗ in place of Λ), so does w ( X c,π,ht ),i.e. w ( X c,π,ht ) ∈ E hk ∗ . By the definitions of P and w , P = P ⊆ H k ∗ and (3.10) is satisfied. As20 c ( m ) > w x > w xx < c ( x, m ) := x ˜ c ( m ) and ¯ π ( x, m ) := µγσ are unique maximizers of the supremums in (3.11), respectively. The supremum in (3.12) is zero, as g ≡ w x >
0. Hence, ¯ h ( x, m ) := 0 trivially maximizes (3.12). The only condition that remainsto be checked is “( c ∗ , π ∗ , h ∗ ) in (3.13) belongs to P and W ∗ t := w ( X c ∗ ,π ∗ ,h ∗ t ) satisfies (2.12)”.Observe that a unique solution X ∗ = X c ∗ ,π ∗ ,h ∗ to (3.2) exists as a geometric Brownian motion dX ∗ t = X ∗ t (cid:18) r + 1 γ (cid:16) µσ (cid:17) − ˜ c ( m ) (cid:19) dt + X ∗ t µγσ dB t , (A.19)This implies that( X ∗ t ) − γ = x − γ exp (cid:18) (1 − γ ) (cid:18) r + 12 γ (cid:16) µσ (cid:17) − ˜ c ( m ) − (1 − γ )2 γ (cid:16) µσ (cid:17) (cid:19) t + (1 − γ ) µγσ B t (cid:19) , (A.20)which is again a geometric Brownian motion that satisfies the dynamics dY t Y t = (1 − γ ) (cid:18) r + 12 γ (cid:16) µσ (cid:17) − ˜ c ( m ) (cid:19) dt + (1 − γ ) µγσ dB t , Y = x − γ . Consequently, e − Λ ∗ t E (cid:20) e − γ ( ψ − − ζ − γ − γ mt ( X ∗ t ) − γ (cid:21) = x − γ e ( C − Λ ∗ ) t , (A.21)where C := (1 − γ ) (cid:18) r + 12 γ (cid:16) µσ (cid:17) − ˜ c ( m ) (cid:19) − γ ( ψ −
1) 1 − ζ − γ − γ m. Remarkably, by the definitions of ˜ c ( m ) and Λ ∗ in (3.18) and (4.2), a direct calculation shows that C − Λ ∗ = − ˜ c ( m ) <
0, where the inequality follows from ˜ c ( m ) >
0. It follows from (A.21) thatlim t →∞ e − Λ ∗ t E (cid:20) e − γ ( ψ − − ζ − γ − γ mt ( X ∗ t ) − γ (cid:21) = 0 . (A.22)On the other hand, we can rewrite (A.20) as( X ∗ t ) − γ = x − γ exp (cid:18) (1 − γ ) (cid:18) r + 12 γ (cid:16) µσ (cid:17) − ˜ c ( m ) (cid:19) t (cid:19) · Z t , (A.23)where Z is a geometric Brownian motion with the dynamics dZ t = Z t (1 − γ ) µγσ dB t , Z = 1. As Z isa martingale, we can apply the Burkh¨older-Davis-Gundy inequality to get E (cid:20) sup s ∈ [0 ,t ] ( X ∗ s ) − γ (cid:21) ≤ Kx − γ e (cid:16) | − γ | (cid:12)(cid:12)(cid:12) r + γ ( µσ ) − ˜ c ( m ) (cid:12)(cid:12)(cid:12)(cid:17) t | − γ | µγσ E (cid:34)(cid:18) (cid:90) t Z s ds (cid:19) / (cid:35) , (A.24)for some constant K >
0. By Jensen’s inequality and Fubini’s theorem, E (cid:20)(cid:18) (cid:90) t Z s ds (cid:19) / (cid:21) ≤ (cid:18) (cid:90) t E [ Z s ] ds (cid:19) / = (cid:18) (cid:90) t e (1 − γ )2 µ γ σ s ds (cid:19) / = γσ | − γ | µ (cid:18) e (1 − γ )2 µ γ σ t − (cid:19) / . We then conclude from the above two inequalities that E (cid:20) sup s ∈ [0 ,t ] ( X ∗ s ) − γ (cid:21) < ∞ , ∀ t ≥ . (A.25)21y (A.22) and (A.25), ( X ∗ ) − γ satisfies (2.11) (with Λ ∗ in place of Λ), and so does the process W ∗ t := w ( X ∗ t ) = δ θ ˜ c ( m ) − θψ ( X ∗ t ) − γ − γ , i.e. W ∗ ∈ E h ∗ k ∗ . By applying Itˆo’s formula to W ∗ t and noting E (cid:20) sup s ∈ [0 ,t ] π ∗ s ( X ∗ s ) − γ (cid:21) < ∞ for all t ≥ , (A.26)a consequence of (A.25) and π ∗ t ≡ µγσ , we argue as in the proof of Theorem 3.1 that W ∗ t is asolution to (2.9). Moreover, W ∗ t = δ θ ˜ c ( m ) − θ +(1 − γ ) ( X ∗ t ) − γ − γ = δ θ ˜ c ( m ) − θ ( c ∗ t ) − γ − γ By (4.8), this shows that W ∗ satisfies (2.12) with k = k ∗ . Hence, ( c ∗ , h ∗ ) is k ∗ -admissible, so thatwe can conclude ( c ∗ , π ∗ , h ∗ ) ∈ P . Theorem 3.1 is then applicable, asserting that w ( x, m ) = v ( x, m )and ( c ∗ , π ∗ , h ∗ ) optimizes (3.5). A.4 Proof of Theorem 4.1
Define w ( x, m ) := δ θ x − γ − γ u ∗ ( m ) − θψ for ( x, m ) ∈ R . To apply Theorem 3.1, we need to verify allits conditions. It can be checked, as in (3.15)-(3.17), that w ∈ C , ( R + × R + ) solves (3.8). By thedefinitions of P and w , P ⊆ H k ∗ and (3.10) is satisfied for any ( c, π, h ) ∈ P . As w x > w xx < g satisfies Assumption 1, ¯ c , ¯ π , and ¯ h in (3.19) are unique maximizers of the supermums in(3.11) and (3.12). It remain to show (i) for any ( c, π, h ) ∈ P , w ( X c,π,ht , M ht ) ∈ E hk ∗ ; (ii) ( c ∗ , π ∗ , h ∗ ),defined using ¯ c , ¯ π , and ¯ h as in (3.13), belongs to P and W ∗ t := w ( X c ∗ ,π ∗ ,h ∗ t , M h ∗ t ) satisfies (2.12). (i) Take any p = ( c, π, h ) ∈ P , and set W t := w ( X p t , M ht ) for t ≥
0. We will prove W ∈ E hk ∗ . • Case (i)-1: γ ∈ ( ψ , u ∗ ( m ) ≥ ˜ c ( m ) ≥ ˜ c (0) = k ∗ >
0. As θ > γ ∈ ( ψ , < W t = δ θ ( X p t ) − γ − γ u ∗ ( M ht ) − θψ ≤ δ θ ( X p t ) − γ − γ ( k ∗ ) − θψ ∀ t ≥ , Since ( X p ) − γ satisfies (2.11) (as p ∈ P = P ), the above implies that W also satisfies (2.11). • Case (i)-2: γ > ζ <
1. As p ∈ P = P , there exists η ∈ (1 − γ ,
1) such that (4.3) holds.Consider α := − η γ ( ψ − − γ ( ζ − γ − > , α (cid:48) := − (1 − η ) γ ( ψ − − γ ( ζ − γ − > , (A.27) F t := (cid:16) u β ( M ht ) (cid:17) − θψ exp (cid:18) − α (cid:48) (cid:90) t M hs ds (cid:19) for t ≥ . (A.28)First, we claim that the process F is bounded from above; more specifically,sup t ≥ F t ≤ u β (cid:18) − θα (cid:48) ψ β (cid:19) − θ/ψ < ∞ . (A.29)Observe that dF t dt = − (cid:18) α (cid:48) M ht + θψ u β ( M ht ) − u (cid:48) β ( M ht ) dM ht dt (cid:19) F t = − (cid:18) α (cid:48) M ht + θψβ ( β − g ( h t )) (cid:2) u β ( M ht ) − ˜ c ( M ht ) (cid:3)(cid:19) F t , (A.30)22here the second equality follows as u β solves (4.6) with q = β . For each ω ∈ Ω, consider S ( ω ) := (cid:26) t ≥ M ht ( ω ) = − θα (cid:48) ψβ ( β − g ( h t )) (cid:0) u β ( M ht ) − ˜ c ( M ht ) (cid:1) ( ω ) (cid:27) . We deduce from (A.30) thatlocal maximizers of t (cid:55)→ F t ( ω ) must occur at time points in S ( ω ) . (A.31)Also, by g ≥ L t ( ω ) := − θα (cid:48) ψβ ( β − g ( h t )) (cid:0) u β ( M ht ) − ˜ c ( M ht ) (cid:1) ( ω ) ≤ − θα (cid:48) ψ β, ∀ t ≥ . (A.32)This particularly implies that M ht ( ω ) = L t ( ω ) ≤ − θα (cid:48) ψ β, for each t ∈ S ( ω ) . (A.33)Now, there are three distinct possibilities: 1) There exists t ∗ ≥ M ht ( ω ) < L t ( ω )for all t > t ∗ . Then, S ( ω ) ⊆ [0 , t ∗ ] and (A.32) implies M ht ( ω ) < − θα (cid:48) ψ β for all t > t ∗ . It thenfollows from (A.31) and (A.28) thatsup t ≤ t ∗ F t ( ω ) = sup t ∈ S ( ω ) F t ( ω ) ≤ sup t ∈ S ( ω ) u β (cid:0) M ht ( ω ) (cid:1) − θψ ≤ u β (cid:18) − θα (cid:48) ψ β (cid:19) − θ/ψ , (A.34)where the last inequality follows from (A.33). Moreover,sup t>t ∗ F t ( ω ) ≤ sup t>t ∗ u β (cid:0) M ht ( ω ) (cid:1) − θψ ≤ u β (cid:18) − θα (cid:48) ψ β (cid:19) − θ/ψ , i.e. (A.29) holds. 2) There exists t ∗ ≥ M ht ( ω ) > L t ( ω ) for all t > t ∗ . By (A.30), F t ( ω ) is strictly decreasing for t > t ∗ . Thus, sup t ≥ F t ( ω ) = sup t ≤ t ∗ F t ( ω ) = sup t ∈ S ( ω ) F t ( ω ).By the estimate in (A.34), (A.29) holds. 3) Neither 1) nor 2) above holds. This entailssup { t ≥ t ∈ S ( ω ) } = ∞ . Hence, sup t ≥ F t ( ω ) = sup t ∈ S ( ω ) F t ( ω ), so that (A.29) holds bythe estimate in (A.34). Now, since u ∗ ≤ u β (by (4.11)), − θ/ψ >
0, and 1 − γ < ≥ e γ ( ψ − − γ ( ζ − γ − (cid:82) t M hs ds W t ≥ δ θ (cid:16) u β ( M ht ) (cid:17) − θ/ψ e γ ( ψ − − γ ( ζ − γ − (cid:82) t M hs ds ( X p t ) − γ − γ = δ θ F t e − α (cid:82) t M hs ds ( X p t ) − γ − γ ≥ δ θ u β (cid:18) − θα (cid:48) ψ β (cid:19) − θ/ψ e − α (cid:82) t M hs ds ( X p t ) − γ − γ , where the equality follows from (A.28) and (A.27), and the last inequality is due to (A.29).Recalling that p ∈ P = P , we conclude from (4.3) and the above inequality thatlim t →∞ e − Λ ∗ t E (cid:20) e γ ( ψ − − γ ( ζ − γ − (cid:82) t M hs ds W t (cid:21) = 0 . On the other hand, since M ht ≤ me βt , E (cid:20) sup s ∈ [0 ,t ] | W t | (cid:21) ≤ δ θ | − γ | u β ( me βt ) − θ/ψ E (cid:20) sup s ∈ [0 ,t ] ( X p s ) − γ (cid:21) < ∞ , ∀ t ≥ . where the finiteness is a direct consequence of p ∈ P .23 Case (i)-3: γ > ζ = 1. In view of (4.5), u q ≡ k ∗ > q >
0. It then followsfrom (4.11) that u ∗ ≡ k ∗ >
0. The required properties then follow directly from p ∈ P = P . (ii) Now, we show that ( c ∗ , π ∗ , h ∗ ) ∈ P and W ∗ t := w ( X c ∗ ,π ∗ ,h ∗ t , M h ∗ t ) satisfies (2.12). Observethat a unique solution M ∗ = M h ∗ to (2.2) exists. As h ∗ by definition only depends on u ∗ , g , andthe current mortality rate, M ∗ is a deterministic process. Thanks to (4.13), t (cid:55)→ M ∗ t is strictlyincreasing. Also, a unique solution X ∗ = X c ∗ ,π ∗ ,h ∗ to (3.2) exists, which admits the formula( X ∗ t ) − γ = x − γ exp (cid:18) (cid:90) t (1 − γ ) (cid:18) r + 12 γ (cid:16) µσ (cid:17) − u ∗ ( M ∗ s ) − h ∗ s − − γ γ (cid:16) µσ (cid:17) (cid:19) ds + (1 − γ ) µγσ B t (cid:19) . (A.35) • Case (ii)-1: γ ∈ ( ψ , M ∗ t is strictly increasing, u ∗ ( M ∗ t ) ≥ u ∗ ( m ) ≥ ˜ c ( m ), where thesecond inequality follows from (4.11) and (4.7). With this and h ∗ t ≥
0, we deduce from (A.35)that (A.20) holds with “=” therein replaced by “ ≤ ”. As k ∗ > c ( m ) > X ∗ ) − γ satisfies (2.11).With this, we can argue as in Case (i)-1 to show that W ∗ t := w ( X ∗ t , M ∗ t ) belongs to E h ∗ k ∗ . • Case (ii)-2: γ > ζ (cid:54) = 1. As u ∗ solves (3.17) and h ∗ maximizes the supremum in (3.17), u ∗ ( M ∗ t ) − ˜ c ( M ∗ t ) − ( ψ − h ∗ t = M ∗ t ( u ∗ ) (cid:48) ( M ∗ t ) u ∗ ( M ∗ t ) ( β − g ( h ∗ t )) > ∀ t > , where the inequality follows from (4.13). This gives h ∗ t < ψ − ( u ∗ ( M ∗ t ) − ˜ c ( M ∗ t )), so that u ∗ ( M ∗ t ) + h ∗ t < ψψ − u ∗ ( M ∗ t ) − ψ − c ( M ∗ t ) ≤ ψψ − u β ( M ∗ t ) − ψ − c ( M ∗ t ) , (A.36)where the last inequality follows from u ∗ ( m ) ≤ u β ( m ) (see (4.11)). For any η ∈ (1 − γ , α, α (cid:48) > u β ( m ) can be written as u β ( m ) = β e − m ψθβ (1 − ζ − γ ) (cid:16) m ψθβ (1 − ζ − γ ) (cid:17) − k ∗ /β Γ (cid:16) − k ∗ β , m ψθβ (1 − ζ − γ ) (cid:17) where Γ is the upper incomplete gamma function Γ( s, z ) := (cid:82) ∞ z t s − e − t dt . Similarly to theargument in [14, (A.6)-(A.7)], by using the fact lim z →∞ Γ( s,z ) e − z z s − = 1,lim m →∞ ψ − ψ (cid:0) α + ( ζ − γ − (cid:1) m ( γ − u β ( m ) = ψ − ψ α + ( ζ − γ − ψ − ζ − γ −
1) = α + ( ζ − γ − ψ ( ζ − γ − > , (A.37)where the inequality follows from the definition of α and η > − γ . This, together with M ∗ being a strictly increasing deterministic process, implies the existence of s ∗ > α + ( ζ − γ − M ∗ s > ψ ( γ − ψ − u β ( M ∗ s ) for s > s ∗ . (A.38)Consider the constant 0 ≤ K := max t ∈ [0 ,s ∗ ] (cid:8) ψψ − u β ( M ∗ t ) − α +( ζ − γ − γ − M ∗ t (cid:9) < ∞ . In view of(A.35), (A.36), and ˜ c ( m ) = k ∗ + (1 − ψ ) ζ − γ − − γ m (see (4.8)), e − α (cid:82) t M ∗ s ds ( X ∗ t ) − γ ≤ x − γ exp (cid:18)(cid:90) t (1 − γ ) (cid:18) r + 12 γ (cid:16) µσ (cid:17) + k ∗ ψ − − ψψ − u β ( M ∗ s ) − α + ( ζ − γ − − γ M ∗ s (cid:19) ds (cid:19) · Z t ≤ x − γ e (1 − γ ) (cid:16) r + γ ( µσ ) + k ∗ ψ − − K (cid:17) s ∗ e (1 − γ ) (cid:16) r + γ ( µσ ) + k ∗ ψ − (cid:17) ( t − s ∗ ) Z t , Z is the driftless geometric Brownian motion defined below (A.23), and the secondinequality follows from (A.38). It follows that e − Λ ∗ t E (cid:104) e − α (cid:82) t M ∗ s ds ( X ∗ t ) − γ (cid:105) ≤ x − γ e (cid:16) (1 − γ ) (cid:16) r + γ ( µσ ) + k ∗ ψ − − K (cid:17) − Λ ∗ (cid:17) s ∗ e (cid:16) (1 − γ ) (cid:16) r + γ ( µσ ) + k ∗ ψ − (cid:17) − Λ ∗ (cid:17) ( t − s ∗ ) = x − γ e (cid:16) (1 − γ ) (cid:16) r + γ ( µσ ) + k ∗ ψ − − K (cid:17) − Λ ∗ (cid:17) s ∗ e − ( γ + γ − ψ − ) k ∗ ( t − s ∗ ) → t → ∞ , where the equality follows from a direct calculation using the definition of Λ ∗ in (4.2), and theconvergence is due to k ∗ >
0. Namely, X ∗ satisfies (4.3). On the other hand, by (A.36) and M ∗ t ≤ me βt , we obtain from (A.35) that( X ∗ t ) − γ ≤ x − γ exp (cid:18)(cid:90) t (1 − γ ) (cid:18) r + 12 γ (cid:16) µσ (cid:17) − ψψ − u β ( me βs ) (cid:19) ds (cid:19) · Z t , where Z is again the driftless geometric Brownian motion defined below (A.23). By theBurkh¨older-Davis-Gundy inequality, we obtain the estimate in (A.24) with − ˜ c ( m ) thereinreplaced by ψψ − u β ( me βt ). This then implies E (cid:2) sup s ∈ [0 ,t ] ( X ∗ s ) − γ (cid:3) < ∞ , by the inequalitypreceding (A.25). Finally, under E (cid:2) sup s ∈ [0 ,t ] ( X ∗ s ) − γ (cid:3) < ∞ and (4.3), the same argument asin Case (i)-2 shows that W ∗ t := w ( X ∗ t , M ∗ t ) belongs to E h ∗ k ∗ . • Case (ii)-3: γ > ζ = 1. By (4.5), u β ( m ) ≡ k ∗ >
0. As M ∗ t is strictly increasing,˜ c ( M ∗ t ) ≥ ˜ c (0) = k ∗ . The estimate (A.36) then becomes u ∗ ( M ∗ t ) + h ∗ ≤ ψψ − k ∗ − ψ − k ∗ = k ∗ ,so that we can deduce from (A.35) that( X ∗ t ) − γ ≤ x − γ exp (cid:18)(cid:90) t (1 − γ ) (cid:18) r + 12 γ (cid:16) µσ (cid:17) − k ∗ − (1 − γ )2 γ (cid:16) µσ (cid:17) (cid:19) ds + (1 − γ ) µγσ B t (cid:19) . The arguments in Proposition 4.1 can then be applied to show that ( X ∗ ) − γ satisfies (2.11).Then, we may argue as in Case (i)-3 to show that W ∗ t := w ( X ∗ t , M ∗ t ) belongs to E h ∗ k ∗ . Finally, by applying Itˆo’s formula to W ∗ t and using (A.26), a consequence of (A.25) and π ∗ t ≡ µγσ ,we argue as in the proof of Theorem 3.1 that W ∗ t is a solution to (2.9). Also, W ∗ t = δ θ u ∗ ( M ∗ t ) − θ +(1 − γ ) ( X ∗ t ) − γ − γ = δ θ u ∗ ( M ∗ t ) − θ ( c ∗ t ) − γ − γ ≤ δ θ ˜ c ( M ∗ t ) − θ ( c ∗ t ) − γ − γ , (A.39)where the inequality follows from u ∗ ≥ ˜ c (by (4.11) and (4.7)) and the fact that θ > γ ∈ ( ψ , θ < γ >
1. By (4.8), this shows that W ∗ satisfies (2.12) with k = k ∗ . Hence, ( c ∗ , h ∗ ) is k ∗ -admissible, and we can now conclude ( c ∗ , π ∗ , h ∗ ) ∈ P . By Theorem 3.1, v ( x, m ) = w ( x, m ) and( c ∗ , π ∗ , h ∗ ) optimizes (3.5). A.5 Proof of Proposition 4.2
Define w ( x, m ) := δ θ x − γ − γ u β ( m ) − θψ for ( x, m ) ∈ R . To apply Theorem 3.1, we need to verify allits conditions. It can be checked directly that w ∈ C , ( R + × R + ) solves (3.8), as u β is a solutionto (4.4) (Lemma 4.1). By the definitions of P and w , P ⊆ H k ∗ and (3.10) is satisfied for any( c, π, h ) ∈ P . Following part (i) of the proof of Theorem 4.1, we get w ( X c,π,ht , M ht ) ∈ E hk ∗ for any( c, π, h ) ∈ P ; the proof is much simpler here, as M ht = me βt in the current setting. As w x > xx <
0, ¯ c ( x, m ) := xu β ( m ) and ¯ π ( x, m ) := µγσ are unique maximizers of the supremums in (3.11),respectively. The supremum in (3.12) is zero, as g ≡ w x >
0. Hence, ¯ h ( x, m ) := 0 triviallymaximizes (3.12). It remains to show that ( c ∗ , π ∗ , h ∗ ), defined using ¯ c , ¯ π , and ¯ h as in (3.13), belongsto P and W ∗ t := w ( X c ∗ ,π ∗ ,h ∗ t , M h ∗ t ) satisfies (2.12).Observe that M h ∗ t = me βt as h ∗ ≡
0, and a unique solution X ∗ = X c ∗ ,π ∗ ,h ∗ to (3.2) exists,which satisfies the dynamics (A.19) with ˜ c ( m ) replaced by u β ( me βt ). This implies( X ∗ t ) − γ = x − γ exp (cid:18)(cid:90) t (1 − γ ) (cid:18) r + 12 γ (cid:16) µσ (cid:17) − u β ( me βs ) − (1 − γ )2 γ (cid:16) µσ (cid:17) (cid:19) ds + (1 − γ ) µγσ B t (cid:19) . (A.40) • Case 1: γ ∈ ( ψ , − γ > u β ( m ) ≥ ˜ c ( m ) (see (4.7)), we deduce from (A.40)that (A.20) holds with “=” therein replaced by “ ≤ ”. As k ∗ > c ( m ) >
0, the samearguments in Proposition 4.1 can be applied to show that ( X ∗ ) − γ satisfies (2.11). With this,we can argue as in Case (i)-1 of the proof of Theorem 4.1 to obtain W ∗ t := w ( X ∗ t , M ∗ t ) ∈ E h ∗ k ∗ . • Case 2: γ > ζ (cid:54) = 1. For any η ∈ (1 − γ , α > z →∞ Γ( s,z ) e − z z s − = 1 yieldslim m →∞ αm ( γ − u ( m ) = α ( ψ − ζ − γ − > . (A.41)where the inequality follows from the definition of α and η > − γ . This implies that thereexists some s ∗ > αme βs ≥ ( γ − u ( me βs ) for all s ≥ s ∗ . (A.42)Consider 0 ≤ K := max t ∈ [0 ,s ∗ ] (cid:110) ˜ u ( me βt ) − αme βt γ − (cid:111) < ∞ . Now, by M t = me βt and (A.40) , e − α (cid:82) t M s ds ( X ∗ ) − γ = x − γ exp (cid:18)(cid:90) t (1 − γ ) (cid:18) r + 12 γ (cid:16) µσ (cid:17) − u β ( me βs ) − αme βs (1 − γ ) (cid:19) ds (cid:19) · Z t ≤ x − γ e (1 − γ ) (cid:16) r + γ ( µσ ) − K (cid:17) s ∗ e (1 − γ ) (cid:16) r + γ ( µσ ) (cid:17) ( t − s ∗ ) Z t , where Z t is the driftless geometric Brownian motion defined below (A.23), and the inequalityfollows from (A.42). It follows that e − Λ ∗ t E (cid:104) e − α (cid:82) t M s ds ( X ∗ t ) − γ (cid:105) ≤ x − γ e (cid:16) (1 − γ ) (cid:16) r + γ ( µσ ) − K (cid:17) − Λ ∗ (cid:17) s ∗ e (cid:16) (1 − γ ) (cid:16) r + γ ( µσ ) (cid:17) − Λ ∗ (cid:17) ( t − s ∗ ) = x − γ e (cid:16) (1 − γ ) (cid:16) r + γ ( µσ ) − K (cid:17) − Λ ∗ (cid:17) s ∗ e − γk ∗ ( t − s ∗ ) → , as t → ∞ , where the second line follows from a direct calculation using the definition of Λ ∗ in (4.2), andthe convergence is due to k ∗ >
0. On the other hand, similarly to (A.23), we rewrite (A.40) as( X ∗ t ) − γ = x − γ exp (cid:18)(cid:90) t (1 − γ ) (cid:18) r + 12 γ (cid:16) µσ (cid:17) − u β ( me βs ) (cid:19) ds (cid:19) · Z t , where Z is again the driftless geometric Brownian motion defined below (A.23). By Burkh¨older-Davis-Gundy’s inequality, we obtain the estimate in (A.24) with − ˜ c ( m ) therein replaced by u β ( me βt ). This implies E (cid:2) sup s ∈ [0 ,t ] ( X ∗ s ) − γ (cid:3) < ∞ , by the inequality preceding (A.25). Under E (cid:2) sup s ∈ [0 ,t ] ( X ∗ s ) − γ (cid:3) < ∞ and (4.3), the same argument as in Case (i)-2 of the proof ofTheorem 4.1 shows that W ∗ t := w ( X ∗ t , M ∗ t ) belongs to E h ∗ k ∗ . Case 3: γ > ζ = 1. By (4.5), u β ( m ) ≡ k ∗ >
0. Then, in view of (A.40), we can applythe same arguments as in Proposition 4.1 to show that ( X ∗ ) − γ satisfies (2.11). With this, wemay argue as in Case (i)-3 in the proof of Theorem 4.1 to obtain W ∗ t := w ( X ∗ t , M ∗ t ) ∈ E h ∗ k ∗ . Finally, by applying Itˆo’s formula to W ∗ t and using (A.26), a consequence of (A.25) and π ∗ t ≡ µγσ ,we argue as in the proof of Theorem 3.1 that W ∗ t is a solution to (2.9). Also, the same calculationas in (A.39), with u ∗ therein replaced by u β , can be carried out, thanks to u β ≥ ˜ c by (4.7). Thisshows that W ∗ satisfies (2.12) with k = k ∗ . Hence, ( c ∗ , h ∗ ) is k ∗ -admissible, and we can conclude( c ∗ , π ∗ , h ∗ ) ∈ P . By Theorem 3.1, v ( x, m ) = w ( x, m ) and ( c ∗ , π ∗ , h ∗ ) optimizes (3.5). References [1]
J. Aurand and Y.-J. Huang , Epstein-Zin utility maximization on random horizons , (2019). Preprint,available at https://arxiv.org/abs/1903.08782.[2]
R. Bansal , Long-run risks and financial markets , Review - Federal Reserve Bank of St.Louis, 89 (2007),pp. 283–299.[3]
R. Bansal and A. Yaron , Risks for the long run: A potential resolution of asset pricing puzzles , TheJournal of Finance, 59 (2004), pp. 1481–1509.[4]
J. Beeler and J. Y. Campbell , The long-run risks model and aggregate asset prices: An empiricalassessment , Critical Finance Review, 1 (2012), pp. 141–182.[5]
L. Benzoni, P. Collin-Dufresne, and R. S. Goldstein , Explaining asset pricing puzzles associatedwith the 1987 market crash , Journal of Financial Economics, 101 (2011), pp. 552–573.[6]
H. S. Bhamra, L.-A. Kuehn, and I. A. Strebulaev , The levered equity risk premium and creditspreads: A unified framework , The Review of Financial Studies, 23 (2010), pp. 645–703.[7]
P. Briand, B. Delyon, Y. Hu, E. Pardoux, and L. Stoica , Lp solutions of backward stochasticdifferential equations , Stochastic Processes and their Applications, 108 (2003), pp. 109 – 129.[8]
D. Duffie and P.-L. Lions , PDE solutions of stochastic differential utility , J. Math. Econom., 21(1992), pp. 577–606.[9]
I. Ehrlich , Uncertain lifetime, life protection, and the value of life saving , Journal of health economics,19 (2000), pp. 341–367.[10]
I. Ehrlich and H. Chuma , A model of the demand for longevity and the value of life extension ,Journal of Political economy, 98 (1990), pp. 761–782.[11]
S. Fan , Bounded solutions, L p ( p > solutions and L solutions for one dimensional BSDEs undergeneral assumptions , Stochastic Process. Appl., 126 (2016), pp. 1511–1552.[12] B. Gompertz , On the nature of the function expressive of the law of human mortality and on a newmode of determining the value of life contingencies , Philosophical Transactions of the Royal Society ofLondon, 115 (1825), pp. 513–583.[13]
M. Grossman , On the concept of health capital and the demand for health , Journal of Political economy,80 (1972), pp. 223–255.[14]
P. Guasoni and Y.-J. Huang , Consumption, investment and healthcare with aging , Finance andStochastics, 23 (2019), pp. 313–358.[15]
R. E. Hall and C. I. Jones , The value of life and the rise in health spending , The Quarterly Journalof Economics, 122 (2007), pp. 39–72.[16]
L. P. Hansen, J. Heaton, J. Lee, and N. Roussanov , Intertemporal substitution and risk aversion ,in Handbook of Econometrics, J. J. Heckman and E. E. Leamer, eds., vol. 6A, Elsevier, 1 ed., 2007,ch. 61, pp. 3967–4056. J. Hugonnier, F. Pelgrin, and P. St-Amour , Health and (other) asset holdings , The Review ofEconomic Studies, 80 (2013), pp. 663–710.[18]
F. Imperial , Modelling Stock Prices and Stock Market Behaviour using the Irrational Fractional Brow-nian Motion: An Application to the S&P500 in Eight Different Periods , PhD thesis, June 2018.[19]
J. Jacod and A. N. Shiryaev , Limit theorems for stochastic processes , vol. 288 of Grundlehren derMathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag,Berlin, second ed., 2003.[20]
H. Kraft, T. Seiferling, and F. T. Seifried , Optimal consumption and investment with Epstein-Zin recursive utility , Finance Stoch., 21 (2017), pp. 187–226.[21]
H. Kraft, F. T. Seifried, and M. Steffensen , Consumption-portfolio optimization with recursiveutility in incomplete markets , Finance Stoch., 17 (2013), pp. 161–196.[22]
Y. Melnyk, J. Muhle-Karbe, and F. T. Seifried , Lifetime investment and consumption withrecursive preferences and small transaction costs , Mathematical Finance, forthcoming, (2017). Availableat http://dx.doi.org/10.2139/ssrn.2970469.[23]
OECD/European Union , Health at a Glance: Europe 2018: State of Health in the EU Cycle , OECDPublishing, Paris/European Union, Brussels, 2018.[24]
S. F. Richard , Optimal consumption, portfolio and life insurance rules for an uncertain lived individualin a continuous time model , Journal of Financial Economics, 2 (1975), pp. 187–203.[25]
S. Rosen , The value of changes in life expectancy , Journal of Risk and uncertainty, 1 (1988), pp. 285–304.[26]
R. L. Schilling , Measures, integrals and martingales , Cambridge University Press, Cambridge, sec-ond ed., 2017.[27]
M. Schroder and C. Skiadas , Optimal consumption and portfolio selection with stochastic differentialutility , J. Econom. Theory, 89 (1999), pp. 68–126.[28]
S. Sch¨utte, P. Acevedo, and A. Flahault , Health systems around the world—a comparison ofexisting health system rankings , J Glob Health, 8 (2018), p. 010407.[29]
T. Seiferling and F. T. Seifried , Epstein-zin stochastic differential utility: Existence, uniqueness,concavity, and utility gradients , (2016). Preprint. Available at https://dx.doi.org/10.2139/ssrn.2625800.[30]
D. S. Shepard and R. J. Zeckhauser , Survival versus consumption , Management Science, 30 (1984),pp. 423–439.[31]
C. Skiadas , Recursive utility and preferences for information , Economic Theory, 12 (1998), pp. 293–312.[32]
A. Tandon, C. Murray, J. Lauer, and D. Evans , Measuring overall health system performancefor 191 countries , World Health Organization, Geneva, (2000).[33]
A. Vissing-Jørgensen and O. Attanasio , Stock-market participation, intertemporal substitution,and risk-aversion , American Economic Review, 93 (2003), pp. 383–391.[34]
X. Wang and S. Fan , A class of stochastic gronwall’s inequality and its application , Journal of In-equalities and Applications, (2018).[35]
H. Xing , Consumption-investment optimization with Epstein-Zin utility in incomplete markets , FinanceStoch., 21 (2017), pp. 227–262.[36]
M. E. Yaari , Uncertain lifetime, life insurance, and the theory of the consumer , The Review of Eco-nomic Studies, 32 (1965), pp. 137–150.[37]
M. Yogo , Portfolio choice in retirement: Health risk and the demand for annuities, housing, and riskyassets , Journal of Monetary Economics, 80 (2016), pp. 17–34., Journal of Monetary Economics, 80 (2016), pp. 17–34.