Optimal Investment, Consumption and Life Insurance under Mean-Reverting Returns: The Complete Market Solution

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1 Opimal Invesmen, Consumpion and Life Insurance under Mean-Revering Reurns: The Complee Marke Soluion Traian A. Pirvu Dep of Mahemaics & Saisics McMaser Universiy 180 Main Sree Wes Hamilon, ON, L8S 4K1 Huayue Zhang Dep of Finance Nankai Universiy 94 Weijin Road Tianjin, China, May 14, 01 Absrac This paper considers he problem of opimal invesmen, consumpion and life insurance acquisiion for a wage earner who has CRRA (consan relaive risk aversion) preferences. The marke model is complee, coninuous, he uncerainy is driven by Brownian moion and he sock price has mean revering drif. The problem is solved by dynamic programming approach and he HJB equaion is shown o have closed form soluion. Numerical experimens explore he impac marke price of risk has on he opimal sraegies. Keywords: Porfolio allocaion; Life insurance; Mean revering drif. 1 Inroducion The goal of his paper is o provide opimal invesmen, consumpion and life insurance acquisiion sraegies for a wage earner who uses an expeced uiliy crierion wih CRRA ype preferences. Exising works on opimal invesmen, consumpion and life insurance acquisiion use a financial model in which risky asses are modelled as geomeric Brownian moions. In order o make he model more realisic, we allowed he risky asses o have sochasic drif parameers. Le us review he lieraure on opimal invesmen consumpion and life insurance. The invesmen/consumpion problem in a sochasic conex was considered by Meron [7] and [8]. His model consiss in a risk-free asse wih consan rae of reurn and one or more socks, he prices of which are driven by geomeric Brownian moions. The horizon T Work suppored by NSERC gran , SSHRC gran , MITACS gran and he Naural Science Foundaion of China ( ). We hank he anonymous referee for valuable commens and suggesions. 1

2 is prescribed, he porfolio is self-financing, and he invesor seeks o maximize he expeced uiliy of ineremporal consumpion plus he final wealh. Meron provided a closed form soluion when he uiliies are of consan relaive risk aversion (CRRA) or consan absolue risk aversion (CARA) ype. I urns ou ha for (CRRA) uiliies he opimal fracion of wealh invesed in he risky asse is consan hrough ime. Moreover for he case of (CARA) uiliies, he opimal sraegy is linear in wealh. Richard [11] added life insurance o he invesor s porfolio by assuming an arbirary bu known disribuion of deah ime. In he same vein Pliska and Ye [10] sudied opimal life insurance and consumpion for an income earner whose lifeime is random and unbounded. Ye [13] exended [10] by considering invesmens ino a financial marke. Huang and Milevsky [4] solved a porfolio choice problem ha includes moraliy-coningen claims and labor income under general hyperbolic absolue risk aversion (HARA) preferences focusing on shocks o human capial and financial capial. Horneff and Maurer [3] considered he problem in which an invesor has o decide among shor and long posiions in moraliy coningen claims. Their analysis revealed when he wage earners demand for life insurance swiches o he demand for annuiies. Duare e al. [1] exends [13] o allow for muliple risky asses modeled as geomeric Brownian moions. More recenly Kwak e al. [6] looked a he problem of finding opimal invesmen, consumpion and life insurance acquisiion for a family whose parens receive deerminisic labor income unil some deerminisic ime horizon. The novely of our work is ha we allow for sochasic drif parameer. More precisely he sock price is assumed o have mean revering drif. This sock price model was used by Kim and Omberg [5] who managed o obain closed form soluions for he opimal invesmen sraegies. Laer, Wacher [14] added ineremporal consumpion o he model considered by Kim and Omberg. Furhermore, opimal sraegies are obained in closed form and an empirical analysis has shown ha under some assumpions he sock drif is mean revering for realisic parameer values. We solve wihin his framework he Hamilon Jacobi Bellmann (HJB) equaion associaed wih maximizing he expeced uiliy of consumpion, erminal wealh and legacy and his in urn provides he opimal invesmen consumpion and life insurance acquisiion. By looking a he explici soluions we found ou ha : 1) when he wage earner accumulaes enough wealh (o leave o his/her heirs), he/she considers buying pension annuiies raher han life insurance ) when he value of human capial is small he wage earner buys pension annuiies o supplemen his/her income 3) when he wage earner becomes older (so he hazard rae ges higher) he wage earner has a higher demand (in absolue erms) for life insurance/pension annuiies. In our special model, due o he same risk preference for invesmen, consumpion and legacy, he opimal legacy (wealh plus insurance benefi) is relaed o he opimal consumpion. More precisely he raio beween he opimal legacy and opimal consumpion is a deerminisic funcion which depends on he hazard rae, risk aversion, premium insurance raio, he weigh on he beques, and is independen of he financial marke characerisics. In a fricionless marke (i.e. hazard rae equals he premium insurance raio) i urns ou ha he opimal legacy equals he opimal consumpion. Our main moivaion in wriing his paper is o explore he impac sochasic marke price of risk has on he opimal invesmen sraegies. This could no be observed by previous works which assumed a geomeric Brownian moion model for he sock prices. We found ou ha he opimal invesmen sraegy (in he sock) is significanly affeced by he marke price of risk (MPR). Moreover, he opimal invesmen in he sock is increasing in he MPR. As for he opimal insurance we found wo paerns depending on he risk aversion. A risk averse wage earner pays less for life insurance if MPR is increasing up o a cerain hreshold; when he MPR

3 exceeds ha hreshold, he amoun he wage earner pays for life insurance sars o decrease. A risk seeking wage earner has a differen behaviour; hus he/she pays less for life insurance when he MPR increases. Organizaion of he paper: The remainder of his paper is organized as follows. In Secion we describe he model and formulae he objecive. Secion 3 performs he analysis. Numerical resuls are discussed in Secion 4. Secion 5 concludes. The paper ends wih an appendix conaining he proofs. The Model In his paper, we assume ha a wage earner has o make decisions regarding consumpion, invesmen and life insurance/pension annuiy purchase. Le T > 0 be a finie benchmark ime horizon, and {W ()} [0,T ] a 1 dimension Brownian moion on a probabiliy space (Ω, {F } [0,T ], F, P). The filraion {F } is he compleed filraion generaed by {W ()} [0,T ]. Le E denoe he expecaion wih respec o P. The coninuous ime economy consiss of a financial marke and an insurance marke..1 The financial marke The financial marke conains a risk-free asse earning ineres rae r 0 and one risky asse. By some noaional changes we can address he case of muliple risky asses. The asse prices evolve according o he following equaions: db() = rb()d, B(0) = 1, ds() = S() [µ() d + σdw ()]. Moreover, he marke price of risk {θ()} [0,T ] = { µ() r σ } [0,T ] is a mean revering process, i.e., dθ() = k(θ() θ)d σ θ dw (). Here σ, σ θ, k, θ are posiive consans. This model for he sock price was considered by [5] and [14]. In addiion, we assume ha he wage earner receives income a he random rae i which follows a geomeric Brownian moion wih posiive consans ν i and σ i.. The life insurance di() = i()(ν i d + σ i dw ()), We assume ha he wage earner is alive a = 0 and his/her lifeime is a non-negaive random variable τ defined on he probabiliy space (Ω, F, P) and independen of he Brownian moion {W ()} [0,T ]. Le us inroduce he hazard funcion λ () : [0, T ] R +, ha is, he insananeous deah rae, defined by 3

4 P( τ < + ε τ ) λ() = lim. ε 0 ε From he above definiion i follows ha: P(τ < s τ > ) = 1 exp{ s λ(u) du}, (.1) and T P(τ > T τ > ) = exp{ λ(u) du}. (.) Denoe by f(s; ) he condiional probabiliy densiy for he deah a ime s condiional upon he wage earner being alive a ime s. Thus f(s; ) = λ(s) exp( s λ(v)dv). (.3) Here F (s; ) denoes he condiional probabiliy for he wage earner o be alive a ime s condiional upon being alive a ime s; consequenly F (s; ) = exp( s λ(v)dv). (.4) In our model he wage earner purchases erm life insurance/pension annuiy wih he erm being infiniesimally small. Premium is paid (life insurance) or received (pension annuiy) coninuously a rae p() given ime. In compensaion, if he wage earner dies a ime when he premium paymen rae is p(), hen eiher he insurance company pays an insurance amoun p() p() η() if p() is posiive, or he amoun η() should be paid by wage earner s family if p() is negaive. Here η : [0, T ] R + is a coninuous, deerminisic, prespecified funcion ha called premium-insurance raio, and 1 η() is referred o as loading facor. In a fricionless marke η() = λ(), bu due o commission fees η() > λ(). In order o simplify he analysis we assume ha η() = λ(). The insurance marke model was pioneered by [1] who considered he problem of opimal financial planning decisions for an individual wih an uncerain lifeime. Laer, ha model was exended by [11]. Many works followed [1] and [11] by considering insananeous erm life insurance; i means ha he invesor can only purchase life insurance for he nex insan; if surviving he nex insan he invesor has o buy again he insananeous erm life insurance and so on. The purchase of an annuiy could be reversed by buying a life insurance ha guaranees he paymen of annuiy s face amoun on he deah of he holder. Indeed here are siuaions of life annuiies when premium is paid only a deah (e.g. reverse morgage). This brings up he noion of insananeous pension annuiy as he reverse of insananeous erm life insurance. One has o admi ha insananeous erm life insurance producs can no be purchased in he real world and one can no find he exac same payoff srucure in erm life insurance producs. These ficiious insananeous erm life insurance producs are used in porfolio managemen because hey offer a simplified and racable model which can be used as a benchmark. Anoher drawback of his modelling approach comes from he irreversibiliy of pension annuiies. In real world, erm or lifelong survival coningen producs pay a cashflow for he lengh of he erm or as long as he buyer is alive. However, he inroducion of life long paymens may necessiaes complicaed numerical reamens and hese models may no be racable. 4

5 .3 The Objecive The wage earner sars wih wealh x R + and receives income a a predicable rae i() during he period [0, min{t, τ}]; i means ha he income will be erminaed by he invesor s deah or benchmark ime T, whichever happens firs. A every ime, he wage earner has wealh X() and makes he following decisions: he/she chooses π(), he fracions for he wealh process invesed in he risky asse, c() he consumpion rae, and p(), he proporion of he wealh o be paid (received) for life insurance (pension annuiy). The wealh process {X π,c,p ()} [0,T ] = {X()} [0,T ] saisfies on [0, min{t, τ}] he self-financing equaion dx() = [X()(r + (µ() r)π() c() p()) + i()] d + X()π()σdW (). (.5) The iniial wealh X(0) = x is a primiive of he model. The wage earner oal legacy if he/she dies a ime is he wealh plus insurance amoun ( I() = X() 1 + p() ). (.6) η() In order o evaluae he performance of an invesmen-consumpion-insurance sraegy he wage earner uses an expeced uiliy crierion. For a sraegy α = {π, c, p} and is corresponding wealh process, le us define [ T τ J(, x; α) = E,x U (s, c(s)x(s)) ds + βu (τ, I(τ))1 {τ T } + U (T, X(T ))1 {τ>t } ], where E,x is he condiional expecaion operaor, given X() = x, and U is he wage earner uiliy. I is furher assumed ha ime preferences are exponenial and risk preferences are of CRRA ype, i.e., U (s, x) = e ρ(s ) x wih < 1, and ρ > 0. Here 1 sands for he coefficien of relaive risk aversion. The posiive consan β is he weigh on he wage earner legacy s uiliy. A high β reflecs he fac ha he uiliy of he legacy is more imporan. A more realisic model would perhaps allow for differen consumpion and legacy uiliies, bu in such a case one may loose racabiliy. The following lemma shows ha he above expeced uiliy risk crierion wih random ime horizon is equivalen o one wih a deerminisic planning horizon; for he proof see []. Lemma.1 The funcional J of (.7) equals J(, x; α) E,x [ T (.7) ] [F (s; )U (s, c(s)x(s)) ds + βf(s; )U (s, I(s))] ds + F (T ; )U (T, X(T )), where F (s; ) and f(s; ) are given by (.4) and (.3) respecively. The wage earner s objecive is o find he sraegy α = {π, c, p } which maximizes he risk crierion, i.e., α = arg max J(, x; α). (.9) (.8) 5

6 3 The Analysis Due o he marke compleeness we can compue D(, θ(), i()), he presen value of he income flow a rae i(s) on [, τ], as expecaion under he maringale measure Q of he discouned fuure income payoff. Imagine ha he wage earner buys life insurance a rae i(s) on [, τ], hen a deah ime τ i will pay off i(τ) η(τ). This observaion leads o he following resul. Lemma 3.1 The ime value, D(, θ(), i()), of he fuure [, τ] income is given by and D(, θ(), i()) = E Q [ E Q R s [e σ i = i() θ(u)du ] = e σ i((θ() k θ δ Here δ = k σ θ, and ζ(x) = 1 δ (1 e δx ). Appendix A gives he Proof. e r(s ) i(s)e R s λ(u)du ds] e (νi r)(s ) e R s λ(u)du E Q R s [e σ i θ(u)du ]ds, (3.1) k θ )ζ(s )+ δ (s )) e σ i σ R θ s ζ (s v)dv. Given, he curren ime, in ligh of his Lemma we can assume ha he wage earner does no receive income bu his/her wealh is X() + D(, θ(), i()). The amoun D is refer o as he value of human capial. The HJB equaion associaed wih maximizing (.8) is he following second-order PDE: wih he boundary condiion v (s, x, θ, i) (ρ + λ(s))v(s, x, θ, i) + sup H(s, x, θ, i; α) = 0 (3.) s α Here, he Hamilonian funcion H is given by v(t, x, θ, i) = x. (3.3) H(s, x, θ, i; α) = [(r + σθπ c p)x + i] v x + 1 σ π x v v + ( k(θ θ)) x θ + 1 v σ θ θ + ν ii v i + 1 σ i i v i σσ θπx v x θ + σσ iπxi v x i σ iσ θ i v i θ + (cx) px (x + λ(s) + βλ(s) ). (3.4) The firs order condiions (FOC) are π = θ v x σ θ v θ x + σ ii v i x σx v x, c = ( v x ) 1 1 x 6, p = λ ( 1 β v x ) 1 1 x 1. (3.5)

7 We inroduce he following PDE which is inimaely relaed o (3.): + σ θ h s + wih erminal condiion ρ + λ(s) (r + βλ(s) + 1 θ (1 ) )h + ( k(θ θ) 1 θσ θ) h θ h + K(s) = 0. (3.6) θ h(t, θ) = 1. Here K(s) = 1 + βλ(s). We are able o solve his PDE in closed form by using Feyman-Kac Theorem. Lemma 3. The soluion of (3.6) is given by R T T h(, θ) = [e q(u)du e θa()+b() R s + K(s)e q(u)du e θa(s)+b(s) ds] Appendix B gives he Proof. Theorem 3.3 The funcion v(s, x, θ, i) = h 1 (s, θ) (x + D(s, θ, i)), (3.7) solves he HJB equaion (3.)-(3.3). The opimal admissible invesmen-consumpion-insurance sraegy α (s) = {π (s), c (s), p (s)} s [,T ] is given by π (s) = (X (s) + D(s, θ(s), i(s)))θ(s) σ(1 )X (s) σ θ D θ (s, θ(s), i(s)) σx (s) + σ ii(s) D i σ θ(x (s) + D(s, θ(s), i(s))) h θ ((s, θ(s)) σ(1 )X (s)h(s, θ(s)) (s, θ(s), i(s)) σx, (s) (3.8) c (s) = [h(s, θ(s))] 1 X (s) + D(s, θ(s), i(s)) X, (3.9) (s) [ ] p (s) = λ(s) β 1 [h(s, θ(s))] 1 X (s) + D(s, θ(s), i(s)) X 1. (3.10) (s) Appendix C gives he Proof. 7

8 3.1 Opimal Consumpion versus Legacy Theorem 3.3 yields he raio beween he opimal legacy I (s) and opimal consumpion C (s) = c (s)x (s). Corollary 3.4 For every < 1, we have I (s) C (s) = β1. This resul says ha he legacy/consumpion raio is independen of he financial marke characerisics. I depends on he invesor s primiives: he coefficien of risk aversion 1, he beques weigh β. If a loading facor is assumed i will also depend on i as well as on he hazard rae. The higher he he weigh β he higher he he legacy/consumpion raio. This fac is somehow expeced. Furhermore, a more risk averse wage earner consumes less in favour of he legacy. If he weigh β = 1, hen he opimal legacy equals he opimal consumpion. Le us now examine he relaionship beween wealh, consumpion and life insurance/pension annuiy. If he wage earner buys life insurance hen he/she consumes more; his may be explained by he legacy being supplemened by he insurance paymen in case of deah (so he/she may consume more and save less for his/her heirs). On he oher hand if a pension annuiy is acquired he/she consumes less so ha here will be enough money a he deah ime for legacy and paying off he pension annuiy. In real world for he case of pension annuiies, hose who survive share in he moraliy credi, money ha are added o invesmen reurns from he pension annuiy. Le us poin ou ha he finding of his Corollary is coningen on he special ype of risk preferences considered (CRRA ype wih he same coefficien of risk aversion for consumpion and legacy). 4 Numerical resuls In his secion, a 35 years old wage earner (he iniial ime = 35), has an iniial wealh of 50,000 dollars and a benchmark ime T = 65, invess (in he financial marke) consumes and buys life insurance (pension annuiies) as o maximize his/her expeced uiliy. The financial marke parameers are r = 0.01, i() = 4000, ν i = 0.0, σ i = 0.04, σ = , σ θ = , k = 0.06 and θ = The weigh β = 1, ρ = 0.03, and λ() = x exp 10.54, (Gomperz hazard funcion). We are mainly ineresed in he impac MPR has on he opimal insurance sraegy. 35 Sock invesmen versus MPR 1 Insurance amoun versus MPR 30 = = π(θ) 15 p(θ) θ θ 8

9 70 Sock invesmen versus MPR 1 Insurance amoun versus MPR 60 = = π(θ) 30 p(θ) θ θ A risk averse wage earner ( = 1) invess less in he sock han a risk seeking wage earner ( = 0.1) and his invesmen is increasing in he MPR θ. A risk averse wage earner pays less for life insurance as long as he MPR θ is small. However when MPR θ is large enough he wage earner pays more for life insurance. On he oher hand, if he wage earner is risk seeking hen he/she pays less for life insurance when MPR θ ges large. 5 Conclusion The purpose of life insurance is o provide financial securiy for he holders of such conracs and heir families. Life insurance acquisiion can be considered in connecion wih consumpion and invesmen in he financial marke. The problem of opimal invesmen, consumpion and life insurance acquisiion for a wage earner who has CRRA risk preferences is analysed in his paper. The wage earner receives income a a sochasic rae, consumes, invess in a sock and a risk free asse and buys life insurance. The MPR of he sock is assumed o follow a mean revering process. The opimal sraegies in his model are characerized by HJB equaion which is solved in closed form. Numerical resuls explore he effec of he MPR on he opimal sraegies. 6 Appendix A Proof of Lemma 3.1 Define he maringale measure Q, such ha W = W + 0 θ(s)ds is a Brownian moion under probabiliy measure Q. Under Q, D(, θ(), i()) can be compued as follows: D(, θ(), i()) = E Q i(τ) [e r(τ ) λ(τ) ]. Thus D(, θ(), i()) = E Q [ = e r(s ) i(s)e R s λ(u)du ds] e r(s ) e R s λ(u)du E Q [i(s)]ds. (6.1) 9

10 The dynamics of sochasic income under Q measure is: di() = i()[(ν i σ i θ())d + σ i d W ()], so i(s) = i()e (ν i σ i )(s ) e σ i( W (s) W ()) e σ R s i θ(u)du. The dynamics of MPR θ() is dθ() = (k σ θ )θ()d + k θd σ θ d W (). Recall ha δ = k σ θ, so d(e δ θ()) = e δ (k θd σ θ d W ()). Consequenly Inegraing, we ge: s θ u = e δ(u ) θ + k θ δ (1 e δ(u ) ) σ θ θ u du = (θ() k θ δ where ζ(x) = 1 δ (1 e δx ). Thus D(, θ(), i()) = i() k θ )ζ(s ) + (s ) δ u s e δ(u v) d W (v), σ θ ζ(s v)d W (v), (6.) e (νi r)(s ) e R s λ(u)du E Q R s [e σ i θ(u)du ]ds, where E Q R s [e σ i θ(u)du ] = e σ i((θ() k θ δ k θ )ζ(s )+ δ (s )) e σ i σ R θ s ζ (s v)dv. B Proof of Lemma 3. For every s [, T ], we define he following process: dy(s) = ( k wih he iniial value y() = θ, where 1 σ θ)y(s)ds + σ θ dŵ (s), Ŵ (s) = k θ σ θ s + W (s). Le G(s, y(s)) = er s 0 r(u,y(u))du h(s, y(s)), where wih q(s) = r(s, Θ) = q(s) + ΓΘ, ρ + λ(s) (r + βλ(s) ), 1 10

11 and Γ = (1 ). Applying Iô formula o G(s, y(s)) leads o ) R s dg(s, y(s)) = e 0 ( K(s)ds r(u,y(u))du h + σ θ (s, y(s))dŵ (s). y Inegraing from o T and aking expecaion condiioned on y() = θ, one ges Iô formula gives where â = (k + Le R T h(, θ) = e q(u)du R T T Ê[e Γy (u)du R s ] + K(s)e q(u)du R s Ê[e Γy (u)du ]ds. (6.3) 1 σ θ), ˆb = dy (s) = â(ˆb y (s))ds + ĉy(s)dŵ (s), σ θ (k+ 1 σ θ), ĉ = σ θ. Le Y (s) = y (s), so ha dy (s) = â(ˆb Y (s))ds + ĉ Y (s)dŵ (s). P (, y) = Ê[e R T ΓY (u)du ], where Y () = y = θ. The funcion P saisfies he following PDE wih he erminal condiion Le us guess ha P + ΓyP + â(ˆb y) P y + ĉ y P = 0, (6.4) y P (T, Y (T )) = 1. (6.5) P (, y) = exp (a()y + b()). Plugging his guess ino (6.4) we ge he following Riccai sysem for he funcions a(), b(): a () = ĉ a () + â a() Γ, wih he boundary condiions Define δ = â Γĉ, Case 1: If δ > 0, le = δ, and b () = âˆb a(), a(t ) = b(t ) = 0. a() = Γ(e (T ) 1) + (â + )(e (T ) 1), 11

12 Case : If δ = 0, le Case 3: If δ < 0, le = δ, and â+ (T ) b() = âˆb ĉ log[ e + (â + )(e (T ) 1) ] a() = ĉ (T â ) + â ĉ, T b() = âˆb a(u)du a() = ( ĉ an (T ) + an 1 ( â ) ) + â ĉ, Finally, Therefore T b() = âˆb a(u)du R T Ê[e Γy (u)du ] = e θa()+b(). R T T h(, θ) = [e q(u)du e θa()+b() R s + K(s)e q(u)du e θa(s)+b(s) ds]. C Proof of Theorem 3.3 By Lemma 3.1 one can hink of he wage earner as having an iniial wealh X() + D(, θ(), i()) and no income insead of having an iniial wealh X() and income sream a rae i. Following Proposiion in [9], he soluion of he HJB (3.) is v(s, x, θ, i) = h 1 (s, θ) (x + D(s, θ, i)), s [, T ]. The funcion h is chosen such ha V (s, x, θ) = h 1 (s, θ) x, solves he HJB where V s (s, x, θ) (ρ + λ(s))v (s, x, θ) + sup H(s, x, θ; α) = 0, (6.6) α H(s, x, θ; α) = (r + σθπ c p)x V x + σ π x V V + ( k(θ θ)) x θ px (x + + σ θ V θ σσ θπx V x θ + (cx) + βλ(s) λ(s) ). (6.7) 1

13 Based on he FOC, we obain c (s) = ( V x ) 1 1, x π V x (s) = θ V θ x σ θ, σx V x p (s) = λ(s) ( 1 V β x ) 1 1 x 1. Subsiuing (c, π, p ) ino he HJB equaion (6.6) leads o he following PDE: V V (ρ + λ(s))v + (rx + βλ(s)x) V + ( k(θ θ)) s x θ + σ θ V θ V (θ x σ θ V x θ ) + 1 (1 + βλ(s))( V x ) 1 = 0, (6.8) V x wih he erminal condiion V (T, x, θ) = x. Nex, guess he soluion of his equaion o be of he form V (s, x, θ) = h 1 (s, θ) x. Le K(s) = 1 + βλ(s). Then h solves he following PDE + σ θ h s + wih erminal condiion ρ + λ(s) (r + βλ(s) + 1 θ (1 ) )h + ( k(θ θ) 1 θσ θ) h θ h + K(s) = 0. (6.9) θ Funcion h was found in Lemma 3.. References h(t, θ) = 1. [1] Duare, I., Pinheiro, A., Pino, A, and Pliska, S. (011) An Overview of Opimal Life Insurance Purchase, Consumpion and Invesmen Problems, Dynamics, Games and Science 1,

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