Stochastic Optimal Control Problem for Life Insurance

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1 Sochasic Opimal Conrol Problem for Life Insurance s. Basukh 1, D. Nyamsuren 2 1 Deparmen of Economics and Economerics, Insiue of Finance and Economics, Ulaanbaaar, Mongolia 2 School of Mahemaics, Mongolian Universiy of Science and echnology, Ulaanbaaar, Mongolia 1 Inroducion Beginning in he 196 s, many researchers consruced models o analyze for life insurance and he behavior of invesmen for an individual under an uncerain lifeime. Yaari 1965 is a saring poin for he modern research on he demand for life insurance. Yaari considered he problem of life insurance under an uncerain lifeime for an individual. Following Meron s 1969,1971 work, Richard 1975 used he Yaari s seing for an uncerain lifeime and dynamic programming o consider a life-cycle life insurance and consumpion invesmen problem. Richard employed he dynamic programming echnique o aack his problem and o obain explici soluions for Consan Relaive Risk Aversion CRRA case. 2 Formulaion of he model he coninuous ime economy consiss of a financial marke and an insurance marke. We assume ha here is a risk-free securiy in he financial marke whose ime- price is denoed by S. I evolves according o Dynamic of financial asses for individuals ds S = rd where S = s is given posiive consan and r : [ τ, ] R + is a coninuous deerminisic funcion. τ is a beginning ime o earn wage of a consumer, -is a planning horizon for he wage earner. Suppose he wage earner is endowed wih he iniial wealh W τ and will receive he wage a rae Y during he period [ τ, ]. is an age of reire. Here he specified funcion Y : [ τ, ] R + is Borel measurable and saisfies Y d <. τ Define c-consumpion rae a ime, P -amoun of insurance purchased a age, θ- insurance premium rae a ime and I τ -indicaor funcion of ime se τ. Given he consumpion process c, he wage process Y, he premium rae process θp, 39

2 Mongolian Mahemaical Journal 4 he wealh process W for consumer s on [ τ, ] is defined by dw = rw d + Y I { τ d cd θp d, τ. 1 We suppose he wage process Y is defined by dy = µ y d + σ y db, τ. 2 Here µ y is an average wage funcion, σ y is a volailiy wage funcion, B is a Brownian moion. Subsiuing 2 ino 1, we have dw = rw + µ y I { τ c θp d + σ y I { τ db Objecive funcion We denoed lifeime, a nonnegaive random variable defined on he probabiliy space {Ω, F, P. Now suppose ha he random variable τ has a probabiliy disribuion wih underlying probabiliy densiy funcion π and disribuion funcion given by F = P τ < = πd. he funcion S, which is called he survivor funcion, is defined o be he probabiliy ha he survival ime is greaer han or equal o S = P τ = P τ P τ < = 1 F = πd, where S = 1, S =, and S = π <. he hazard funcion represens he insananeous deah rae for wage earner who has survived o ime and i is defined by P τ + δ τ λ = lim. δ δ From his definiion we can obain some useful relaionships beween π and λ, namely I hen follows ha λ = π S. 4 λ = π S = S S = d ln S. d he case in which he survivor funcion is given by S = e λτdτ. Given an uncerain lifeime wih a densiy funcion π, consumer s expeced uiliy is E π ψ = [ πψd = π ] e τ ρvdv U τ, cτ dτ + Φ, Z d

3 Mongolian Mahemaical Journal 41 = π e τ ρvdv Ucτdτ d + π Φ, Z d. Where U τcτ is he uiliy for consumpion, Φ, Z is he uiliy for he beques, e τ ρvdv is he discoun funcion. Using inegraion by pars as before, we can obain objecive funcion as E π ψ = 2.2 Life insurance model max {c, P [πφ, Z + Se ] ρτdτ U, c d. We suppose ha problem for life insurance can be wrien bellow { [πφ, Z + Se ] ρτdτ U, c d Subjec o: dw = rw + µ y I { τ c θp d + σ y I { τ db, Z = W + P, W τ = W τ, 5 where Z is he coningen beques. 3 Expliciy soluion s necessary condiion Le he presen value of he indirec uiliy funcion J, W a ime be { J, W = [πφ, Z + Se ] ρτdτ U, c d max {c, P Now we can wrie HJB equaion for problems 5. HJB equaion is.. = max {λsφ, Z + Se ρτdτ U, c + J {c, P + rw + µ y I { τ c θp J W σ2 yi { τ J W W Where J, J W, and J W W are parial derivaives. For he simpliciy, le he value funcion be of he forms 6 J, W = Se ρτdτ F, W = e ρτ+λτdτ F, W, Φ, Z= 1 φz, e ρτdτ. λ

4 Mongolian Mahemaical Journal 42 For F, W funcion, he above equaion is = max {φz, + U, c ρ + λ F + F {c, P + rw + µ y I { τ c θp F W σ2 yi { τ F. W W Assuming ha ρ + λ = ρ, ϑ = 1 τ in he above equaion, we have = max {φz, + U, c {c, P ρ F + F + rw + µ y I { τ c 1 τ P F W σ2 yi { τ F W W. 7 Hence he firs order condiions are 4 Main resul { U c = F W φ Z = 1 τ F W. 8 heorem 1. In he case he uiliy funcion for consumpion is of he form CARA, U, c = α e c, and he uiliy funcion for beques is of he form Φ, Z = A 1 λ e ρτdτ e KZ, and F, W = be aw hen he soluion of he problem 5 and is parameers are deermined by followings: Opimal conrols are: c = 1 aw + ln α a, P = ab K 1 W and parameers are: [ { a = exp rτ + 1 τ τ dτ π K τ b = exp ab K τ, K τ τ { { exp aτ exp dτ { τ D = µ y I { τ ln α a rs + 1 s τ ds dτ ] 1 A + π Dτ { s exp as 1 K τ 1 2 σ2 yi { τ a ds dτ, where A = a + ρ.

5 Mongolian Mahemaical Journal 43 Proof. We shall guess he value funcion F o be of he form F, W = be aw. 9 Here a, b are deerminisic funcions. By using he firs equaion of he firs-order condiion 8, we have αe c = abe aw. his implies c = 1 aw + ln α ab. 1 Similarly by using he second equaion of he firs-order condiion 8, we have I follows Consequenly we ge and we have φz, = Ae KZ. φ Z = AKe KZ. AKe KW +P = ab τ e aw, A = ab K τ, P = a 1W. 11 K Subsiuing hese resuls and F W, F W W ino he HJB equaion 7, we have obained he following equaion ab = K τ ab + rw + µ y I { τ σ2 yi { τ a 2 b. + ρ b + Hence we can find he Bernoulli equaion for a a + r + 1 a = τ I follows and for b cons A a = K τ τ ba W b aw + ln a ab [ { exp rτ + 1 exp ln b a 1 τ K τ { τ τ dτ τ a K 1 W ab a 2. rs ds dτ], S τ ln b = D. 12

6 Mongolian Mahemaical Journal 44 Here D = µ y I { τ ln α a From 12, we have { { aτ b = exp exp dτ cons B + Using he boundary condiion, we can ge and which complees he proof. 5 CONCLUSION 1 K τ 1 2 σ2 yi { τ a Dτ cons A = π K τ, cons B = A π a + ρ. s { as exp ds dτ. In his paper, we se up a new model o invesigae he problem of opimal life insurance purchase, consumpion for sochasic wage under an uncerain lifeime. An analyical soluion of above porfolio opimizaion problem was invesigaed. Following menioned resul has been obained by assuming he uiliy funcion o be of he form CARA: opimal consumpion and opimal insurance premium are linear in wealh wih ime variable coefficien. 6 REFERENCE 1. Fwu-Rang Chang 24, Sochasic Opimizaion in Coninuous ime, Cambridge universiy press, Unied Kingdom. 2. Jinchun, Ye. 26, Opimal Life Insurance Purchase, Consumpion and Porfolio Under an Uncerain Life, hesis for he degree of Docor of Philosophy in he Graduae College of he Universiy of Illinois a Chicago. 3. Meron, R.C. 1969, Lifeime Porfolio Selecion Under Uncerainy: he Coninuous- ime Case, Review of Economics and Saisics, 51, Meron, R.C. 1971, Opimal Consumpion and Porfolio Rules in a Coninuous ime Model, Journal of Economic heory, 3, Richard, S,F., Opimal consumpion, porfolio and life insurance rules for an uncerain lived individual in a coninuous ime model, journal of financial economics 2, Yaari, M.E, 1965, Uncerain Lifeime, Life Insurance, and he heory of he Consumer, Review of economic Sudies 32,

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