Asymptotic behavior of an optimal barrier in a constraint optimal consumption problem

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1 Asympoic behavior of an opimal barrier in a consrain opimal consumpion problem Peer Grandis Insiu für Wirschafsmahemaik TU Wien Wiedner Haupsraße 8-1,A-14 Wien Ausria Augus 214 Keywords: opimal consumpion, singular conrol problem, free boundary value problem, asympoic behavior AMS subjecs classificaions. 49J2,35R37,45J5 Absrac In he aricle [5] we have invesigaed he following problem: For a given upper bound of he ruin probabiliy, maximize he expeced discouned consumpion of an invesor in finie ime [,T]. The endowmen of he agen is modeled by Brownian moion wih posiive drif. I was shown here ha he opimal sraegy is a barrier sraegy wih coninuous ime-dependen barrier funcion b(). In he presen paper we show ha, under some regulariy condiions, he asympoic behavior for he barrier funcion, if ime ends o mauriy, is given by T b() σ (T )ln(t ) = 3, where σ denoes he volailiy of Brownian moion. 1 Inroducion In a recen paper [5] we considered he following problem. The endowmen of an economic agen before consumpion is described by Brownian moion wih posiive drif µ, i.e. by he process x+µ+σw, where x is he iniial endowmen, W sandard Brownian moion and σ he volailiy. Furhermore, we describe his accumulaed consumpion by a nondecreasing process C, which allows he possibiliy of lump sum consumpion. The goal of he agen is o maximize his expeced deflaed consumpion over a finie ime horizon [, T]. Moreover, an upper bound on he ruin probabiliy is given. In erms of acuarian mahemaics his can also be seen as an opimal dividend problem of an insurance company (see, e.g., [1] or [6] for he problem wihou consrain and an infinie ime horizon, resp. [3] and [4] for he unconsrain problem wih finie ime horizon). Le us also noe ha, by subsiuing he endowmen level zero wih an arbirary posiive level, one easily ges he pgrand@fam.uwien.ac.a, el , fax

2 soluion of he problem maximize your consumpion, while saying above a cerain level wih a cerain posiive probabiliy, by solving our original problem. I was shown in [5] ha he soluion of he described problem is a so called barrier sraegy, which means ha we consume jus as much as i is necessary o keep he endowmen below a cerain ime-dependen barrier funcion. Moreover, i was shown ha his barrier funcion is coninuous and can be approximaed by C 1 -soluions of cerain inegral equaions. In he presen paper we wan o invesigae he asympoic behavior of he opimal barrier funcion b(), if T. I urns ou ha, under some regulariy assumpions, we have T b() σ (T )ln(t ) = 3. Beside he inrinsic ineres in finding he asympoic behavior of he barrier funcion, his resul can also help o improve he numerical calculaion of he barrier, as i was done in [5], Fig Le us finally noe ha he asympoic behavior of he opimal barrier funcion in he unconsrain problem is given by T b() σ (T )ln(t ) = 1, as we have shown in he second par of [4]. Since his paper is only a shor noe, we have ried o keep his inroducion as concise as possible. The reader can find more background and references in he inroducion of he aricle [5]. 2 The model and preinary resuls We model he endowmen of he agen a ime by he following process X = x+µ+σw C, T. (1) Here x, σ, µ are posiive consans, describing he iniial endowmen, he volailiy and he drif of he process. W denoes sandard Brownian moion on a filraion IF, which fulfills he usual condiions of righ coninuiy and compleeness. We furher assume ha C is an admissible process, i.e. C A, where A consiss of all IF-adaped, nondecreasing, caglad processes, fulfilling C X. Since C describes he accumulaed consumpion, his means ha we are no allowed o consume a lump sum larger han our curren endowmen. Our exremal problem wih consrain is he following one J (,x,c) := IE [ τ e βs dc s +e βτ X τ ] max, IP(τ < T), where τ denoes he ime of ruin, i.e. τ := inf{s > X s = } T, and a consan wih < 1 < < < 1, where 1 is he ruin probabiliy of an agen no consuming a all. Noe ha he symbol τ is mean o include a possible jump a ime s =, and ha, for definieness, we se X s :=,C s := C τ+, for s > τ. Moreover, concerning he consan, we noe he following: If we consider he same problem as (2), bu wihou consrain on he ruin probabiliy, he soluion can be found in [3] and [4]. I urned ou ha a barrier sraegy wih ime dependen barrier is opimal for his problem. I is shown in he inroducion of [5] ha, due o he asympoic behavior of his opimal barrier for T, which has been provided in [4], an agen, using his opimal sraegy, has a sric posiive survival probabiliy, denoed by 1. Clearly, his sraegy solves our consrain problem wih, and we ge he relevan inerval for as 1 < <. } (2) 2

3 We now formulae a new problem wih penaly [ τ ] J 1 (,x,c) := IE e βs dc s +e βτ ( γ +X τ )1 {τ=t} max, (3) where 1 {} denoes he indicaor funcion, whereas γ is some posiive consan, conrolling he weigh for he penalizaion of premaure ruin (resp. a reward for saying alive ). Finally we define he value funcion of problem (3) as V(,x) := sup J 1 (,x,c), (4) C A where J 1 (,x,c) is defined analogously as J 1 (,x,c) wih he saring poin of inegraion insead of. Now, he algorihm provided before Theorem 2.2 in [5] shows ha, choosing an appropriae weigh of penalizaion γ, he soluion of problem (3) is also he soluion of he problem (2). Hence, Theorem 2.2 of [5] implies Theorem 2.1 The soluion of problem (2) (as well as of problem (3) for an appropriae chosen γ) is given by a barrier sraegy wih a coninuous barrier funcion b(), T wih b(t) =, i.e. he opimal consumpion process and he opimal sae process are given by C = max s b(s)] s +, >, C =, X = x+µ+σw C. (5) Moreover, we have ǫk b (ǫ k) () = b() poinwise, where he b (ǫ k) are C 1 soluions of he inegral equaion (6.2) in [5], and he sequence ǫ k can be consruced explicily (see Lemma 7.1 here). 3 The resul In his secion we describe and prove our main resul, namely he asympoic behavior of he opimal barrierfuncion, ifime endsohemauriy ime T. Inheformulaion andheproofofhis resul we shall simulaneously use he original ime, as well as he backwards running ime ν := T, depending on he siuaion. Moreover, we will denoe by R he funcions of regular variaion (for ν ); see, e.g., [2] for an inroducion ino his opic. Theorem 3.1 a) Le be he opimal barrier funcion for he problem (2), and le g(ν) := σ νlnν, hen we have inf ν g(ν) = 3. b) If we assume in addiion ha R holds, we have ν g(ν) = 3. Proof. In he proof of Lemma 7.3a of [5] one finds Claim 2, which implies inf ν for all δ wih < δ < 3. This gives immediaely inf ν g(ν) δ, g(ν) 3. (6) 3

4 Hence i is sufficien o show Claim 1. inf ν g(ν) 3. We argue by conradicion: So le us assume ha here exis δ,ν > wih g(ν) 3+δ, (7) for all ν ν. The idea of he proof is he following. In [5] we found a formula, expressing he value funcion a he free boundary as funcional of he funcion b(s),s [,T]. Basically his is done by a smooh fi procedure. We show now ha, if he opimal barrier funcion would fulfill propery (7), hen one could find a differen barrier(sraegy) giving a beer arge funcional han he value funcion for ν small enough, hence he assered conradicion. In [5] we found V(ν,) = e β (b()+d()) = e β ( b()+ µe βs βe βs b(s) ) ds+γ = e β b()+ µ ( e β e βt) T β e βs b(s)ds+γ. (8) β For he firs equaliy we used he formula appearing in he paragraph afer eq. (9.49) in [5], for he second one (9.49) iself, as well as he fac ha H ends o zero for ǫ by he sanding assumpions of secion 6 in [5], and finally he definiion γ := e βt γ. Le us noe ha he employed expressions for V(, x), b(), d() are given in [5] acually for an approximaing problem described by he perurbaion parameer ǫ. Bu since we have V (ǫ) (,x) V, b (ǫ) b() and d (ǫ) d(), for ǫ poinwise (see eq. (7.8), resp. Lemma 7.4 and Proposiion 7.5 here), we can use hese formulas for our value and barrier funcion. We now evaluae our arge funcional wih he barrier sraegy C, which is a barrier sraegy wih he barrier funcion b (ν) := ( 3+)g(ν) and < δ 2 : J(ν, =: f(ν)g(ν),c ) = µ β ( e β IE [e βτ]) [ τ +e β f(ν)g(ν) βie e βs X s ds ] +γip(τ = T). (9) Here we have used τ for he ruin ime, if sraegy C is chosen, and eq. (3.5) in he proof of Proposiion3.2of[5]. Weonlyhadoreplacehefuncionf n (τ)herebyγ1 {τ =T} here. Subracing (9) from (8) yields V(ν,) J(ν,,C ) = µ ( [ IE e βτ] e βt) +γip(τ < T) β [ ] [ τ ] T +βie e βs (X s b(s))ds βie e βs b(s)ds. (1) τ We sar wih he esimaion of he firs erm and ge ( IE µ β [ e βτ] e βt) = µ β e βt ( IE [ ] ) e β(t τ ) 1 DνIP(τ < T) Dνν ( 3+) 2 2 lnν Dν 5 2 +, (11) 4

5 for ν small enough. Here D denoes a generic, model parameer depending, consan, possibly varying from place o place. Moreover, Proposiion 9.8, equaion (9.1) of [5] is used in he second inequaliy. Le us noe ha he proof of (9.1) here for he exponen 3 2 works as well for a general parameer a, which we se here equal o ( 3+) 2 2. For he second erm of he r.h.s. of (1) we ge easily he upper esimae Dν 3 2 +, and for he fourh ha i is an o(ν 2). 5 Alogeher we arrive a [ ] τ r.h.s.(1) Dν βie τ Dν βie Dν β 2 Dν β 2 e βs (X s b(s))ds e βs( ( ) 3+)g(s) b(s) ds ( e βs g(s) ( 3+) b(s) ) ds g(s e βs g(s)( δ) ds Dν βδ e βs g(s)ds 4 <, (12) for ν small enough. Here we have used in he second inequaliy he fac ha - apar from possibly he saring poin - he process X s lies below he barrier b, in he hird one he smallness of he probabiliy IP(τ < T), for he fourh one our assumpion (7) and finally for he fifh one he fac ha we have chosen < δ 2. (12) gives us he desired conradicion, proves herefore Claim 1 and hence our asserion a). The proof of b) is again by conradicion. So - using a) - le us assume ha here exiss a δ > and a sequence ν i, s.. f(ν i ) = b(ν i) g(ν i ) = 3+δ, (13) and le us assume w.l.o.g. ha δ < 1/1 holds as well. The uniform convergence heorem for slowly varying funcions L (see, e.g., [2], Theorem 1.2.1) implies f(cν) ν f(ν) = 1, uniformly for all c [δ,1]. This gives he exisence of a ν >, s.. holds for all ν ν, uniformly in c [δ,1]. Hence, for all ν [δν i,ν i ] and all i IN. Defining G(ν i ) := ν i f(cν) f(ν) > 1 δ2 (14) f(ν) f(ν i )(1 δ 2 ) = ( 3+δ)(1 δ 2 ), (15) δν i f(ν)g(ν)dν g(ν)dν, his gives δν i ( 3+δ)(1 δ 2 )g(ν)dν = ( 3+δ)(1 δ 2 )(G(ν i ) G(δν i )) = ( ( 3+δ)(1 δ 2 )G(ν i ) 1 G(δν ) i), (16) G(ν i ) and we noe ha he las bracke ends by our assumpion g R 1/2, which implies by Karamaas heorem G R 3/2, o (1 δ 3/2 ), for i. 5

6 Using now he hird inequaliy of (12), as well a he backwards running ime, yields V(ν i,b(ν i )) J(ν i,b(ν i ),C ) Dν i + β 2 We asser now Claim 2. Dν i + β 2 e β(t ν)( ( 3+)g(ν) ) dν <, for i large enough, which would be clearly he desired conradicion. We sar wih he following definiion and obvious asympoic equivalence H(ν i ) := β 2 for i. Hence, o show Claim 2, i is enough o prove e β(t ν)( ( ) 3+)g(ν) dν. (17) e β(t ν) ( 3+)g(ν)dν β 2 e βt ( 3+ )G(ν i ), (18) Dν i +H(ν i ) β 2 for i large enough. Sufficien for his in urn is or Using (16), we have o show Dν i +H(ν i ) β 2 e βt (1 δ 2 ) e β(t ν) dν <, dν <, Dν i +H(ν i ) β 2 e βt (1 δ 2 ) f(ν)g(ν)dν <. δν i Dν i +H(ν i ) β 2 e βt (1 δ 2 )( 3+δ)(1 δ 2 )G(ν i )(1 2δ 3/2 ) <, for i large enough. Dividing his by G(ν i ), we end up wih he asserion ha i suffices o prove Dν i G(ν i ) + H(ν i) G(ν i ) β 2 e βt ( 3+ δ) ) <, (19) 2 for i large enough. Now, as he firs erm ends o and he second one o β 2 e βt ( 3 + ), for i by (18), i is enough o chose = δ 4 o prove (19). This gives he desired conradicion and concludes herefore he proof of asserion b) and he heorem.. Acknowledgemens: Suppor by he Ausrian Science Foundaion (Fonds zur Förderung der wissenschaflichen Forschung), Projec nr. P26487, is graefully acknowledged. References [1] S. Asmussen and M. Taksar, Conrolled diffusion models for opimal dividend pay-ou, Ins. Mah. Econom., 2 (1997), pp [2] N. H. Bingham, C. M. Goldie and J.L. Teugels, Regular variaion, Cambridge Universiy press, Cambridge, [3] P. Grandis, Opimal consumpion in a Brownian model wih absorpion and finie ime horizon, Applied Mah. and Opimizaion 67, pp , (213). [4] P. Grandis, Exisence and asympoic behavior of an opimal barrier for an opimal consumpion problem in a Brownian model wih absorpion and finie ime horizon, Applied Mah. and Opimizaion 69, pp , (214). 6

7 [5] P. Grandis, An opimal consumpion problem in finie ime wih a consrain on he ruin probabiliy, o appear in Finance and Sochasics (214). [6] S.E. Shreve, J.P. Lehoczky and D.P. Gaver, Opimal consumpion for general diffusions wih absorbing and reflecing barriers, SIAM J. Conrol Opim., 22 (1984), pp