Risk Sensitive Portfolio Management With CoxIngersollRoss Interest Rates: the HJB Equation


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1 Risk Sensitive Potfolio Management With CoxIngesollRoss Inteest Rates: the HJB Equation Tomasz R. Bielecki Depatment of Mathematics, The Notheasten Illinois Univesity 55 Noth St. Louis Avenue, Chicago, IL USA Stanley R. Pliska Depatment of Finance, Univesity of Illinois at Chicago 6 S. Mogan Steet, Chicago, IL USA ShuennJyi Sheu Institute of Mathematics, Academia Sinica Nankang, Taipei, Taiwan 59 ROC Octobe, 3 Abstact This pape pesents an application of isk sensitive contol theoy in financial decision making. The investo has an infinite hoizon objective that can be intepeted as maximizing the potfolio s isk adjusted exponential gowth ate. Thee ae two assets, a stock and a bank account, and two undelying Bownian motions, so this model is incomplete. The novel featue hee is that the inteest ate fo the bank account is govened by CoxIngesollRoss dynamics. This is significant fo isk sensitive potfolio management because the facto pocess, unlike in the Gaussian and all othe cases teated in the liteatue, cannot be negative. Keywods: model isk sensitive contol, optimal potfolios, CIR inteest ates, incomplete AMS Subject Classification: 6J, 9A9, 9C4, 93E The eseach of the fist autho was patially suppoted by NSF Gant DMS99737, the thid autho was patially suppoted by NSC 95M35
2 Risk Sensitive/CIR Potfolio Management JEL Classification: C6, C63, G, E
3 T.Bielecki/S.Pliska/S.Sheu 3 Contents Intoduction 3 Fomulation of the Optimal Risk Sensitive Asset Management Poblem 4 3 Analysis of the HamiltonJacobiBellman Equation 6 4 Poof of Theoem Poofs of Theoems 3. and Intoduction Beginning with the pioneeing wok by Meton [], [], [] and continuing though the ecent books by Kaatzas and Sheve [7] and Kon [8], some vey sophisticated stochastic contol methods have been developed fo potfolio management. Vitually all of these studies make use of an expected utility citeion. But ecently a new citeion has emeged fom the contol theoy liteatue. Called the isk sensitive citeion, this was oiginally used (see, fo example, Whittle [5] fo a decision make seeking to maximize some (andom cash ewad (o minimize some cash payment while simultaneously being concened about the isk o uncetainty in the size of the ewad. Essentially, this citeion equals the expected value of the ewad minus a penalty tem that is popotional to the vaiance of the ewad. The constant of popotionality is a paamete whose value can be chosen in ode to achieve fo the decision make an appopiate tadeoff between the expectation of the ewad and its vaiance. Recognizing its elevance to potfolio management, Bielecki and Pliska [5] applied the isk sensitive idea to a vesion of Meton s [] intetempoal capital asset picing model. The esult was an infinite hoizon citeion that they called the isk adjusted gowth ate and viewed as being analogous to the classical Makowitz singlepeiod appoach except that instead of tading off singlepeiod citeia the investo is tading off the potfolio s long un gowth ate vesus its aveage volatility (see Bielecki and Pliska [9] fo a detailed study of vaious economic and mathematical popeties of this citeion. Bielecki and Pliska also showed in [5] and subsequent wok (see [], [3], [4], [6], [7], [8], and [] that the esulting models usually have the vitue of being moe tactable than coesponding models which use taditional expected utility citeia. Othe studies of the isk sensitive citeion fo potfolio management include Bagchi and Kuma [], Fleming and Sheu [3], [4], [5], Kuoda and Nagai [9], Nagai [3], and Nagai and Peng [4]. Kaise and Sheu [6] discuss the solution of a geneal equation (in R n that is elated to the HJB equation in this pape. Thoughout all this wok on isk sensitive potfolio management the undelying facto pocess, if any, was taken to be Gaussian o, at least (see Nagai [3], a pocess whose domain is all of some Euclidean space. The aim of this pape is to povide some initial esults on isk sensitive potfolio management fo a case whee this kind of condition does not hold. Since inteest ate pocesses
4 4 Risk Sensitive/CIR Potfolio Management ae commonly taken as facto pocesses and since the socalled CoxIngesollRoss [] inteest ate pocess (a popula one in finance liteatue cannot be negative, this model of the facto pocess was chosen fo ou object of study. The esult is a isk sensitive potfolio optimization model having a facto pocess whose domain is the nonnegative potion of the eal line. Since this is a model of inteest ates, it is moe ealistic than, say, Gaussian models, but it comes with a pice: the esulting analysis is exceptionally lengthy, complex, and technical. This is tue even though ou model is athe simple, having just this scalavalued facto pocess, two assets (the usual bank account and a isky stock, and two undelying Bownian motions. Consequently, this pape will study only the associated HamiltonJacobiBellman equation, saving the veification of optimality and elated issues fo a sepaate pape. Afte fomulation of ou model in Section, the main esults ae pesented in Section 3. Chief among these is Theoem 3., which assets the HJB equation has a unique solution. Needed fo its poof and of sepaate inteest ae some esults petaining to a elated, tuncated poblem: fo some fixed numbe M the investo is equied to keep all of his o he money in the bank account wheneve the inteest ate exceeds M. Existence of a unique solution to the HJB equation fo this tuncated poblem is established by Theoem 3.. Intuitively, one should expect the solution of the tuncated HJB equation to convege to the solution of the oiginal one as M ; this is indeed the case, as stated in Theoem 3.3. The est of the pape is devoted to the poofs of these thee theoems. Theoem 3. is poved in Section 4, wheeas the othe two ae poved in Section 5. Fomulation of the Optimal Risk Sensitive Asset Management Poblem In this section we fomulate an optimal dynamic asset management poblem featuing a isk sensitive optimality citeion. Let (Ω, {F t } t, F, P be the undelying pobability space. The secuities maket involves a single facto, namely, an inteest ate that is subject to the socalled Cox IngesollRoss [] dynamics d t = c( t dt + λ t dw t, ( whee c,, and λ ae thee specified positive scala paametes. In ode to ensue that the inteest ate pocess is always stictly positive, we make the following (see Felle [] Assumption: c > λ. Thee ae two assets. One is the customay bank account: ds (t S (t = tdt; ( hee S (t epesents the time t amount of money in the bank account assuming none is added o withdawn afte time. The othe asset is a stock (o stock index whose pice pocess satisfies ds (t S (t = µ( tdt + σdw t + ρd W t. (3
5 T.Bielecki/S.Pliska/S.Sheu 5 Hee W t and W t ae two independent Bownian motions, σ and ρ ae two specified scala paametes, and µ( := µ + µ, (4 whee µ and µ ae two specified scala paametes. Note that with µ we can allow the level of inteest ates to affect the etun popeties of the stock, and with σ the esiduals of the inteest ate pocess will be coelated with the esiduals of the stock s etun pocess. Fo instance, with suitable values of σ and ρ this coelation is negative. Tading stategies will be adapted ealvalued stochastic pocesses that ae denoted h. We shall intepet h t as the popotion of the investo s timet wealth that is invested in the stock. In geneal, fo each timet we allow h t to be any eal numbe, that is, we do not impose any shot selling estictions, etc. Additional assumptions about admissible tading stategies will be povided below. The investo s timet wealth will be denoted V t. Unde the tading stategy h, the coesponding wealth pocess V will satisfy dv t V t = [( h t t + h t µ( t ]dt + h t (σdw t + ρd W t. (5 By standad esults, thee exists a unique, stong, and almost suely positive solution to this equation; it is given by ( t V t = V exp h t σdw t + t h t ρd W t + t [ (σ + ρ h t + ( h t t + h t µ( t ]dt. (6 In this pape we conside the following family of isk sensitized optimal investment poblems, labeled as P θ : fo θ (,, maximize the isk sensitized expected gowth ate J θ (v, ; h := lim inf t ( /θt ln E h [e (θ/lnvt V = v, = ] (7 ove the class of all admissible investment pocesses h, whee E h is the expectation with espect to P. The notation E h emphasizes that the expectation is evaluated fo the wealth pocess V coesponding to the investment stategy h. The paamete θ hee is intepeted as the measue of the investo s attitude towad isk; the bigge the value of θ, the moe isk avese the investo. This is because the citeion can be intepeted, at least appoximately, as the potfolio s exponential gowth ate minus a penalty tem which equals θ/4 times the potfolio s asymptotic vaiance. A compehensive intepetation of this isk sensitive objective fo potfolio management can be found in Bielecki and Pliska [9]. We note that the techniques used in this pape can also be used to study poblems P θ fo negative values of θ, coesponding to isk seeking investos. The isk null case, fo θ =, can be studied independently o as the limit of the isk avese situation when the isksensitivity paamete θ goes to zeo. Howeve, we shall not conside the cases whee θ in this pape. Fo much of what follows we find it convenient to intoduce the scala paamete γ := θ/. (8
6 6 Risk Sensitive/CIR Potfolio Management Since θ is always stictly positive, the paamete γ should always be egaded as stictly negative. Moeove, the eade should keep in mind that coesponding to any appeaance of the paamete γ is θ = γ. 3 Analysis of the HamiltonJacobiBellman Equation In this section we fomulate ou model and pesent ou main esults concening the HamiltonJacobi Bellman equation coesponding to the investo s potfolio optimization poblem P θ. We not only establish existence and uniqueness of a solution, we also establish some impotant popeties of this solution. This analysis is athe involved, and so the balance of this pape is devoted to the poof of the esults in this section. In view of ou isk sensitive objective, we ae inteested in computing the expectation of quantities like V γ t fo some γ <. Since by equation (6 V γ t = V γ exp (γ t h t σdw t + γ t h t ρd W t + t γ[ (σ + ρ h t + ( h t t + h t µ( t ]dt, (9 we ecognize that it is convenient to make a Gisanovtype change of pobability measue. In paticula, it is staightfowad to show fo each tading stategy h and T > that [ T ] E[V γ T ] = Ẽ V γ (γ exp L( t, h t dt, ( whee we have intoduced the notation Ẽ fo expectation unde the new pobability measue and the additional functions L(, u := ( γ(σ + ρ u + µ(u +, ( and µ( := µ(. ( Moeove, unde this new pobability measue the dynamics fo the inteest ate pocess ae given by d t = ( c( t + γσλ t h t dt + λ t d W t, (3 whee W denotes a (scala valued Bownian motion unde this new pobability measue. Using standad methods of isk sensitive contol theoy (see, fo example, [8], [4], and [9], it is now staightfowad to specify the HamiltonJacobiBellman dynamic pogamming equation. This is Λ = λ d Φ c( dφ d d + λ ( dφ d [ + inf γσλ u dφ ] {u R} d + γl(, u. (4 We seek a solution in tems of the scala Λ and the bias function Φ such that Λ is the optimal isk adjusted gowth ate in poblem P θ and such that the minimal selecto identifies an optimal (o, at least, an ɛoptimal tading stategy. It is convenient to tansfom this equation into a simple fom. Since the stock popotion h t is unesticted, we see that the minimizing value of u in the HJB equation must satisfy the fist ode
7 T.Bielecki/S.Pliska/S.Sheu 7 condition γσλ Φ + γ[ ( γ(σ + ρ u + µ(] =. In othe wods, ou candidate h fo the optimal tading stategy will satisfy the expession h t = u ( t, whee u ( := γ ( σ + ρ Substituting this value of u in the HJB equation, intoducing the function g := dφ d, µ( + σλ dφ. (5 d and doing a little algeba enables one to see that the oiginal HJB equation is equivalent to Λ = λ dg d + ( λ + γ σ γ σ + ρ g + b(g + d(, (6 whee we have intoduced fo convenience the functions b( := c( + γ σλ µ( (7 γ σ + ρ and d( := γ γ σ + ρ [ µ(] + γ. (8 Hee is ou main esult about the HJB equation: Theoem 3. The HJB equation (6 has a unique solution (Λ, g satisfying the following two popeties: lim g ( = [ Λ γ µ ] c γ σ + ρ (9 and eithe g ( ( lim = + γ σ γ σ + ρ fo µ o ( lim g ( = + γ γ [ µ λ γ γ σ + ρ + γ γ σ ( ( c c σ + ρ λ λ 4 γ ( λ + γ γ ] σ µ σ + ρ λ ( σ σ + ρ ( fo µ =. Moeove, Λ is chaacteized as the smallest Λ such that the HJB equation has a solution defined fo all. In ode to study poblem P θ, as well as to investigate a elated poblem of sepaate inteest, conside exactly the same poblem except that now, fo some abitay positive numbe M, we impose the tading stategy constaint that h t = if t > M. Analogous to the unconstained poblem, the dynamic pogamming equation fo this constained, tuncated poblem is Λ M = λ dg d + ( λ + γ σ γ σ + ρ g + b(g + d(, M, (
8 8 Risk Sensitive/CIR Potfolio Management Λ M = λ dg d + λ g c( g + γ, > M. (3 In addition, ou candidate fo the optimal tading stategy is now given by u M ( := ( γ σ + ρ µ( + σλ g(, M, (4 u M ( :=, > M. Moeove, fo this constained poblem we have the following impotant esult: Theoem 3. The HJB equation (, (3 fo the constained poblem has a unique solution (Λ M, g M satisfying the following two popeties: and lim g M ( = [ Λ M c γ γ µ ] σ + ρ (5 lim g M ( = c c λ λ 4 γ λ. (6 As ou next main esult indicates, the solutions of the two kinds of investment poblems ae elated in an intuitive manne. Theoem 3.3 The following hold: and lim M g M ( = g ( (7 lim M Λ M = Λ. (8 Remak. Fo the equation (6, thee is a smallest Λ such that (6 has a smooth solution W. This follows fom the agument in [6]. We can show that Λ in Theoem 3. is the smallest Λ mentioned above. See 5.3. The agument in [6] is applicable to the equations in multidimensional spaces, and theefoe, can be applied to a model with seveal assets and multiple facto pocesses. Howeve, thee is difficulty to obtain W and to undestand its behavio. These thee theoems ae poved in the following two sections. We now conclude this section by suggesting a pocedue fo computing the solution (Λ, g of (6. Suppose fo any numbe Λ we can solve fo the function g satisfying (6 and (9. It tuns out that Λ is chaacteized as the smallest Λ such that this solution g is finite fo all >. Theefoe, if some value Λ gives a finite solution then Λ Λ. On the othe hand, if some value Λ does not coespond to a finite g, then Λ < Λ. Hence a suitable iteative pocedue should convege to (Λ, g. 4 Poof of Theoem 3. We begin by making a tansfomation of the constained HJB equation (. Denoting A := + γ σ γ σ + ρ,
9 T.Bielecki/S.Pliska/S.Sheu 9 Λ := AΛ, d( := Ad(, and ḡ := Ag, (9 we see by simple substitution that ( is equivalent to Λ = λ dḡ d + λ ḡ + b(ḡ + d(, M. (3 We would like to know that this equation has a (possibly unique solution ḡ fo an abitay Λ, but establishing this is not so easy because the second tem on the ight hand side is nonlinea and the coefficient of the fist deivative tem is degeneate at =. Ou appoach will be to addess these issues by studying the function g( := ḡ(e( whee e( := c λ ( exp c λ + γσ µ(s ( γλ(σ + ρ ds. s This is because ḡ satisfies equation (3 if and only if g satisfies and this latte diffeential equation will be easie to analyse. d g d + e( g = λ e([ Λ d(], (3 Lemma 4. If the diffeential equation (3 has a solution g on (, ] fo some g( as. >, then Poof. We fist pove that fo any c > thee ae n, n =,,, which tend to as n tends to infinity and which satisfy g( n > c. If not, thee is > such that g( c fo all <. Since we have d g d g + e( = λ e([ Λ d(] g ( g( g( = e(s ds + λ s e(s[ Λ d(s] g ds. (3 (s Fo small >, the fist tem on the ighthand side is bounded above by c c λ + fo some c > and the second tem is bounded above by c 3 /c fo some c 3 >. Fom this, the ight hand side conveges to as, in which case g( as, a contadiction. By this esult we can take a sufficiently small > such that g( > c. Next we show that g( > c,. (33 To see this, suppose this is not tue. Then thee is some < such that g( = c and g( > c, < <.
10 Risk Sensitive/CIR Potfolio Management By (3 d g d ( = e( c + λ e( [ Λ d( ] <. This contadicts the specified popety of c. Finally, it suffices to show that fo any c > thee is some 3 > such that g( c, 3, (34 because it is easy to see that ou lemma follows fom (33 and (34. To pove (34, suppose thee is 4 > such that g( 4 > c. Then d g d ( 4 e( 4 c + λ e( 4 [ Λ d( 4 ] <. 4 Theefoe, g( is deceasing at 4. This agument also shows that g is deceasing on the set { g( > c }. Then we must have g > c on (, 4 ]. This leads to a contadiction since using (3 with = 4 and small we have the ight hand side tending to while the left hand side is bounded. This completes the poof of the Lemma. Theoem 4. If ḡ is a solution of (3 defined on (, ] fo some >, then eithe lim ḡ( = c λ + (35 o lim ḡ( = ( Λ c A γ µ γ σ + ρ. (36 Remak. Note that (36 is equivalent to (5. Poof. By (3 we have so by Lemma 4. we have g( d g d λ e([ Λ d(], λ s e(s[ Λ d(s]ds. Substituting fo e(s and so foth, it is appaent this implies fo some numbe c > that The next main step is to show fo some numbe c > that We conside two cases, depending upon whethe o not g( c c λ, < <. (37 g( > c c λ, < <. (38 Λ A γ µ γ σ + ρ. (39 Fist we suppose inequality (39 is tue, in which case Λ d( > fo all positive in some neighbohood of zeo. We have one of the following two possibilities: thee ae infinitely many
11 T.Bielecki/S.Pliska/S.Sheu n >, n =,,, which tend to as n tends to infinity and ae such that g( n < ; o thee is > such that g( fo <. We conside the fist possibility. Then thee is sufficiently small such that g( <. It is easy to see by (3 that g( < fo all. The conditions (39 and (3 imply d g d g e(, in which case g( g( e(s ds, that is, g( g( + e(s ds. It follows that fo some constants c, c, we have g( > c c λ, (4 since By (3 again we have e(s ds c c λ +. g( g( = e(s ds + λ s e(s[ Λ d(s] g ds. (4 (s We use this to study the limiting behavio of g(. Fo the fist tem on the ight hand side we have by L Hospital s ule lim c λ e(s ds = c λ. Fo the second tem on the ight hand side of (4 we have λ s e(s[ Λ d(s] g (s ds c Hee we use (4. So by (4 we have lim s c c + λ λ + ds c c λ +. g( = c +. (4 λ c λ Hence fo the fist possibility ( i.e, g( < fo some small > and when (39 holds, we have poved (38 and (35. Now we conside the second possibility: thee is > such that g( fo all. Since g( = e(s g (sds + then it follows by L Hospital s ule and (37 that λ s e(s[ Λ d(s]ds, (43 g( lim = ( Λ c λ c A γ µ γ σ + ρ. (44
12 Risk Sensitive/CIR Potfolio Management This with the elation g( = ḡ(e( and the definition of e( imply (36. We summaize what we have shown. Assuming the condition (39, we have (4 o (44. They ae equivalent to (35 and (36, espectively. They also imply (38. Fo the emainde of this poof we conside the opposite case, namely, whee inequality (39 does not hold. We choose > such that Λ d( < fo <. Now (4 and the definition of e( imply g( g( e(s ds c ( 3 c + λ fo some numbe c 3 >. The second inequality holds fo > small. By (43, g( < fo < < if is small. In paticula, g( <. So by the above inequality we can pove (38. We now asset that g( > c λ δ fo some small > and some positive δ with δ < implies If not, we can find < < such that g( > c λ δ,. f( <, < and f( =, whee f( := g( + c λ δ. Since df d = d g ( c d + λ δ c λ δ 3 g ( = e( + λ [ Λ d(]e( ( c + λ δ c λ δ 3, it is easy to see this is stictly positive at = by using f( =. This contadicts the popety of, and so the assetion is established. Let < δ < / and > be small, and conside two situations, depending upon whethe g( > c λ +δ, <. (45 Fo the fist situation, assume (45 does not hold fo infinitely many which tend to. Then by the peceding assetion thee is > such that g( c λ +δ,. Note that we have aleady established the popety g( < fo small. Fom these, by (3 we have g (s g( = e(s ds + e(s λ s [ Λ d(s]ds c c λ +δ c c λ c c λ +δ, since δ < /. That is, g( c c λ +δ.
13 T.Bielecki/S.Pliska/S.Sheu 3 Continuing in an iteative fashion one obtains g( c c λ +mδ, if m is such that m δ <, whee c may depend on m and δ. Especially, this holds fo m = m, m δ < m+ δ. Apply this same pocedue once moe to obtain g( c c λ + δ c c λ c c λ, (46 whee δ = m δ. The last step is due to δ >. We shall now show (44 by the following calculation. In view of (37 and (38 we have c e(s lim λ λ s [ Λ d(s]ds = ( Λ c A γ µ γ σ + ρ. (47 In addtion, by (37 and (46 we have c lim λ e(s g(s ds =. This with (47 and (43 imply (44, which is equivalent to (36. Now we conside the opposite situation, namely, thee is < δ < / such that (45 does hold fo some. Then g( g( = e(s ds + Coesponding to the two tems on the ight hand side we have and Hence lim c λ e(s ds = c λ lim c λ lim λ s e(s[ Λ d(s] g (s ds. λ s e(s[ Λ d(s] g ds =. (s g( = c +, (48 λs c λ so by the definition of e( and the elationship between ḡ and g we see that (35 holds. completes the poof of this theoem. This Fom now on we shall focus on solutions of (3 that satisfy (36 athe than (35. The eason will become appaent below. In paticula, see Coollay 4. which gives special popeties of the solution satisfying (36. In the poof of Theoem 3. it will be seen that the smallest Λ such that (3 has a finite solution fo all < M coesponds to a ḡ which satisfies (36. See 5.3. Suppose a solution ḡ of (3 and (36 exists, and conside the coesponding solution g of (3, so that g also satisfies (43. Denote g ( ( := λ s e(s[ Λ d(s]ds
14 4 Risk Sensitive/CIR Potfolio Management and Then Since (36 implies fo small, we have, g ( := g( g ( (. g ( = g (s e(s ds. g( c c λ g ( c c λ +. Moeove, g( = g ( ( = g ( ( g ( (s e(s ds [ g ( (s + g (s] e(s ds g ( (s g (s e(s ds Continuing this pocedue, we may define g (n ( ecusively by t g (n+ ( = g ( ( e(s g(n (s ds. We can show by induction that the following holds : This gives an asymptotic expansion of g( fo small >. g (s e(s ds. g( = g (n ( + O ( c λ +n+. (49 Lemma 4. Fix Λ. Fo small enough > thee exists a unique g satisfying fo all (, ] both (3 and (43. It also satisfies g( c c λ,, (5 fo a positive numbe c, as well as (44(this is equivalent to (36 if we take ḡ( = g(/e(. In addition, we have (49 and g( λ s e(s[ Λ d(s]ds,. (5 Remak. Since thee is a one to one coespondence between solutions of ( satisfying (5 and solutions of (3 satisfying (36 (see the beginning of this section, it follows fom Lemma 4. that thee exists a solution g of ( satisfying (5, at least a solution in some neighbohood of =. Poof. Fo some suitable positive numbes and c (to be decided late, conside the opeato T defined fo f F c, whee T f( := f (s e(s ds + λ s e(s[ Λ d(s]ds
15 T.Bielecki/S.Pliska/S.Sheu 5 and F c := {f : f( c c λ, }. In ode to show that T f F c we need to estimate T f(. The fist tem in the definition of T is bounded by c c s δ ds = c c δ+, whee δ := c λ and /e(s c s δ, c = c /( + δ. The second tem is bounded by c ( Λ c 3 s δ ds = c ( Λc 3 δ, whee c 3 = c 3 /δ and e( c 3 δ fo small, and whee c ( Λ := max < λ [ Λ d(]. Theefoe T f( [c c + c ( Λc 3 ] δ [c c + c ( Λc 3 ] δ if. It now follows by taking c = c ( Λc 3 and = /[4c c 3 c ( Λ] that and so T : F c F c. On the othe hand, fo T f ( T f ( f f c T f( c c λ, whee denotes the supnom on [, ] and f (s + f (s f (s f (s e(s ds s c λ e(s ds c c f f,, c := max e( c λ. Hence by taking small enough so that c c < we see that T will be a contaction mapping fom F c into F c. Hence T has a unique fixed point, say g, which means that g satisfies (43 and thus (3. If we define ḡ( = g(/e(, then ḡ is a solution of (3 defined on (, ]. Theefoe, in view of Theoem 4., eithe one of (35 o (36 holds. Since g is in F c, we have (5. Then (35 cannot be tue. Since (36 is equivalent to (44, (44 holds. Finally, (5 is a consequence of (43. This completes the poof of the Lemma. Lemma 4.3 Let g and g be the two solutions of (3 (equivalently, (43 coesponding to Λ and Λ, espectively, as defined in Lemma 4.. If Λ < Λ then g < g.
16 6 Risk Sensitive/CIR Potfolio Management Poof. Since g and g both satisfy equation (3 (with thei espective values of Λ we can subtact one equation fom the othe to obtain d d ( g g + e( [ g + g ][ g g ] = λ e([ Λ Λ ]. We thus have g ( g ( = e(s λ s [ Λ Λ ( ] exp s g (u + g (u du ds. e(u This is stictly negative if Λ < Λ, so Lemma 4.3 is established. Coollay 4. Fo each Λ, (3 has only one solution satisfying (36. Fo fixed Λ, assume ḡ defined on (, ] is a solution of (3 satisfying (36. Let y < ḡ (, and suppose ḡ is the solution of (3 such that ḡ( = y. Then ḡ( exists fo (, ] and ḡ satisfies (35. Remak. While uniqueness of this solution is tue fo geneal, existence of a solution has only been established fo small enough >. We note that if g( is finite, then g is well defined fo >, up to a (possibly infinite point denoted ( Λ whee lim ( Λ g( = (If g explodes, then by (5 it explodes in the negative diection. Fo each M > we now define Λ (M := inf{ Λ : the coesponding solution g of (3 satisfying (36 is finite fo all M}, and note solutions of (3 satisfying (36 ae given by (43. The peceding esults now imply the following: Coollay 4. Fix abitay < M <. Then Λ (M < and, fo each Λ > Λ (M, the coesponding solution g of (43 is finite fo all M. Moeove, g(m as Λ and g(m as Λ. Remak. Recall that a solution ḡ of (3 is well defined if and only if a solution g of (43 is well defined. Since ḡ = Ag, the peceding coollaies tell us when the constained dynamical pogamming equation ( has a unique solution fo M. In paticula, since g( = ḡ(/a and Λ = AΛ, we see that (36 implies (5. Also, note that in Lemma 5. below we pove that Λ (M >. Poof. We pove Λ (M <. The est is a consequence of eithe Lemma 4.3 o a simila agument. We fist take a small enough and a finite Λ, and g ( = λ s e(s( Λ d(sds e(s g (sds,. We know that g ( is finite fo [, ]. Now fo θ > we conside g θ ( = λ s e(s( Λ + θ d(sds e(s g θ(sds,.
17 T.Bielecki/S.Pliska/S.Sheu 7 This has solution g θ ( in a neighbohood of as given in Lemma 4. with Λ = Λ + θ. We know that g θ ( is finite fo [, ] and that g θ ( g (,. Let us fix < <, a lage K >, K > g [, ], the maximum of g (, [, ]. Thee is a θ such that fo θ > θ, g θ ( is inceasing fo, if g θ ( K. This is due to the following calculation: d d g θ( = λ e(( Λ + θ d( e( g θ( λ inf {e(( Λ + θ d [, [ ]}, ] e( [, ]K >, whee the last inequality holds if θ is lage enough. Fom this, fo a fixed K, we must have g θ ( > K if θ is lage enough. Suppose not. Then using the fact that g θ ( > g (, and the above monotonicity esult we can conclude that Thus g θ ( < K,. d d g θ( ( λ inf {e(( Λ + θ d [, [ ]}, ] e( [, ]K is lage than a given numbe (say L fo if θ is lage enough. Fo such θ, d g θ ( = g θ ( + d g θ(d g ( + L(, and this is lage than K if L is lage enough. This gives a contadiction. Next, fo a fixed K > if θ is lage enough, then g θ ( > K implies g θ ( > K, M. This follows by using the popeties that and the estimate inf M e( >, sup M e( < d d g θ( λ inf { M e(( Λ + θ d [,M]} e( [,M]K > fo a M satisfying g θ ( = K if θ is lage enough. We conclude fom the above analysis that fo a K > g [, ], thee is a θ sufficiently lage such that g θ (M > K. This implies Λ (M Λ + θ <. As a consequence of this agument, we also have that g θ (M tends to as θ tends to. This ends the poof. We now tun to the study of the solution of the constained dynamical pogamming equation fo > M, that is, equation (3. But the solution of this diffeential equation must satisfy the bounday condition at = M that has g(m taking the value that comes fom the solution of ( and (5 fo M.
18 8 Risk Sensitive/CIR Potfolio Management Lemma 4.4 Given a specified value of g(m, equation (3 has a unique solution g on [M,, whee := sup{ > M : g( > }. Also, thee exists some K <, which does not depend on, such that g( K on [M, (but may depend on g(m. Poof. It is sufficient to pove the existence of K. The existence of follows fom the theoy of odinay diffeential equations. Fist note that (3 can be ewitten as dg d c [ ] g + g λ = λ [Λ M γ], M. (5 It suffices to show that if a solution is such that g( < c, then g( < c fo all >, whee and c hee ae lage. Suppose, on the contay, thee is some > such that g( < c fo < and g( = c. Then by diffeential equation (5 we must have dg d ( <. But this is a contadiction, so Lemma 4.4 is established. Fo fixed M and any Λ > Λ (M/A we know by Coollay 4. that ( with Λ M = Λ has a solution g on (, M] with g(m finite. So coesponding to each such Λ we can, as in the following lemma, conside the solution g of (3 on [M, that takes this coesponding value of g(m at = M. In othe wods, fo each Λ > Λ (M/A we have a solution of ( and (3 that is continuous on [, fo some > M. Lemma 4.5 Let g and g be two solutions of ( and (3 coesponding to Λ and Λ, espectively, whee Λ, Λ > Λ (M/A. Then Λ < Λ implies g ( < g ( if g is defined at. Poof. Fist conside two diffeential equations (5, one satisfied by (g, Λ M = Λ and the othe by (g, Λ M = Λ. Subtacting one fom the othe gives d d (g g + [ c λ ( / + g ] + g (g g = λ [Λ Λ ], in which case ( g ( g ( = exp M ( + M λ s (Λ Λ exp ( c λ ( /s + g (s + g (s ds [g (M g (M] Since g (M g (M >, Lemma 4.5 follows fom this. s ( c λ ( /u + g (u + g (u du ds. Fo each M >, we now define Λ M := inf{λ M : the coesponding solution g of (, (3 satisfying (5 is finite fo all }, and we obseve that Λ M Λ (M/A. The peceding esults imply the following: Coollay 4.3 Fo each fixed numbe M < we have Λ M <.
19 T.Bielecki/S.Pliska/S.Sheu 9 Poof. We need to pove the existence of a Λ M such that g( is finite fo all >, whee g is the solution fo (, (3 and (5. By (5, if g(m > and Λ M >, then g( > fo all > M. We can show by the agument in the poof of Lemma 4.5 that g(m > if Λ M is sufficiently lage. This completes the poof. By Coollay 4.3 we know fo Λ M Λ M that thee exists a solution of (, (3 and (5. We now investigate the limiting behavio of this solution g as. Theoem 4. Fix M < and abitay Λ M Λ M, and conside the solution g = g M of (, (3 satisfying (5. Then exactly one of the following two conditions will hold, that is, eithe o lim g( = c λ c λ 4 γ λ (53 lim g( = c c λ + λ 4 γ λ. (54 Poof. Denote α := c λ and note that α is negative and satisfies c λ 4 γ λ, α c λ α + γ λ =. Next, define ḡ( = g( α, and note that, in view of (3, we must have dḡ ( c d + λ 4 γ λ + c λ We now claim thee is c lage enough such that ( ḡ + ḡ ΛM = λ c λ α, > M. (55 ḡ( c /, M. (56 If not, then not only is thee some > M such that ḡ( < c /, but thee is some > M such that ḡ( < c /,. (57 To see this, suppose ḡ( < c / but (57 is false. Then thee is some such that ḡ( = c / and ḡ( < c / fo <. It then follows fom (55 that dḡ c d ( [ c λ 4 γ ( λ + ΛM λ c λ α ]. This implies df d ( <, whee f( := ḡ( + c /. But this is a contadiction, so we see that if (56 is false, then thee exists some > M such that (57 is tue. The ḡ used in this poof must not be confused with the ḡ that is the solution of diffeential equation (3.
20 Risk Sensitive/CIR Potfolio Management Using (55 and (57 one can show that This, in tun, implies dḡ d + ḡ <,. ḡ( + ḡ( + ( <,. But this cannot be tue fo all, so (56 must be tue. We now conside two cases, depending upon whethe o not Λ M λ c α >. (58 λ If (58 is tue and ḡ( > fo some > M, then ḡ( > fo all. Fom this, we conclude that one of the following two possibilities holds: eithe thee is > M such that ḡ( >,, (59 o thee is > M such that ḡ( <,. (6 We have the same conclusion if the opposite of (58 holds, so it suffices to conside (59 and (6 sepaately. Fist we assume (6. This togethe with (56 implies In othe wods, lim ḡ( =. lim g( = α, (6 which is equivalent to (53 in this case. Fo the est of this poof we shall assume (59 and show that fo small c > and some > M then eithe ( c ḡ( + λ 4 γ λ + c λ > c,, (6 o ( c ḡ( + λ 4 γ λ + c λ < c,. (63 Indeed, it is easy to see that in ode to pove this assetion it suffices to show that if ( ḡ( + c λ 4 γ λ + c λ > c (64 holds fo some c > and fo = >, then (64 in fact holds fo all. To pove this last statement, assume the contay: thee is some > such that (64 holds fo < and equality holds in (64 fo =. In this case df ( c d ( = c c λ 4 γ λ + c ( ΛM λ + λ c λ α c λ >,
21 T.Bielecki/S.Pliska/S.Sheu whee we have defined ( c f( := ḡ( + λ 4 γ λ + c λ. But this is a contadiction, so (64 must be tue fo all. Ou next step is to show (63 cannot hold. This is because it and (55 would imply since ḡ( >, in which case dḡ ( d > c ΛM ḡ + λ c λ α,, ḡ( exp{c ( }ḡ( + ( ΛM λ c λ α s exp{c ( s}ds. But this contadicts Lemma 4.4, so we conclude that if ḡ( > fo all (i.e., if (59 holds, then (6 holds fo any c > povided is lage enough. Having established (6, we now pove that fo a fixed lage thee is a c lage enough such that To pove this, we conside the function ( c ḡ( + λ 4 γ λ + c λ < c,. (65 ( c f( := ḡ( + λ 4 γ λ + c λ. We choose c such that f( < c / (66 fo =. Ou next objective in this poof is to show that this implies (66 is tue fo all. Othewise, thee is some > such that (66 is tue fo < and equality holds in (66 fo =. By (55 again it is easy to see that df ( d ( + c / ΛM = ḡ( ( c / + λ c λ α + c < if c is lage enough. But this is anothe contadiction, so (66 must be tue fo all. Now (6 and (65 imply lim f( =. This is equivalent to (54. This togethe with (6 completes the poof of Theoem 4.. It tuns out that the limit (53 is the one we want; (54 will now be ignoed. See Lemma 4.7. Hee again we ae inteested in the smallest Λ M such that (, (3, and (5 have a solution fo all. The following lemma says thee is at most one value of Λ M giving a solution of (, (3, and (5 that also satisfies (53. Lemma 4.6 Suppose g and g ae two solutions of (, (3, and (5 coesponding to Λ M = Λ, Λ, espectively. If both g and g satisfy (53, then g = g and Λ = Λ.
22 Risk Sensitive/CIR Potfolio Management Poof. Subtacting the equation fo g fom the equation fo g gives By (53 we then have d d (g g + ( c λ + c λ + g + g (g g = λ (Λ Λ. (g g (ē( = λ s (Λ Λ ē(sds, (67 whee we have intoduced the function { ( ē( := exp c λ + c } λ s + g (s + g (s ds. Hee > M is fixed. The integal on the ight hand side of (67 is finite by (53. Moeove, (67 implies g g > if Λ Λ <. But we also have g g < if Λ Λ <. Theefoe Λ = Λ and g = g, that is, the poof of Lemma 4.6 is completed. It emains to pove the solution g coesponding to Λ M = Λ M satisfies (53. In othe wods, with Lemma 4.6 establishing uniqueness, it emains to establish existence. This is a consequence of the following lemma, because if fo Λ M = Λ the coesponding limit is (54, then fo all Λ M < Λ in some neighbohood of Λ the coesponding limits also satisfy (54. Hence if thee exists a solution fo Λ M = Λ M (the infimum of Λ M fo which thee exists a solution; Λ M is finite by Coollay 4. above and Lemma 5. below, and the infimum is attained by the same kind of agument used below in the poof of Theoem 5., this solution must satisfy the othe limit, namely (53. Lemma 4.7 If Λ M = ˆΛ is such that the coesponding solution of (, (3 satisfies (5 and (54, then thee exists some δ > such that fo any Λ M > ˆΛ δ the solution of (, (3 exists fo all. Poof. Let ĝ be the solution coesponding to ˆΛ, so dĝ d c λ ( ĝ + ĝ = λ (ˆΛ γ, > M. With g being the solution of ( and (3 coesponding to Λ, wite ḡ := g ĝ so dḡ d + dĝ d c λ ( (ḡ + ĝ + (ḡ + ĝ = λ (Λ γ. This implies dḡ d + ( c λ ( + ĝ ḡ + ḡ = λ (Λ ˆΛ. (68 We now seek the solution of (68 such that ḡ δ, whee ḡ := sup ḡ(. M Hee δ will be chosen late in a manne which depends on δ, whee Λ ˆΛ < δ. Note that (68 can be ewitten as ḡ( = ḡ(m ē( ē(ḡ ē(s (sds + ē(s (Λ ˆΛ M λ M ē( s ds,
23 T.Bielecki/S.Pliska/S.Sheu 3 whee we intoduced the function { ( ē( := exp c } λ ( /s + ĝ(s ds. We use again the fixedpoint agument to get a solution g. Denote M F := {f : [M, R, f δ, f(m = ḡ(m}, whee ḡ(m = g(m ĝ(m, and whee g is the solution of ( coesponding to Λ M = Λ. We denote fo f F T f( := ḡ(m ē( ē(s M ē( f (sds + ē(s (Λ ˆΛ λ M ē( s ds. We know ḡ(m if Λ ˆΛ. We conside whee δ is small. Then take δ > satisfying Note fo δ small enough we can take Λ ˆΛ < δ, δ = ḡ(m max M ē(, δ ē(s sup M M ē( ds + δ + δ δ = ( δ + δ λ sup λ sup M M { M ē(s { M ē( s ds} < δ. ē(s ē( s ds}. Then it is not difficult to show that the opeato T : F F. Moeove, fo abitay f, f F we have T f T f δ sup ē(sds f f = K f f. M ē( M By taking δ small enough one has the numbe K <. Then T is a contaction with a unique fixed point in F, which is the unique solution of (3. This completes the poof of Lemma Poofs of Theoems 3. and 3.3 Ou fist esult shows that the solutions of Theoem 3. convege as M to a solution of the HJB equation (6 that also satisfies conditions (9 and ((if µ, ( if µ =. Late in this section we will show uniqueness, theeby completing the poofs of both Theoems 3. and Theoem 3.3. Theoem 5. Let Λ M and g M ( be as in Theoem 3.. Then Λ M Λ and g M ( g( as M, whee Λ and g( satisfy (6 and g( also satisfies (9 and eithe ( (in the case µ o ( (in the case µ =.
24 4 Risk Sensitive/CIR Potfolio Management To pove Theoem 5. we need the following fou lemmas. The fist two of these ae based upon the following equation: Λ = λ dg d + λ ( + γ γ σ σ + ρ g + b(g + d(, R +. (69 Hee R >, and note this equation is essentially the same as (, which is pat of the dynamical pogamming equation in Theoem 3.. Lemma 5. Let < R be abitay. Then thee is K >, depending on, R, and Λ, such that if (69 has a solution g, then g( K, R. Moeove, K can be chosen to be inceasing in Λ. Theefoe, fo, R, Λ fixed, the set { g( : R ; g satisfies (69} is bounded. Remak. If the value of the ODE solution g is specified at < R, say, then the values of g fo all will be detemined. The most inteesting pat of this lemma is the conclusion that egadless of the initial value we choose fo g at, if g( is finite in (, R + ], then g( K fo all R, whee K is independent of g, although it may depend on R. Consequently, in ode to have a solution of (68 we cannot abitaily assign a value of g at. Regading the dependence of K on Λ, an expession fo K is povided afte (7 below. This dependence will be used in the poof of Theoem 5.. Poof. Let g satisfy (69. Take φ : [, [, ] smooth such that φ( =, R, =,, R + <. (7 Without loss of geneality, we can take φ such that it satisfies the following popety: dφ( φ( d K. Hee K is some numbe that may depend on and R. To see this, we denote h( by so h( := φ( = + We then choose h( such that h( is bounded and dφ( φ( d, R, R h(udu, > R. R+ h(udu =. R Moeove, we choose h( =, R +. The deivatives of h of any ode at R and R + ae. Thus φ satisfies the equied popety on [R,. We can apply a simila agument fo (, ].
25 T.Bielecki/S.Pliska/S.Sheu 5 Conside f( = φ(g (. Then f( takes a maximum at some satisfying / R +. Denote Then that is, X = f( = max f(. df d ( =, φ( g( dg d ( = g ( dφ d (. (7 We multiply (69 at = by φ( g( and use (7 to get Λφ( g( = 4 λ g ( dφ d ( + λ ( + γ σ γ σ +ρ φ( g 3 ( +b( φ( g ( + d( φ( g(. Assume g(. Then divide the above elation by g( to obtain Λφ( = 4 λ g( dφ d ( + λ ( + +b( φ( g( + d( φ(. γ γ σ σ +ρ φ( g ( Then whee Fom (7, in which case α = α( = λ ( + γ γ β = β( = λ ( + X + αx = β, (7 σ σ +ρ (b( φ( dφ 4 λ φ( d (, γ γ σ σ +ρ ( d( φ( + Λφ(. X = α ± α + β, X α + α + β. We see X K, whee so that K depends on, R, and Λ. Since K = max{ α( + α( + β(, R + } max R g( X, the esult follows. The next lemma says that the set of all Λ such that (69 has a solution is bounded below. Lemma 5. Fo a fixed R, thee is a Λ(R such that if (69 has a solution g, then Λ Λ(R.
26 6 Risk Sensitive/CIR Potfolio Management Poof. We take < < R and a smooth function φ as in (7. We can define ˆb( fo all > such that ˆb( = b(, R and such that the diffusion pocess defined by has an invaiant density that we denote by ˆp(. Denote whee Then dˆ(t = ˆb(ˆ(tdt + λ ˆ(tdB(t Φ( = exp(( + W ( = γ γ σ σ W (, + ρ g(udu. ˆLΦ( = ˆΛΦ( ˆd(Φ(, R, whee ˆΛ = ( + γ γ σ σ +ρ Λ, We have ˆd( = ( + ˆLΦ(φ( ˆp(d = γ γ σ σ +ρ d(, ˆLf( = λ d f df d ( + ˆb( d (. ˆΛΦ(φ(ˆp(d ˆd(Φ(φ( ˆp(d. The equation fo the invaiant density ˆp is: d (λ ˆp( d d (ˆb(ˆp( =. d In one dimension, we have d d (λ ˆp( ˆb(ˆp( =. Fom this and the integation by pats fomula we then have ˆLΦ(φ(ˆp(d = λ dφ d (dφ d (ˆp(d. Consequently, ˆΛ Φ(φ( ˆp(d = ˆd(Φ(φ(ˆp(d By Lemma 5. thee is some numbe K, which depends on Λ, such that K Φ( K, dφ d ( K λ dφ d (dφ (ˆp(d. (73 d
27 T.Bielecki/S.Pliska/S.Sheu 7 fo R. Fom this and (73, we have ˆΛ Φ(φ(ˆp(d K( ˆd + λ R dφ d. The left hand side is lage than ˆΛ K φ( ˆp(d. Fom these, ˆΛ has an uppe bound depending only on R. This completes the poof. Fo the following lemma and subsequent use we shall make us of a quantity that was defined in Section 4, namely, Λ M := inf{λ M : ( and (3 has a solution satisfying (5}. Lemma 5.3 Fo each M >, the set {Λ M ; M M } is bounded above. Poof. It is enough to show that thee is Λ such that ((3 has a solution g = g M satisfying (5 such that g( > fo all, fo Λ M = Λ with M M, since this will imply Λ M Λ by the definition of Λ M. We take Λ lage enough such that ( has a solution g satisfying lim g( = c (Λ γ µ ( γ σ + ρ >. By (, it is easy to see that g( > fo < M, since in < M, g is inceasing at the zeos of g. This agument also applies to M. That is, g( cannot be fo finite. Theefoe, we get a unique solution of ((3 satisfying (5. Lemma 5.4 Let g = g M be a solution of ( and (3 satisfying (5 with Λ = Λ M Λ M. Then thee is some M > such that fo M M, g( <, > M. Poof. By (53, g( will be negative if is lage enough. Fom equation (3 and the fact that Λ is bounded below (see Lemma 5., it is easy to see that g( < fo > M if M is lage enough, since g( fo > M is inceasing at zeos of g. This agument also applies to M M. Theefoe, g( < fo > M if M is lage enough. This completes the poof. Amed with these lemmas, we can now pove Theoem 5.. Poof of Theoem 5. By Lemmas 5. and 5.3, fo a fixed M >, {Λ M, M M } is bounded above and below. We can take a sequence M n as n such that Λ M n conveges to some Λ. Boundedness of {Λ M n } also implies the unifom boundedness of { gm n ( } on compact sets by Lemma 5.. This futhe implies the unifom boundedness of { dg Mn d ( } on compact sets, by using ( and (3.
28 8 Risk Sensitive/CIR Potfolio Management Theefoe, we can take a subsequence of {M n } (still denoted by {M n }, such that gm n ( conveges to g( unifomly on compact sets. We know Λ, g satisfy (6 and g satisfies (9. In fact, we only need to ule out the possibility that g satisfies (35. But since the gm n ( satisfy (5 fo c and independent of n (see the poof of Lemma 4., it follows that (35 cannot hold fo g. It emains to pove that ( o ( (depending on the case holds fo g, because then (Λ, g = (Λ, g satisfies the popeties in Theoem 3. (see also 5.3 below. Fom this it follows that the limit of (Λ M, g M as M is unique, and so Theoem 5. will be poved. We now pove that ( holds fo g when µ. By Lemma 5.4, thee is M such that g( <, M. (74 We need to know the behavios of the solutions of (6 as. This will be given in Theoem 5.. Now g given above is a solution of (6. Define ḡ = Ag, A = + γσ /( γ(σ + ρ. Accoding to this theoem, eithe (75 o (76 holds. Fom (8, we can conclude the following. If (75 holds, then g( < fo lage. If (76 holds, then g( > fo lage. Since (74 holds, we must have (75. This in tun implies ( by a simple calculation. The case µ = is teated in a simila manne. This completes the poof. Theoem 5. Let (Λ, g be a solution of (6 fo < <. Then exactly one of the following elations holds: eithe o Hee lim (ḡ( ḡ ( = 8 ( (ḡ( ḡ ( = ( λ λ ( γ γ σ σ + ρ σ + ρ lim γ µ, µ, γ λ (76 ( c lim ( = λ 4 γ ( λ + γ σ, µ γ σ + ρ =. (77 ḡ ( := ( b( λ ( + γ γ σ σ +ρ d( λ + b( λ 4, (75 while b(, d(, and ḡ( ae defined by (7, (8, and (9, espectively. In ode to pove Theoem 5. we need thee moe lemmas. Fo these we conside a function g( that is finite fo all and satisfies (6 and (9. Using this and ḡ ( as specified in Theoem 5., we then define ĝ := ḡ ḡ. Since ḡ satisfies λ ḡ ( + b(ḡ ( + d( =,
29 T.Bielecki/S.Pliska/S.Sheu 9 it follows that whee λ d d ĝ( + λ ĝ( + b(ĝ( = L(, L( := Λ λ d d ḡ( and b( := b( + λ ḡ ( = λ ( d( λ Notice this equation can be ewitten as + b( λ 4 = ( λ d( + b(. dĝ d + ĝ + b( λ ĝ = L( λ. (78 In ode to investigate the asymptotic popeties of (78 we calculate d( λ = µ( ( + b( λ 4 γ γ cσλ λ 4 σ + ρ = γ ( λ + γ γ + λ 4 ( c( + ( σ + ρ + γ γ = γ γ σ + ρ λ µ( cσλ λ 4 Since it follows when µ that Moeove, ḡ ( = γ λ γ b( λ ( = ( σ + ρ γ σ γ γ σ σ + ρ σλ σ + ρ µ( ( γ λ + γ µ( + c λ 4 ( γ λ ( + γ γ σ + ρ σ + ρ µ( + σ (σ + ρ λ σ σ + ρ µ( + c λ 4 ( γ λ ( + γ γ b( ( λ = d( λ + b( λ 4, γ γ σ + ρ σ σ + ρ (µ ( γ γ On the othe hand, if µ = then σ σ + ρ. µ + O( as. (79 λ σ + ρ λ µ + O( as. (8 and b( λ ( c = λ 4 γ ( λ + γ σ ( + O γ σ + ρ ḡ ( = c ( c λ λ 4 γ ( λ + γ σ ( + O γ σ + ρ as (8 as.
30 3 Risk Sensitive/CIR Potfolio Management Fom this we see that if µ then L( = λ ( ( γ 4 γ σ + ρ + λ λ γ γ σ σ + ρ µ ( + O as, (8 wheeas if µ = then L( ( = Λ + O as. We ae now eady fo the fist of the thee lemmas that will be used in the poof of Theoem 5.. Lemma 5.5 Thee exist positive numbes c and such that ĝ( > c, all. (83 Poof. This poof is by contadiction. Suppose it is false. Then fo any c > and > thee exists some > such that ĝ( c /. Fom this we shall pove that ĝ( c, all. (84 But if this is not tue, then without loss of geneality thee is some > such that ĝ( = c / and ĝ( < c /, < <. Denoting f( := ĝ( + c /, we then see that df( d = ĝ( + L( λ b( λ ĝ( c if we take c lage enough. This is a contadiction; (84 must be tue if this lemma is false. By (84 and (78, we have dĝ d + ĝ <. Then dĝ d ĝ + <, which implies ĝ( ĝ( + ( < fo all >. This cannot be tue fo all, so (84 leads to a contadiction. The poof is complete. < Lemma 5.6 Suppose fo some lage > that with = we have c < ĝ( < b( λ + c. If c is lage, then this inequality also holds fo all.
31 T.Bielecki/S.Pliska/S.Sheu 3 Poof. By (78 and (8, ĝ c / is inceasing at such that ĝ c / =. Theefoe, Denote c < ĝ(,. { := inf > : ĝ( b( λ + c }. We have shown ĝ( > c / fo < <, so it suffices to show we have =. Assume not. Then ĝ( = b( /(λ + c / and We now conside ĝ( < b( λ + c, <. = L( λ f( := ĝ( + b( λ c and show that d d f( <, which leads to a contadiction. We have ( d d f( = d d ĝ( + d b( d λ ( c ( c ( b( λ + c + d d b( λ ( c. Fom this and (79,(8 we can show d d f( <. This completes the poof. Lemma 5.7 Let c > be small. With, c as in the peceding lemma such that c is lage enough, thee exists some > such that ĝ( + b( λ c fo all. Poof. We fist show that thee is some > satisfying Othewise, ĝ( + b( λ c. ĝ( + b( λ < c, all >. (85 By (78, we have dĝ d c ĝ + L( λ c ĝ c c ĝ if c is lage enough. Then ĝ( exp{ c ( }ĝ(. But this contadicts (85. With as above, we now show that ĝ( + b( λ c, all >.
32 3 Risk Sensitive/CIR Potfolio Management Othewise, thee is some > such that ĝ( + b( λ In this case we conside and thus d d f( = d d ĝ( + d d = c and ĝ( + b( λ > c, all < <. f( := ĝ( + b( λ ( ( b( ( λ = c c b( λ + L( λ + d d ( b( λ (. By (79 and (8, it is easy to see that the ight hand side is positive. But this is anothe contadiction, so this poof is complete. We ae now eady fo the poof of Theoem 5.. Recall that ĝ := ḡ ḡ. Poof of Theoem 5. By Lemmas , fo positive numbes and c, c ( c small, c lage eithe o c / < ĝ( < c /, (86 c < ĝ( + b( λ < c /,. (87 We fist suppose that (86 holds. Denote so that we have fo ĝ( = By (79 and L Hospital s Rule we then have and lim ( b(s e( := exp λ s ds, L(s e(s λ s e( ds + L(s e( λ s e(sds = ( 8 lim e( λσ λ ĝ(s e(sds =. ĝ(s e(s e( ds. ( γ ( γ σ + ρ This implies (75. On the othe hand, suppose (87 holds. Ou next step is to show that b( c < ĝ( + λ < c /,. (88
33 T.Bielecki/S.Pliska/S.Sheu 33 We do this by fist showing that thee is some numbe > satisfying (88 fo =. This is tue, fo if not then ĝ( + b( λ c,. Then (78 implies dĝ d c ĝ + L( λ c ĝ, whee c is given in (87. Integating this we obtain ĝ( ĝ( exp( c ln / = ĝ( ( c/. But this contadicts the assetion that ĝ( + b( λ < c / fo all, so we know (88 holds fo some >. Fo the final step, we use an agument by contadiction, as at the beginning of this poof, to show that the inequalities in (88 hold fo all. This also implies (88 by choosing a lage c, if necessay. Finally, (76 follows diectly fom (88, so this poof is completed. We now have fully established Theoem 5.. Thus to complete the poofs of Theoems 3. and 3.3 it only emains to establish uniqueness of the solution of the HJB equation. This is accomplished by the following lemma. Lemma 5.8 Let g and g be solutions of (6 satisfying (9 coesponding to Λ and Λ, espectively. Let ĝ = ḡ ḡ and ĝ = ḡ ḡ with ḡ defined as in Theoem 5., and suppose ĝ and ĝ both satisfy limit (75. Then g = g and Λ = Λ. Poof. Denote Λ = ( + γ σ γ σ + ρ Λ, Λ = ( + γ σ γ σ + ρ Λ. We subtact the equation fo ĝ fom the equation fo ĝ, theeby obtaining Denote Then and so d ( b( d (ĝ ĝ + λ + ĝ + ĝ (ĝ ĝ = Λ Λ λ. ( ẽ( := exp d ( (ĝ ( ĝ ( ẽ( d ( ( ĝ ( ĝ ( ẽ( = ( b(s λ s + ĝ (s + ĝ (s ds. = Λ Λ λ ẽ(, Λ Λ λ ẽ(sds. s Without loss of geneality, suppose Λ Λ, in which case ĝ ( ĝ (. But Lemma 4.3 implies ĝ ( ĝ (. Theefoe, ĝ ( = ĝ (, Λ = Λ, and this poof is completed. Ramak. The following esult, not cucial fo the poofs of Theoems 3. o 3.3, says that Λ is the smallest numbe such that (6 has a solution defined on [,. Fo Λ = Λ, (6 has a unique solution. A moe geneal esult of this kind is given in the pape by Kaise and Sheu [6].
34 34 Risk Sensitive/CIR Potfolio Management Theoem 5.3 Let Λ be given in Theoem 3.. Then thee is only one solution fo (6 on [, with Λ = Λ. If (6 has a solution on [,, then Λ Λ. Poof. We conside only µ. The agument fo the case µ = is simila. Assume Λ = Λ and g is a solution of (6 on [,. Assume g g given in Theoem 3.. Then (35 holds fo g. Since g satisfies (36, a simple compaison agument fo ODE shows that g( < g ( fo all. But g satisfies eithe one of (75 o (76. Since g satisfies (75, theefoe g < g implies that g also satisfies (75. Now by Lemma 5.8, we conclude g = g, a contadiction. We now conside Λ such that (6 has a solution g defined on [,. Then (6 must have a solution g defined on [, satisfying (36. If g also satisfies (36, then g = g. See Coollay 4.. If g satisfies (35, then we have g( > g ( fo small >, and hence fo all. This implies g is also defined fo all >. Now if Λ < Λ, then g( < g ( fo all by Lemma 4.3. By Theoem 5., g eithe satisfies (75 o (76. We know g satisfies (75. Togethe with g < g, we conclude that g satisfies (75. By Lemma 5.8, we have g = g, Λ = Λ, a contadiction. This completes the poof. Refeences [] Bagchi, A., and Kuma, K.S., (, Dynamic Asset Management: Risk Sensitive Citeion with Nonnegative Factos Constaints, Recent Developments in Mathematical Finance, edited by J. Yong, Wold Scientific, Singapoe, pp. . [] Bielecki, T.R., Hais, A., Li, Z., and Pliska, S.R., (, Risk Sensitive Asset Management: Two Empiical Examples, Poceedings of the Octobe,, Confeence on Mathematical Finance in Konstanz, Gemany, edited by M. Kohlmann, Bikhaüse, Basel, Switzeland, pp [3] Bielecki, T. R., HenandezHenandez, D., and Pliska S.R., (999, Risk sensitive contol of finite state Makov chains in discete time, with applications to potfolio management, Math. Meth. Ope. Res., vol. 5, pp [4] Bielecki, T. R., HenandezHenandez, D., and Pliska S.R., (, Risk sensitive Asset Management with Constained Tading Stategies, Recent Developments in Mathematical Finance, edited by J. Yong, Wold Scientific, Singapoe, pp [5] Bielecki, T.R., and Pliska, S.R., (999, Risk Sensitive Dynamic Asset Management, Appl. Math. Optim., vol. 39., pp [6] Bielecki, T.R., and Pliska, S.R., (, Risksensitive dynamic asset management in the pesence of tansaction costs, Finance and Stochastics, vol. 4, pp. 33.
Ilona V. Tregub, ScD., Professor
Investment Potfolio Fomation fo the Pension Fund of Russia Ilona V. egub, ScD., Pofesso Mathematical Modeling of Economic Pocesses Depatment he Financial Univesity unde the Govenment of the Russian Fedeation
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