2002（SCI）OPTIMAL CONSUMPTION AND PORTFOLIO WITH BOTH FIXED AND PROPORTIONAL TRANSACTION COSTS

OPTIMAL CONSUMPTION AND PORTFOLIO WITH BOTH FIXEDAND PROPORTIONAL TRANSACTION COSTSBERNT KSENDALAND AGNES SULEMSIAM J. CONTROL OPTIM. c 2002 Society for Industrial and Applied MathematicsVol. 40, No. 6, pp. 17651790Abstract. We consider a market model with one risk-free and one risky asset, in which thedynamics of the risky asset are governed by a geometric Brownian motion. In this market weconsider an investor who consumes from the bank account and who has the opportunity at any timeto transfer funds between the two assets. We suppose that these transfers involve a xed transactioncost k0, independent of the size of the transaction, plus a cost proportional to the size of thetransaction.The objective is to maximize the cumulative expected utility of consumption over a planninghorizon. We formulate this problem as a combined stochastic control/impulse control problem, whichin turn leads to a (nonlinear) quasi-variational HamiltonJacobiBellman inequality (QVHJBI). Weprove that the value function is the unique viscosity solution of this QVHJBI. Finally, numericalresults are presented.Key words. portfolio selection, transaction cost, impulse control, quasi-variational inequalities,viscosity solutionsAMS subject classications. Primary, 93E20, 91B28; Secondary, 60H30, 49L25, 35R45PII. S03630129003760131. Introduction. Let (,F,P) be a probability space with a given ltrationFtt0. We denote by X(t) the amount of money the investor has in the bank attime t and by Y(t) the amount of money invested in the risky asset at time t.Weassumethatintheabsenceofconsumptionandtransactionstheprocess X(t)growsdeterministically at exponential rate r, while Y(t) is a geometric Brownian motion;i.e.,dX(t)=rX(t)dt, X(0)=x,(1.1)dY(t)=Y(t)dt+Y(t)dW(t),Y(0)=y,(1.2)where W(t) is one-dimensional Ft-Brownian motion and r0 and negationslash= 0 areconstants.Suppose that at any time t the investor is free to choose a consumption ratec(t) 0. This consumption is automatically drawn from the bank account holdingwithnoextracosts. Moreover,atanytimetheinvestorcandecidetotransfermoneyfromthebankaccounttothestockandconversely. Assumethatapurchaseofsizelscriptofstocksincursatransactioncostconsistingofasumofaxedcostk0(independentofthesizeofthetransaction)plusacost lscript proportionaltothetransaction( 0).These costs are drawn from the bank account. Similarly a sale of size m of stocksincursthexedcost K0plustheproportionalcost m ( 0). Forsimplicitywewillassumethat K =k and =. InthiscontexttheinvestorwillonlychangehisReceived by the editors July 28, 2000; accepted for publication (in revised form) July 3, 2001;published electronically February 14, 2002. This work was partially supported by the French-Norwegian cooperation project Aur 99050./journals/sicon/40-6/37601.htmlDepartment of Mathematics, University of Oslo, P. O. Box 1053 Blindern, N0316 Oslo, Nor-way (oksendalmath.uio.no) and Norwegian School of Economics and Business Administration,Helleveien 30, N5045 Bergen, Norway.INRIA, Domaine de Voluceau-Rocquencourt B.P. 105, F78153 Le Chesnay Cedex, France(Agnes.Suleminria.fr).1765Downloaded 01/27/16 to 02. Redistribution subject to SIAM license or copyright; see /journals/ojsa.php1766 BERNT KSENDAL AND AGNES SULEMportfolio nitely many times in any nite time interval. The control of the investorwillconsistofacombinationofaregularstochasticcontrol c(t)andanimpulsecontrolv =(1,2,.;1,2,.). Here 0 10, 0 bracketleftbiggr+(r)222(1)bracketrightbigg.(1.23)See,e.g.,DN,section2.Fromnowonweassumethat(1.23)holds.Itiseasytoseethat(x,y)0(x,y).(1.24)ThisisalsopointedoutinCorollary2.2,tobeprovedlater.DNandSSconsiderthecasewithproportionaltransactioncostsonly(k=0),inwhichcasetheproblemcanbeformulatedasasingularstochasticcontrolproblem.It is proved in DN and SS that under some conditions there exist two straightlines1,2throughtheorigin,boundingacone NT,suchthatitisoptimaltomakeno transactions if (X(t),Y(t) NT and make transactions corresponding to localtimeat(NT),resultinginreectionsbacktoNT everytime(X(t),Y(t) (NT).Dependingontheparameters,theMertonlinemayormaynotgobetweenthelines1,2(seeFigure1.2andthediscussioninAMS, section7.2). Foranextensionoftheseresultstomarketswithjumps,seeFS1andFS2.xy =1x21NTThe Merton lineyFig.1.2. The no-transaction cone when k =0.The rst paper to model markets with xed transaction costs k0 by impulsecontroltheoryseemstobeEH,buttheydonotconsideroptimalconsumption.Perhaps the paper which is closest to ours is K. Here optimal consumption inmarkets with xed transaction costs is considered, but consumption is allowed onlyat the discrete times of the transactions. This makes it possible to put the problemwithintheframeworkofimpulsecontrolandquasi-variationalinequalities.In our paper we allow consumption to take place at any time, independent ofthe (discrete) times chosen for the transactions. As explained above, we model thisDownloaded 01/27/16 to 02. Redistribution subject to SIAM license or copyright; see /journals/ojsa.phpOPTIMAL PORTFOLIO WITH FIXED TRANSACTION COSTS 1769asacombinedstochasticcontrolandimpulsecontrolproblem,oracombinedcontrolproblem,forshort.Insection2weintroducequasi-variationalHamiltonJacobiBellmaninequalities(QVHJBI) associated with this combined control problem. We point out that if afunction(x,y)satisestheseQVHJBI(andsomeadditionalsmoothnessconditions),then coincideswiththevaluefunction,denedby(1.16). (SeeTheorem2.1).Insection3weprovethatthevaluefunctionistheuniqueviscositysolutionoftheQVHJBIformulatedinsection2.Finally in section 4 we present some numerical estimates for and the optimalconsumption-investmentpolicy w=(c,v).Forotherrecentpapersonimpulsecontrolandcombinedcontrolsee,e.g.,B,M, CZ1, CZ2, and BP and the references therein. We refer to BL and Sforacomprehensivetreatmentofthegeneraltheoryofimpulsecontrolanditsquasi-variationalinequalities.Remark1.1. AnothernaturalchoiceofsolvencyregionwouldbethesetS+:=0,)0,).(1.25)Thischoicemodelsasituationinwhichnoborrowingorshort-sellingisallowed. Wewill mostly use the choice S given by (1.9) in this paper, but we point out that theproofscarryovertotheS+casewithonlyminormodications. (UsuallytheS+caseissimpler.)2. Quasi-variationalHamiltonJacobiBellmaninequalities(QVHJBI).LetAcbethegeneratoroftheprocessZc(t)=(s+t,Xc(t),Yc(t)whentherearenotransactions;i.e., Acisthepartialdierentialoperatorgivenby(Acf)(s,x,y)=fs+(rxc)fx+yfy+122y22fy2(2.1)for any f : R3 R and (s,x,y) such that the derivatives exist. In particular, iff(s,x,y)=esg(x,y),then(Acf)(s,x,y)=esLcg(x,y),whereLcg(x,y)=g+(rxc)gx+ygy+122y22gy2.(2.2)For (x,y)Sand negationslash=0setxprime=xprime()=xk|,yprime=yprime()=y+.(2.3)Wedenetheinterventionoperator M byMh(x,y)=suph(xprime,yprime); R0,(xprime,yprime)S(2.4)foralllocallybounded h:SR+,(x,y)S.If(xprime,yprime)negationslash S forall R0,weset Mh(x,y)=0. Ifforall(x,y)Sthereexists(xprime,yprime)=(xprime(),yprime()SsuchthatMh(x,y)=h(xprime,yprime),Downloaded 01/27/16 to 02. Redistribution subject to SIAM license or copyright; see /journals/ojsa.php1770 BERNT KSENDAL AND AGNES SULEMthenwesethatwide(x,y)=hatwideh(x,y)=(xprime,yprime).(2.5)(More precisely, we lethatwide(x,y) denote a measurable selection of the map (x,y) (xprime,yprime).)If is the value function for our problem, then for each s we can interpretM(s,x,y)asthemaximalvaluewecanobtainbymakinganadmissibletransactionat(s,x,y).Following B we call a locally bounded function h :tildewideSR+stochasticallyC2with respect to Zcif (Ach)(z) exists for almost all z =(s,x,y) with respect totheGreenmeasure (expectedoccupationtimemeasure) G(z0,),andthegeneralizedDynkinformulaholdsfor h,i.e.,Ez0h(Zc(prime)=Ez0h(Zc()+Ez0bracketleftbiggprimeintegraldisplay(Ach)(Zc(t)dtbracketrightbiggforallstoppingtimes ,primesuchthat prime TR:=inft0,|Zc(t)|RR forsome R0;Zc(t)negationslashtildewideSand XH(z)=1ifz H, XH(z)=0ifz negationslash H.If h isafunctionon S,wedeneLh(x,y)=supc0braceleftbiggLch(x,y)+cbracerightbigg, (x,y)S,(2.7)andL0h(x,y)=L0h(0,y)=h+yhy+122y22hy2(2.8)forallpoints(x,y)wherethepartialderivativesof h involvedin Lch exist.Wethenset(see(1.11)(1.13)fordenitionsof lscript1,lscript2,and P)L1h(x,y)=braceleftBiggLh(x,y) for (x,y)S(lscript1lscript2)0,P,L0h(x,y) for (x,y)0,P.(2.9)Notethatat0,Ptheonlyadmissibleconsumptionis c=0.ByadaptingTheorem3.1inBtooursituation,wegetthefollowingsucientQVHJBI.Theorem2.1. Let S andtildewideS be as dened in (1.9) and put U = S(lscript1lscript2),tildewideU =0,)U.Downloaded 01/27/16 to 02. Redistribution subject to SIAM license or copyright; see /journals/ojsa.phpOPTIMAL PORTFOLIO WITH FIXED TRANSACTION COSTS 1771(i) Suppose we can nd a locally bounded function : SR+such that C1(U)and(s,x,y):=es(x,y) isstochastically C2withrespectto Zc(t)(2.10)forallMarkovcontrols c=c(x,y);L1 0 a.e. withrespectto G(z0,)ontildewideU forall z0tildewideU;(2.11)(x,y)M(x,y) forall(x,y) U.(2.12)Then(x,y)(x,y) forall(s,x,y)tildewideU.(ii)DenethecontinuationregionD=(x,y) U;(x,y) M(x,y).SupposeL1(x,y)=0 on D(2.13)andthathatwide(x,y)=hatwide(x,y)(denedin(2.5)existsforall(x,y)S. Denec(x,y)=braceleftBigg(x)11for (x,y) U 0,P,0 for (x,y)0,P,anddenetheimpulsecontrolv:=(1,2,.;1,2,.)asfollows.Put 0=0andinductivelyk+1=inftk;(X(k)(t),Y(k)(t)negationslash D,(2.14)k+1=hatwide(X(k)(k+1),Y(k)(k+1),(2.15)wherehatwide isasdenedin(2.5)and(X(k),Y(k)istheprocessobtainedbyapplyingthecombinedcontrolw(k):=(c,(1,.,k;1,.,k),k=1,2,.Suppose w:=(c,v)Wandthatet(X(w)(t),Y(w)(t)0 as t a.s.(2.16)andthatthefamilye(X(w)(),Y(w)(); stopping time(2.17)isuniformlyintegrable. Then(x,y)=(x,y)(2.18)Downloaded 01/27/16 to 02. Redistribution subject to SIAM license or copyright; see /journals/ojsa.php1772 BERNT KSENDAL AND AGNES SULEMand wisoptimal.Proof. ThisfollowsbytheproofofTheorem3.1inBwithonlyminormodi-cations. NotethattheHamiltonJacobiBellmaninequality(HJBI)(3.7)inBhasthefollowingforminourcase,if(x,y) U 0,P:L(x,y)=supc0braceleftbigg+(rxc)x+yy+122y22y2+cbracerightbigg0.This can only hold ifx0, and then the supremum of this expression is obtainedwhenc=c=parenleftbiggxparenrightbigg 11.(2.19)If (x,y) 0,P, then only the zero consumption c = c=0 is admissible, so againbytheHJBIweget L0(0,y)=0.Wecanusethistoprovetheclaim(1.24),asfollows.Corollary2.2.(i) Asin(1.21)(1.22)let0(x,y)=C1(x+y)(2.20)bethevaluefunctionfortheMertonproblem(k=0). Then(x,y)0(x,y) forall(x,y)S.(2.21)(ii) Let b beaconstantsuchthat1 b 1+.(2.22)Suppose.(2.23)Thenthereexists K0,x+byk(1b) for +(1)(K)11.(2.27)If(2.23)holds,then(2.27)holdsforK largeenough. Thus(2.24)followsfromTheo-rem2.1(i).Remark2.3. Corollary2.2provesinparticularthatthevaluefunctionisnite.Moreover,(x,y)isbounded oneverystraightlinein S oftheformx+by=constantforeveryconstant b 1,1+.Remark2.4(Some comments on the boundary behavior). Suppose the currentposition of the investor is a point (x,y) S. If we make a transaction of size atthatinstant,thenafterthetransactionthenewpositionisgivenbybraceleftbiggxprime=x|k,yprime=y+.(2.28)Hencexprime+(1)yprime=x+(1)yk(|+)(2.29)andxprime+(1+)yprime=x+(1+)yk(|).(2.30)Downloaded 01/27/16 to 02. Redistribution subject to SIAM license or copyright; see /journals/ojsa.php1774 BERNT KSENDAL AND AGNES SULEMx0lscript2(xprime,yprime)QP(xprime,yprime)(x,y)(x,y)lscript1yFig.2.1. Examples of transactions (buying and selling).In particular, if we sell stocks (0),thenxprime+(1+)yprime= x+(1+)y k,so(x,y) will move to a point (xprime,yprime) on the lineparallelto lscript2lyingk1+unitsbelowtheparallelof lscript2through(x,y).WenowusethistodeducetheboundarybehaviorofthevaluefunctiononS.(a) If (x,y) lscript1, then we have to make an immediate transaction to avoid thediusionY(t)takingusoutofS. Bytheaboveweseethattheonlypossibilityistosell stocksofsuchanamountthat(xprime,yprime)=(0,0). Weconcludethat(x,y)=M(x,y)=0 for(x,y) lscript1.(2.31)(b) If(x,y) lscript2,wearguesimilarly: Theonlyadmissibleactionistobuy stocksimmediatelyofsuchanamountthat(xprime,yprime)=(0,0). Hence(x,y)=M(x,y)=0 for(x,y) lscript2.(2.32)(c) Onthesegment0liminfnMu(n)+=liminfnu(n)+forsome0.Since u iscontinuous,thereisaneighborhood G of 0suchthatu(prime)liminfu(n)+ forall prime G.Butifnisbigenoughwehavelscript(n)G negationslash=,sosincenisamaximumpointofuonlscript(n)wehaveu(n) u(prime) for n bigenough.Thiscontradictionshowsthat()holds,andtheproofiscomplete.Lemma3.3.(i)Let u:SR. Then Mu Mu.(ii)Let :SR besuchthat M. Then M.Proof. (i) Choose 0,nS, n =1,2,.,such that n 0and Mu(n) Mu(0)asn . ThenbyLemma3.2(i)appliedtotheuscfunction u,Mu(0)= limnMu(n)limsupnMu(n)Mu(0).(ii) Choose 0,nS, n =1,2,.,such that n 0and M(n) M(0)as n . Then(0)M(0)= limnM(n)liminfnM(n)M(0).Corollary3.4. Suppose u : SR is usc and u(0) Mu(0)+ for some0S, 0. Then u(0) Mu(0)+.Proof. u(0) Mu(0)+=Mu(0) Mu(0)+ byLemma3.3(i).Asin(2.7)welet L bethedierentialoperatorLh(x,y)=supc0braceleftbiggh+(rxc)hx+yhy+122y22hy2+cbracerightbigg,(3.1)andasin(2.2)wesetMh(x,y)=supnegationslash=0h(xprime,yprime);(xprime,yprime)S for (x,y)S,(3.2)wherexprime=xk|,yprime=y+.(3.3)Downloaded 01/27/16 to 02. Redistribution subject to SIAM license or copyright; see /journals/ojsa.phpOPTIMAL PORTFOLIO WITH FIXED TRANSACTION COSTS 1777Theinequalities(2.11),(2.12),and(2.13)ofTheorem2.1togetherwiththeboundarybehavior(2.36)canbecombinedintooneequationasfollows:F(D2(),D(),)=0 forall =(x,y)S,(3.4)whereF:R22R2RSR2 RisdenedbyF(A,p,g,)=max(A,p,g,),(Mgg)(),S0,(A,p,g,),0,Q,0(A,p,g,) 0,P,(Mgg)() lscript1lscript2,(3.5)where(A,p,g,)=g+r1p1+2p2+12222A22+maxc0parenleftbiggcp1+cparenrightbigg(3.6)and0(A,p,g,)=g+2p2+12222A22,A=bracketleftbigAijbracketrightbig1i,j2.(3.7)Note that F is not a local operator: The value of F at (A,p,g,) depends on thevalueof g onthewholespace S.AlsonotethatF(A,p,g,)=max(A,p,g,),(Mgg)() forall S(3.8)andthatF(A,p,g,)=F(A,p,g,) (i.e., F islsc).(3.9)Following Barles B, we now give the denition of the viscosity solution of ellipticequationsoftype(3.4).Definition3.5.(i)Afunction u USC(S)isaviscositysubsolutionofF(D2u(),Du(),u,)=0 forall =(x,y)S(3.10)ifforeveryfunctionf whichisC2inaneighbourhoodofS andforeverypoint0Ssuchthat f u on S and f(0)=u(0)wehaveF(D2f(0),Df(0),u,0)0.(3.11)(ii) A function u LSC(S) is a viscosity supersolution of (3.10) if for everyfunction f whichis C2inaneighbourhoodof S andforeverypoint 0Ssuchthatf u on S and f(0)=u(0)wehaveF(D2f(0),Df(0),u,0)0.(3.12)Downloaded 01/27/16 to 02. Redistribution subject to SIAM license or copyright; see /journals/ojsa.php1778 BERNT KSENDAL AND AGNES SULEM(iii) We say that a function u:SR is a viscosity solution of (3.10) if u islocally bounded and u is a viscosity subsolution and u is a viscosity supersolution of(3.10).Anequivalentdenitionofviscositysolutionswhichisusefulforprovingunique-nessresultsisthefollowing(seeCIL,section2).Definition3.6.(i)Afunction u USC(S)isaviscositysubsolutionof(3.4)ifF(A,p,u,)0 forall (p,A)J2,+Su(), S.(3.13)(ii)Afunction u LSC(S)isaviscositysupersolutionof(3.4)ifF(A,p,u,)0 forall (p,A)J2,Su(), S.Here the second order “superjets” J2,+S, J2,Sand their “closures”J2,+S,J2,SaredenedbyJ2,+Su()=braceleftBigg(p,A) R2R22;limsupSbraceleftbigu()u()p()12()TA()|2bracerightbig0bracerightBigg(3.14)(where()Tdenotesmatrixtransposed),J2,+Su()=(p,A) R2R22; (n,pn,An)SR2R22,with(pn,An) J2,+Su(n)and(n,u(n),pn,An)(,u(),p,A), when n ,(3.15)andJ2,Su=J2,+S(u),J2,Su=J2,+S(u).(3.16)Wearenowreadyfortherstmainresultofthissection.Theorem3.7. Supposethat(2.23)holds. Thenthevaluefunctionisaviscositysolutionof(3.4).Proof. We rst make some useful observations. Suppose w =(c,v) Wis anadmissible control with v =(1,2,.;1,2,.), where 1 0 a.s. Then by theMarkovproperty wehave,with Jwasin(1.14),Jw(z)=Ezbracketleftbiggintegraldisplay0e(s+t)c(t)dt+Jw(Z(w)()bracketrightbigg(3.17)forallstoppingtimes 1.Notethat()M() forall S.(3.18)Toseethis,supposeonthecontrarythatthereexists 1suchthat(1)0 a.s.(A)Weprovethatisaviscositysubsolution. Tothisend,letf beaC2functionin a neighborhood of S and let 0Sbe such that f onS and f(0)=(0).Weconsiderthefollowingtwocasesseparately.Case1.(0)M(0).Thenby(3.8)F(D2f(0),Df(0),f(0),0)(M)(0)=0,andhence(3.11)holdsat 0for u=.Case2.(0) M(0).It suces to prove that Lf(0) 0. We argue by contradiction: Suppose 0=(x0,y0) Sand Lf(0) 0. Hence by continuity,fx() 0 in a neighborhood G of 0. But then,with =(x,y),Lf()=f()+(rxhatwidec)fx+yfy+122y22fy2+hatwidecwith hatwidec=hatwidec()=(fx)11forall GS.HenceLf()iscontinuousonGSandsothereexistsa(bounded)neighborhoodGof 0suchthat G=(x,y);|xx0| 0andLf() M(0)+,wecanbyCorollar