ABSTRACT. RUBTSOV, ALEXEY VLADIMIROVICH. Stochastic Control in Financial Models of Investment and Consumption. (Under the direction of Dr. Min Kang.

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1 ABSTRACT RUBTSOV, ALEXEY VLADIMIROVICH. Sochasic Conrol in Financial Models of Invesmen and Consumpion. Under he direcion of Dr. Min Kang. An exension of he classical Meron s model of opimal invesmen and consumpion is considered. We consider a problem of opimal porfolio managemen under uncerainy in uiliy funcion. In some research, i is claimed ha he uiliy of goods depends no only on he goods hemselves, bu also on qualiy of he goods. However, he qualiy of he goods is subjec o random changes for example, he echnological progress. Alhough here is much debae on he dynamics of echnological progress, i is no uncommon in he lieraure ha he progress in echnology in some areas grows exponenially. This implies ha i makes sense o model hese random changes by a Geomeric Brownian moion. Thus, i is a naural problem of ineres o find ou how he opimal policy changes when some uncerainy in he uiliy funcion is inroduced. The definiions and heoreical resuls used in he research are provided and reviewed. Once he problem of sochasic conrol is defined, he classical Meron s model of opimal invesmen and consumpion is presened. The imporance of he proposed uncerainy in uiliy is also discussed. Once he required heoreical resuls are saed, he problem of expeced uiliy maximizaion wih fully observed uncerainy in uiliy is solved for a specific uiliy funcion of hyberbolic absolue risk aversion class. To obain he opimal soluion, he Hamilon- Jacobi-Bellman HJB equaion is derived and he soluion is obained. To verify ha he viscosiy soluion o he HJB equaion is he value funcion, a so-called Verificaion Theorem is proved. As a resul, he opimal invesmen and consumpion are found for he problems of maximizing he expeced uiliy of consumpion and final wealh, only he expeced uiliy of consumpion, and only he expeced uiliy of final wealh. Having obained he soluion o he fully observed case, he problem of expeced uiliy maximizaion is solved under he assumpion ha uncerainy in uiliy is no fully observed. Afer he corresponding HJB equaion is derived and solved, he Verificaion Theorem is used o verify ha he soluion is he value funcion. Thus, he opimal invesmen and consumpion under parial observaions are obained for he problems of maximizing he expeced uiliy of consumpion and final wealh, only he expeced uiliy of consumpion, and only he expeced uiliy of final wealh.

2 c Copyrigh 1 by Alexey Vladimirovich Rubsov All Righs Reserved

3 Sochasic Conrol in Financial Models of Invesmen and Consumpion by Alexey Vladimirovich Rubsov A disseraion submied o he Graduae Faculy of Norh Carolina Sae Universiy in parial fulfillmen of he requiremens for he Degree of Docor of Philosophy Operaions Research Raleigh, Norh Carolina 1 APPROVED BY: Dr. Min Kang Chair of Advisory Commiee Dr. Salah Elmaghraby Dr. Julie Ivy Dr. Tao Pang

4 DEDICATION To my parens. ii

5 BIOGRAPHY Alexey Rubsov was born on January 8, 1983 in Saraov, Russia. Afer five years of sudies a Saraov Sae Social-Economic Universiy, he received a diploma in Saisics, which is usually considered o be equivalen o Maser s Degree in Saisics. In 8 he received his Masers Degree in Financial Mahemaics a Norh Carolina Sae Universiy. From year 8 o 11, he has been pursuing he Ph.D. degree in Operaions Research a Norh Carolina Sae Universiy. iii

6 ACKNOWLEDGEMENTS Firs of all, I would like o hank my academic advisor for all her help. I also wan o hank he commiee members for heir helpful suggesions and advice. I appreciae he suppor of he Operaions Research deparmen provided during my sudies. iv

7 TABLE OF CONTENTS Lis of Figures vii Lis of Noaions viii Chaper 1 Inroducion Sochasic Opimizaion and is Applicaions Mahemaical Preliminaries The Problem of Sochasic Conrol Classical Model of Opimal Invesmen and Consumpion Maximizing he Uiliy of Consumpion and Final Wealh Maximizing he Uiliy of Consumpion Maximizing he Uiliy of Final Wealh Uncerainy in he Uiliy Funcion Chaper Fully Observed Case Formulaion of he Problem Fully Observed Uiliy Randomness Process Derivaion of he HJB Equaion Verificaion Theorem Maximizing he Uiliy of Consumpion and Final Wealh Soluion o he HJB equaion Verificaion Maximizing he Uiliy of Consumpion Soluion o he HJB equaion Verificaion Maximizing he Uiliy of Final Wealh Soluion o he HJB equaion Verificaion Analysis of he Resuls and Numerical Experimens Conclusions Chaper 3 Parially Observed Case Parially Observed Uiliy Randomness Process Condiional Disribuion Reward Funcional and Value Funcion Derivaion of he HJB Equaion Maximizing he Uiliy of Consumpion and Final Wealh Soluion o he HJB equaion Verificaion v

8 3.6 Maximizing he Uiliy of Consumpion Soluion o he HJB equaion Verificaion Maximizing he Uiliy of Final Wealh Soluion o he HJB equaion Verificaion Analysis of he Resuls and Numerical Experimens Conclusions Chaper 4 Summary and Fuure Research Summary Fuure Research References Appendices Appendix A Proofs for Chaper A.1 Theorem A. Theorem A.3 Lemma Appendix B Proofs for Chaper B.1 Lemma B. Theorem B.3 Lemma vi

9 LIST OF FIGURES Figure.1 Consumpion C,1 per uni wealh as a funcion of ime and β σ =.4; b funcion of ime and σ β = Figure. Consumpion C, per uni wealh as a funcion of ime and β σ =.4; b funcion of ime and σ β = vii

10 LIST OF NOTATIONS R k - k-dimensional Euclidean space. C[, T ]; R k - he space of all coninuous funcions defined on [, T ] and aking values in R k. x = x 1... x k - a vecor made of absolue values of each componen of vecor x R k. k 1/, x - norm of x, unless oherwise indicaed, if x R k hen x = x i if k m x R k m hen x = 1/. xij i=1 j=1 BU - he Borel sigma-algebra generaed by all he open ses in a meric space U he smalles sigma-algebra conaining all he open ses of U. F 1 F - direc produc of sigma-algebras F 1 and F he sigma-algebra generaed by ses A B, A F 1, B F. B m [, T ] C[, T ]; R m - he space of all coninuous R m -valued funcions defined on [, T ]. B m C[, ; R m - he space of all coninuous R m -valued funcions defined on [, wih meric ˆρb 1, b = j 1 b j 1 b C[,j];R m 1, b 1, b B m. Under he meric ˆρ he space B m is a Polish space complee separable meric space. L p F, T ; Rk - he space of all {F s } s [,T ] -adaped, R k -valued processes X such ha E T X p d <, where x p = x 1 p x k p, x R k, p 1. A n T U - he space of all {B +C[, T ]; R k } -progressively measurable processes ψ : [, T ] C[, T ]; R k U, where B m [, T ] = {X X B m [, T ]}, [, T ], B B m [, T ] = σbb m [, T ], [, T ], B + B m [, T ] = s> B s B m [, T ], [, T. i=1 viii

11 Noe ha B B m [, T ] is he sigma-algebra in B m [, T ] generaed by BB m [, T ], and hus i conains B m [, T ]. The fac ha B + B m [, T ] B B m [, T ] is proved in [1], p.1. If he inerval [, is considered hen we wrie A k U. X U - porfolio wealh process when he conrol U is used. S - n-dimensional column vecor of sock prices risky asses. N - he value of he riskless asse. Z - uiliy randomness process. L - naural logarihm of he uiliy randomness process. P - he observed process he uiliy randomness process is no fully observed. Π - n-dimensional row vecor ha represens he fracions of wealh invesed in he risky asses. C - consumpion per uni ime. U - he conrol process in his disseraion he conrols are Π and C. U - he space of conrol values. B,1 - n-dimensional Brownian moion column vecor. B, - one-dimensional Brownian moion. B,3 - one-dimensional Brownian moion. ix

12 Chaper 1 Inroducion 1.1 Sochasic Opimizaion and is Applicaions The model of opimal invesmen and consumpion under uncerainy in uiliy funcion is considered in his disseraion. The maximized crierion in he model is he expeced uiliy. To maximize he expeced uiliy, he mehods of sochasic opimizaion are used o find he opimal soluion. Sochasic opimizaion plays an imporan role in he design, analysis, and operaion of modern sysems. Sochasic opimizaion mehods are used in models ha are inappropriae for classical deerminisic mehods of opimizaion. Algorihms ha ake advanage of sochasic opimizaion echniques find heir applicaions in problems in saisics, science, engineering, and business. Classical deerminisic opimizaion is based on he assumpion of perfec informaion abou he minimized maximized funcion and is derivaives, if necessary. This informaion is hen used o deermine he direcion of search in a deerminisic manner a every sep of he algorihm. However, in many pracical problems, his informaion is no available. In conras o deerminisic opimizaion, sochasic opimizaion mehods are used when randomness appears in he formulaion of he opimizaion problem iself random objecive funcion, random consrains, ec.. Sochasic opimizaion also includes mehods wih random ieraes. Some sochasic opimizaion mehods use random ieraes in solving sochasic problems. Random real daa arise in such problems as real-ime esimaion and conrol, simula- 1

13 ion based opimizaion where Mone Carlo simulaions are run as esimaes of an acual sysem, and problems where here is experimenal random error in he measuremens of he crierion. In such cases, knowledge ha he funcion values conain random noise leads naurally o algorihms ha use saisical inference ools in esimaion of he rue values of he funcion and/or make saisically opimal decisions abou he nex seps of he opimizaion algorihm. Mehods of his class include: sochasic approximaion, sochasic gradien descen, finie-difference sochasic approximaion, simulaneous perurbaion sochasic approximaion, ec. On he oher hand, even when he daa se consiss of exac measuremens, some mehods inroduce randomness ino he search-process o accelerae progress. Such randomness can also make he mehod less sensiive o modeling errors. Furher, he injeced randomness may enable he mehod o escape a local minimum and evenually o approach a global opimum. Indeed, his randomizaion principle is known o be a simple and effecive way o obain algorihms wih almos cerain good performance uniformly across many daa ses, for many sors of problems. Sochasic opimizaion mehods of his kind include: simulaed annealing, reacive search opimizaion, cross-enropy mehod, random search, ec. Sochasic opimizaion is closely conneced o sochasic conrol. The heory of sochasic conrol provides a vas array of heoreical and compuaional ools ha find heir applicaions in many areas dealing wih decision-making under uncerainy. These areas include indusrial processes, roboics, insurance, economics, and finance. A common feaure of sochasic conrol problems is ha a conrolled dynamical sysem is subjec o random perurbaions and he goal is o opimize some performance crierion. One of he mos ineresing applicaions of he heory of sochasic conrol in finance is porfolio opimizaion problem in which an agen invess wealh ino risky and riskless asses and chooses a rae of consumpion wih he goal of maximizing he expeced uiliy of consumpion. Under some assumpions, in [15] Meron solved he problem of expeced uiliy maximizaion. In ha work i was assumed ha he uiliy funcion is a power funcion and he marke includes he riskless asse wih consan rae of reurn and a risky asse wih consan mean rae of reurn and volailiy parameer. Some of he assumpions were relaxed laer. The resricion o power uiliy funcions was removed in [11], and in [1, 18], he marke coefficiens were allowed o be non-consan. Afer ha iniial paper, he Meron s model was generalized in many direcions.

14 One generalizaion is he inroducion of ransacion coss [] which are he coss incurred when wealh is moved from one asse o he oher. The presence of ransacion coss makes he model more realisic. Anoher exension is including pas sock prices in he decision-making process. In he Meron s model he invesor makes invesmen decisions based on curren informaion and does no consider he pas sock prices. However, in he real world, invesors ake ino accoun he hisoric performance of he risky asses. This approach is called sochasic porfolio opimizaion wih memory and is reaed in [19]. In many cases, he invesor does no possess complee informaion abou he sock prices, model parameers values, ec. This implies ha he opimal invesmen and consumpion should be obained under he assumpion ha some of he required informaion is parially observed. The models wih parial observaions are discussed in [3, 4]. These generalizaions are no he only ones and here are many oher possible exensions which make he model more realisic, and his disseraion hesis suggess one more way of exending he model o include such facor as echnological progress in invesmen decision-making under uncerainy. In some research [14, ], i is claimed ha he uiliy of goods depends no only on he goods hemselves bu also on qualiies of he goods. Tha is why i makes sense o exend he classical model of opimal invesmen and consumpion [7, 16, 1] o incorporae his feaure. When he suggesed uncerainy in uiliy is inroduced, i is ineresing o find he difference in he opimal policies of he new model and he classical model. I is naural o assume ha when we buy differen hings we are no buying jus objecs, we are buying he qualiies ha hose objecs possess and we need hose qualiies. For example, if we consider a diamond ha coss as much as a house hen clearly i has uiliy which is differen from ha of he house. However, using he price of a merchandise as he only argumen o he uiliy funcion, he uiliies of he diamond and he house are he same. Since he echnology is changing, new producs keep coming ou and subsiue he old ones giving he increase in uiliy. For example, having a compuer oday gives much more opporuniies o is user compared o he compuers and echnologies available 3 years ago. Therefore, preferences migh change because of beer characerisics of new goods. I is imporan o noe ha his change does no have o enail he change of prices. On he oher hand, preference change migh also be due o worsened qualiy of he 3

15 producs or some oher reason for example, buying he same produc over and over migh decrease is uiliy and ends up in saiaion wih he produc. The described process is no deerminisic because i is no known how he marke will change. Therefore, i makes sense o model he uncerainy in uiliy by a sochasic process. Alhough here is much debae on he dynamics of echnological progress [1, 5], i is no new in lieraure ha in some areas i is growing exponenially [8]. A model ha assumes exponenial growh can also be used o model linear behavior because of he represenaion e x = 1 + x + ox and, hus, changing he parameers of he model accordingly, will help analyze he resuls when he growh is close o linear. Apar from big echnological advancemens here are minor improvemens in producs ha people use every day. Therefore, he Geomeric Brownian moion can be used o model he uncerainy in uiliy. 1. Mahemaical Preliminaries Le Ω be a nonempy se and F be a sigma-algebra on Ω, hen Ω, F is called a measurable space. Le P be a probabiliy measure defined on he ses from F, hen Ω, F, P is called a probabiliy space. The probabiliy space is said o be complee if for any se A F such ha PA = null se a se B A is also in F, i.e. B F. Definiion 1. Le a measurable space Ω, F be given. A monoone family of sub-sigmaalgebras F F, [, is called a filraion if F 1 F, 1,, 1 <. Definiion. A filraion is called righ lef coninuous if F = F + s> F s F = F σ s< F s, <. Definiion 3. A filered probabiliy space Ω, F, {F s } s [,, P is said o saisfy he usual condiion if i is complee 1, F conains all he P-null ses in F, and he filraion {F s } s [, is righ coninuous 3. 1 I is necessary o require he filered probabiliy space be complee because, for example, if ξ is a random variable F-measurable funcion and η ξ almos everywhere hen η is no necessarily measurable. Thus, by compleing he probabiliy space we exend he space of measurable and, herefore, inegrable funcions. The Io s sochasic inegral wih finie upper limi migh no be a maringale if he filraion is incomplee in he sense ha all P-null ses of F should be included ino F. See [13], p In Lemma see below ha plays an imporan role in appropriaely formulaing sochasic opimal conrol problems, he obained process φ is only {B + B m [, T ]} [,T ] -progressively measurable, no necessarily {B B m [, T ]} [,T ] -progressively measurable, see [9], p.. 4

16 Definiion 4. Le Ω, F and E, E be wo measurable spaces. A funcion X : Ω E is an F/E measurable funcion, or random elemen, or E-valued random variable if {ω : Xω B} F, B E. Definiion 5. If Ω is a opological space hen he smalles sigma-algebra BΩ conaining all open ses of Ω is called he Borel sigma-algebra of Ω. Definiion 6. Le I be a subse of he real line. A family of random variables {X s, s I} from Ω, F, P o R k is called a sochasic process. For any ω Ω, he map X ω 1 is called a sample pah. Definiion 7. Le Ω, F, {F s } s [, be a filered measurable space and X be a sochasic process aking values in a meric space Q, d. 1 The process X is said o be measurable if he map, ω X ω is B[, F/BQ-measurable. The process X is said o be F -adaped if for all in [, he map ω X ω is F /BQ-measurable. 3 The process X is F -progressively measurable if for all in [, he map s, ω X s ω is B[, ] F /BQ-measurable or {s, ω : s, ω Ω, X s A} B[, ] F, for all A BQ. Remark 1. a Noice ha in he above definiion 3 = 1,. b If he process X is F -adaped i does no mean he process Y = X s ds is F -adaped. However, in many cases i is required ha he process Y be adaped for example, in proving he Bellman s Principle of Opimaliy. By Fubini s heorem see for example, [13], p.3, he process Y is adaped if he process X is F -progressively measurable. Definiion 8. Le Ω, F, {F s } s [,, P be a filered probabiliy space. An F -adaped R m -valued process B is called an m-dimensional F -Brownian moion over [, if for all s <, B B s is independen of F s and is normally disribued wih mean and covariance si, where I is he m m ideniy marix. If PB = = 1 hen B is called an m-dimensional sandard F -Brownian moion over [,. Le a A k R k, s 1 A k R k m hen we have he following 1 The noaions X ω and X will be used inerchangeably. 5

17 Definiion 9. An equaion of he form { dx = a, Xd + s 1, XdB, X = ξ 1.1 is called a sochasic differenial equaion wih he iniial condiion. Remark. Noice ha he coefficiens a, s 1 hrough X. are no random and depend on ω Ω There are differen noions of soluions o 1.1 depending on differen roles ha he underlying filered probabiliy space Ω, F, {F s } s [,, P and he Brownian moion B are playing. Definiion 1. Le Ω, F, {F s } s [,, P, m-dimensional sandard F -Brownian moion B be given, and ξ is F -measurable. An F -adaped coninuous process X, is called a srong soluion of 1.1 if 1. X = ξ, P-a.s.,. as, X + s 1 s, X ds <, 3. X = X + as, Xds + s 1 s, XdB s,, P-a.s.,, P-a.s. Definiion 11. If for any wo srong soluions X and Y of equaion 1.1 defined on any given Ω, F, {F s } s [,, P along wih any given sandard F -Brownian moion, we have PX = Y, = 1 hen we say ha he srong soluion is unique 1. Definiion 1. A 6-uple Ω, F, {F s } s [,, P, B, X is called a weak soluion of 1.1 if 1. Ω, F, {F s } s [,, P is a filered probabiliy space saisfying he usual condiion,. B is an m-dimensional sandard F -Brownian moion and X is F -adaped and coninuous, 3. X and ξ have he same disribuion, 4. and 3 of he definiion 1 hold. Remark 3. For he srong soluion he filered probabiliy space Ω, F, {F s } s [,, P and he F -Brownian moion B on i are fixed a priori. For he weak soluion, Ω, F, {F s } s [,, P and B are pars of he soluion. 1 Therefore, he soluion is unique if X, Y are indisinguishable processes. 6

18 Definiion 13. If for any wo weak soluions Ω, F, {F s } s [,, P, B, X and Ω, F, { F s } s [,, P, B, X of 1.1 wih PX D = P X D, D BR k, we have PX A = P X A, A BB k, hen we say ha he weak soluion is unique. Now we define anoher ype of SDE which will be used in sochasic conrol problems. Definiion 14. An equaion of he form { dx = a, X, ωd + s 1, X, ωdb, X = ξ 1. is called a sochasic differenial equaion wih random coefficiens a : [, B k Ω R k, s 1 : [, B k Ω R k m which explicily depend on ω Ω X, B, ξ also depend on ω bu i is suppressed o simplify he noaion. Remark 4. Noice ha he case when he coefficiens a, s 1 depend on X insead of X i.e. a : [, R k Ω R k, s 1 : [, R k Ω R k m is a special case of he equaion in he above definiion. Nex, we define wha we mean by a soluion o 1.. Definiion 15. Le he maps a : [, B k Ω R k and s 1 : [, B k Ω R k m and an m-dimensional sandard F -Brownian moion B be given on a given filered probabiliy space Ω, F, {F s } s [,, P. Le ξ be F -measurable. An F -adaped coninuous process X,, is called a soluion of 1. if 1. X = ξ, P a.s.;. as, X, ω + s 1 s, X, ω ds <,, P a.s.; 3. X = ξ + as, X, ωds + s 1 s, X, ωdb s,, P a.s.. 7

19 Definiion 16. If PX = Y, = 1 holds for any wo soluions X, Y of 1. defined on he given filered probabiliy space Ω, F, {F s } s [,, P along wih he given F -Brownian moion B hen we say ha he soluion is unique. Remark 5. Since he coefficiens a, s 1 should be given a priori, equaion 1. should be defined on a given probabiliy space Ω, F, P where a, s 1 are defined. Therefore, for an SDE wih random coefficiens i does no make sense o alk abou weak soluions. Now we sae he condiions under which he SDE 1. admis a unique soluion. Theorem 1. Assume ha for any ω Ω, a,, ω A k R k and s 1,, ω A k R k m and for any x B k, a, x, and s 1, x, are boh {F }-adaped processes. Moreover, here exiss a consan L > such ha for all [,, x, y B k, and ω Ω, a, x, ω a, y, ω L x y B k, s 1, x, ω s 1, y, ω L x y B k, 1.3 a,, + s 1,, L F, T ; R, T >. Then for any ξ L p F Ω; R k p 1, 1. admis a unique soluion X such ha 1 for any T > E max X s p K1 + E ξ p, s T E X X s p K1 + E ξ p s p/, s, [, T ]. 1.4 Moreover, if η L p F Ω; R k is anoher random variable and Y is he corresponding soluion of 1., hen for any T >, here exiss a K > such ha E max X s Y s p KE ξ η p. 1.5 s T The proof of he heorem is given in appendix A.1. The following heorem will be used in defining he problem of sochasic conrol. Theorem. Le Ω, F, {F s } s [,, P be a filered probabiliy space and le X be F - adaped, and lef or righ coninuous. Then X is F -progressively measurable. The proof of he heorem is given in appendix A.. Therefore, if he assumpions of Theorem 1 are saisfied hen he soluion X of 1. is F -progressively measurable. 1 Alhough he consan K depends on T, he noaion K is used. 8

20 These are he basic erminology and resuls ha will be used in formulaion of sochasic conrol problems discussed in he nex secion. 1.3 The Problem of Sochasic Conrol Le < T < be given. Le Ω, F, {F s } s [,T ], P be a given filered probabiliy space saisfying he usual condiion see Definiion 3, on which is defined an m-dimensional sandard Brownian moion B. space. Consider he following sochasic conrolled sysem 1 { wih he reward funcional Le U be a Polish space complee separable meric dx U = a, X U, U d + s 1, X U, U db, [, T ], 1.6 X = x wx, {U s } s [,T ] = E x [ T e ζ f, X U, U d + e ζt gx U T T ], where ζ > is a parameer. We define he space of feasible conrols 3 U s {U : [, T ] Ω U U is F -progressively measurable}. The problem of sochasic conrol is o find U ha maximizes wx, U over he se U s4. One of he approaches o solve he problem is o use he dynamic programming [9]. To guaranee ha he reward funcional is well-defined measurabiliy and inegrabiliy of T e ζ f, X U, U d + e ζt gx U T T, uniqueness of X we will make some assumions specified laer. Noice ha he coefficiens a, X U ω, U ω = ā, X U ω, ω, ω, ω of equaion 1.6 depend on ω no only hrough s 1, X U ω, U ω = s 1, X U X bu also hrough U and, hus, he heory for an SDE wih random coefficiens can be applied. This problem is saed in a srong form he probabiliy space is given and fixed and i is he problem ha we wan o solve evenually he iniial ime is and he 1 Noaion X U means ha X depends on U. E x [ ] E[ X = x ]. 3 To simplify he noaion, U will be used insead of {U s } s [,T ]. 4 The superscrip s in U s means ha he srong formulaion is being considered. See Remark 3. 9

21 iniial sae x is deerminisic. In order o apply he dynamic programming echnique we need o consider a family of problems wih differen iniial imes and saes. In he deerminisic case we can change he iniial ime and sae wihou changing he mahemaical framework of he problem. However, in he sochasic case he saes along a given rajecory become random variables on he original probabiliy space. More specifically, if X is a sae rajecory saring from x a ime in a probabiliy space Ω, F, {F s } s [,T ], P, hen for any ime >, X is a random variable in Ω, F, P raher han a deerminisic poin in R k. Since he conrol U is F -progressively measurable hen a any ime insan he conroller knows all he relevan pas informaion of he sysem up o ime and in paricular abou X. This implies ha X is acually no uncerain for he conroller a ime. In mahemaical erms, X is almos surely deerminisic under a differen probabiliy measure P F. Indeed, his can be made precise and we have he following proposiion. Proposiion 1. Le Ω, F, {F s } s [,T ], P be a filered probabiliy space. Le X be an F -adaped process. Then for any s [, T ] P{ω X ω = X ω} F s ω = 1, P a.s., [, s]. Proof. P{ω X ω = X ω} F s ω = E[I {ω X ω =X ω} F s ]ω = I {ω X ω =X ω}ω = 1, P a.s.. Therefore, under he probabiliy measure P F s ω where ω is fixed, he random variable X is almos surely a deerminisic consan equal o X ω for any [, s]. Thus, he above idea requires us o vary he probabiliy spaces as well in order o employ dynamic programming. Therefore, we consider he weak formulaion 1. 1 The word weak means ha when solving he problem he probabiliy space is allowed o vary which is similar o he case wih obaining a weak soluion o 1.1. The weak formulaion does no make sense if he coefficiens a, s 1 depend on ω explicily. See Remark 5. 1

22 For any, x [, T R k consider he sae equaion { dxs Us = as, Xs Us, U s ds + s 1 s, Xs Us, U s db s, s [, T ], X = x, 1.7 wih he reward funcional 1 w, x, U = E,x [ T e ζs fs, Xs Us, U s ds + e ζt gx U T T ]. 1.8 Nex, we define he space of admissible conrols U w [, T ] on [, T ] as he se of 5- uples Ω, F, P, B, U saisfying he following assumpion. Assumpion B: 1. Ω, F, P is a complee probabiliy space 3 ;. {B s } s [,T ] is an m-dimensional sandard Brownian moion defined on Ω, F, P over [, T ] wih B = almos surely and F s, is he sigma-algebra generaed by B r, r s augmened by all P-null ses in F; 3. U : [, T ] Ω U is an F s, -progressively measurable process on Ω, F, P; 4. Under U for any x R k equaion 1.7 admis a unique soluion in he sense of definiions 15, 16 on Ω, F, {F s, } s [,T ], P; 5. The funcion f, X U, U is in L 1 F, T ; R and he funcion gxu T T is in L1 F T Ω; R. Here he spaces L 1 F, T ; R and L1 F T Ω; R are defined on he given filered probabiliy space Ω, F, {F s, } s [,T ], P. Under his resricion on he space of conrols he reward funcional defined by 1.8 is well-defined. Also, noe ha he expecaion in formula 1.8 is aken wih respec o he probabiliy measure P. The problem of sochasic conrol is o maximize w, x, U over he space of admissible conrols U w [, T ]. 1 E,x [ ] E[ X = x]. The superscrip w means ha he weak formulaion is considered. 3 Alhough denoed similarly, his space should be disinguished from he original filered probabiliy space on which 1.6 is defined. 11

23 To guaranee he uniqueness of soluions o 1.7 we have o impose some assumpions. Le d be he dimension of U R d, he space of conrol values. We make he following assumpion. Assumpion A: The maps a : [, T ] R k U R k, s 1 : [, T ] R k U R k m, f : [, T ] R k U R, g : R k R are coninuous. There exis concave, increasing in each independen variable, coninuous funcions ϕ 1 : R k+d [, and ϕ : R k [, such ha ϕ i =, i = 1,, if any of he independen variables is equal o, and here exiss a consan L > such ha we have 1 1 a, x 1, u a, x, u L x 1 x, [, T ], x 1, x R k, u U, s 1, x 1, u s 1, x, u L x 1 x, [, T ], x 1, x R k, u U, 3 f, x 1, u f, x, u Lϕ 1 u, x 1 x, [, T ], x 1, x R k, u U, 4 gx 1 gx Lϕ x 1 x, x 1, x R k, 5 a,, u + s 1,, u L,, u [, T ] U, 6 E sup U s L, U U w [, T ], [, T ]. s [,T ] Under he assumpions A1,,5, equaion 1.7 admis a unique soluion by Theorem 1 he soluion is coninuous, see Definiion 15. Indeed, if in A1,,5 we consider a, s 1 as maps ā : [, T ] B k Ω, s 1 : [, T ] B k Ω hen he Theorem is applicable. Also, since U is assumed o be F s, -progressively measurable, X is F s, -progressively measurable see Theorem, and a, s 1 are coninuous, we have ha ā,, ω A k R k, s 1,, ω A k R k m, ω Ω. Since f, g are coninuous, he reward funcional 1.8 is well-defined. Thus, we can define he value funcion: v, x = sup U U w [,T ] w, x, U,, x [, T R k, vt, x = gx, x R k. Nex we derive some properies of he value funcion ha will be used in proving Bellman s Principle of Opimaliy. 1 Noice he difference beween and. If x = x 1 x, x 1, x R hen x = x 1 + x bu x = x 1 x. Since X and U are F s, -progressively measurable and f is coninuous, he process T fs, X Us s, U s ds is F T, -measurable see Remark 1. Since g is coninuous hen gx U T T is also F T,-measurable. 1

24 Lemma 1. Le he assumpion A hold. Then for any ε > and [, T ] here exiss δε > such ha if x y δε. w, x, U w, y, U ε, U U w [, T ], v, x v, y ε, Proof. Le T, x, y R k. For any admissible conrol U le X U, Y U represen he saes saring a ime wih values x and y, respecively. Then by Theorem and 1.5 we have E X s Y s K x y. Using his resul, assumpion A, and sup s [,T ] Jensen s inequaliy we obain w, x, U w, y, U [ T = E [ T E e ζs fs, X Us s Lϕ 1 U s, X Us s E[LT ϕ 1 sup U s, sup Xs Us s [,T ] s [,T ], U s ds fs, Y Us s Ys Us ds + Lϕ X U T T E[LT ϕ 1 sup U s 1 d, sup Xs Us s [,T ] s [,T ] LT ϕ 1 E sup U s 1 d, E sup Xs Us s [,T ] s [,T ] LT ϕ 1 E sup U s 1 d, E sup Xs Us s [,T ] s [,T ], U s ds + e ζt gx U T T Y U T T ] Ys Us + Lϕ X U T T Y U T T ] Ys Us 1 k + Lϕ X U T T LT ϕ 1 L1 d, K x y 1 k + Lϕ K x y 1 k, Ys Us 1 k + Lϕ E X U T T ] gy U T T Y U T T 1k ] 1.9 Y U T T 1k Ys Us 1 k + Lϕ E sup Xs Us Ys Us 1 k s [,T ] where 1 d = 1,..., 1 T is a d-dimensional vecor and 1 k = 1,..., 1 T is a k-dimensional vecor. Noice ha in 1.9 value of each independen variable in ϕ i, i = 1, was changed from o and he direcion of he inequaliy follows from he fac ha x i x = x x k, i = 1,..., k, x Rk. Since funcions ϕ i, i = 1, are coninuous, increasing in each variable, and ϕ i =, i = 1,, if any of heir independen variables is equal o, we have ha for any ε > here exiss δε > such ha w, x, U w, y, U ε, if x y δε. 13

25 Taking he supremum over U U w [, T ] we obain sup w, x, U w, y, U U U w [,T ] Therefore, v, x v, y ε. sup U U w [,T ] sup U U w [,T ] w, x, U w, y, U w, x, U = v, x v, y. sup w, y, U U U w [,T ] Lemma. Le Ω, F, P be a complee probabiliy space and U, d be a Polish space. Le B : [, T ] Ω R m be a coninuous process and F B be he sigma-algebra generaed by B s, s. Then U : [, T ] Ω U is {F B } [,T ] -adaped if and only if here exiss φ A m T U such ha U ω = φ, B ω 1, P a.s., [, T ]. The proof of he lemma is given in appendix A.3. Le [, T, θ [, T and ξ be an F θ, -measurable random variable and X be a soluion of { dxs Us = as, Xs Us, U s ds + s 1 s, Xs Us, U s db s, s [θ, T ], X θ ω = ξω. Now we are ready for he nex lemma. 1.1 Lemma 3. Le [, T and U U w [, T ]. Then for any θ [, T and F θ, -measurable random variable ξ wθ, ξ, U = E θ,ξ [ T θ e ζs θ fs, X Us s ], U s ds + e ζt θ gx U T T F θ,, P a.s. Proof. Since he conrol U is F s, -adaped see he assumpion B and F s, is he sigmaalgebra generaed by he Brownian moion B r, r s, hen by Lemma here exiss a funcion φ A m T U such ha U sω = φs, B s ω, P a.s., ω Ω, s [, T ]. 1 The process B is he process B s if s wih values equal o B if s >. 14

26 Therefore, 1.1 can also be wrien as { dxs φ = as, Xs φ, φs, B s ds + s 1 s, Xs φ, φs, B s db s, s [θ, T ], X θ ω = ξω. This equaion has a unique srong soluion because he equaion { dx s = ās, X s, B s ds + s 1 s, X s, B s db s, s [θ, T ], X θ ω = ξω is a special case of 1.1. Indeed, if we wrie dy s = db s and consider Y as a componen of X, hen we have 1.1 and hus, under he assumpions A1,, 5 he equaion 1.1 has a unique srong soluion. On he oher hand, from Proposiion 1 i follows ha Pω : ξω = ξω F θ, ω = 1, P-a.s. This means ha here is an Ω F wih PΩ = 1 such ha for any ω Ω, ξ becomes a deerminisic consan ξω under he new probabiliy space Ω, F, P F θ, ω. In addiion, for any s θ, we have ha U s ω = φs, B s ω = φs, B s ω + B θ ω, where B s = B s B θ is a sandard Brownian moion. Noe ha B θ almos surely equals a consan B θ ω under he probabiliy measure P F θ, ω. I follows hen ha U s is adaped o he filraion generaed by he sandard Brownian moion B s for s θ. Hence by he definiion of admissible conrols Ω, F, P F θ, ω, B s, U s U w [θ, T ]. Thus, if we work under he probabiliy space Ω, F, P F θ, ω and noice he weak uniqueness see Definiion 13 of 1.1 and 1.11 we obain he resul. Now we derive Bellman s Principle of Opimaliy ha will be very imporan in deriving he Hamilon-Jacobi-Bellman equaion used in solving he opimal conrol problem presened in his disseraion. Theorem 3. Le assumpion A hold. Then for any, x [, T R k and for all, θ saisfying θ T we have v, x = [ θ sup E,x U U w [,T ] e ζs fs, Xs Us, U s ds + e ζθ vθ, X U θ θ ] Proof. By definiion of supremum, for any ε > here exiss an admissible conrol U here exiss Ω, F, P, B, U such ha using Lemma 3 15

27 v, x ε < w, x, U = E,x [ T = E,x [ θ + E,x [ T θ = E,x [ θ + e ζθ E θ,x U θ θ = E,x [ θ E,x [ θ ] e ζs fs, Xs Us, U s ds + e ζt gx U T T e ζs fs, Xs Us, U s ds e ζs fs, X Us s e ζs fs, Xs Us, U s ds [ T [ θ sup E,x U U w [,T ] θ ]], U s ds + e ζt gx U T T F θ, e ζs θ fs, X Us s ]], U s ds + e ζt θ gx U T T F θ, ] e ζs fs, Xs Us, U s ds + e ζθ wθ, X U θ θ, U ] e ζs fs, Xs Us, U s ds + e ζθ vθ, X U θ θ e ζs fs, Xs Us, U s ds + e ζθ vθ, X U θ θ ]. Conversely, by Lemma 1 for any ε > here exiss δε such ha if x y δε hen e wθ, ζθ x, U wθ, y, U + vθ, x vθ, y ε 3, U U w [θ, T ] Le {D j } j 1 be a Borel pariion of R k D j BR k, wih diameer diamd j < δε. Choose x j conrol U j here exiss Ω j, F j, P j, B j, U j such ha D j = R k, D i D j =, i j j=1 D j. For each j here is an admissible e ζθ wθ, x j, U j e ζθ vθ, x j ε Hence for any x D j, combining 1.14 and 1.15, we have e ζθ wθ, x, U j e ζθ wθ, x j, U j ε 3 e ζθ vθ, x j ε 3 e ζθ vθ, x ε. 16

28 By he definiion of admissible conrol Ω j, F j, P j, B j, U j see assumpion B and Lemma here is a funcion φ j A m T U such ha U j s ω = φ j s, B j sω, P j a.s., for all s [θ, T ]. For any admissible Ω, F, P, B, U define a new conrol Ū s ω = { U s ω, s [, θ, φ j s, B s ω B θ ω, s [θ, T ] and X s D j. Clearly, Ω, F, P, B, Ū is admissible. Therefore, v, x w, x, Ū = E,x [ T ] e ζs fs, XŪs s, Ūsds ζt + e gxūt T = E,x [ θ + e ζθ E θ,x Ū θ θ = E,x [ θ E,x [ θ e ζs fs, Xs Us, U s ds [ T θ e ζs θ fs, XŪs s, Ūsds + e ζt θ gxūt T F θ, ] e ζs fs, Xs Us, U s ds + e ζθ wθ, XŪθ θ, Ū e ζs fs, Xs Us, U s ds + e ζθ vθ, X U θ θ ]. ε Taking he supremum over U U w [, T ] we obain Equaion 1.13 is very difficul o solve direcly. One of he echniques allowing o obain he value funcion v, x, [, T ], x R k is o use he Bellman s Principle of Opimaliy o derive he second-order parial differenial equaion ha his funcion should saisfy. This equaion is called he Hamilon-Jacobi-Bellman HJB equaion. Alhough he equaion can be obained in general form, i will be derived for he specific sochasic conrol problems considered in he nex secions. ]] 17

29 1.4 Classical Model of Opimal Invesmen and Consumpion In his secion we consider he model of opimal invesmen and consumpion originaed by Meron [15]. Consider an invesor who a each ime has a porfolio valued a X. This porfolio invess in a money marke accoun riskless asse paying rae of ineres r and in n socks risky asses modeled by Geomeric Brownian moion 1. Suppose a each ime, he agen holds H shares of he risky asses, H shares of he riskless asse, and consumes a a rae C per uni ime. We define he corresponding processes below Riskless asse: dn = rn d, where r is a coninuous deerminisic funcion. Le us denoe r r. Risky asses : ds i = S i µ i d + n j=1 σ i,j 1 db j, where σ 1 = σ i,j 1 i,j=1...n is coninuous deerminisic volailiy marix which is inverible for each and, hus, he marke is complee, µ i is coninuous deerminisic expeced reurn. Le us denoe σ 1 σ 1, µ i µ i, i = 1,..., n. Porfolio value : X U = H N + H S, where H is he number of shares of he riskless asse, H is a row vecor of numbers of shares of he risky asses, S = S 1,..., S n T is a vecor of asses prices. Porfolio process: dx U = H dn + H ds C d, where C denoes consumpion per uni ime a ime. Expanding he expression for he porfolio process we obain dx U = rh N d + n HS i µ i i d + i=1 n j=1 σ i,j 1 db j C d = 1 Π 1rX U d + Π X U µd + σ 1 db C d = 1 Π 1rX U + Π X U µ C d + Π σ 1 X U db, See [3], p.147. This model is more general han he model in Meron s paper [15] where, for example, he rae of ineres was considered o be a consan. This generalizaion is necessary for comparing he resuls of his research wih he resuls in he classical model. 18

30 where Π = diaghs X U T represens he fracions of wealh invesed in he risky asses, 1 Π 1 = H N is he fracion of wealh invesed in he riskless asse borrowing and shorselling borrowing and selling asses are allowed, 1 = 1,..., 1 T X U, µ = µ 1,..., µ n T is a vecor of expeced reurns 1, and B = B 1,..., B n T is an n-dimensional Brownian moion. Using he previous noaion a 1 Π 1rX U Π C, U = R n [,. + Π X U µ C, s 1 Π σ 1 X U, U = Once he conrolled sysem 1.16 has been defined, we se up he reward funcional and he value funcion. The problem of sochasic conrol is o find he consumpion and invesmen sraegy ha maximizes he invesor s expeced uiliy. Le ζ > denoe he uiliy discoun rae which may be differen from he risk-free rae r. Remark 6. The reward funcionals, value funcions, and opimal sraegies obained in he following secions will be denoed by he same noaion even hough hey are no necessarily he same and, hus, should be inerpreed in he conex of each secion only Maximizing he Uiliy of Consumpion and Final Wealh Le fs, Xs Us, U s = Cs, gx U T T = XUT T,, 1 be he hyperbolic absolue risk aversion HARA uiliy funcions for consumpion and final wealh, respecively. These uiliy funcions are very general and are ofen used because of heir mahemaical simpliciy. The reward funcional and he value funcion w, x, U = 1 E,x [ T v, x = e ζs C s ds + e ζt X U T T ], sup w, x, U. U U w [,T ] The corresponding HJB equaion for, T and x > is p ζp + sup 1 π1rx + πxµ c p x + 1 π, c R n [, πσ 1x p xx + c 1 In his secion, 1 is an n-dimensional vecor. =. 19

31 The erminal and boundary condiions are { pt, x = 1 x, x >, p, =,, T. The meaning of he erminal condiion is ha if he invesor sars a ime T hen here is no ime for rading and he uiliy is equal o he uiliy of he wealh he sars wih. The boundary condiion means ha if he invesor has no money hen here in nohing o inves and he uiliy is zero. where The soluion o his boundary value problem is p, x = e x T T qτdτ + e τ 1. dτ qydy q = ζ 1 + r 1 + r µt σ1 T I has been verified [15] ha p v. Therefore, he opimal conrol 1 is Π = r µt σ 1 σ1 T 1, C = 1 e T X U qτdτ + T 1.4. Maximizing he Uiliy of Consumpion e τ qydy dτ. Le fs, Xs Us, U s = Cs, gx U T T,, 1 be he uiliy funcions for consumpion and final wealh, respecively. The reward funcional w, x, U = 1 E,x [ T ] e ζs C s ds, and he value funcion v, x = sup w, x, U. U U w [,T ] 1 If some of he enries of he vecor Π are negaive posiive, hen he invesor should shorsell buy he corresponding asses.

32 The corresponding HJB equaion for, T and x > is p ζp + sup 1 π1rx + πxµ c p x + 1 π, c R n [, πσ 1x p xx + c The erminal and boundary condiions are { pt, x =, x >, p, =,, T. The soluion o his boundary value problem is p, x = x T e τ 1, dτ qydy =. where q is defined in I has been verified [15] ha p v. Therefore, he opimal conrol is Π = r µt σ 1 σ1 T 1, C = 1 T X U e τ qydy dτ Maximizing he Uiliy of Final Wealh Le fs, Xs Us, U s =, gx U T T = XUT T,, 1 be he uiliy funcions for consumpion and final wealh, respecively. The reward funcional and he value funcion w, x, U = 1 [ E,x e ζt X U T v, x = T ], sup w, x, U. U U w [,T ] The corresponding HJB equaion for, T and x > is p ζp + sup 1 π1rx + πxµ c p x + 1 π, c R n [, πσ 1x p xx =. The erminal and boundary condiions are { pt, x = 1 x, x >, p, =,, T. 1

33 The soluion o his boundary value problem is p, x = e x T qτdτ 1, where q is defined in I has been verified [15] ha p v. Therefore, he opimal conrol is Π = r µt σ 1 σ1 T 1, C = Uncerainy in he Uiliy Funcion In some papers [14, ], i is claimed ha he uiliy of goods depends no only on he goods hemselves bu also on qualiies of he goods. The classical models of opimal invesmen and consumpion [7, 16, 1] can be exended o incorporae his feaure. Thus, i is of ineres o find ou how he opimal policy changes when he uncerainy in uiliy is inroduced. Indeed, when we buy differen hings we are no buying jus objecs, we are buying he qualiies ha hose objecs possess and we need hose qualiies. Clearly, a diamond ha coss as much as a house has uiliy which is differen from ha of he house. However, if he only argumen o he uiliy funcion is he price of a merchandise hen he uiliies of he diamond and he house are he same. The good, per se, does no give uiliy o he consumer; i possesses characerisics, and hese characerisics give rise o uiliy. 1 Since he echnology is changing, new producs keep coming ou and subsiue he old ones giving he increase in uiliy. For example, having a compuer oday gives much more opporuniies o is user compared o he compuers and echnologies available 3 years ago. Therefore, preferences migh change because of beer characerisics of new goods. I is imporan o noe ha his change does no have o enail he change of prices. The inflaionary increase in prices being relaive o he decrease caused by echnological progress means here is no variaion in he price index. 1 Lancaser, K.J A New Approach o Consumer Theory, Journal of Poliical Economy, Vol.74, N., p.134. Cencini, A Inflaion and Unemploymen: Conribuions o a New Macroeconomic Approach, Rouledge sudies in he modern world economy, Rouledge, p.3.

34 On he oher hand, preference change migh also be due o worsened qualiy of he producs or some oher reason for example, buying he same produc over and over migh decrease is uiliy and ends up in saiaion wih he produc. The described process is no deerminisic because i is no known how he marke will change. Therefore, i makes sense o model he uncerainy in uiliy by a sochasic process. Alhough here is much debae on he dynamics of echnological progress [1, 5], i is no new in lieraure ha in some areas i is growing exponenially [8]. A model ha assumes exponenial growh can also be used o model linear behavior because of he represenaion e x = 1 + x + ox and, hus, changing he parameers of he model accordingly, will help analyze he resuls when he growh is close o linear. Apar from big echnological advancemens here are minor improvemens in producs ha people use every day. Therefore, he Geomeric Brownian moion can be used o model he uncerainy in uiliy. The role played by he uiliy discoun rae ζ in he noaion of his paper used in classical models should no be confused wih he proposed uiliy randomness. The uiliy discoun rae accouns for he preference o obain somehing now insead of waiing and geing i laer. For example, i is preferable o have a lapop oday raher han omorrow. However, his only works if he implied compuer has he same characerisics. In general, i is no rue because compuer in he fuure can be more advanced. 3

35 Chaper Fully Observed Case.1 Formulaion of he Problem The problem of opimal porfolio managemen under uncerainy in uiliy funcion is considered in his chaper. We begin by assuming ha we have a filered probabiliy space Ω, F, {F s } s [,T ], P saisfying he usual condiion and here are wo sochasic processes defined on his space, X U, Z which represen he agen s wealh and uiliy randomness process a ime, respecively here we use he same noaion as in chaper 1 and X U means ha X depends on he conrol U. The wo processes are defined by he following sochasic differenial equaions: dx U = ax U, U d + s 1 X U, U db,1,.1 dz = bz d + s Z db,, where B,1 = B 1,1,..., B n,1 T is an n-dimensional Brownian moion, B, is a one dimensional Brownian moion, B,1, B, are correlaed and db i,1db, = ρ i d, where funcion ρ = ρ 1,..., ρ n T is coninuous and deerminisic ρ < 1 for all [, T ]. To shoren he noaion, le a ax U, U, b bz, s 1 s 1 X U, U, s s Z, and ρ ρ. Noe ha he uiliy randomness process Z does no depend on he conrol U = Π, C. However, he randomness of he sock prices represened by he Brownian moions B,1 can be correlaed wih ha of he uiliy randomness process specified by B,. This assumpion makes sense because i is reasonable o assume ha echnological 4

36 progress migh influence he sock prices. Denoe he uiliy discouning facor by qs = e ζs, where ζ > is he uiliy discoun rae. Define he reward funcional 1 as w, x, z, U = E,x,z [ T and he value funcion as v, x, z = ] qsfs, Xs Us, C s, Z s ds + qt gx U T T, Z T,. sup w, x, z, U..3 U U w [,T ] Therefore, he goal is o find a feasible conrol process {U s } s [,T ] supremum of he reward funcional. ha gives he The sysem.1 can be pu in a form 1.6 and, hus, under cerain assumpions Lipschiz coninuiy in space variable and boundedness a zero has a unique srong soluion. Indeed, changing he Brownian moions db,1, db, ino independen Brownian moions d B,1 = db,1, d B, = db, ρ T db,1 1 ρ, he sysem.1 can be wrien as d X = ã, U X, U d + s 1, X U, U d B, where X = X U, Z T, B = B,1, B, T and he funcions a, b and s 1, s are combined ino ã and s 1, respecively. This suggess he assumpions on a, s 1, b, s required for he resuls of secions 1. and 1.3 in chaper 1 be applicable, namely defining U as he space of conrol values, he funcions a : [, U R, s 1 : [, U R 1 n, b : [, R, and s : [, R are coninuous and saisfy he assumpions A1,, and 5. Similarly, funcions f, and g should saisfy assumpion A3, and 4.. Fully Observed Uiliy Randomness Process In his secion he problem.3 for cerain class of funcions f, g, a, b, s 1, and s is solved under he assumpion ha he uiliy randomness process Z is fully observed. In addiion o he processes defined in secion 1.4, we also define 1 E,x,z [ ] E[ X U = x, Z = z]. I should be aken ino accoun ha hese funcions do no depend on explicily, and b, and s do no depend on he conrol. 5

37 Uiliy randomness process: dz = βz d + σ Z db,, Z = z. As i was menioned in secion 1.5, a Geomeric Brownian moion is an appropriae process o model he uncerainy in he uiliy funcion. Le us use he previous noaion b βz, s σ Z. The parameer β represens he expeced insananeous growh in uiliy and σ > is he uiliy growh volailiy. Since echnological achievemens usually end o raise he uiliy, i is only naural o assume ha β >. We use he uiliy funcions of hyperbolic absolue risk aversion HARA ype, which is UC = C wih, 1. To model he uncerainy in he uiliy, we muliply he uiliy funcion by he uiliy randomness process Z. Therefore, he funcions f and g are f, x, c, z = c z and gx, z = x z, respecively. As i was menioned in secion 1.3, he problem.3 is difficul o solve direcly. One way o solve i is o derive a corresponding second-order parial differenial equaion more precisely, HJB equaion ha he value funcion should saisfy. I is assumed ha 1 he value funcion v C 1,, D C D, D = {, x, z : [, T, x >, z > } and he HJB equaion admis a classical soluion. If he soluion has been obained and i has been verified ha he soluion o he HJB equaion is he value funcion hen he opimal conrol can be found..3 Derivaion of he HJB Equaion In his secion we derive he HJB equaion using he Bellman s equaion defined in Consider he imes, θ [, T, θ > and a consan conrol U u U w [, T ] hen from Bellman s Principle of Opimaliy 1.13, we have he following inequaliy for he value funcion v, x, z E,x,z [ θ ] qsfs, Xs u, C s, Z s ds + qθvθ, Xθ u, Z θ..4 1 Noaion D means he compleion of D, and C 1,, D is he space of funcions coninuously differeniable in firs independen variable and wice coninuously differeniable in he second and hird independen variables. 6

38 Since v C 1,, D C D, Io s formula see [3], p.167 yields d qsvs, X u s, Z s = qs v s ζvds + v x dx u s + v z dz s + v xz d[x u s, Z s ] + 1 v xxd[x u s, X u s ] + 1 v zzd[z s, Z s ] = qs v s ζvds + v x ads + v x s 1 db s,1 + v z bds + v z s db s, + 1 v xxs 1 s T 1 ds + 1 v zzs ds + v xz s 1 ρs ds. Inegraing from o θ, and noing ha q = 1, we ge qθvθ, Xθ u, Z θ = v, X u, Z θ + qs v s ζv + v x a + v z b + 1 v xxs 1 s T1 + 1 v zzs + v xz s 1 ρs ds + θ qsv x s 1 db s,1 + θ qsv z s db s,. The sochasic inegrals in he above expression are local maringales see [1], p.36. h Consider a sequence of sopping imes τ n = inf{h : qsv x s 1 + qsv z s ds n}. Noice ha τ n diverges o infiniy almos surely as n goes o infiniy. Le τ = θ τ n, hen he sochasic inegrals τ qsv x s 1 db s,1 and are maringales. Plugging qτvτ, X u τ, Z τ ino equaion.4 we obain v, x, z E,x,z [ τ τ qs fs, Xs u, C s, Z s + v s ζv + v x a + v z b + 1 v xxs 1 s T v zzs + v xz s 1 ρs ds τ τ ] + qsv x s 1 db s,1 + qsv z s db s, + v, X u, Z. ] Since E,x,z [v, X u, Z = v, x, z, we have E,x,z [ τ qs fs, Xs u, C s, Z s + v s ζv + v x a + v z b + 1 v xxs 1 s T ] v zzs + v xz s 1 ρs ds, qsv z s db s, 7

39 in oher words, E,x,z [ τ ] qs fs, Xs u, C s, Z s ds + Lvs, Xs u, Z s ds,.5 where L = s ζ + a x + b z + 1 s 1s T 1 x + 1 s z + s 1 ρs z x is a differenial operaor. [ T ] Assume ha E,x,z qs fs, Xs u, C s, Z s ds+lvs, Xs u, Z s ds < if < T <. Then aking he limi as n goes o infiniy of.5 and using he Dominaed Convergence Theorem see [3], p.7, we obain, for any < θ < T, E,x,z [ θ ] qs fs, Xs u, C s, Z s ds + Lvs, Xs u, Z s ds..6 Noe ha f is coninuous, v C 1,, D C D, and we divide equaion.6 by θ and ake he limi as θ decreases o o ge f, x u, c, z + Lv, x u, z,, x, z D. Also noice ha his is rue for any consan conrol u U w [, T ] for all [, T, hen we reach sup u U f, x u, c, z + Lv, x u, z,, x, z D..7 On he oher hand, suppose ha U is an opimal conrol hen by definiion of he value funcion v, x, z = E,x,z [ θ Using he same approach as before, we obain ] qsfs, X U s s, C s, Z s ds + qθvθ, X U θ θ, Z θ. f, x U, c, z + Lv, x U, z =,, x, z D..8 Thus, equaions.7 and.8 sugges ha he funcion v should saisfy he following 8

40 equaion sup u U f, x u, c, z + Lv, x u, z =,, x, z D. Therefore, v saisfies solves he HJB equaion wih he boundary condiion given below. sup u U f, x u, c, z + Lv, x u, z =,, x, z D, vt, x, z = gx, z, x >, z >. where L = ζ + a x + b z + 1 s 1s T 1 x + 1 s z + s 1 ρs z x.9 is a differenial operaor..4 Verificaion Theorem Once he HJB equaion has been solved and we have a soluion v, x, z we need o check ha his soluion is indeed he funcion defined by.3. The nex heorem gives sufficien condiions ha he soluion o he HJB equaion should saisfy o be he value funcion. The noaion p and v are used o disinguish a soluion of he HJB equaion from he value funcion, respecively. Theorem 4. Verificaion Theorem Le a funcion p C 1,, D C D and saisfy a quadraic growh condiion such as p, x, z C1 + x + z,, x, z D for some consan C >. 1 Suppose ha sup u U f, x u, c, z + Lp, x u, z,, x, z D,.1 pt, x, z gx, z, x >, z >,.11 where L = ζ + a + b + 1s x z 1s T x s + s z 1 ρs z x Then p v on D. is a differenial operaor. Suppose ha pt, x, z = gx, z and here exiss a measurable funcion U, x, z, 9

41 where, x, z D, and aking values in U such ha sup u U f, x u, c, z + Lp, x u, z = f, x U, c, z + Lp, x U, z =..1 Also assume ha he sochasic differenial equaion dx s = ax s, U s, X s, Z s ds + s 1 X s, U s, X s, Z s db s,1 admis a unique srong soluion denoed by X, given an iniial condiion X = x and he process U U w [, T ]. Then he funcion p is he value funcion v given by.3, in oher words p = v on D, and U is an opimal Markov conrol a ime depends only on X and Z. Proof. 1 Assume ha he funcion p saisfies he saed in he heorem assumpions. h Le τ n = inf{h : qsp x s 1 + qsp z s ds n} be a sequence of sopping imes diverging o infiniy almos surely as n goes o infiniy and le τ = τ n θ, hen by he inegral form of Io s formula, we have for all U in U w [, τ] τ τ τ qτpτ, Xτ Uτ, Z τ = p, X U, Z + qslpds + qsp x s 1 db s,1 + qsp z s db s,. Taking expecaions of boh sides and using he fac ha he sochasic inegrals are maringales, we obain E,x,z [qτpτ, X Uτ τ [ τ ], Zτ Uτ ] = p, x, z + E,x,z qslpds, U U w [, τ]. Since p saisfies inequaliy.1, we have qsfs, Xs Us, C s, Z s + qslps, Xs Us, Z s, U s U, s [, T ]. Hence for all U in U w [, τ], E,x,z [qτpτ, X Uτ τ Noe ha τ qsfs, X Us s [ τ, Zτ Uτ ] p, x, z E,x,z, C s, Z s ds, Z τ C1 + sup Xs Us s [,T ] inequaliies are inegrable see Theorem 1. ha pτ, X Uτ τ T + sup s [,T ] ] qsfs, Xs Us, C s, Z s ds..13 qsfs, Xs Us, C s, Z s ds and also recall Z s and he righ hand sides of hese If we apply he Dominaed Convergence 3

42 Theorem o ake he limi in.13 as n goes o infiniy hen one can obain E,x,z [qθpθ, X U θ θ [ θ, ZU θ θ ] p, x, z E,x,z ] qsfs, Xs Us, C s, Z s ds, U U w [, θ]. By he assumpion ha he funcion p is coninuous in D, by he condiion.11 and by he Dominaed Convergence Theorem, as θ goes o T, we ge ha for all U U w [, T ] he following inequaliy holds or equivalenly, E,x,z [qt gx U T T p, x, z E,x,z [ T p, x, z E,x,z [ T, Z T ] E,x,z [qt pt, X U T T, ZU T T ] ], qsfs, Xs Us, C s, Z s ds ] qsfs, Xs Us, C s, Z s ds + qt gx U T T, Z T..14 Since.14 holds for all U U w [, T ], we can conclude p v. Adoping a way similar o he proof of par 1, we consider a sequence of sopping imes τ n defined as in 1, τ = τ n θ, and apply he Io s formula o he funcion qτpτ, X U τ τ, Z τ. Then we ake expecaion on boh sides and ake limi as n ends o infiniy. Then i is easy o see [ E,x,z [qθpθ, X U θ τ θ, Z θ ] = p, x, z + E,x,z By.1 we have ha [ E,x,z [qθpθ, X U θ θ θ, Z θ ] = p, x, z E,x,z and aking he limi as θ goes o T we obain p, x, z = E,x,z [ T ] qslps, X U s s, Z s ds. ] qsfs, X U s s, C s, Z s ds ] qsfs, X U s s, C s, Z s ds + qt gx U T T, Z T = w, x, z, U, which means ha p, x, z = w, x, z, U v, x, z. This, ogeher wih he resul in par 1 implies ha p = v wih U as an opimal Markov conrol. 31

43 .5 Maximizing he Uiliy of Consumpion and Final Wealh In his secion we consider he problem of maximizing he uiliy of consumpion and final wealh. The value funcion assuming, 1 is v 1, x, z = 1 [ T sup E,x,z qsc s Z s ds + qt XT U Z T ]. U U w [,T ] The corresponding HJB equaion.9 for, T, x >, and z > is p ζp + βzp z + 1 σ z p zz + sup 1 π1rx + πxµ c p x π, c R n [, + 1 πσ 1x p xx + πσ 1 ρσ xzp zx + c z =.15 where for each, T we have π R n is a row vecor ha represens he fracions of wealh invesed in he risky asses, 1 = 1,..., 1 T R n, µ R n is a column vecor of expeced reurns, c is a scalar-valued consumpion rae, and ρ R n is he correlaion column vecor defined in secion.1. The erminal and boundary condiions are pt, x, z = 1 x z, x >, z >, p,, z =,, T, z >, p, x, =,, T, x >. We now discuss he meaning of he erminal and boundary condiions. The erminal condiion pt, x, z = 1 x z means ha if he invesor sars rading a ime T hen here is no ime for invesmen and he uiliy of his wealh is equal o he uiliy of he wealh he sars wih. The boundary condiion p,, z = says ha if he iniial capial is zero hen he value funcion is zero. The hird condiion p, x, = implies ha he invesor is no ineresed in rading because he qualiy of goods he can buy is zero and herefore he value funcion is zero regardless of he amoun of wealh he has. This case seems unrealisic and he probabiliy of his happening is zero. 3

44 .5.1 Soluion o he HJB equaion We look for a soluion in he form of p, x, z = 1 x zh 1 where h is some posiive funcion. This form of soluion is suggesed by he funcions f, x, c, z = c z gx, z = x z,, 1 defined in secion.. Firs, we evaluae he supremum sup 1 π1rx + πxµ c x 1 zh 1 π, c R n [, + 1 πσ 1x 1x zh 1 + πσ 1 ρσ zx h 1 + c z = x zh 1 sup π, c R n [, 1 π1r + πµ c x + 1 πσ πσ 1 ρσ + = x zh 1 sup 1 π1r + πµ + 1 π R n πσ πσ 1 ρσ + sup Consider he funcions c [, c x h 1 c x h 1 c. x and g 1 π = 1 π1r + πµ + 1 πσ πσ 1 ρσ, g c = c x h 1 c x. The Hessian of he funcion g 1 π is Hπ = σ 1 σ T 1 1 and i is negaive definie. Also, d g dc Thus, = 1c x h 1 <. Therefore, he maximum π, c is obained from g 1 π = µ r + σ 1 σ T 1 π T 1 + σ 1 ρσ =, dg dc = 1 x + c 1 =. x h 1 π = r µt ρ T σ1 T σ σ 1 σ1 T 1, 1 c = x h. 33

45 Subsiuing π ino he supremum over π we obain 1 r µt ρ T σ1 T σ σ 1 σ1 T 1 1r + r µt ρ T σ1 T σ σ 1 σ1 T 1 µ r µt ρ T σ1 T σ σ 1 σ1 T 1 r µ T ρ T σ σ T σ σ 1 σ1 T 1 σ 1 ρσ 1 1 = r r µt ρ T σ1 T σ σ 1 σ1 T 1 1r + r µt ρ T σ1 T σ σ 1 σ1 T 1 µ r µt ρ T σ1 T σ σ 1 σ1 T 1 r µ T ρ T σ σ T σ σ 1 σ1 T 1 σ 1 ρσ 1 1 = r r µt ρ T σ1 T σ σ1 T 1. 1 Now we subsiue c ino he supremum over c we have Therefore, he HJB equaion.15 becomes x x h x + h x h 1 = 1 h + 1 h. 1 x zh h ζ x zh 1 + β zx h 1 + x zh 1 r r µt ρ T σ1 T σ σ1 T h + 1 h Dividing boh sides of he equaion.16 by 1 x zh we obain h ζ 1 h + β 1 h + 1 h r r µt ρ T σ1 T σ σ1 T h + 1 h which is equivalen o h ζ 1 h + β 1 h =..16 = + r 1 h + r µt ρ T σ T 1 σ σ T h =. 34

46 In oher words, h + yh + 1 =,.17 where y is defined as y = ζ + β + r 1 + r µt ρ T σ T 1 σ σ T Since y is coninuous, he equaion.17 wih he erminal condiion ht = 1 admis he unique soluion h 1 = e T T yτdτ + e τ yqdq dτ. The funcion h 1 is in C 1 [, T ] and is posiive. Therefore, he soluion o he HJB equaion.15 is.5. Verificaion p 1, x, z = x z e T T yτdτ + e τ 1. dτ yqdq.19 The soluion p 1 o equaion.15 is a C 1,, D C D funcion. The quadraic growh condiion is also saisfied as one can see below. We see ha he funcion h 1 s > is bounded on he inerval [, T ] and since < < 1, x, z we have x < 1 + x which implies x z z + xz < 1 + xz + z < 1 + x + z + z < 31 + x + z. I should also be verified ha he obained conrol is admissible. The wealh process 1.16 when he conrol {Π s, C s } s [,T ] is used, saisfies he following sochasic differenial equaion dxs = 1 Π srxs + Π sxs µ Cs ds + Π sσ 1 Xs db s,1 35

47 = 1 r µt ρ T σ1 T σ σ 1 σ1 T 1 1r + r µt ρ T σ1 T σ σ 1 σ1 T 1 µ X s ds + r µt ρ T σ1 T σ σ 1 σ1 T 1 σ 1 Xs db s,1 h 1 s 1 = r r µt ρ T σ1 T σ σ 1 σ1 T 1 r µ 1 Xs ds 1 h 1 s + r µt ρ T σ1 T σ σ1 T 1 Xs db s,1 1 which is a Geomeric Brownian moion assumpion B4 in Secion 1.3 is saisfied. Since X s is coninuous hen by Theorem X s is F s, -progressively measurable and B3 in secion 1.3 is fulfilled. From he fac ha h 1 s > K > for all s [, T ] for some consan K > and f, x, z, c = c z < c + z, we have 1 C s Z s = Zs 1 X s h 1 < X s K s + Z s and, hus, by Theorem 1 funcion f is inegrable. Similarly, funcion g is also inegrable. This implies ha B5 in secion 1.3 is saisfied. Therefore, he conrol is admissible. Since he soluion p 1, x, z o.15 saisfies he assumpions of he verificaion heorem hen v 1 = p 1 and he opimal conrol is Π s = r µt ρ T σ1 T σ σ 1 σ1 T 1, Cs,1 = X s 1 h 1 s...6 Maximizing he Uiliy of Consumpion In his secion, we will consider a problem of maximizing he uiliy of consumpion. The value funcion is v, x, z = 1 [ T ] sup E,x,z qsc s Z s ds. U U w [,T ] The corresponding HJB equaion.9 for, T, x > and z > is p ζp + βzp z + 1 σ z p zz + sup 1 π1rx + πxµ c p x π, c R n [, + 1 πσ 1x p xx + πσ 1 ρσ xzp zx + c z =..1 36

48 The dimensions and meaning of all he variables and parameers is he same as in secion.5. The erminal and boundary condiions are pt, x, z =, x >, z >, p,, z =,, T, z >, p, x, =,, T, x >. The meaning of he erminal and boundary condiions is he same as in secion Soluion o he HJB equaion The calculaions are he same as in secion.5 and he only difference is ha he erminal condiion for he funcion h is given as ht =. Therefore, we have h + yh + 1 =,. where y is defined by.18. I is easy o verify ha he soluion o equaion. wih he erminal condiion ht = is h = T e τ yqdq dτ. The funcion h is in C 1 [, T ] and is posiive. Therefore, he soluion o he HJB equaion.1 is given as.6. Verificaion p, x, z = x z T e τ 1. dτ yqdq.3 The verificaion is mosly idenical o he one in secion.5 bu he funcion h s goes o as s goes o T due o coninuiy. Hence he condiion h s > K > is no saisfied. To verify ha he funcion f is inegrable, le us denoe A 1 r r µt ρ T σ T 1 σ σ 1 σ T 1 1 r µ 1 and A r µt ρ T σ T 1 σ σ T 1 1 1, hen he equaion for he wealh process assuming s < T becomes dx s = A 1 1 Xs ds + A Xs db s,1. h s 37

49 The soluion o his sochasic differenial equaion wih he iniial condiion X = x is Thus, C s = where Y s = xe X s =xe =xe s A db τ,1 + s s A db τ,1 + s A 1 1 h τ 1 A A 1 1 A dτ dτ e s X s s h s = xe A db τ,1 + s A 1 1 A dτ e s s A db τ,1 + s A 1 1 A dτ 1 h τ dτ h s 1 h τ dτ. e s = Y s 1 h τ dτ h s is a Geomeric Brownian moion wih he iniial condiion Y = x. We will show ha he erm 1 h s e h τ dτ is bounded on [, T ]. Since he funcion ys, s [, T ] is coninuous, i aains is minimum and maximum. Tha is why, for some ε > we may denoe m = min min ys, ε and s [,T ] M = max max ys, ε. The following inequaliies hold s [,T ] and, hence, T s e mτ s dτ h s = Using he above inequaliies, we obain T s s e T τ s yqdq dτ e Mτ s dτ s 1 m emt s 1 h s 1 M emt s 1. 1 e s 1 h τ dτ h s s e 1 h τ dτ s M e MT l 1 dl 1 emt s Ms+ln 1 e MT 1 m emt s 1 e 1 m emt s 1 = e 1 m emt s 1 and afer simlificaion, he las erm in he above inequaliy equals me Ms 1 emt s 1 e MT e mt s 1 = mems 1 e MT s 1 e MT e mt s 1 = which goes o memt M MeMT, or afer simplificaion e MT 1 m e MT 1 mems e MT 1 MT s 1 e 1 emt s as s approaches T from he lef. 38

50 From he fac ha funcion f saisfies f, x, z, c = c z < c + z, we have he Zs inequaliy 1 C s Z s = 1 X s h < s K Y s + Z s, where K = MeMT. Thus, e MT 1 by Theorem 1 he funcion f is inegrable. Therefore, v = p. The opimal conrol is Π s = r µt ρ T σ1 T σ σ 1 σ1 T 1, Cs, = X s 1 h s..4.7 Maximizing he Uiliy of Final Wealh In his secion, we look a he problem of maximizing he uiliy of final wealh. The value funcion is v 3, x, z = 1 sup E,x,z [qt XT U Z T ]. U U w [,T ] The corresponding HJB equaion.9 for, T, x > and z > is p ζp + βzp z + 1 σ z p zz + sup π, c R n [, 1 π1rx + πxµ c p x + 1 πσ 1x p xx + πσ 1 ρσ xzp zx =..5 The dimensions and meaning of all he variables and parameers is he same as in secion.5. The erminal and boundary condiions are pt, x, z = 1 x z, x >, z >, p,, z =,, T, z >, p, x, =,, T, x >. The meaning of he erminal and boundary condiions is he same as in secion Soluion o he HJB equaion We may look for a soluion in he form of p, x, z = 1 x zh 1 where h is some posiive funcion. This form of soluion is suggesed by he funcions f, x, c, z = c z and gx, z = x z,, 1 defined in secion.. Firs we evaluae he supremum in 39

51 he equaion.5 sup π, c R n [, 1 π1rx + πxµ c x 1 zh πσ 1x 1x zh 1 + πσ 1 ρσ zx h 1 1 π1r + πµ c x + 1 πσ πσ 1 ρσ = x zh 1 sup π, c R n [, = x zh 1 sup 1 π1r + πµ + 1 πσ πσ 1 ρσ + sup π R n Consider he funcions c [, c. x g 1 π = 1 π1r + πµ + 1 πσ πσ 1 ρσ, g c = c x. The Hessian of g 1 π is Hπ = σ 1 σ T 1 1 and is negaive definie. Also, he maximum of g c is reached when c =. Therefore, he maximum of g 1 π is obained from g 1 π = µ r + σ 1 σ T 1 π T 1 + σ 1 ρσ =. Thus, π = r µt ρ T σ1 T σ σ 1 σ1 T 1, 1 c =. Subsiuing π, c ino he suprema we obain 1 r µt ρ T σ T 1 σ σ 1 σ T r µt ρ T σ1 T σ σ 1 σ1 T 1 1 1r + r µt ρ T σ1 T σ σ 1 σ1 T 1 µ 1 σ r µ T ρ T σ T 1 σ σ 1 σ T σ 1 ρσ 4

52 = r r µt ρ T σ1 T σ σ 1 σ1 T 1 1r + r µt ρ T σ1 T σ σ 1 σ1 T 1 µ r µt ρ T σ1 T σ σ 1 σ1 T 1 r µ T ρ T σ σ T σ σ 1 σ1 T 1 σ 1 ρσ 1 1 = r r µt ρ T σ1 T σ σ1 T 1. 1 Therefore, he HJB equaion.5 becomes 1 x zh h ζ x zh 1 + β zx h 1 + x zh 1 r r µt ρ T σ1 T σ σ1 T 1 =..6 1 Dividing boh sides of he equaion.6 by 1 x zh, we obain equivalenly, Hence h ζ 1 h + β 1 h + 1 h r r µt ρ T σ1 T σ σ1 T 1 = 1 h ζ 1 h + β 1 h + r 1 h + r µt ρ T σ T 1 σ σ T h =. h + yh =,.7 where y is defined by.18. The soluion o.7 wih he erminal condiion ht = 1 is h 3 = e T yτdτ. The funcion h 3 is in C 1 [, T ] and is posiive. Therefore, he soluion o he HJB equaion.5 is p 3, x, z = x z e T yτdτ

53 .7. Verificaion For his problem, he verificaion is quie similar o he one already done in secion.5 and v 3 = p 3. The opimal conrol here is Π s = r µt ρ T σ1 T σ σ 1 σ1 T 1, Cs,3 = Analysis of he Resuls and Numerical Experimens Here we analyse and inerpre he resuls obained in secions.5,.6, and.7. Throughou his secion, we assume ha all he parameers are consan, < T, and he invesor has he uiliy funcion UC = C,, 1 he uuliy funcion assumed in secions.5,.6, and.7. From.,.4, and.9 we have he following claim. Proposiion. The opimal porfolio is Π = r µt σ 1 σ T 1 1 v x X v xx ρt σ 1 1 σ Z v zx X v xx regardless of he problem maximizing boh he uiliy of consumpion and final wealh, only he uiliy of consumpion, or only he uiliy of final wealh. There are a few hings o discuss abou his proposiion. By comparison wih he soluion of he classical model secion 1.4, one should noice ha he second fracion is he effec of he randomness in he uiliy. For n = 1 here is only one risky asse, if µ < r, hen he invesor will shor borrow and sell some of he risky asses, and if µ > r, hen he will inves in he risky asses. We nex show ha he Muual Fund Theorem holds. Theorem 5. The opimal porfolio is made of hree muual funds. Firs fund Φ 1 consiss of he risk-free asse and he oher wo funds Φ = r µ T σ 1 σ T 1 1, Φ 3 = ρ T σ 1 1 include risky asses. The vecors Φ, Φ 3 represen he second and hird porfolio s weighs of he risky asses a ime. The opimal allocaion of he wealh in each fund is given by λ = vx X v xx, λ 3 = σ Z v zx X v xx, and λ 1 = 1 λ λ 3. 4

54 Proof. The proof follows immediaely from he obained opimal porfolio see Proposiion. The muual fund heorem above saes ha he invesor who wans o maximize his expeced uiliy. will be indifferen beween choosing from he linear combinaion of n + 1 asses or a linear combinaion of he hree muual funds. The firs and second funds are he same as hose in he classical problem whereas he hird fund arises from he correlaion beween B,1 and B,. Le n = 1, and we realize he opimal porfolio is Π = r µ ρσ σ1. 1 σ 1 1 Since dz ds Z S = ρσ 1 σ d, consider Π as a funcion of ρ, hen dπ = σ dρ σ 1 > and we see ha 1 he higher he correlaion beween relaive changes in asse price and uiliy randomness process is, he more he invesor should inves in he asse. Also, comparing he behaviors of he invesor in he classical case and he invesor who akes ino accoun echnological progress, produc improvemens and oher facors, he laer is invesing more in he risky asse when ρ > and less when ρ <. The opimal porfolio does no include he drif parameer β of he uiliy randomness process. This means ha he porion of he wealh invesed in he risky asse does no depend on how fas new producs come ino he marke. I only depends on how volaile hese changes are which is characerized by he variance parameer σ. Also if ρ > ρ <, hen he larger he value of σ is, he more less shares of he risky asse should be included in he opimal porfolio. Proposiion 3. The highes saisfacion he invesor can acquire is from maximizing boh he uiliy of consumpion and final wealh. In oher words, v 1, x, z v, x, z, and v 1, x, z v 3, x, z, for all, x, z D. To see why proposiion 3 is rue one needs o compare.19,.3, and.8. Remark 7. In general, similar conclusions comparing v and v 3 value funcions in he problems of maximizing he uiliy of consumpion and final wealh, respecively canno be made because e T T yτdτ e τ funcion y. For example, if y =, hen e T yqdq dτ can be posiive or negaive depending on he yτdτ = 1 and hus we can clearly see ha v > v 3, if T > 1 and v 3 > v, if T < 1. T e τ yqdq dτ = T and Proposiion 4. If he uiliy uncerainy and he marke risk are uncorrelaed, hen he invesor who akes ino accoun he echnological progress invess as much in he risky 43

55 asse as he invesor who does no consider ha. The only difference is in heir opimal consumpion and he value funcion. This resul readily follows from he definiion of y in.18 and he obained opimal porfolio. One should also noice ha he uiliy discoun rae ζ plays he role opposie o ha of he expeced insananeous growh rae β in he uiliy. Nex we sae a raher obvious observaion abou he dependence of he value funcions on he echnological progress. Proposiion 5. The more rapid he echnological progress is, he higher he value funcions v 1, v, and v 3 are. To verify his proposiion we need o find how sensiive he value funcions are o he change in he parameer β he higher he value of β is, he faser producs imporve in he marke. Thus, we may consider he value funcions as funcions of β. Since he signs of dv 1, dv, and dv 3 are he same as he signs of dh 1, dh, and dh 3 respecively, we dβ dβ dβ dβ dβ dβ consider h 1, h, and h 3 as funcions of β and recall ha h = eyt 1, h 3 = e yt, and dy = 1. Then we reach dβ 1 dh dβ = dh dy dy dβ = T eyt y e yt 1 1 y 1 = eyt yt 1 + 1, 1 y dh 3 dβ = dh 3 dy dy dβ = T 1 eyt. From he above expressions, we see ha dh and dh 3 >. Indeed, o show ha dβ dβ, i is enough o check ha he minimum of fx, y = exy xy 1+1, y > is equal dh dβ o. We have f = x xy e xy = if x =. Le z = xy hen fx, y = gz = e z z Since dg = dz zez = if z = and dg is posiive when z > and negaive when z <. dz So we have ha he minimum of gz is reached a z =. Therefore, he minimum of fx, y is equal o zero. Since h 1 = h + h 3 we have dh 1 > as well. dβ Alhough he signs of he second derivaives for he funcions v 1, v are quie difficul o deermine in general, for he funcion v 3 i is fairly easy. y Indeed, from he above expression for he derivaive of h 3 wih respec o β and from.8, we have dv 3 dβ = x z T e yt 1 44

56 and afer differeniaing again, we obain d v 3 dβ = x z T e yt 1. Since he second derivaive wih respec o β is posiive, he funcion v 3 is convex in β. Therefore, we have he following proposiion. Proposiion 6. When he objecive is o maximize he expeced uiliy of final wealh, he corresponding value funcion v 3 is convex in β. This resul means ha he value funcion increases a a higher rae as he parameer β increases. For example, o double he agen s expeced uiliy, he expeced insaneneous growh rae β has o increase by less hen is curren value. This makes sense because, as i was menioned in he inroducion, he echnological progress is assumed o increase he uiliy exponenially. Proposiion 7. If y >, hen as he erminal ime increases, he value funcions grow a an exponenial rae. To verify his proposiion, one needs o consider he value funcions as funcions of he final ime T. Since h 1 = h + h 3, we find ha he derivaives are posiive dh dt = eyt, dh 3 dt = yeyt. For example, if he agen wans o double he value of his value funcion obained over ime inerval, T, hen he needs o increase T by he amoun smaller han T. Remark 8. If y, he proposiion 7 is no rue in general. Nex, o simplify he analysis, we assume ha n = 1, hen we have Proposiion 8. If µ r and ρ >, hen he value funcions are increasing in he volailiy consan σ and correlaion ρ. For he same reason as above, i is enough o deermine he signs of he derivaives 45

57 of h 1, h, and h 3 wih respec o σ and ρ. Recall ha Then dy = µ r + ρσ 1σ ρ, dσ 1 σ 1 dy dρ = µ r + ρσ 1σ σ. 1 σ 1 dh = dh dy = T eyt y e yt 1 µ r + ρσ 1 σ ρ, dσ dy dσ y 1 σ 1 dh 3 = dh 3 dy = T µ r + ρσ 1σ ρ e yt, dσ dy dσ 1 σ 1 dh dρ = dh dy dy dρ = T eyt y e yt 1 µ r + ρσ 1 σ σ, y 1 σ 1 dh 3 dρ = dh 3 dy dy dρ = T µ r + ρσ 1σ σ e yt. 1 σ 1 Since h 1 = h + h 3, we also have dh 1 dσ > and dh 1 >. The assumpion ha µ r dρ is no unrealisic, because his is usually he case he expeced reurn on a risky asse is usually higher han he riskless ineres rae. The fac ha under he assumpions ha were made he value funcions are increasing in σ makes sense, because in his case here is a high probabiliy ha he producs are o be improved significanly and his, in urn, will increase he agen s saisfacion. I is worh noing ha his inerpreaion is in accordance wih ha of an opion price sensiiviy o he change in he underlying sock volailiy vega in greeks. I is well known ha he larger he volailiy of he underlying is, he higher he opion price becomes because in his case he probabiliy ha he opion will be in-he-money is bigger han for he opion wrien on a sock wih small volailiy. Remark 9. If ρ <, he derivaives can ake on posiive or negaive values depending on he values of he oher parameers. As can be easily verified, he necessary and sufficien condiion for he value funcions o be increasing in σ is ρ σ 1 σ > r µρ and o be increasing in ρ is ρσ 1 σ > r µ. Proposiion 9. For a given wealh process X, he opimal raes of he consumpion per uni ime are decreasing funcions of β. Furhermore, if µ r and ρ >, hen hey are also decreasing boh in σ and ρ. 46

58 As i was found in.,.4 he opimal raes of he consumpion per uni ime are C,i = X h i, i = 1, depending on he objecive. We can use he evaluaed derivaives of he funcions h i, dh i, dh i dβ Indeed, dσ, i = 1, o find he derivaives of he funcions C,i, dc,i dβ, dc,i dσ. dc,i dβ = X h i dh i dβ, dc,i = X dh i, dσ h i dσ dc,i dρ = X dh i h i dρ. Therefore, under he assumpions ha were made, he derivaives above are negaive. This proposiion agrees wih he proposiions, 5, and 6, because i says ha when he producs improve fas and he relaive change in he asse s price is posiively correlaed wih ha of he echnological improvemens, he invesor should inves more in he risky asse and, hus, decrease he consumpion. To help he readers undersanding we consider he following numerical example wih parameers values chosen only for demonsraion. Le he correlaion be ρ =.4 and he risky asse volailiy be σ 1 =.1. The risk-free ineres rae is r =.5. The risky asse s insananeous mean rae of reurn is µ =.. The uiliy discoun rae is ζ =.5. The invesor s iniial wealh is x = 1. The relaive risk aversion is =.5, and he erminal ime is T = 1. The parameers β and σ vary in he inerval [,.5]. The consumpion per uni wealh as a funcion of ime and β, ime and σ, when he uiliy of consumpion and final wealh is maximized, is shown in Figure.1. As one can see from Figure.1, he consumpion rae is increasing over ime. The consumpion per uni wealh as a funcion of ime and β, ime and σ, when only he uiliy of consumpion is maximized, is shown in Figure.. One can observe ha as approaches erminal ime T, he opimal rae of consumpion C, per uni wealh approaches infiniy. However, i shouldn be inerpreed as an infinie rae of consumpion. Raher more proper explanaion may be he following. Since here is no uiliy associaed wih he wealh for > T, he consumpion rae should increase, hus, making X approach as approaches T. A similar explanaion of his behavior in he classical model can be found in [16]. The graphs of he opimal consumpions C,1 and C, per uni wealh as funcions 47

59 Figure.1: Consumpion C,1 per uni wealh as a funcion of ime and β σ =.4; b funcion of ime and σ β =.4. Figure.: Consumpion C, per uni wealh as a funcion of ime and β σ =.4; b funcion of ime and σ β =.4. of ime and correlaion ρ look very similar o he graphs in Figure.1 and Figure., respecively..9 Conclusions In his chaper so far we mainly exended he classical model of opimal invesmen and consumpion by adding uncerainy in he uiliy funcion. I was shown ha he Bellman s Principle of Opimaliy also holds for his new randomized model and as a resul, we derived he Hamilon-Jacobi-Bellman equaion associaed wih he value funcion. Since no all he soluions of he HJB equaion are he value funcions, we proved he heorem ha can be used o verify ha he obained soluion becomes he 48