OPTIMAL INVESTMENT AND CONSUMPTION PORTFOLIO CHOICE PROBLEM FOR ASSETS MODELED BY LEVY PROCESSES

Transcription

1 OPTIMAL INVESTMENT AND CONSUMPTION PORTFOLIO CHOICE PROBLEM FOR ASSETS MODELED BY LEVY PROCESSES By RYAN G. SANKARPERSAD A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 211

2 c 211 Ryan G. Sankarperad 2

3 ACKNOWLEDGMENTS I would like to thank my diertation advior Dr Liqang Yan for hi help throughout the reearch proce, your guidance wa crucial to my growth a a mathematician. Alo Dr Yan coure in tochatic calculu and mathematical finance helped develop my interet in the field of finance. I would alo like to individually thank all member of my diertation committee. Dr Michael Jury year long cla in Meaure Theory wa fundamental to my undertanding of more advanced concept required for thi diertation. Dr Murali Rao coure in Probability theory helped olidify my undertanding of meaure theory and wa a great introduction to the theory of random variable. Dr Stan Uryaev coure on fixed income derivative, portfolio theory and rik management technique introduced much of the financial theory that I have ued in thi diertation and throughout the job application proce. Dr Joeph Glover poignant quetion during my Oral Exam and inight into the field of Levy Procee were helpful in guiding my reearch over the lat few year of reearch. I would alo like to thank Dr Jay Ritter from the Department of Finance for teaching me the claical theory of corporate finance during hi year long coure. Lat but not leat, I would like to thank my family (Mom, Dad, Rianna and Rehma) for all the love and upport throughout the diertation proce, you were all intrumental to my ucce. 3

4 TABLE OF CONTENTS page ACKNOWLEDGMENTS LIST OF TABLES LIST OF FIGURES ABSTRACT CHAPTER 1 GENERAL FRAMEWORK AND PROBLEM SETUP Introduction Stochatic Calculu Preliminarie Ito Diffuion and it Generator STATEMENT OF MAIN PROBLEM Optimal Control Theory Dynamic Programming Methodology Hamilton Jacobi Bellman Verification Theorem MERTON PROBLEM Claic Merton Solution Infinite Horizon Power Utility Infinite Horizon Log Utility Finite Horizon Power Utility Finite Horizon Log Utility LEVY PROCESS Why ue a Levy Proce with Jump? Preliminarie of Levy Proce Ito-Levy Diffuion and it Generator Verification Theorem for Levy Proce with Jump OPTIMAL CONTROL PROBLEM IN THE JUMP CASE Claic Merton Problem with Jump Jump Cae with Log Utility Finite Horizon Power Utility with Jump Generalization of Bequet Function in the Jump Cae

5 6 NUMERICAL RESULTS AND CONCLUSION Jump diffuion and Levy triplet Optimal θ for Power Utility Γ(1, 1) Proce Compound Poion Proce with Exponential Denity Explicit Value Function Concluion REFERENCES BIOGRAPHICAL SKETCH

6 Table LIST OF TABLES page 2-1 Table with comparion of two world ued to olve optimal control problem Maximum difference between finite horizon and infinite horizon for different value of T Table for θ a a function of σ, where α =.16, r =.6, γ = Table for θ a a function of γ, where α =.16, r =.6, σ =

7 Figure LIST OF FIGURES page 3-1 Value function for power utility with no jump with parameter, α =.16, r =.6, δ = 1, γ =.5, σ = Wealth proce for power utility with no jump with parameter, w = 1, α =.16, r =.6, δ = 1, γ =.5, σ = Value function for log utility with no jump with parameter, α =.16, r =.6, δ = 1, γ =.5, σ = Wealth proce for Log utility with no jump with parameter, w = 1, α =.16, r =.6, δ = 1, σ = Plot of the value function for value of T = 1, 2, 1 compared to the plot of the value function for infinite horizon Plot of the value function for value of T = 1, 2.5, 5, 7.5, 1 compared to the plot of the value function in the infinite horizon cae Plot of the value function for value of λ = 1,.5,.25,.125, 1e-9 compared to the plot of the value function for λ =

8 Abtract of Diertation Preented to the Graduate School of the Univerity of Florida in Partial Fulfillment of the Requirement for the Degree of Doctor of Philoophy OPTIMAL INVESTMENT AND CONSUMPTION PORTFOLIO CHOICE PROBLEM FOR ASSETS MODELED BY LEVY PROCESSES Chair: Liqang Yan Major: Mathematic By Ryan G. Sankarperad May 211 We conider an extenion of Merton optimal portfolio choice and conumption problem for a portfolio in which the underlying riky aet i an exponential Levy proce. The invetor i able to move money between a rik free aet and a riky aet and conume from the rik free aet. Given the dynamic of the total wealth of the portfolio we conider the problem of finding portfolio weight and a conumption proce which optimize the invetor expected utility of conumption over the invetment period. The problem i olved in both the finite and infinite horizon cae for a family of hyperbolic abolute rik averion utility function uing the technique of tochatic control theory. The general cloed form olution are found for for the cae of a power utility function and then for a more generalized utility. We conider a variety of Levy procee and make a comparion of the optimal portfolio weight. We find that our reult are conitent with expectation that the greater the inherent uncertainty of a given proce lead to a maller fraction of wealth inveted in the riky aet. In particular an invetor i more careful when the riky aet i a dicontinuou Levy proce when compared to the continuou cae uch a thoe found in a geometric Brownian motion model. 8

9 CHAPTER 1 GENERAL FRAMEWORK AND PROBLEM SETUP 1.1 Introduction Given a portfolio of aet the problem of finding optimal weight for each of the aet ha been extenively tudied and ha many real world application. The firt major breakthrough in thi area wa made by Henry Markowitz in hi 1952 JF paper [17 in which he wa able to find optimal portfolio weight for each of the aet in the portfolio in a dicrete time etting. Markowitz analyi wa a one period model that aimed to minimize the rik under a contraint on the expected return of the portfolio. He wa able to find a o called efficient frontier which wa the interection of the et of all admiible portfolio with a portfolio with minimum rik and maximum expected return. One major computational retriction of thi model i that one need a large data et to compute the neceary covariance between each of the aet in the portfolio. A ucceful attempt to extend Markowitz model to a continuou time etting wa made by Robert Merton in hi 1972 [18 work for which he received the Nobel prize. In thi work Merton howed that if aet in the portfolio were modeled uing geometric Brownian motion, the portfolio of n aet could be reduced to a portfolio of only two aet uing hi o called mutual fund theorem. The mutual fund portfolio coniting of two aet i made up of a rik free aet and a riky aet. The rik free aet i typically i a U.S. treaury bill while the riky aet i a linear combination of the n 1 remaining aet in the portfolio. Thi idea mirror the dicrete time idea of Markowitz market portfolio which i made up of a combination of aet. Once thi implification ha been made Merton conider the problem of finding optimal conumption and portfolio weight for the family of hyperbolic abolute rik averion utility function. Once the et of optimal control are found Merton then find the portfolio weight for each of the n aet in the portfolio o that a comparion can be made with the optimal Markowitz mean-variance portfolio. Thi procedure will not be the 9

10 a major focu of our paper and we refer the intereted reader to [18 for a dicuion in thi direction. There have been many tudie which have conidered relaxing the condition of Merton original paper. Davi and Norman [1 199 paper relaxe the condition of no tranaction cot and preent more generalized reult to include convex tranaction cot function. Fleming and Pang [11 conider the Merton problem with a Vaieck interet rate model and preent reult uing a ubolution/uperolution methodology. Paper uch a [16 and [27 introduce illiquid aet into the market and preent reult to the original Merton problem. Each of thee paper preent the claic Merton problem and it olution in ome context, a trend that we will continue in thi paper. However we will move in a new direction by conidering the Merton problem with a exponential Levy proce o that the price proce i allowed to have jump (in particular i allowed to be dicontinuou). Although we preent ome reult in the geometric Brownian cae (claic Merton cae), the main reult of our paper are for price procee with dicontinuitie uch a thoe conidered in Merton 1976 paper [19. In thi paper Merton conider price procee which are allowed to have Poion jump and derive a cloed form olution for the price of vanilla option. Thi paper wa a major breakthrough in the ue of dicontinuou price procee, and although will not be concerned directly with option pricing formula we will ue the idea preented in thi paper. The main reult of our paper come from modeling the riky aet in our portfolio by a Levy proce with jump and finding control which optimize expected utility of conumption. Thi work cloely follow reearch performed by Bernt Okendal which are ummarized in hi book applied tochatic control of jump diffuion [21. Okendal olve the Merton problem in the cae when the underlying price proce i an exponential Levy proce, a olution which we preent for comparion and completene. Once we have preented the olution in cae covered under Okendal preentation we proceed to generalize the utility function and derive cloed form olution to the correponding problem. A 1

11 derivation of the cloed form olution i provided along with numerical reult to verify the accuracy of the olution provided. The formulation and olution of our problem require the tudy of tochatic calculu and other theorie which require knowledge of tochatic calculu. A complete urvey of tochatic calculu i not provided and it i aumed the reader i familiar with the general theory found in uch book a [2 and [22. In particular we aume the reader ha had ome introductory expoure to tochatic differential equation and the aociated probability theory required to olve thee equation. We begin with a preentation of the tochatic calculu neceary for our analyi, tarting with ome introductory definition and theorem. 1.2 Stochatic Calculu Preliminarie Thi ection lay the foundation for much of the paper, many of thee reult are found in introductory tochatic calculu book uch a [2 and [22. We aume the reader i familiar with a probability pace (Ω, F, P) where Ω i the et of all poible outcome, F i the σ-algebra, and P i a probability meaure. A tochatic proce (X t ) t i a equence of random variable defined of the probability pace (Ω, F, P) taking value in R. In particular for fixed time t the map X t (ω) i a random variable for each ω Ω, while for a fixed realization of the event ω Ω the map X t (ω) i a real valued function for each t. A i tandard we will uppre the dependence on ω throughout the paper and write the random variable a X t. To tudy a tochatic proce (X t ) t defined on (Ω, F, P) we need a way of encompaing the information the proce X t ha accumulated up time t o that deciion can be made about the proce. Information about random variable i claified uing the concept of a σ-algebra, hence for a tochatic proce we will need to conider a equence of increaing σ-algebra typically referred to a a filtration. 11

12 Definition 1.1. Given a probability pace (Ω, F, P), a filtration i a family F = (F t ) t of σ-algebra uch that F F t < t The four tuple (Ω, F, F, P) i called a filtered probability pace. A proce i a equence of random variable, in order to work with each of thee random variable we need a way of deciding whether a given event probability can be computed. To make deciion about a random variable up to time t we need that the random variable X t be F t -meaurable. A tochatic proce i jut a equence of random variable o we need the following definition for the meaurability of the entire equence of random variable. Definition 1.2. A tochatic proce (X t ) t i called F t -adapted, if X t i F t -meaurable for all t. An important filtration we will be conidering i the filtration generated by a proce X, it i defined by Definition 1.3. The filtration generated by the proce X denoted F X i F X = σ(x ; [, t) Thi filtration i the mallet σ-algebra for which all the X, t are meaurable. In particular we have that the proce X i F X -adapted. To define a tochatic differential equation we will need to conider procee with non zero quadratic variation which extend the differential equation from Newtonian Calculu. To include term with non zero quadratic variation in our diffuion equation we define a tochatic proce of thi type called Brownian motion, we will extend thi definition later on to include more general procee (Levy Proce). Definition 1.4. Let (Ω, F, {F t } t, P) be a filtered probability pace. An F t adapted tochatic proce (B t ) t i called a Brownian motion if the following condition hold (i) B = a. 12

13 (ii) for t < t 1 < < t n the random variable B t1 B t,, B tn B tn 1 are independent. (Independent increment) (iii) B t+h B t i normally ditributed for all t and h, in particular B t+h B t ha a N(, t) ditribution. (iv) For every ϵ > and h (tochatic continuity) lim P( B t B h > ϵ) = t h For much of our work we will be conidering the filtration generated by Brownian motion, in particular we will be working with the filtration F B = σ(b ; [, t). When conidering our tochatic differential equation a a diffuion proce the Brownian motion term contain all ource of randomne. Hence the filtration σ(b ; [, t) will contain all information neceary to make deciion about the random component of the diffuion proce. Once we are able to encapulate the information of a proce in the form of a σ-algebra we are able to develop the idea of the conditional expectation given a σ-algebra. Thi concept allow u to find a ubet of tochatic procee whoe expected future value baed on given information i equal to it current value called a martingale. The mathematical formalitie of which are captured in the following definition. Definition 1.5. A tochatic proce (X t ) t i called a martingale with repect to the filtration (F t ) t if for all t E[ X t < i.e. X t i integrable X t i F t -meaurable for all t for any < t, E[X t F = X We will be intereted in two pecific example of martingale in thi paper, thee are Brownian motion and the compenated Levy meaure Ñ. The fact that Brownian motion i a martingale proce, may be one of the mot widely ued and ueful propertie in mathematical finance. The martingale nature of the compenated Levy meaure allow it to inherit many of the ueful propertie that Brownian motion poe, and we will 13

14 ue thee extenively while proving reult for general Levy procee later on. We would like to tudy the et of all Ito procee which can be written in term of thee two martingale o that we may replicate the modern portfolio theory. To do thi we mut write down the tochatic diffuion equation which model tock price of our portfolio, hence we mut give a precie mathematical definition of a diffuion equation. 1.3 Ito Diffuion and it Generator To tudy a portfolio of aet in a continuou time etting we need to decribe the dynamic of the given aet, thi can be done by uing the theory of tochatic differential equation. Each aet will be modeled uing a diffuion proce which can be thought of a a particle whoe trajectory i influenced by an external ource of randomne. Thi randomne may come from be attributed to one or many external ource, for the remainder of the paper we will aume that there will be only one ource of randomne. Thi mean that our theorem will be tated in the one dimenional cae, ince a one dimenional Brownian motion will be the driving force of randomne in our diffuion equation. The multidimenional verion of the theorem we ue in thi ection can be found in an introductory tochatic calculu book uch a [2 or [22. The ource of randomne i modeled by adding a Brownian motion term to a tandard diffuion proce, more preciely Definition 1.6. An Ito diffuion i a tochatic proce (X t ) [,T atifying a tochatic differential equation of the form dx t = α(t, X t ) dt + σ(t, X t ) db t t [, T ; X = x R (1 1) where B t i an one-dimenional Brownian motion and α : [, T R R, σ : [, T R R atify the following global Lipchitz and at mot linear growth condition for all x, y R, t [, T α(t, x) α(t, y) + σ(t, x) σ(t, y) C x y 14

15 α(t, x) + σ(t, x) D(1 + x ) Moreover if the olution X t (ω) i adapted to the filtration σ(b ; t) then E,x [ T X t 2 dt < The unique olution of Equation (1 1) will be denoted by X,x t for all t, if = we will ue the notation Xt x. We note that the drift coefficient α and the diffuion coefficient σ depend on the time parameter t, with a tranformation we can reduce thi cae to the time dependent cae. The reduction will be performed later on in the paper, hence many of the definition and theorem will be tated without explicit time dependence. The olution X,x t of Equation (1 1) i referred to a time homogeneou the following reaon, X,x +h = x + +h +h α(r, Xr,x ) dr + σ(r, Xr,x ) db r (1 2) Making the change of variable u = r we may write Equation (1 2) a X,x +h = x + h = x + h α(u +, Xu+),x du + α(u +, X,x u+) du + h h σ(u +, X,x u+) db u+ σ(u +, X,x u+) db u where db u = B u+ B i a Brownian motion with the ame P-ditribution a B u. Since (B u ) u and (B u ) u have the ame P-ditribution a tochatic differential equation of the form dx t = α(t, X t ) dt + σ(t, X t ) db t ; X = x whoe olution X,x h can be written a h h X,x h = x + α(u, Xu,x ) du + σ(u, Xu,x ) db u (1 3) we have that the equation governing both of thee Ito procee are eentially the ame from a probability tandpoint. In particular, comparing Equation (1 2) and (1 3) 15

16 we ee that the latter i jut the former equation with = o that X,x h ha the ame P-ditribution a X,x +h i.e. (X t) t i time homogeneou. We will henceforth refer to both tochatic differential equation interchangeably. An important property of Ito diffuion which will be of great ue i the o called Markov property which allow u to ue the preent behavior of a diffuion to make deciion about the future without conidering pat behavior. In order to tate the Strong Markov property we need the following definition of a random time Definition 1.7. Let (F t ) t be a filtration, a function τ : Ω [, ) i called a topping time with repect to (F t ) t if {ω : τ(ω) t} F t A topping time i a random variable for which the et of all path(event) ω Ω with {τ(ω) t} can be decided given the filtration F t. The definition allow u to conider only the random time for which we may decided whether or not the time ha been reached given the information up to time t i.e. given F t. Once we have defined thi random time we may tate the Strong Markov property Theorem 1.1. (Strong Markov property for Ito diffuion) Let (X t ) t be an Ito diffuion, τ a topping time and f : R n R be a Borel meaurable function, then for ω Ω, h E x [f (X τ+h ) F τ (ω) = E X τ (ω) [f (X h ) Throughout much of the theory of mathematical finance we aume that the price proce of a given aet i the olution of a tochatic differential equation, hence a tochatic proce. We would like to conider function on thee tochatic proce, o the natural quetion i to ak whether a function on a tochatic proce i itelf a tochatic proce. In order to anwer thi quetion we need to be able to write down the differential equation aociated with thi new proce. Thi mean that the claical theory of Newtonian calculu mut be extended to include tochatic term. Thi i done 16

17 by Ito formula which i a generalization of the chain rule from Newtonian calculu, we will extend to a more general verion later in the paper. Theorem 1.2. (1- dimenional Ito formula) Given an Ito proce X t of the form dx t = α(t, X t ) dt + σ(t, X t ) db t where B t R i a Brownian motion. If g(t, x) C 1,2 ([, ), R) then Y t = g(t, X t ) i an Ito proce with dy t = g t (t, X t) dt + g x (t, X t) dx t Proof. A proof of Ito lemma can be found on page 46 of [2 g 2 x (t, X t) dx 2 t dx t A major component of the theory of tochatic optimal control i the infiniteimal generator of an Ito diffuion. Thi can be thought of a and extenion of the derivative from the determinitic calculu, where the limit definition ha a imilar form with extenion to include to tochatic nature of an Ito diffuion. Definition 1.8. Let (X t ) t be an Ito diffuion in R of the form dx t = α(t, X t ) dt + σ(t, X t ) db t X = x then the infiniteimal generator A of X t i defined by Aϕ(, x) = lim t E,x [ϕ(t, X t ) ϕ(, x) t for all x R, [, ) and ϕ C 1,2 ([, ), R). The ubet of L2 ([, ) R) for which the limit Aϕ(, x) exit for all, x will be called D A. It turn out that thi derivative in the tochatic ene i related to the claical derivative in an inherent way, which can be found by applying Ito Lemma. The following theorem how that the infiniteimal generator of an Ito diffuion turn out to be a 17

18 econd order differential operator in the cae of diffuion driven by Brownian motion. It will alo be hown later that in the cae of diffuion driven by a more general Levy proce the infiniteimal generator will end up being an integro-differential operator. Theorem 1.3. Let X t be an Ito diffuion of the form dx t = α(t, X t ) dt + σ(t, X t ) db t X = x and conider the differential operator A on C 1,2 ([, ), R) given by A = t + α x σ2 2 x 2 and let ϕ C 1,2 ([, ), R) be uch that for all t, [, ) and x R [ t [ t ( ) 2 ϕ E,x Aϕ(r, X r )dr <, and E,x x (r, X r)σ(r, X r ) dr < then ϕ D A and Aϕ(, x) = Aϕ(, x). Proof. Since ϕ C 1,2 ([, ), R) we may apply Ito formula to compute dϕ(t, X t ) = ϕ t (t, X t) + ϕ x (t, X t)dx t ϕ 2 x (t, X t)dx 2 t dx t = ϕ t (t, X t) + ϕ x (t, X t) [α(t, X t ) dt + σ(t, X t ) db t σ2 (t, X t ) 2 ϕ x (t, X t)dt 2 = Aϕ(t, X t ) dt + ϕ x (t, X t)σ(t, X t ) db t Integration of both ide of the equation with repect to the proper meaure we find that ϕ(t, X t ) ϕ(, X ) = t Aϕ(r, X r ) dr + t ϕ x (r, X r)σ(r, X r ) db r uing the fact that X = x and taking the expectation E,x of both ide of the equation we are able to compute the numerator of the limit E,x [ϕ(t, X t ) ϕ(, x) = E,x [ t [ t Aϕ(r, X r ) dr + E,x ϕ x (r, X r)σ(r, X r ) db r 18

19 Given that the function ϕ (r, X x r)σ(r, X r ) i B F-meaurable, F r -adapted, and [ E,x t ( ϕ (r, X x r)σ(r, X r ) ) 2 dr < a tandard reult from tochatic calculu give that E,x [ t ϕ x (r, X r)σ(r, X r ) db r = hence we have that E,x [ϕ(t, X t ) ϕ(, x) = E,x [ t Aϕ(r, X r ) dr dividing both ide by t and taking limit give E,x [Φ(t, X t ) Φ(, x) Aϕ(, x) =: lim = lim t t [ 1 t = lim E,x Aϕ(r, X r ) dr t t = E [lim,x 1 t Aϕ(r, X r ) dr t t [ ( d t ) = E,x Aϕ(r, X r ) dr dt = E,x [Aϕ(t, X t ) =: E [Aϕ(t, X t ) X = x = Aϕ(, x) t E,x [ t Aϕ(r, X r) dr t where the fourth equality follow from dominated convergence theorem which we may [ apply uing the aumption that E,x t Aϕ(r, X r)dr < o that the term in the expectation i eentially bounded. Given the form of the infiniteimal generator of an Ito diffuion we may ue the following theorem to compute expectation of firt exit time of the diffuion from a given region. We will not explicitly compute any expectation of exit time, however we will need Dynkin formula to prove reult about exit time of controlled Ito diffuion. 19

20 Theorem 1.4. (One dimenional Dynkin Formula) Let X t be a olution of an Ito diffuion of the form dx t = α(t, X t ) dt + σ(t, X t ) db t X = x and τ be a topping time with E,x [τ <. If ϕ C 1,2 ([, ) R), then [ τ E,x [ϕ(τ, X τ ) = ϕ(, x) + E x Aϕ(r, X r )dr Proof. Since ϕ C 1,2 ([, ) R) we may apply Ito formula a in the proof of theorem to get E,x [ϕ(t, X t ) ϕ(, x) = E,x [ t Letting t = τ we have that E,x [ϕ(τ, X τ ) = ϕ(, x) + E,x [ τ [ t Aϕ(r, X r ) dr + E,x ϕ x (r, X r)σ(r, X r ) db r [ τ Aϕ(r, X r ) dr + E,x To finih the proof we need to now how that E,x [ τ ϕ x (r, X r)σ(r, X r ) db r σ(r, X r) ϕ x (r.x r)db r =. To implify notation let g(r, X r ) = σ(r, X r ) ϕ x (r, X r), then we have that g i a bounded Borel F r -meaurable function i.e. g M for ome M >. Conider the family of { τ k function g(r, X r ) db r }k, I claim thi family i uniformly integrable with repect to the meaure P x. To how thi family i uniformly integrable it i enough to apply theorem C.3 from [2, hence we need to how the following integral i finite, [ ( τ k ) 2 [ τ k E,x g(r, X r )db r = E,x g 2 (r, X r ) dr M 2 E,x [τ k M 2 E 2 [τ < where the firt equality follow from the Ito Iometry. The finitene of thi integral from the ue of a quadratic tet function give u that the family i uniformly integrable o that [ τ k lim E,x g(r, X r )db r k [ = E,x lim k τ k g(r, X r )db r 2

21 hence combing the reult above we have that [ τ E,x g(r, X r )db r [ τ k [ τ k = E,x lim g(r, X r )db r = lim E,x g(r, X r )db r k k [ k = lim E,x χ {r<τ} g(r, X r )db r = k where the lat equality follow from the fact that k reult now follow from the fact that Aϕ(, x) = Aϕ(, x). χ {r<τ}g(r, X r )db r i a martingale, the 21

22 CHAPTER 2 STATEMENT OF MAIN PROBLEM 2.1 Optimal Control Theory We have now etup a foundation for our problem, thi ection will tate the main problem and begin to talk about how we may go about finding a olution uing optimal control theory. To begin thi ection we give ome detail on control and how they are implemented in our problem. Definition 2.1. Given a Borel meaurable et U R a control proce i a B F- meaurable tochatic proce (u t ) t taking value in U i.e. u : [, ) Ω U Definition 2.2. Given a control proce (u t ) t a controlled proce X t i a olution of the tochatic differential equation dx t = α(t, X t, u t )dt + σ(t, X t, u t )db t X = x > (2 1) where α : R R U R, σ : R R U R and B t i an one-dimenional Brownian motion and for all x, y R, t, u, v U we have α(t, x, u) α(t, y, v) + σ(t, x, u) σ(t, y, v) C 1 x y + C 2 u v α(t, x, u) + σ(t, x, u) D(1 + x + u ) The parameter u t U in Equation (2 1) i ued to control the proce X t, given a Borel et U R we vary the parameter u t U to control the behavior of the proce X t. The deciion on how to vary the control u t i baed on the information up to time t, o that u t i F t meaurable. Since the control proce depend on the underlying ω we have amended our definition of an Ito diffuion to include appropriate condition for the exitence and uniquene of a controlled diffuion. In order to do thi we have et it up o that the global Lipchitz and linear growth condition hold for all control u t U. To apply the theory of diffuion we need Equation (2 1) to be an Ito diffuion, thi i poible if we retrict the control we ue to a ubet of control called Markov control 22

23 Definition 2.3. Let X t be a olution to Equation (2 1), a Markov control i a control which doe not depend on the initial tate of the ytem (, x) but only depend on the current tate of the ytem at any time t i.e. there exit a function u : R n+1 U R k uch that u(t, w) = u(t, X t (w)) The Markov control i important in our tudy becaue the retriction to thi ubet of all control allow u to turn the equation governing the wealth proce into an Ito diffuion. The fact that the wealth proce i a diffuion make ue of important theorem from the theory of diffuion poible, without thi we would not be able to proceed uing our methodology. 2.2 Dynamic Programming Methodology Conider the following following controlled proce on [, τ G dx t = α(t, X t, u t ) dt + σ(t, X t, u t ) db t X = x (2 2) with given performance function [ τg J u (x ) = E x f (r, X r, u r )dr + g(τ G, X τg )χ {τg < } (2 3) where τ G = inf{t > : (t, X t ) / G} i a topping time, and the et G [, τ G R i called the olvency et. The continuou function f : [, τ G R R R i called the utility function and the continuou function g : R R R i called the bequet function. We would like to find an optimal control u from a et of admiible control o that thi performance function i maximized. To find our optimal control we mut ue the technique of dynamic programming, a the name ugget we need to conider a family of optimal control problem from which we will elect an optimal control. In order to do thi we need to conider different initial time and tate along a given trajectory of the controlled diffuion. If we were to conider a diffuion X t tarting at x a in Equation (2 2) then an admiible control u i (F t ) t meaurable which mean that the controller ha all information about the ytem up to time t o that X t i almot urely determinitic 23

24 under a probability meaure P( F t ). Thi mean for all tate of the ytem we already know the initial time point and the initial value at thi time point. To be able to ue the method of dynamic programming we mut vary the initial time and tate of the ytem and chooe the bet poible control from a et of admiible control. To do thi we conider controlled diffuion of the form dx t = α(t, X t, u t ) dt + σ(t, X t, u t ) db t X = x (2 4) where t [, τ G and the performance function ha the form [ τg J u (, x) = E,x f (r, X r, u r )dr + g( τ G, X τg )χ { τg < } (2 5) where τ G = inf{t > : (t, X t ) / G}. Before moving on to the tatement of the optimal control problem we need to dicu exactly what mathematical propertie an admiible control hould have. We give the general mathematical formulation of an admiible control over the et [, τ G here, we will ue a lightly different et later on when we write the time homogeneou verion of the controlled proce. Definition 2.4. We ay that the control proce u i admiible and write u A[, τ G if u i meaurable, (F t ) t -adapted with E,x [ τg u r 2 dr < Equation (2 4) ha a unique trong olution (X t ) t for all X = x and [ E,x up X t 2 < t τ G J u (, x) i well defined i.e. [ τg E,x f (r, X r, u r )dr + g( τ G, X τg )χ { τg < } < The tochatic control problem may now be tated, given continuou utility function f and bequet function g we would like to find optimal control u and value function Φ(, x) 24

25 uch that Φ(, x) = up J u (, x) = J u (, x) (2 6) u A[, τ G Φ( τ G, x) = g( τ G, x) (2 7) ubject to the contraint of Equation (2 4). To begin the analyi of the optimal control problem in Equation (2 6), we introduce ome notation that will make our problem a time homogeneou control problem. We firt begin by rewriting Equation (2 4) uing the ubtitution Y t = + t X +t t which give u a time homogeneou controlled proce Y t atifying dy t = α(y t, u +t )dt + σ(y t, u +t )db +t Y = (, x) =: y (2 8) We may alo recat the performance function in Equation (2 5) uing our new notation, letting r = + t we have τg f (r, X r, u r )dr = τg f ( + t, X +t, u +t )dt = τg f (Y t, u +t )dt where τ G = τ G, hence the performance function become [ τg J u +t (y) = E y f (Y t, u +t )dt + g(y τg )χ {τg < } (2 9) Since db +t ha the ame ditribution a db t we may make thi replacement in the controlled proce equation. We note that the control from Equation (2 9) i a time hifted verion of the control from Equation (2 11), after the ubtitution for Y t we ee that we are now working with the control proce u +t in each equation. The olution to the optimal control problem i found by working in two different world, thi idea i at the center of our entire analyi hence we hall pend ome time dicuing how to go back and forth between each world. We ummarize each of the equation and the 25

26 vital information in each world in Table 2 1, thi table i very ueful when comparing equation between the different world. The main idea of working between the two Table 2-1. Table with comparion of two world ued to olve optimal control problem Two world Original X t pace dx t = α(t, X t, u t ) dt + σ(t, X t, u t ) db t X = x [ τg J(, x) = E,x f (t, X t, u t ) dt + g( τ G, X τg ) Tranformed Y t = ( + t, X +t ) pace dy t = α(y t, u +t ) dt + σ(y t, u +t ) db t Y = (, x) =: y [ τg J(y) = E y f (Y t, u +t ) dt + g(y τg ) u t = u(t, X t ) u +t = u( + t, X +t ) = u(y t ) u t A[, τ G u A[, τ G world i that we will be olving the optimal control problem in the Y t pace where the control i u +t and then we will tranform back to the X t pace where the control i u t. The connection between the two world that we will need i the fact that in the Y t world we have that u = u (, x) = u (, X ) ince X = x in the X t world. Now renaming the variable we have that the optimal control in the X t world (denoted imply u t ) i given by u t = u (t, X t ) where u i the optimal control we find in the Y t world. Hence, the optimal control problem i olved by tranforming into the Y t world where an optimal control u (, w) will be found, then uing that optimal control we are able to tranform back to the X t world to olve for the optimal control u t (t, X t ) which i a olution of the control problem in the X t world. However for now we make no ditinction between the different verion of α, σ and u and tate the control problem in the new notation. Given utility function f and bequet function g we would like to find optimal control u and value function Φ(y) uch that Φ(y) = up J u (y) = J u (y) (2 1) u A[,τ G where [ τg J(y) = E y f u (Y t, u +t )dt + g(y τg )χ {τg < } (2 11) 26

27 ubject to the contraint dy t = α(y t, u +t )dt + σ(y t, u +t )db t Y = (, x) =: y (2 12) In our new notation a Markov control (we make no ditinction between u and u) i one uch that u +t = u( + t, X +t ) = u(y t ) hence we may rewrite Equation (2 12) a dy t = α(y t, u(y t ))dt + σ(y t, u(y t ))db t Y = (, x) =: y (2 13) We now have the right hand ide of Equation (2 13) only depend on the tate of the ytem Y t at time t and doe not depend on time explicitly, i.e. the olution Y t of Equation (2 13) i a time homogeneou Ito diffuion. Given that the tate of our ytem ha thi form we may now apply the theory of diffuion to olve our optimal control problem. Before tating the main theorem neceary to olve the optimal control problem we mention that the notation between the two world will be ued interchangeably for the remainder of the paper, any confuion with the notation can be reolved by conulting Table 2 1. The main theorem of the paper will be tated in the Y t world ince much of the work will be done in thi etting and the reult will then be tranformed back to the X t world to provide a olution to the original problem. 2.3 Hamilton Jacobi Bellman The Hamilton Jacobi Bellman theorem i at the heart of much of the analyi that will be performed throughout the paper, it ue will be of an indirect nature but it i of great importance hence we provide the complete tatement and proof. Theorem 2.1. Let J u (y) be a in Equation (2 11), where u = u(y ) i a Markov control and Φ(y) = up J u (y) u 27

28 Suppoing the function Φ C 2 G C(G) i bounded for all finite topping time τ G a.. P y and all y G. If an optimal Markov control u exit and G i regular for Y u t then up [f u (y) + A u Φ(y) = for all y G (2 14) u A[,τ G and Φ(y) = g(y) for all y G (2 15) Proof. Since G i regular for Y u t we have τ G = for any y G, o that Φ(y) = g(y τg )χ {τg < } = g(y) hence Equation (2 15) hold for all y G. To prove Equation (2 14) fix y = (, x) G and take u to be any Markov control. If τ τ G i a topping time then we may compute E y [J u (Y τ ) = E y [ E Yτ [ τg = E y [ E y [ τg τ f u (Y r ) dr + g(y τ ) f u (Y r ) dr + g(y τ ) F τ [ [ τg τ = E y E y f u (Y r ) dr + g(y τg ) f u (Y r ) dr F τ [ τg τ = E y f u (Y r ) dr + g(y τg ) f u (Y r ) dr [ τg [ τ = E y f u (Y r ) dr + g(y τg ) E y f u (Y r ) dr [ τ = J u (y) E y f u (Y r ) dr Thi how that J u (y) = E y [ τ f u (Y r ) dr + E y [J u (Y τ ) (2 16) which i an equality ued to prove Bellman principle of optimality which we do not directly prove here, but we will ue the above equality to finih the proof. Firt we mut define the proper continuation region, let U G of the form U = {(r, x) G : r < t 1 } 28

29 where < t 1. Then if τ = τ U i the firt exit time from U and u (y) = u (r, x) i the optimal control, defining a control v u(r, x) = u (r, x) if (r, x) U x G\U where v A[, τ G i an arbitrary control. With thi continuation region we have that Φ(Y τ ) = J u (Y τ ) = J u (Y τ ) (2 17) Uing the fact that ϕ(y) i the upremum and Equation (2 16) we have [ τ Φ(y) J u (y) = E y f u (Y r ) dr + E y [J u (Y τ ) [ τ = E y f u (Y r ) dr + E y [ϕ(y τ ) (2 18) Since we aumed Φ(y) C 2 (G) and τ i a topping time we may apply Dynkin formula to get E y [Φ(Y τ ) = Φ(y) + E y [ τ A u Φ(Y r )dr (2 19) If we now plug Equation (2 19) into Equation (2 18) we get Φ(y) E y [ τ [ τ f u (Y r ) dr + Φ(y) + E y A u Φ(Y r ) dr Combining expectation and ubtracting Φ(y) from both ide we have E y [ τ f u (Y r ) dr + A u Φ(Y r ) dr Now we let t 1 and uing the fact that f ( ) and A u Φ( ) are continuou at y we may perform the integration to get that (f u (y) + A u Φ(y))E y [τ 29

30 dividing out E y [τ we have that f u (y) + A u Φ(y) for all topping time u A[, τ G. The upremum i obtained at the optimal control u where Φ(y) = J u (y) = E y [ τg f u (Y r ) (Y r ) dr + g(y τg ) (2 2) i a olution of the combined Dirichlet-Poion problem o that A u (y) Φ(y) = f u (y) (y) for all y G Thi theorem give a neceary condition that tate if an optimal control u exit then the function F (u) = f u (y) + A u ϕ(y) attain it maximum value at u = u. However it doe not addre the quetion of ufficiency of the optimal control u i.e. if for every point y G we find a u (y) uch that F (u (y)) =, will u (y) be an optimal control? Thi quetion can be anwered by the following convere Theorem 2.2. Let ϕ C 2 (G) C(G) atify the following condition f u (y) + A u ϕ(y) y G and for all u A[, τ G lim t τg ϕ(y t ) = g(y τg )χ {τg < } a.. P y {ϕ(y τ )} τ τg i uniformly P y integrable for all Markov control and all y G. then ϕ(y) J u (y) for all Markov control u A and y G. Moreover if for all y G we find a Markov control u = u (y) uch that f u (y) (y) + A u (y) ϕ(y) = (2 21) then u t = u (Y t ) i optimal and ϕ(y) = Φ(y) = J u (y). Proof. Let R < and define the topping time T R = min{r, τ G, inf{t > ; Y t R}} 3

31 o that we have lim R T R = τ G. We may apply Dynkin formula and ue the hypothei that A u ϕ(y) f u (y) for all y G to get that [ TR E y [ϕ(y TR ) = ϕ(y) + E y A u ϕ(y r ) dr [ TR ϕ(y) E y f u ϕ(y r ) dr Rearranging of thi equation and combining term with expectation give u that ϕ(y) E y [ TR f u (Y r ) dr + ϕ(y TR ) (2 22) Applying Fatou Lemma to thi equation and uing the remaining hypothei we have [( TR ϕ(y) lim inf E y R E y [lim inf R ( TR [ T = E y f u (Y r ) dr + lim ) f u (Y r ) dr + ϕ(y TR ) ) f u (Y r ) dr + ϕ(y TR ) ϕ(y TR ) R [ T = E y f u (Y r ) dr + lim ϕ(y t ) t τg [ T = E y f u (Y r ) dr + g(y τg )χ {τg < } = J u (y) To complete the proof, if we are able to find a control u (y) uch that f u (y) (y) + A u (y) ϕ(y) = then we may apply the ame argument a above to u t = u (Y t ) o that the inequality become equality i.e. ϕ(y) = J u (y) and u i an optimal control. 2.4 Verification Theorem The Hamilton Jacobi Bellman equation ha many technical condition that are generally difficult to check explicitly, in particular we do not know the olution Φ a priori. Each verion of the Hamilton Jacobi Bellman equation require that we know that the olution atify Φ C 2 (G) C(G). Thi make olving the optimal control problem uing the Hamilton Jacobi Bellman theorem directly rather difficult, in order to olve 31

32 thi problem we introduce the idea of verification theorem. Uing the Hamilton Jacobi Bellman theorem would require u to know the olution Φ and check all the required technical condition, once thi i done then we are able to determine if a control u i an optimal control and how that Φ i actually a olution to the Hamilton Jacobi Bellman equation. The verification theorem revere the order of thi proce. We firt begin with a olution ϕ to the Hamilton Jacobi Bellman equation from which we are able to alo get a candidate for the optimal control u. Once we have hown that ϕ atifie the condition of the verification theorem we can then how that ϕ u = Φ o that the optimal control problem i olved. Theorem 2.3. Let u A[, τ G and (, x) G and uppoe the following condition are atified for all [, τ G and x R ϕ C 1,2 ([, τ G ) R) i continuou on [, τ G R and atifie the quadratic growth condition ϕ(, x) C ϕ (1 + x 2 ) ϕ atifie the Hamilton Jacobi Bellman equation up [f u (, x) + A u ϕ(, x) = [, τ G ) u A[,τ G ϕ(τ G, x) = g(τ G, x) f u i continuou with f u (, x) C f (1 + x 2 + u 2 ) for ome contant C f >. σ u (, x) 2 C σ (1 + x 2 + u 2 ) for ome contant C σ >. then ϕ(, x) Φ(, x) for all (, x) G. Moreover if u o (, x) i the max of u f u (, x) + A u ϕ(, x) and u = u (, X ) i admiible then ϕ(, x) = Φ(, x) for all (, x) G and u i and optimal trategy i.e. Φ(, x) = J u (, x). Proof. Fix [, τ G and x R, to get that the proce X i bounded we define the topping time τ n = τ G inf{t > : X t X n} Let u A[, τ G be an admiible control and X = x then by Ito lemma we have that ϕ(τ n, X τn ) = ϕ(, x) + τn A u r ϕ(r, X r ) dr + τn ϕ x (r, X r )σ u r (r, X r ) db r (2 23) 32

33 Since ϕ x (, X ) i continuou on the et [, τ n we have that there exit a contant C ϕx uch that ϕ x (, x) 2 C ϕx. Uing the fact that u r i admiible and that X r i bounded on [, τ n we find that hence τ n E,x [ τn [ τn ϕ x (r, X r )σ ur (r, X r ) 2 dr E,x ϕ x (r, X r ) 2 σ ur (r, X r ) 2 dr [ τg E,x C ϕx C σ (1 + X r 2 + u r 2 ) dr < ϕ x (r, X r )σ ur (r, X r ) db r i a martingale which give that [ τn E,x ϕ x (r, X r )σ u r (r, X r ) db r = Taking expectation on both ide of Equation (2 23) and uing the lat calculation we get that E,x [(τ n, X τn ) = E,x [ϕ(, x) + τn A ur ϕ(r, X r ) dr adding E,x [ τ n ingle expectation we find that E,x [ τn f ur (r, X r ) dr to both ide of thi equation and combining term into a f u r (r, X r ) dr + ϕ(τ n, X τn ) = E,x [ϕ(, x) + τn (f u r (r, X r ) + A u r ϕ(r, X r )) dr E,x [ϕ(, x) = ϕ(, x) (2 24) where we have ued that fact that for any control u r A we have that f ur (r, X r ) + A u r ϕ(r, X r ). If we can now how that the left hand ide goe to J u (, x) a n the proof will be complete. In order to do thi we need to let n o that τ n τ G, our main concern here will be bringing the limit inide of the expectation operator. To accomplih thi we would like to ued dominated convergence theorem, hence we need to check that the hypothei are met i.e. τn τn f u r (r, X r ) dr + ϕ(τ n, X τn ) f u r (r, X r ) dr + ϕ(τ n, X τn ) τg C f (1 + X r 2 + u r 2 ) dr + C ϕ (1 + X τg 2 ) L 1 33

34 an application of dominated convergence give and the fact that ϕ(τ G, X τg ) = g(τ G, X τg ) E,x [ τn [ τg f u r (r, X r ) dr + ϕ(τ n, X τn ) E,x f u r (r, X r ) dr + g(τ G, X τg ) = J u (, x) Letting n on both ide of Equation (2 24) and uing thi reult give u that J u (, x) ϕ(, x) for all u A ince the right hand ide doe not depend on u we take the upremum of both ide over the et of all admiible control to get Φ(, x) = up J u (, x) ϕ(, x) u A If we were now able to find a maximizer u the only part of the proof above that would be different i the fact that intead of an inequality in the argument we would have equality in Equation (2 24) o that we may continue the argument to get that Φ(, x) = J u (, x) = ϕ(, x) completing both part of the proof. Hence the verification theorem give u a way of finding a olution to the optimal control problem without having to check the complicated hypothee of the Hamilton Jacobi Bellman equation. The verification methodology of olving the optimal control problem uing the verification theorem i a follow Write down the Hamilton Jacobi Bellman equation and the appropriate boundary condition. Take the firt order derivative with repect to the control variable in the Hamilton Jacobi Bellman equation, and olve for the optimal control candidate u. Plug the optimal control back into the Hamilton Jacobi Bellman equation to get a non linear partial differential equation which ha to be olved for a candidate of the value function ϕ ubject to the boundary condition Show that ϕ atifie the condition of the verification theorem and that u = u (, X ) i an admiible control for all [, τ G, then ϕ = Φ and u i an optimal control trategy. 34

35 CHAPTER 3 MERTON PROBLEM 3.1 Claic Merton Solution Thi ection formulate and olve a verion of the claic optimal control problem that Merton olved in hi 1971 paper [18. In hi paper Merton aumed a general family of utility function called the Hyperbolic abolute rik averion utilitie. I will preent a ubet of thi family which avoid the complete generality a Merton originally preented but which till capture the eence of hi reult. The original problem conider a portfolio of n aet, however with the aumption of log normally ditributed aet we may ue a o called mutual fund theorem which allow u to intead conider a portfolio of 2 aet. The aet under conideration are a rik free aet (uch a a U.S. treaury bond) and a riky aet (uch a a hare of tock) which may be written a a linear combination of the n aet. To begin the analyi let (Ω, F, F [, ), P) be a filtered probability pace. The rik free R t aet evolve according to the differential equation dr t = rr t dt; R = 1 (3 1) where r i a contant which repreent the rik free rate of interet, while the riky aet S t evolve according to a Geometric Brownian motion ds t = αs t dt + σs t db t ; S = (3 2) where α and σ repreent the rate of the return and volatility of the aet S t repectively. Definition 3.1. (Trading trategy)a trading trategy i a two dimenional tochatic proce {π t = (πt, πt 1 )} t [, ), uch that π t i B F-meaurable and F t -adapted. Financially we interpret πt a the number of hare in the rik free aet R t, while πt 1 i the number of hare in the riky aet S t at time t. Hence the total wealth at time t of 35

36 the portfolio of aet may be written a W t = π t R t + π 1 t S t (3 3) Let c t be an F t adapted proce that the invetor i able to chooe at time t, which repreent the rate at which money can be moved from the rik free aet to the riky aet within the portfolio. Definition 3.2. Self financing A trading trategy (πt, πt 1 ) i called elf financing if the correponding wealth proce (W t ) t i continuou and adapted uch that W t = w + t π udr u + t π 1 uds u t c u du (3 4) The aumption of a elf financing portfolio of aet baically tate that no ource of external capital can be added to the portfolio, any capital gain mut be reinveted into the portfolio. We alo have underlying aumption that there are no tranaction cot to reditribute capital between the two aet. Let θ t be the fraction of total wealth of the portfolio inveted in the riky aet, then we may write θ t = π1 t S t W t while the fraction of wealth in the rik free aet i π t Rt W t. Auming that W t > for all t we may divide Equation (3 5) by W t to ee that 1 θ t = π t Rt W t. With thee aumption and the notation we above we may write the change i the wealth proce uing Equation (3 4) a dw t = πt dr t + πt 1 ds t c t dt = rπt R t dt + πt 1 [αs t dt + σs t db t c t dt = (1 θ t )rw t dt + αθ t W t dt + σθ t W t db t c t dt = (α r)θ t W t dt + [rw t c t dt + σθ t W t db t (3 5) 36

37 Equation (3 5) along with initial condition W = πr + πs 1 =: w will erve a the controlled diffuion for the optimization problem. The control proce in thi problem i given by the vector u t = (θ t, c t ), where θ t i the fraction of wealth in the riky aet and c t i the conumption proce. The freedom of choice for θ t can be thought of a an invetor chooing the fraction of total wealth of the portfolio he/he would like inveted in the riky aet. At each point in time the invetor much chooe thi fraction along with the conumption proce c t a to maximize ome performance function. The reallocation of portfolio weight θ t at time t i the continuou time verion of the problem olved by Henry Markowitz [17, which wa done in a dicrete time etting and i famouly referred to a Modern Portfolio Theory. We will aume a trader ha complete information in the market up to time t o that the control proce u t = (θ t, c t ) are adapted with repect to the tandard filtration. We may now define the value function for the optimization problem in the claical cae, we look at everal cae. In thi chapter we will conider cot functional on an infinite horizon with utility function of the form f (t, W t, u t ) = e δt U(c t ) where there i no explicit dependence on W t. However, ince c t i a tochatic proce there will be a built in dependence on W t which will how up when we find the control proce explicitly. Given thi form of the utility function the cot functional become J θ,c (, w) = E,w [ e δt U(c t ) dt = e δ E y [ e δt U(c +t ) dt (3 6) where the (utility) function U i increaing, differentiable and concave in c t. The proce c t i the conumption rate that the invetor mut chooe when rebalancing the portfolio o a to maximize the cot functional. So the optimal control problem we are conidering in thi ection i one in which the invetor trie to maximize the expected utility of conumption over the given trading period by chooing the appropriate control. 37