SIAM J. CONTROL AND OPTIMIZATION (C) 1991 Society for Industrial and Applied Mathematics Vol. 29, No. 3, pp. 702-730, May 1991 011 MARTINGALE AND DUALITY METHODS FOR UTILITY MAXIMIZATION IN AN INCOMPLETE MARKET* IOANNIS KARATZAS?, JOHN P. LEHOCZKY, STEVEN E. SHREVE, AND GAN-LIN XU Abstract. The problem of maximizing the expected utility from terminal wealth is well understood in the context of a complete financial market. This paper studies the same problem in an incomplete market containing a bond and a finite number of stocks whose prices are driven by a multidimensional Brownian motion process W. The coefficients of the bond and stock processes are adapted to the filtration (history) of W, and incompleteness arises when the number of stocks is strictly smaller than the dimension of W. It is shown that there is a way to complete the market by introducing additional "fictitious" stocks so that the optimal portfolio for the thus completed market coincides with the optimal portfolio for the original incomplete market. The notion of a "least favorable" completion is introduced and is shown to be closely related to the existence question for an optimal portfolio in the incomplete market. This notion is expounded upon using martingale techniques; several equivalent characterizations are provided for it, examples are studied in detail, and a fairly general existence result for an optimal portfolio is established based on convex duality theory. Key words, incomplete markets, portfolio processes, stochastic control, convex duality, utility maxi- mization AMS(MOS) subject classifications, primary 93E20; secondary 60G44, 90A16, 49B60 1. Introduction. This paper studies the problem of an agent who receives a deterministic initial capital, which he must then invest in an incomplete market so as to maximize the expected utility of his wealth at a prespecified final time. The market consists of a bond and m stocks, the latter being driven by a d-dimensional Brownian motion. In such a model, incompleteness arises when m is strictly smaller than d. The market coefficients, i.e., the interest rate, the rates of stock appreciation, and the stock volatility coefficients, are random processes adapted to the full d-dimensional Brownian motion. When m < d, it is typically not possible to construct a portfolio consisting of the bond and the m available stocks, so as to completely hedge the risk associated with these coefficient processes. Our model is not Markov, and so the Bellman equation of dynamic programming is inadequate for its analysis. Using the Bellman equation, Svensson [21] has treated an infinite-horizon, incomplete, Markov model with an income process. He derives first-order conditions and obtains explicit solutions when the utility for consumption has constant absolute risk aversion. Duffle and Jackson [4] provide a similar analysis of a finite-horizon, incomplete, Markov model. Received by the editors October 25, 1989; accepted for publication (in revised form) July 5, 1990. An abridged version of this paper was presented at the Workshop on Mathematical Theory of Modern Financial Markets, Mathematical Sciences Institute, Cornell University, July 1989, and at the Ninth International Conference "Analysis and Optimization of Systems," Antibes, France, June 1990. ? Department of Mathematics, Rutgers University, New Bruswick, New Jersey 08903. The author is on leave from Columbia University. This research was supported by National Science Foundation grant DMS-87-23078. $ Department of Statistics, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213. This research was supported by National Science Foundation grant DMS-87-02537. Department of Mathematics, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213. This research was supported by National Science Foundation grant DMS-87-02537. Global Portfolio Research Department, Daiwa Securities Trust Company, Jersey City, New Jersey 07302. 702 UTILITY MAXIMIZATION IN AN INCOMPLETE MARKET 703 The principal result of this paper, Theorem 12.5, provides conditions under which an optimal portfolio exists in an incomplete market, and characterizes this optimal portfolio in terms of the solution to a dual optimization problem. In 2-5, we define the utility maximization problem faced by the agent. In 6 we present the solution when the market is complete (m d), and complete hedging is possible. This solution proceeds in three steps. First, on the underlying probability space we determine a new measure which discounts the growth inherent in the market; under this measure, the expected value of the final wealth attained by any reasonable portfolio is equal to the initial endowment. Second, among all random variables whose expectation under the new measure is equal to the initial endowment, a most desirable one is determined. Third, it is shown that a portfolio can be constructed that attains this most desirable random variable as its terminal wealth; this portfolio is optimal. A complete market is one in which the agent can construct a portfolio that attains as final wealth any random variable with expectation under the new measure .equal to the initial endowment. Because such a construction is possible, it is said that the agent can hedge against the risk associated with this market. Mathematically, the construction of a portfolio uses the fact that any martingale with respect to a Brownian filtration can be represented as a stochastic integral with respect to the Brownian motion; the integrand in this representation leads to the portfolio we are seeking. However, if there are fewer than d stocks, this line of argument fails. In 7 we introduce a convenient way of thinking about an incomplete market: fictitious completion. When there are fewer than d stocks, then we augment the stocks with certain fictitious ones so as to create a complete market. If the fictitious stocks have a high appreciation rate, then under an optimal portfolio the agent will hold a long position in them, but if they have a low (even negative) appreciation rate, then he will hold a short position. Thus we would expect to be able to adjust the appreciation rates of the fictitious stocks so that the agent, by optimal choice, does not invest in them at all. These judiciously chosen fictitious stocks allow us to write down the complete market solution for the utility maximization problem but are superfluous in the actual implementation of the optimal portfolio, which must then also be optimal for the original incomplete market. The fictitious completion with the above property is the least advantageous to the agent, because the portfolio which is optimal under this completion is available to him under every other fictitious completion. We thus have the notion of a least favorable fictitious completion: for every fictitious completion we compute the portfolio which maximizes the expected utility of final wealth, and then we choose the completion which makes this maximum expected utility as small as possible. As explained in 7, a convenient way to parametrize fictitious completions of an incomplete market is by a certain space of continuous local martingales, each local martingale being the Radon-Nikodym derivative process of the new measure alluded to in the earlier discussion of complete markets. This kind of parametrization is studied in 8, and several pertinent results are established. It is also desirable to characterize the local martingale corresponding to the least favorable fictitious completion, and to show that it gives rise to an optimal portfolio in the original incomplete problem; this program is carried out in 9, where various such equivalent characterizations are provided. Section 10 studies two examples in which the least favorable fictitious completion can be computed fairly explicitly. In the first example the utility function is logarithmic, and it is discovered that the fictitious stocks in the least favorable completion should have rates of appreciation equal to the interest rate of the bond. This is a very general result, insensitive to the nature of the dependence of market 704 I. KARATZAS, J. P. LEHOCZKY, S. E. SHREVE, AND G.-L. XU coefficients on the driving Brownian motion. In the second example it is assumed that the utility function is of the power form, and that the driving Brownian motion splits into two independent parts; the first part drives the stock processes, whose coefficients are adapted solely to the second part. The least favorable local martingale is exhibited as the solution to a martingale representation problem, and the optimal portfolio is found to be given by the formula already known to be correct for deterministic model coefficients. In 11 we introduce an auxiliary optimization problem involving the family of local martingales which characterize fictitious completions; this problem is "dual" to the "primal" utility maximization question of 5, in the sense of convex duality. We study the relation between the primal and dual problems, and explain how a solution to the latter induces one for the former. The question of existence in the dual problem is tantamount to the existence of a least favorable fictitious completion; it is dealt with in 12, by the use of methods from convex analysis. Our model for the financial market can be traced back to Merton [16], [17] and Samuelson [20]. The modern mathematical approach to portfolio management in complete markets, built around the ideas of equivalent martingale measures and the creation of portfolios from martingale representation theorems, began with Harrison and Kreps [6] and was further developed by Harrison and Pliska [7], [8], in the context of option pricing. Pliska [19], Cox and Huang [2], [3], and Karatzas, Lehoczky, and Shreve [13] adapted the martingale ideas to problems of utility maximization. Much of this development appears in 5.8 of Karatzas and Shreve [14], from which 6 of the present paper is drawn (see also the review article of Karatzas [12] for a survey of financial economics problems in complete markets). An extension of the papers above to infinite horizon problems is reported by Huang and Pages [11]. A first step toward a martingale analysis of incomplete markets was taken by Pages [18], who considered a Brownian model in which the number of stocks was strictly less than the dimension of the driving Brownian motion. However, the coefficients of the bond and stock prices in this model were allowed to depend on the underlying Brownian model only through the bond and stock prices themselves. Thus, the vector of bond and stock prices formed a Markov process. This specialization created an essentially complete market, and thus it avoided the more interesting case of a market with genuinely unhedgeable risk. However, Pages did characterize the class of equivalent martingale measures which could arise in an incomplete model, and this laid the groundwork for further developments (e.g., Lemma 8.2 in this paper). A more substantial step was taken by He and Pearson [9] in a discrete-time, finite probability space model, where the authors proposed finding the optimal intermediate consumption and terminal wealth corresponding to each of the equivalent martingale measures, and then searching over those policies to find a pair yielding the minimum expected total utility. Using separating hyperplane arguments, they were able to show that the total utility obtained by this two-step "minimax" process is the optimal total value for the incomplete problem. He and Pearson have also studied the incomplete problem in a continuous-time, Brownian model. In an early version of He and Pearson [10], the authors consider Pags’s characterization of the family of equivalent martingale measures and search over this family for a "minimax" equivalent martingale measure, which would lead them to the optimal consumption and portfolio processes just as in a complete market. The martingale associated with this measure would create the "Arrow-Debreu" state prices in the incomplete model. However, the continuous-time model is more subtle than one might expect, and although it is now clear that Arrow-Debreu state prices 705 UTILITY MAXIMIZATION IN AN INCOMPLETE MARKET exist for the incomplete model under some assumptions, it is not clear that they are associated with a martingale. The present paper uses local martingales rather than martingales to address the issue of market incompleteness in continuous-time models. This work was motivated by the aforementioned previous version of He and Pearson [10], and by the use of local martingale methods introduced by Xu [22] in the study of incompleteness induced by a prohibition on the short-selling of stocks. Using the stochastic duality theory of Bismut [1], Xu formulated a dual problem whose solution could be shown to exist and could then be used to obtain existence and characterization of the solution of the original problem. As this paper shows, such duality methods can also be used in the traditional incomplete Brownian market model. While we still do not know if the minimax equivalent martingale measure sought by He and Pearson exists in any generality (see, however, the note following the references), we show here that the solution to Bismut’s dual problem is a "least favorable local martingale" which can be used to generate a sequence of equivalent measures. The existence of this least favorable local martingale is sufficient for the study of many models. A notable exception is the incomplete market model in which the agent’s endowment is a stochastic process; we do not know how to obtain the existence and a characterization of the optimal policy for such a model in terms of a least favorable local martingale, unless it is actually a martingale. He and Pearson [10] have incorporated Xu’s local martingale techniques into their original work. They report the existence of an optimal portfolio for the terminal wealth utility maximization problem when the index of relative risk aversion is everywhere greater than or equal to one, and they report similar results for the problem with intermediate consumption and consumption at the terminal time when the index of relative risk aversion is everywhere less than or equal to one. Our paper deals only with the case of terminal wealth utility maximization when the index of relative risk aversion is everywhere less than or equal to one; the generalization to also allow for intermediate consumption is straightforward. Whereas He and Pearson [10] assume that some augmentation of the market model will result in Markov prices, we allow general It6 price processes. He and Pearson 10] do not provide the full set of equivalent conditions contained in our Theorem 9.4, nor do they use our assumption (4.8). This assumption and the introduction of the set Kl(O’) play a fundamental role in our proof of Theorem 9.4. Remark on Notation: We denote by "standing assumption"those conditions that are always in force; they will not be cited in the theorems. The Standing Assumptions are 2.1, 2.2, 2.3, 4.1, and 5.1. We denote by "assumption" those conditions which are in force only when theorems specifically cite them. 2. The market model. We adopt a model for the financial market consisting of one bond with price Po(t) given by (2.1) dPo( t) r( t)Po( t) at, and m stocks with prices per share Pi(t), i- 1,. (. , e( e( b( , Po(0) 1, ., m, satisfying the equations + 2 (W.( j=l ,..., m. Here W W1, We)* is a d-dimensional Brownian motion on a probability space (f, P), and we denote by {ot} the P-augmentation of the filtration generated by W. It is assumed throughout that d ->_ m, i.e., the number of sources of uncertainty in the model is at least as large as the number of stocks available for investment. 706 I. KARATZAS, J. P. LEHOCZKY, S. E. SHREVE, AND G.-L. XU The interest rate r(t), the vector b(t)=(bl(t),..., bin(t))* of stock appreciation rates, and the volatility matrix or(t) {trij(t)}l__<i=,. -<_j-<_d are the coefficients of the model. They are taken to be progressively measurable with respect to {,}. r Standing Assumption 2.1. Ilb(t)ll at <, Ir(t)l dt<- hold almost surely, for some given real constant L > O. The positive constant T in Standing Assumption 2.1 is the terminal time for the problem. All processes are defined on [0, T]. Standing Assumption 2.2. The matrix tr(t) has full rank for every t. As a result of Standing Assumption 2.2, the matrix (o’(t)tr*(t)) -1 and the relative risk process O(t) a-- ty*(t)(r(t)tr*(t))-l[b(t)-r(t)lm] (2.3) o are defined. Throughout this paper, we denote by 1 k the k-dimensional vector whose every component is one. In addition to Standing Assumptions 2.1 and 2.2, the following assumption will be made throughout. Standing Assumption 2.3. IlO(t)ll 2 dt < o, almost surely P. We shall have occasion to use the so-called discount process Po(t- exp r( s ds as well as the process Wo(t)& W(t)+ (2.5) O(s) as and the exponential local martingale (2.6) Zo(t)&exp O*(s) dW(x)- IIo(x)ll dx DEFINITION 2.4. A financial market as above will be called complete if m d, and incomplete if m < d. 3. Prtfl mt elt roeesses. A porolio process (t)=((t),..., (t))* is an N-valued, {}-adapted process satisfying I]*(t)(t)l 2 dt < a.s.P. (3.1) We regard (t) as the propoion of an agent"s wealth invested in stock at time t; the remaining propoion 1-*(t)l 1-= (t) is invested in the bond. We do not constrain these propoions to take values in the interval [0, 1]; in other words, we allow both sho-selling of stocks, and borrowing at the interest rate of the bond. For a given, nonrandom, initial wealth x>0, let X ’ denote the wealth process corresponding to a poafolio defined by X’(0)= x and dX’(t) r(t)X’(t) dt+X’(t)*(t)[(b(t)-r(t)l) dt+(t)dW(t)] r(t)x ’ (t) dt + X ’ t)*( t)(t)dWo(t). In other words, (3.) (t)X’(t) =x exp *(s)(s) dWo()- I1*()(11 (3.3) =x+ (sX’(s*(s(s Wo (s, d 707 UTILITY MAXIMIZATION IN AN INCOMPLETE MARKET Remark 3.1. An application of It6’s rule to the product of the processes /3X of (2.6), (3.3) leads to Zo and x’ (3.4) l(t)Zo(t)XX’(t) x+ (s)Zo(s)X"(s)(o’*(s)r(s)-O(s)) * dW (s). This shows, in particular, that the process hence a supermartingale. flZo Xx’= is a nonnegative local martingale, 4. Utility functions. The agent in our model has a utility function U:(0, c)- E for wealth. We make the following assumption throughout. Standing Assumption 4.1. U is strictly increasing, strictly concave, continuous and continuously differentiable, and satisfies U’(0) __a lim U’(x) o, (4.1) U’(c) a_ lim U’(x) O. xoo x$O The (continuous, strictly decreasing) inverse of the function U’ will be denoted by I’(0, oe), (0, oe); by analogy with (4.1), it satisfies I(oe) A lim I(y) O. I(0) A lim I(y) oe, (4.2) y-c y$0 We introduce also the function /](y) max U(x)-xy] (4.3) 0< y < U(I(y))-yI(y), x>O which is the Legendre transform of- U(-x), with U extended to be -oe on the negative real axis. The function U is strictly decreasing, strictly convex, and satisfies (4.4) (4.5) l]’(y)=-I(y), 0<y<ee, U(x)=min[l](y)+xy]= l](U’(x))+xU’(x), 0< x <oe. y>O The useful inequalities U(I(y))>= U(x)+y[I(y)-x] Vx>0, y>0, U( U’(x)) <-_ U(y)-x[U’(x)-y] Vx>0, y>0 then follow directly from (4.3), (4.5). The monotonicity of U and /] guarantees that the limits U(o) =a lim U(x), U(0) __a lim U(x), (4.6) (4.7) (0) - xee x$O lim (oe) A lim (y), (y) yoo y,[,0 exist in the extended real-number system. LEMMA 4.2. U(0)= U(oe), U(0)= U(oe). Proof. It follows from (4.3) that U(ee)_-<limy_ U(I(y))= U(O), as well as U(oo)_->lim U whence -e =U(O)-e Ve>O, O(oe) U(O). Similarly, it follows from (4.5) that U(oe) _-> lim_,oo/(U’(x)) (0), as well as U(oe) <-_ lim whence U(oe)= U(O). [ fJ() + e] (J(O) + e Ve>O, 708 I. KARATZAS, J. P. LEHOCZKY, S. E. SHREVE, AND G.-L. XU We will sometimes impose the following condition on the utility function U. Assumption 4.3. For some c (0, 1), y (1, oo), we have ,U’(x) >- u’( yx) Vxe(O,). (4.8) Quite obviously, Assumption 4.3 is satisfied by the utility functions U(x)--log x and U(x)= x/8, with 8 < 1, 8 S0. Upon replacing x by I(y) in (4.8), and then applying I to both sides of the resulting inequality, we see that Assumption 4.3 is equivalent to the condition I(cey) <= yI(y) Vy (0, oo), (4.8)’ for some a statement (4.9) (0, 1), y (1, c). By iterating (4.8)’, we obtain the apparently stronger Va(O, 1), =:iye(1, oo), suchthat I(oy)<=yI(y) Vy(O, oo). 5. The utility maximization problem. For a given utility function U and a given initial capital x> 0, the stochastic control problem considered in this paper is the following: to maximize the expected utility from terminal wealth EU(X"’( T)), over the class (x) of portfolio processes 7r that satisfy (5.1) E( U(X’( T)))-< oo, where a -=a max {-a, 0}. The value function of this problem is denoted by (5.2) V(x) sup EU(XX’"(T)). s(x) To be sure this problem is meaningful, we make the following assumption throughout. Standing Assumption 5.1. V(x) < oo, for all x (0, oo). A portfolio process r sf(x) which attains the supremum in (5.2) is called optimal In 9, 11, and 12 we provide conditions that ensure the existence of optimal portfolios, as well as various characterizations of optimality. Some examples, in which optimal portfolios can be computed explicitly, appear in 10. Remark 5.2. In the case of a market model for which the relative risk process 0(. of (2.4) satisfies the condition rl10(t)112 at<-C a.s. for some given real constant C > 0, a sufficient condition for Standing Assumption 5.1 is (5.4) U(x) <- k + k2 x6 /x (O, oo) kl > 0, k2 > 0, 8 (0, 1). Indeed, the process Zo of (2.6) is then a martingale, and W0(’) is a Brownian motion under the probability measure Po(A)--E[Zo(T)IA] on oft (the Girsanov theorem; cf. Karatzas and Shreve [14, 3.5]). For any p (1, 1/8] and suitable constants cl > 0, c2 > 0, we have for some UP+(x) c --t- c2 X6p [x (0, 709 UTILITY MAXIMIZATION IN AN INCOMPLETE MARKET from (5.4), where U/(x)a_ max { U(x), 0}. Also, we have (XX’(T)) ap= xaP exp p (5. - exp r(s) ds- p *(s)(s) dWo(x)- (ex) p. exp p l from (3.3), and EoZ-q(t) (5.7) p Eo for II*(s)(s)ll as 2 II*(s)(s)ll ds *(s)(s) dWo (s) II*(s)(s)ll = as exp q O*(s) dWo (s)-exp 2 II0(s)ll ds IIo(s)ll as --<exp{ q(q-1)2 c} from (2.6) with 1/p+ I/q= 1. Now (5.5)-(5.7), in conjunction with the H61der inequality, give .EU(X’=(T)) Eo[Zff’(T) U(XX’=( T))] <= (EoZq( T))’/q(EoUP+(XX’( T))) ’/p =<exp{ (q-l)2 C} (c’+c2(eLx)p)l/p<cx3 for every 7r M(x), justifying Standing Assumption 5.1. 6. The complete market solution. The utility maximization problem of 5 admits a simple solution in the case m d of a complete market; this solution was derived by Karatzas, Lehoczky, and Shreve [13] and independently by Cox and Huang [2], [3]. In this section we briefly review the pertinent results, both for easy reference and for later usage in the treatment of the incomplete market case. For the purposes of this discussion, we need the following assumption. Assumption 6.1. E[/3(T)Zo(T)I(y(T)Z0(T))] < c, for all y e (0, c). Under it, the function ;To" (0, ) (0, c) defined by (6.1) - ;To(y) a- E[,8(T)Zo(T)I(y[3(T)Zo(T))], O<y<oo inherits from I the property of being a continuous, strictly decreasing mapping of from (0, oo) onto itself, and so ;To has a (continuous, strictly decreasing) inverse (0, co) onto itself. We define o _a I(o(X)/3 T)Zo(T)). Note that for every portfolio process r (x), the supermartingale flZoX x’ of (3.4) (6.2) satisfies (6.3) E [fl (T) Zo( T) X x’ (T) <= x. 710 I. KARATZAS, J. P. LEHOCZKY, S. E. SHREVE, AND G.-L. XU LEMMA 6.2. The random variable satisfies E[fl( T)Zo(T) :] x, E(U(G))- < oo, (6.4) (6.5) and for every portfolio r ,91 (x), we have EU(XX’=(T))<=EU((). (6.6) Proof Equation (6.4) follows directly from the definitions of (4.6) we have : and o. From U() -> U(1)+ o(X)(T)Zo(T)[- 1] >- -I U(1)I- o(X)fl( T)Zo( T). (6.7) But fl is nonnegative and bounded almost surely and Zo is a nonnegative local martingale, thus a supermartingale. Therefore E[/3(T)Zo(T)] < oo, and (6.5) follows. Now let r be a portfolio satisfying (5.1). From (4.6), (6.3), and (6.4) we have EU(:) >= E{ U(X"=(T)) + o(X)fl T)Zo( T)[- X"= T)]} >-_EU(X"=(T)). From Lemma 6.2 it develops that if there exists a portfolio -k such that XX’(T), then is optimal. So far we have not used the assumption of market completeness; this assumption is used only in the construction of the portfolio which finances :, : a question that we now broach. We begin with the positive martingale (6.8) M(t) Being adapted to the Brownian filtration {t}, M admits the stochastic integral representation (6.9) O*(s) dW(s) M(t)=x+ o for some {-,}-adapted process satisfying ]]O(s)[] 2 ds<oo almost surely (e.g., Karatzas and Shreve [14, p. 184]). According to It6’s lemma d (t) --Zo(t)(0(t)+M(t)0(t))*dW(t)’ and thus fl T) M(T) x+ Zo( T Ior 1 (6(t)+ M(t)O(t))* dWo (t). Zo( We define (6.10) [ fo 1 M(t) x+ f((t) fl(t)Zo(t)-fl(t) (t) (6.11) and 1 1 (O(s)+M(s)O(s))* dW(s) Zo(s) (t)Zo(t)f(t) (o’*(t)) .(0) x, (T)- --1 (O(t)+M(t)0(t)), d((t)f(t))= as well as hold. A comparison with (3.3) shows that X(.) is the wealth process corresponding to the portfolio .(. )- XX’( ). We have proved the following result. verify that (t)f(t)r*(t)cr(t) dWo(t) -: - 711 UTILITY MAXIMIZATION IN AN INCOMPLETE MARKET THEOREM 6.3. Let an initial wealth x > 0 be given. In a complete market d m) under Assumption 6.1, the portfolio given by (6.11) is optimal. The resulting optimal terminal wealth is given by (6.2). Example 6.4 (Logarithmic utility function). Suppose U(x) log x. Then So(y) l/y, o(X)- 1Ix and (6.12) Let r(t)/-zllO(t)ll x exp dt+ O*(t) dW(t). be given by ,k(t) (or(t) or* (t))-’[ b(t) r(t) 1,, ]. From (3.4) we have Xx’s(T)= x/(T)Zo(T)= :, so is optimal and (6.13) (6.14) V(x)=E[logXX’(T)]=logx+E r(t)+llo(t)l[ dt, provided that this last expectation is finite (cf. Karatzas [12, 9.3, 9.6]). Example 6.5 (Power utility function and deterministic model coefficients). Suppose that U(x)= x/6, where 6 < 1, 6 S0, and suppose that the processes r and 0 are deterministic. Then exp {(/(1 3)) o O*(s) dW (s)- (32/2(1- 3) 2) ]10(s)ll 2 ds} is a martingale with expectation equal to one (Karatzas and Shreve 14, Cor. 5.13, p. 199]), and from (6.1) to t o(y)= yl/(_l) exp =yl/(-l) exp { for( r(s)/- IIo(s)ll ) 3 1- 1 ds, E exp { 3 1-3 O*(s) dW(s) m(s) ds 1-3 where m(t)Ar(t)+ (6.15) It follows that o(X)= x (6.16) =xexp 2(1-3) m(s) ds}, and I r(t)+2(6)[10(t)ll 1 -1 exp {8 at+ 1-3 O*(t) dW(t) Taking (6.17) (t) A 1 1-3 (r(t)cr*(t))-’[b(t)-r(t)l,] in (3.4), we obtain [3(t)Zo(t)XX’(t)=x exp -2(1_3) 2 IIO(s)ll 2 ds+ O*(s) dW(s) from which follows Xx’(T)= sc and thereby the optimality of 7. Fictitious completions of an incomplete market. The utility maximization problem of 5 for an incomplete market (d > m) will be studied by the method of "fictitious completion." We will perform, in other words, the thought experiment of introducing d m additional stocks driven by the d-dimensional Brownian motion W, thus creating a fictitious complete market in which the utility maximization problem can be solved as in 6. We will then try to determine appreciation rates for these additional stocks, 712 I. KARATZAS, J. P. LEHOCZKY, S. E. SHREVE, AND G.-L. XU so that the optimal portfolio in the resulting complete market does not invest in the additional stocks at all, i.e., is in s(x). Following this program, we introduce an { ot}-progressively measurable, uniformly bounded, (d-m)xd matrix-valued process p(t) whose rows, thought of as vectors in d, are orthonormal and in the kernel of or(t), i.e., cr(t)p*(t)= 0. We also introduce an {ot}-progressively measurable, (d- m)-dimensional vector process satisfying II(t)ll (7.1) dt < a.s. We create fictitious stocks with prices Si(t) governed by j=l The matrix-valued process p will be held fixed throughout the remainder of the paper, but the process a will be considered as a parameter. For the augmented stock appreciation rate vector /_a_[b] and the augmented volatility matrix [o we can define an augmented relative risk process - (t) _a_ ,,( t)(( t)*( t))-I [/( t)- r( t)lm] 0(t) + v(t) (7.3) by analogy with (2.3), where ,(t) ___a p,( t)[z( t) r(t) ld_,, ]. (7.4) Note that O*(t),(t)=O, and thus Ilff(t)ll 2= IlO(t)ll2+ll,(t)ll 2. It will be assumed that io (7.5) ]lu(t)ll z dt < oo holds almost surely, so that (by analogy with (2.6) and (6.1)) we may define the exponential local martingale Z(t) a--exp (O*(s)+ (llO(s)ll2/ll (s)ll =) ds dW (7.6) 1 Z(s)(O(s) + ,(s))* dW (s) and the function (7.7) (y) __a E[J3( T)Z( T)I(y[3( T)Z( T))], 0<y<oo. If the condition (7.8) prevails, we may define (7.9) (y) < to be the inverse of x and set g I((x)(T)Z(T)) by analogy with (6.2). Remark 7.1. If the fictitious stocks introduced in this section were really available, then EU(X) would be the maximal expected utility of final wealth (Theorem 6.3). Since these stocks are not available, we have V(x) a_ sup EU(XX’=( T)) <- EU(), (7.10) (x) and equality holds if there exists a portfolio process r (x) such that (7.11) Xx’(T) sx a.s., UTILITY MAXIMIZATION IN AN INCOMPLETE MARKET 713 i.e., if the terminal wealth can be financed without investment in the fictitious stocks. In light of (7.10), such a r would be optimal for the problem of utility maximization in the incomplete market. In 9 we shall discuss properties which r and u must have in order to be related by (7.11). 8. A family of exponential local martingales. Let us denote by S[0, T] the class of {t}-adapted, Ed-valued processes 0 satisfying io (8.1) 0(t)ll dt < c almost surely, and decompose S[0, T] into the orthogonal subspaces (8.2) (8.3) K(tr){uS[0, T]; tr( t) u( t) 0, ’v’t[0, T],a.s.}, K+/-(tr){0 S[0, T]; o(t)Range(tr*(t)),/t[O, T], a.s.}. Remark 8.1. The process 0 of (2.3) belongs to K+/-(tr), whereas the process u of (7.4) belongs to K (r). On the other hand, if u K (tr) is given, then (7.4) can be solved for the appreciation rate vector of the fictitious stocks, by taking this vector equal to a,( t) p( t)u( t) + r( t)ld_,. (8.4) Thus, the class K(r) provides a parameter space for fictitious completions of the incomplete market. We will denote by 3 the fictitious completion of the financial market by the additional stocks of (7.2), with p(. fixed and (. )-= (. ), u K(cr). The associated family of exponential local martingales {Z}K(, given by (7.6), will play a fundamental role in what follows. LEMMA 8.2. Consider the discounted stock price processes a-- ( t)Pi( t), Qi( t) i-1,...,m. Then for every u K (or), the processes ZQi are local martingales under P. Proof. It is seen from (2.2), (2.4) that dQi(t)= Qi(t)[(bi(t)-r(t)) dt+tri(t) dW (t)], where tri(t) is the ith row vector of the matrix tr(t). It follows from this, (7.6), It6’s rule, and tr(t) u(t) 0, that d(Z(t)Qi(t)) for Z(t)Qi(t)[o’(t)-(O(t)+ u(t))*] dW (t). PROPOSITION 8.3. For any given every u K (r); in particular, (8.5) (8.6) 7r M(x), flZX x’E is a local martingale under P E[fl(T)Z(T)XX’(T)]<=x Vu K(cr). Proof. From (3.3), (2.5), and (7.6) follows the analogue fl(t)Z(t)X(t)=x+ (s)Z(s)X(s)[o’*(s)(s)-(O(s)+u(s))]* dW(s) of (3.4) for the process X X ’=. This representation shows that flZX ’ is a positive local martingale, hence a supermartingale, and (8.5) follows. Remark 8.4. Suppose that 7r is a portfolio process, and that X is a continuous, {t}-adapted process which satisfies (8.6) almost surely, for some u K(tr). Then X is the wealth process corresponding to the initial endowment x and the portfolio 714 I. KARATZAS, J. P. LEHOCZKY, S. E. SHREVE, AND G.-L. XU process zr, i.e., X X ’. Indeed, apply It6’s rule to the product of the processes ZX and A, where A- Z is easily seen from (7.6) to satisfy dAy(t) A(t)[(O(t)+ u(t))* dW(t)+(l[O(t)[[:+ u(t)[[:) dt], and obtain (3.3). The following result provides a kind of "converse" to Proposition 8.3. THEOREM 8.5. Consider a positive, 7.-measurable random variable B, there exists a process h K (or) with for which E[(T)Z(T)B]<=x E[(T)Zx(T)B] Vv K(o). Then there exists a portfolio 7r i(x), such that X X’(T) B, almost surely. (8.7) Define a positive, {t}-adapted process X via Proof. 0 <- t<= T. fl(t)Z;,(t)X(t)= M(t) A- E[fl(T)Z(T)BIt], Certainly X(0)=x, X(T)= B almost surely, and the positive martingale M in (8.8) has M(0)= x. From the martingale representation theorem (Karatzas and Shreve [14, Problem 3.4.16, p. 184]), M(t)=X+o (s) dW (s) for some {fft}-adapted process satisfying II(t)ll = at < oo almost surely. Since M is continuous and M(t)> 0 for all (8.8) t [0, T], we may define h S[0, T] by (t)=-p(t)/M(t). Then p*(s) dW M(t) x exp (8.9) (s)-- IIg,(s)ll as r x- Jo M(s)b*(s) dW (s), 0<= <- T. Decomposing @ as @= @1+ @: with @1 K-(cr), @: K(cr) and comparing (8.8), (8.9) with (8.6), it transpires that proving the theorem amounts to finding a portfolio 7r such that -M(t)(tl(t)+ 2(t))= (t)Za(t)X(t)[o’*(t)Tr(t)-(O(t)+ A (t))]. (8.10) This will certainly be possible, provided that (8.11) qt:(t)=h(t) dtxdP a.e. on [0, T]xf, because we can take then 7r to satisfy cr*Tr 0-1 K+/-(r). Consequently, we have to show that (8.7) implies (8.11). To this end, consider an arbitrary but fixed v K(g) and introduce the sequence of stopping times {,}_ given by z,= rinf te[0, r]; M(t)en, or ([[l(s)ll2+ll2(s)ll=+[lA(s)[] =) dsen, (8.12) or II(s)ll = as n, or n-> 1. Obviously, lim._ z. T almost surely, and v(t)l[o. .](t). Clearly, A + ev. K(o’) and for every (8.13) Zx+.(t)= Za(t) exp for every e (8 14) -e *(s) dW (s) we denote v*(s)(dW (s)/ (s) ds)-- >n} v.(t) II(s)ll = ds (-1, 1), n => 1. On the other hand, the definition of z, in (8.12) gives e-3nl I<_ Z+.(T) -<e 3hie Zx( T) -l<e<l. UTILITY MAXIMIZATION IN AN INCOMPLETE MARKET 715 It follows then quite easily (from (8.12)-(8.14) and the dominated convergence theorem) that (8.7) implies 0=--E ( T)Z+( T)B e=0 (8.15) =-E fl(T)Za(T)B or equivalently, in the notation of (a. M(- =E[fl(T) O-- Z,+( oe T) v*(s)(dW(s)+A(s) ds) (8.8)" ,,*(s(dVC(s+,x( ds =0 Vn-->l. Now It6’s rule, in conjunction with (8.9), gives (8.17) From the definition of ’n in (8.12) we see that the expectations of the two stochastic integrals in (8.17) are equal to zero. Substituting back into (8.16), we obtain (8.18) M(t)v*(t)(A(t)-Oz(t))dt=O Vn>=l. E The arbitrariness of v K(tr) in (8.18) leads to (8.11). 9. Equivalent optimality conditions in an incomplete market. The conclusions of 7 were predicated on the assumption (7.8) (y) < oo Vy (0, o), but this condition often will not hold for all v K(r) (cf. 13 (Appendix)). Accord- ingly, we restrict ourselves to the class Kl(O-) ---a {v K(cr); (9.1) v satisfies (7.8)} in what follows. Remark 9.1. If Assumption 4.3 holds, and v K(cr) satisfies (y) <oe for some y (0, oe), then Kl(O-). This can be verified easily, using (4.9). For a fixed initial capital x > 0, let -k M(x) be given, and consider the statement that -k is optimal for the incomplete market maximization problem of 5" (A) OPTIMALITY OF 7’. EU(X’(T)) <-_ EU(X’S(T)), for all 7r M(x). We will characterize condition (A) with the help of the following conditions (B)-(E). For a given A Kl(O-) recall the notation of (4.3), (7.9) and consider the following statements. There exists a portfolio .k M(x) such that XX’S( T) (B) FINANCIBILITY OF , , :. almost surely. (C) LEAST-FAVORABILITY OF A. EU(Xa) <= Eg(x), for all v Kl(cr). (D) DUAL OPTIMALITY OF A. For all v Kl(Cr), , El( (x)( T)Za (T)) <= El( (x)( T)Z( T)). (E) PARSIMONY OF A. E[fl(T)Z(T)sC] -< x, for all t, K,(o’). 716 I. KARATZAS, J. P. LEHOCZKY, S. E. SHREVE, AND G.-L. XU Our principal result of this section, Theorem 9.4, states that conditions (B)-(E) are equivalent and they imply the existence of satisfying (A), provided that Assumption 4.3 and U(0)>- hold. This latter restriction is rather severe, for it excludes the important special case of the logarithmic utility function U(x)= log x. For this reason we also develop a somewhat more modest result, Theorem 9.3, which suffices for a complete treatment of the logarithmic case (Example 10.1). But first, let us try to motivate the developments that follow by discussing the significance of conditions (B)-(E). While we do not present any proofs for the claimed equivalences in the .discussion that follows, we offer some plausible arguments to the effect that conditions (A)-(E) are connected to one another. Discussion 9.2. For any given Kl(r), ( is the optimal level of terminal wealth When will it also be optimal in the original, in the fictitiously completed market incomplete market? Presumably, only when there exists a portfolio -k which invests in the original m stocks only (i.e., ,k (x)), such that X x’ In other words, condition (B) has then to hold, and condition (E) follows directly from (8.5). In particular, (E) is at least as large in the fictitiously says that the value of the contingent claim v Kl(o-). Note in this connection as in any other market completed market that, according to the definitions, E[fl(T)Z(T)(] x, for all K(r). Furthermore, the terminal wealth can be financed by investing in the stocks of any other market (since, in fact, it can be financed by investing in the original m stocks). Thus we obtain the condition (C), which captures the "least favorable" character of Let us derive finally the condition (D), at least in the case U(0)>-c (in which / is bounded from below, thanks to Lemma 4.2, and thus the expectations in (D) are well defined). Indeed, by writing (4.7) with x replaced by and y replaced by (x)(T)Z(T), and taking expectations, we obtain on . , . . E( 2--Y, (x) fl T)Z,( T) >= E&( , (x),8 T)Z, (T)) + o)j, (x) E[fl( T)Z,,( T) ]} {E[fl( T)Z, (T) >= EO( qY, (x)fl( T)Z, T) : from condition (E). THEOREM 9.3. Conditions (B) and (E) are equivalent, and imply (C). Furthermore, if (B) holds, then the portfolio in (B) satisfies (A). Proof (B)(E) follows from Proposition 8.3. in Theorem 8.5. Note that this theorem remains (E)(B) follows by letting B valid if in it K(cr) is replaced by Kl(Cr). In order to see this, it suffices to observe that the processes h +evn (appearing in (8.13)) belongs to K(cr) for every e(-1, 1), n _-> 1, because from (8.14) and the fact that h Kl(o-): : +,,.(y)<=eanll(ye-3’ll)<oo Vy (0, The last statement of the theorem follows from Remark 7.1. (B)(C) holds because the previous implication and (7.10) imply V(x) <= EU(X) Vv K,(o). THEOREM 9.4. Assume that U(O) >-00 holds. Then (i) Conditions (B)-(E) are equivalent, and if (B) holds, then the portfolio "k in (B) satisfies (A); (ii) Conversely, if r sd(x) satisfies (A), then there exists a A K(o’) for which (B)-(E) hold, provided that Assumption 4.3 is also in force. EU(X) 717 UTILITY MAXIMIZATION IN AN INCOMPLETE MARKET Proof In view of Theorem 9.3, we need discuss only the implications (C)(D) (B) and assertion (ii) under the appropriate conditions. (C)(D) For any given y > 0 and ,, Kl(O-), the convexity of U yields -(- I(J((y+e)fl(T)Z,(T))- II(yfl(T)Z,(T))I<=fl(T)Z(T)I (9.2) - fl(T)Z(T) in conjunction with (4.4), for e >-y/2, e #0. From the assumption u K(), the random variable on the right-hand side of (9.2) has expectation equal to (y/2) and the dominated convergence theorem shows that d y E(yfl( T)Z( T)) (9.3) -E[fl( T)Z( T)I(yfl( T)Z( T))] Therefore, for any given x> 0, u K(), the convex function (9.4) f(y)& E(yfl(T)Z(T))+xy, 0<y<, attains its minimum at (x), since f’(y)= x-(y). But thanks to now have for any y > 0 that (9.5) (y). (C) and (4.3), we L(y) L( (x)) E{ F((x)# T)Z(T)) + (x)#( T)Z(T). I((x)#( T)Z( T))] EU(I( (x)( T)Z( T))) EU() EU() L((x)) ve K,() and thus, EU( (x)fl( T)Z T)) E[ U()- (x)fl( T)Z T)(] (x) L( (x)) x (x) =L (x)) E( (x)#( T)Z(T)). (D)(B) Repeat the proof of Theorem 8.5 up to (8.14), with K() replaced by Everything then boils down to showing that the K(), (8.7) by (D), and B by . analogue x of (8.15) can be obtained from the consequence of (D) (9.7) since I + e K() for every e (-1, 1), n 1 (recall the argument in the proof of implication (E)(B) in Theorem 9.3). Indeed, (9.6) follows formally from (9.7) by differentiating inside the expectation sign, and using (4.4), (7.9), (8.13). For a rigorous justification, recall (8.12) and use the convexity of to obtain, for any given y > 0: (9.a y(r)(e z(r((r e-z. (r)) 718 I. KARATZAS, J. P. LEHOCZKY, S. E. SHREVE, AND G.-L. XU K,,-a supo<<l (e3n-l)/e. The expectation of the right-hand side of (9.8) is (ye-3"), a finite quantity equal to yK, le times E[/3( T)Z (T)I(ye-3"B( T)Z (T)] by the assumption A Kl(Cr). On the other hand, the mean-value theorem implies that for each e (-1, 1)\{0} there is a random variable y with values in [0, 1], such that where Z+.(T)-Z(T) U’(yfl(T){Z(T)+ y(Z+.(T)-Z(T)})) From this and (9.7), the conclusion (9.6) follows, thanks to the dominated convergence theorem, by letting e$0. Proof of (ii). Step 1. Let X be the wealth process corresponding to the optimal We have from (3.3) portfolio . fl( t)2( t) x + fl(S)2(s)(cr*(s)(s))* dWo (s) (9.9) =x exp -*(s)cr(s) dWo ()- II*(s)()ll d Now take a bounded, {t}-progressively measurable portfolio process Nm, and perform a small random perturbation of -k according to r/ with values in (9.10) - where-l<e<l, e#0, and T inf ^ / [0, T]; (9.11) (see (9.19), (9.20) below for the definitions of the processes N, A, and ). We also X (.) via define the process (t)X(t) x exp r*(s)o’(s) dWo (9.12) x+ (s)X(s) (s)(s) dWo(s), and note that X(. ) X’=( ). Consequently, (A) gives (9.13) 0 Oe EU(X(T))I =o=0. ds UTILITY MAXIMIZATION IN AN INCOMPLETE MARKET 719 A comparison of (9.9), (9.12) yields (9.14) X(t) 2(t) exp rl*(s)tr(s) dW (s)-- e where (9.15) If’C(t) &- Wo( t) r*(s)’fr(s) ds W(t) + (O(s) o’*(s) (s)) ds. Then, at least formally, (9.13) and (9.14) lead to e ’(2(r2(r (9. n*(s(s dW(s) =0 n_->. Step 2. In order to justify (9.16) rigorously, obsee from (9.14) and (9.11) that e 11, almost surely, and from the concavity of U e-3nll N X T)/(T) 1 e U(X( T))- U(2( T))I U’(min {X( T), X( T)}) U’(e-3"IIX( T))( T) (9.17) X(T)-2(T) e 3hIll- 1 I1 [U(e-3II(T)) U(0)] e3"K, e"n v(2(r))- (0)], with K, as in (9.8). The right-hand side of (9.17) has finite expectation, namely, e3K( V(x)- U(0)). On the other hand, the mean value theorem implies the existence of a random variable y with values in [0, 1], such that [U(X(T))- U(2(T))]=!(X(T)-2(T)) - U’(2(T)+T{X(T)-2(T)}) U’((T)+T{X(T)-(T)})(T) 1 exp e n*(s)(s) d(s) II*(s)n(s)ll: as 1 It is clear now that (9.16) follows from this expansion, (9.13), and the dominated convergence theorem, by letting e0. Step 3. Now proving (B) amounts to finding A e K() such that (T)= I (% (x) (T) Zx (T)), or equivalently (9.18) U’(X( T)) (x)( T)Zx (T). We will show that (9.16) leads to a "natural" candidate process A e K(), which is actually in KI( and for which (9.18) is then shown to hold (Step 4). Consider the process (9.19) n*(s(sW(s+ n*(s(s[O(s-(s(s]as, 720 I. KARATZAS, J. P. LEHOCZKY, S. E. SHREVE, AND G.-L. XU as well as the positive martingale N(t)& E[U’(2(T))2(T)It]=Yo+ (9.20) N(s)O*(s) dW (s), where Yo EN(T) and is some process in S[0, T], constructed by the argument preceding (8.9). Using Standing Assumption 5.1 and U(0)>-oo, note that yo EN(T)<= E[U((T))- U(0)] <oo. Obviously (9.16) amounts to E[N(-,)A(-,)]=O; on the other hand, we have from (9.19) and (9.20) that N(t)*(t)(t)[O(t)+ O(t)-*(t)(t)] dt N(r)A(n) ([*(v( + a( ( ]* + (. From the definition of in (9.11), the stochastic integral has zero expectation, and thus [N(r)A(.)] =0 leads to (v*(([(+ 0(-*((1 =0 n e (9. for arbitrary that , as described above. Because a (9. * T almost surely as n , we obtain (+ 0 belongs to K(). For this choice of I, the exponential local maingale Z of (7.6) becomes (O(s)+(s))*dW(s)- Za(t)=exp (9.23) Io’ (O(s)-*(s)(s))*dW(s- =exp U’(X( T)) fl( T) N(T) fl( T)( T) xp ( 0() dW ()- (T) 2 x xp =Y2 fl( T)Z X ii()-*((sll ds K() and (9.18). From (9.20) and (9.9) and I =0, Step 4. Finally, we justify I it follows that (9.4) 0()+()11 d ((s)(s)) dWo (s)- 0(s)l ds I(s)(s)ll ds T), thanks to (9.23) and (9.22). It remains to show that A K() and yo xx(x). In order to see this, apply I(. to both sides of (9.24), take expeCtations, and use (9.24) aain to obtain ( r)z r)I ( r)z (r) ___x E[ U’(( T))( T)] =__x EN(T) Yo [()z ()x()] x< Yo From Remark 9.1 we have A Kl(r), and yo xx(x) follows. 721 UTILITY MAXIMIZATION IN AN INCOMPLETE MARKET 10. Examples in an incomplete market. In the examples of this section, we assume and the process A Kl(cr) satisfying m <d and produce the optimal portfolio conditions (B), (C), and (E). In Example 10.2, A also satisfies (D). Example 10.1 (Logarithmic utility function). Suppose U(x)=log x. Then (y) 1/y, (x)= 1/x, for all vK(o’), and =x/(T)Z,(T). The process h --0 satisfies (E), because II(s)ll=ds u*(s) dw(s)- E[fl(T)Z,(T)]=xE exp x VEK(). ’o The last inequality follows from the fact that exp {-’o (s) dW (s)- II(s)ll bein a nonneative local mainale, must be a supermainale. Accordin to Theorem and this was deter9.3, the optimal pofolio process must satisfy X’(T)= mined in (6.13) of Example 6.4. The value function V(x) is Biven by the expression (6.14), and it is finite for every x E(0, )if E II0(t)ll at< (ra] Standin 0 corresponds to completion of the market Assumption 5.1). From (?.4) we see that by stocks whose appreciation rates are equal to the interest rate. With a loxarithmic utility unction, the aent will not use such stocks euen hedin purposes. Example 10.2 (Power utility unction and totally unhedeable market coecients). Suppose U(x)=x/, where <1, #0. Suppose that the volatility matrix (t) has the form c(t) #(t), 0], where #(t) is an m x m, nonsinBular matrix for all E [0, T], almost surely. Decompose W into (t) W(t)) and W(t),. (t)=(W+(t),..., Wa(t)) and let {} and {} be the auBmentations under P and of the (independent) filtrations enerated by respectively. Assume that the processes r, b, and # are adapted to {}, a situation we refer to as totally unhedeable market coecients because the stock prices are driven solely by , or , e( e( b( , ,. ., + 2 e(a( j=l m. We show that under these conditions, the pofolio process given by (6.17) is optimal. In the present context, this process is random and {}-adapted, rather than deterministic as in Example 6.5. To verify the above asseion, we note first that 0*(t)= 0*(t), 0], where ( ( (( le*( -[ b( r( ]. We note also that the processes I eK() are of the form 1*(t)=[0, *(t)], where X(t) is (d-m)-dimensional. With m(.) given by (6.15), we define Aexp m(t) dt {,}-maingale N(t) [exp { I re(s)ds}l,} has e the a representation as dN( t) -N( t)i *( t) d (t), where I i K() (see argu- The differential of the positive ment leading to (10.1) exp (8.9) for a justification). Therefore m( =N(r=aexp We may assume without loss of generality that is the coordinate mapping process (t, ) (t) defined on C([0, r], N), the space of continuous functions 722 I. KARATZAS, J. P. LEHOCZKY, S. E. SHREVE, AND G.-L. XU . , ---a from [0, T] to R m, and I is the coordinate mapping process on C([0, T], Ra-,). and P is the product of m-dimensional Wiener measure/5 on and Then f/= x (d- m)-dimensional Wiener measure fi on Abusing notation slightly, we regard the {,}-adapted process as a process on For /kalmost every o3 we have *(s, a3)l[ 2 ds < e, and thus for fixed 03, the process o (t, ff)-exp 6 (s, o3)112 ds (s, o3) dff" (s, o3)-2(1_6) 2 6 1 , is an {,}-martingale on under /5, with expectation equal to one (Karatzas and Shreve [14, Cor. 5.13, p. 199]). Consequently, E exp 1 (10.2) 6 exp =exp *(s, a3) dff" (s, 1-6 2(1--6) a3)}P(do3) IIt(s’ )11- ds 2 From (10.2) and (10.1) we have a(y) =y 1/(8-1)E exp 1-6 1-6 *(s) aft" (s) =y /(-I)E exp 1-6 A6/(1-6)yl/(-I)E exp 6 r(s)+-ll(s)ll=+llX(s)ll = E exp 1-6 m(s) ds + 1 2 ds 0*(s) dff" (s) IIi(s)11 ) ds+ m(s) ds It follows that h Kl(Cr), /Ja(x)--Ax -1, and using (10.1) we obtain (x)( T)Za (T)) 1/(’-1) x exp 1-6 ) 1-26 r(t) +2 II(t)ll= dt 2(6- 1) O*(t) dW(t) Just as in Example 6.5, we conclude from (3.4) that by (6.17). Xx’e(T)= , where - is given Remark 10.3. An important unresolved question is whether there are simple, widely applicable conditions that guarantee that for the process A satisfying conditions (B)-(E) of 9, the nonnegative local martingale Za is actually a martingale. (See, however, the note following the references.) In Example 10.1 we have /x(t)= go(t)=exp O*(s) dw IIo(s)ll = ds 723 UTILITY MAXIMIZATION IN AN INCOMPLETE MARKET we must assume at least that Zo is a martingale in order to conclude that Example 10.2 a computation similar to (10.2) reveals that SO EZ(T)=E exp =e exp i*(t) dfV(t)-- ([[0(t)[[2d-IIi(t)ll 2) dt i*(t) d(- IIi(11 d Z is. In Taking expectations in (10.1) and recalling the definition of A, we see that EZ (T) 1. This is enough to ensure that Z is a maingale (Karatzas and Shreve [14, p. 198]). We have so far been unable to produce an example in which Zo is a maingale but Z is not. 11. Dlty. We hencefoh impose the following assumption. Assumption 11.1. U(0) > In addition to the original, or primal," optimization problem -. (. V(x= sup of 5, we shall consider in what follows the dual optimization problem for y (0, ), namely, This problem will have a value Nnction V" (0, ) N under the following assumption, which will also remain in force for the remainder of the paper. Assumption 11.2. For every y(0, ), there exists p K() such that (11.3) J(y; See Remark 11.9 in connection with this assumption. For arbitrary x>0, y>0, e (x), and p K() it follows from (4.3) that with equality if and only if By taking expectations in (11.4) and recalling Proposition 8.3, we obtain (11.6) J(x; ) j(y; ) + xy, with equality prevailing if and only if (11.5) and x from (11.6) that (y) hold. In paicular, it follows (.7) V(x) v(y)+xy x>0, y>0. Remark 11.3. Suppose that for some given x > 0 and y > 0, there exist and I K() such that (ll.a) Then (x) J(x; x)= J(y; achieves the supremum in (11.1), and a achieves the infimum in (11.2). 724 I. KARATZAS, J. P. LEHOCZKY, S. E. SHREVE, AND G.-L. XU PROPOSITION 11.4. Under Assumptions 11.1 and 11.2, suppose that, for a given y >0, there is an optimal process Ay Kl(o) for the dual problem of (11.2). Then there exists an optimal portfolio sd(x) for the primal problem of (1 1.1) with x- gy(y), and we have rx (y) (11.9) sup V() -y]. >o /(. is convex. The optimality of Ay gives In particular, Proof (11.10) E[(21y(x)fl(T)Z(T))]<- E[((x)fl(T)Z(T))] for this particular x (y). This is Condition (D) of 9. Theorem 9.4(i) shows the existence of a portfolio sg(x), which is optimal for the primal problem and satisfies x Xx’( T) I(yfl( T)Z( T)). We conclude that (11.8) prevails (i.e., (11.6) holds as an equality with r and thus x, ’ Ay), (y) (y; Ay)=J(x; rx)-xy= V(x)-xy<-_sup V(()-y(]. #>o The inequality in the opposite direction follows directly from (1 1.7), and the duality relationship (11.9) is established. Assumption 11.5. Suppose that the dual problem of (11.2) admits an optimal process Ay K1 (o’), for every y > O. A sufficient condition (Theorem 12.3) for the fulfillment of Assumption 11.5 will be given in the next section. Under the Assumption 11.5, (11.9) holds for all y > 0, and the following question arises. Under what conditions can we guarantee that, for every given x>0, there exists an optimal portfolio "2rx for (11.1)? According to Proposition 1 1.4, this will happen if for every x > 0 we can find a real number y(x) > 0 such that (11.11) x= zy(x(y(x)). . PROPOSITION 11.6. Suppose that the conditions of Proposition 1 1.4 hold, as well as Then for every x > O, there exists a real number Assumptions 11.5 and 4.3 and U(eo) y(x) > 0 that achieves infy>o l(y) + xy]; this number satisfies (1 1.1 1) as well. Proof From (11.3), Jensen’s inequality, the supermartingale property of Z, and the decrease of U, we have ](y; ,)_-> (J(yE[fl(T)Z,(T)])>= (yeLEZ,(T))>-_ (J(ye L) V, K(cr) > l(ye L) holds for the constant L>0 of Standing Assumption 2.1. Therefore, iS"(y)_for every y (0, oo), and I(0)= limy+o l’(y)_->/(0) U(oe)= oe (Lemma 4.2). Consequently, for any given x > 0, the convex functionf(y) l(y) + xy, 0< y < (11.12) satisfies f(0+)=f(oe)=, and thus attains its infimum on (0, ) at some point y(x)>0. Now by the Assumption 11.5, there exists a process Ay( Kl(tr) such that V(y(x)) J(y(x); Ay(), and we have inf [Ty(x)x + (Ty(x); Ay(x))] inf [xy+.(y; Zy(x)) y>O 0 _-> inf [xy+ (y)] xy(x)+ (y(x)). y>O In other words, with the notation (11.13) Gy(u)Y(uy;Ay)=E(uy(T)Z(T)), 0<u<, UTILITY MAXIMIZATION IN AN INCOMPLETE MARKET 725 the function Dx(u)uxy(x)+Gy(x)(U), (11.14) 0<u<o achieves its infimum at u- 1. From these considerations and Lemma 11.7 below, it transpires that D’x(1)- xy(x)+ G(x)(1)= xy(x)- y(x)W,((y(x)) 13 is equal to zero, and thus (11.11) holds. LEMMA 11.7. Under the conditions of Proposition 11.6, the function Gy( and satisfies is well defined and finite on (0, oo) for any given 0< y < , of (11.13) G(1) -yW(y). (11.15) Proo Since ()= U(O)- by assumption and by Lemma 4.2, we have from (4.4) that (y)- () Thus, for any given a ’() d(= I() d(, 0< y (0, 1), it follows with the help of (4.9) that (ay)- ()= I() d= a ay for a suitable constant 7 (1, ). Consequently, for any given y (0, ), E(ayfi( T)Z,,( T)) ayE&(yfi( T)Zh( T)) + (1 a) () Since a (0, 1) is arbitrary, E(uy( T)Zy( T)) < (0, 1]. But the function (.) is decreasing, (11.16) , holds for every u so (11.16) also holds for every u > 1. Now use the convexity of the dominated convergence theorem, (4.4), and the fact that I K(), to justify the computations - [ r)z., (r) I( (r) z., r))] -., (), which leads to (11.15). It just remains now to put Propositions 11.4 and 11.6 together, in order to obtain the following existence result for the primal problem (11.1). TuoM 11.8. Suppose that Assumptions 4.3, 11.1, 11.2, and 11.5 hold and en for any given level x > 0 of initial capital, there exists an optimal por(olio U() (x) for the utility maximization problem of 5. In other words, under appropriate conditions, in order to obtain the existence of an optimal pofolio it is sufficient to deal with the existence of a solution for the dual problem (11.2). We will do that in the next section. . 726 !. KARATZAS, .I.P. I+I’Itl()(’ZKY, g. SltRi".Vi,I, ANI) (.-!_ XtJ I,i. Remark 11.9. Assumption 11.2 is satisfied if (5.3) and (5.4) hold. Indeed, it is not hard to check that condition (5.4) and definition (4.3) lead to (11.17) U(y)k+k.y with ce-8/(1-8), k3=(l-8)(k_’) / ’,and .(.v; v) --< k, + k.,y ( .8) V0<v<oo e thus - ’’ EZ,,’(T), where L is the constant of Standing Assumption 2.1. But now (O(t)+ v(t))* dW (t)--R- Z,,"(T)=exp exp (ll0(t)ll/ II.(t)ll ) dt (11 o(.,.)I1- + i1,,(.,)11 ) 2 and if we take v c K (r) to satisfy I," ,-’(.,)11 as <= c a.s. (by analogy with (5.3)) we obtain EZ,,"(T) <= e ’’’’’’’. Back into (11.18), this estimate shows that (11.3) is satisfied. 12. Existence in the dual problem. We will establish here the following existence result for the dual optimization problem of (11.2). This theorem is the final step in the solution to the problem of optimal investment in an incomplete market; see Theorem 12.5. Assumption 12.1. E II0(t)ll dt<oo. Assumption 12.2. x,-xU’(x) is nondecreasing on (0, oo). THEOREM 12.3. Suppose that Assumptions 4.3, 11.1, 11.2, 12.1, and 12.2 hold. Then .]br every y c (0, oo), there exists a process A.,, K(o-) which achieves the in.]imum in J’i (11.2). E] Remark 12.4. Assumption 12.2 is equivalent to (12.1) y--yI(y) is nonincreasing on (0, If U is of class C-(0, oe), Assumption 12.2 amounts to the statement that the Arrow-Pratt measure of relative risk aversion does not exceed one: (12.2) xU"(x) U’(x) ---< Vx (0, oo). On the other hand, it follows from Assumption 12.2 that U’(x)_>- U’(1)/x for all U(x) U(1) + U’(1) log x. Consequently, x => 1, whence (12.3) U(oo) =oo. From this remark and Theorems 12.3, 11.8, and 9.4, we deduce then the fundamental result of 11 and 12 as follows. THEOREM 12.5. Under the assumptions of Theorem 12.3, corresponding to every x > O, there exist" (i) An optimal portfolio M(x) for the utility maximization problem of 5; and (ii) A process A Kl(o) (A depending on x) which achieves the infimum in (11.2) with y= A(x); this A satisfies the equivalent conditions (B)-(E) of 9, and the appearing in (B) is 727 UTILITY MAXIMIZATION IN AN INCOMPLETE MARKET We carry out the proof of Theorem 12.3 in a series of lemmas that take up much of the remainder of the section. Let us start with a rather simp.le observation. LEMMA 12.6. Under Assumptions 11.1 and 12.2, we have U(O)=o, U() >-, and z- (e ) (12.4) is convex on Proof. The first two claims follow directly from Lemma 4.2 and (12.3). As for (12.4), observe from (4.4) and (12.1) that (d/dz)U(eZ)=-eI(e z) is a nondecreasing function of z. Introduce now the Hilbert space H(tr) ---a (12.5) I1 ( )11 ds<oo with inner product (/z, ,)E. /z*(s)u(s) ds and norm [,] V’(u, ,). For a fixed y > 0, we consider the functional Jy" H(tr)--> LJ {+oo} given by (11.2), namely, Yy(,) E(J(y( T) e -(T)) (12.6) with the notation (t)& (12.7) fo lfo (O(s)+(s))* dW()+ (ll0( )ll=+ll (s)ll =)ds, LEMMA 12.7. Under Assumptions 11.1, 12.1, and 12.2, on H (), which satisfies (12.8) lim is a convex functional fly( for every y (0, ). Proo Jy(. K(). From the convexity of the Euclidean norm in d, the decrease of and (12.4), we have ](I+I)NN (12.9) = exp logyexp r(s) X logy+ I log y r(s) ds- (r) in H() and A1 0, A20 with AI+ A2 1. On the other hand, with L for any 1, as in Standing Assumption 2.1, we obtain from (12.4) and Jensen’s inequality Jy() EU(exp [log y + L- (r)]) (12.10) 0(exp [log y + L- E(r)]) 0(exp [log y+ 728 I. KARATZAS, J. P. LEHOCZKY, S. E. SHREVE, AND G.-L. XU LEMMA 12.8. Under Assumptions 11.1, 12.1, and 12.2, we have Jy( u) K(cr)\H(o’) and y (0, 0o). Proof. Fix y(O, 0o), K(cr)\H(cr) and define stopping times 0o, for every , r, [0, T]; T^ inf (x)ll = dx for n 1,2,.... With L as in Standing Assumption 2.1, it follows from Jensen’s inequality, the supermartingale property of Z, and the decrease of U, that y( u) EO(yfl E[ E { (J(yfl T)Z( T))I o%.} T)Z,( T) >= E[ l](yeLE{Z( T)] @.})] -> El)(yeLZ.(%,)) >-- yexp L-I E (l10(s)l12+ II (s)ll 2) ds [3 for every n => 1. The conclusion follows by letting n-* 0o. Proof of Theorem 12.3. Fix y (0, 0o). The convex functional .y(. of (12.6) is lower-semicontinuous in the strong topology of H(cr), by Fatou’s Lemma. Therefore, it is also lower-semicontinuous in the weak topology (Ekeland and Temam [5, Cor. 2.2, Chap. 1]). Thanks to the coercivity property (12.8) of Lemma 12.7, Jy(.) attains its infimum over H(r) at some /y H(r) [5, Prop. 1.2, Chap. 2]. In light of Lemma 12.8 and Assumption 11.2, (12.11) inf yK(o’) J(u)= Jy(A) < 0o. It remains to show that Aye Kl(r). From the decrease of U and/, (4.4) and (4.8)’, we obtain for some a (0, 1), 3’ e (1, 0o)" J()- J(0o) >= J()- J I(u) du >= 1 I (12.12) ->_ f.t() Replacing vf (o, ). c by y(T)Zy(T) in (12.12) and taking expectations, we obtain ay Ay (y) E[fl( T)Z,xy( T)I(y( T)Z,( T))]<= y(1-a) lEO(y/3 (T)Zy( T))- (0o)] < 0o, thanks to (12.11), and thus hy Kl(r) by Remark 9.1. 13. Appendix. We provide in this section an example with a well-behaved utility function U for which we do not have ,(y) < 0o for all y (0, 0o) and u K(r). The existence of such an example necessitates the introduction of the set Kl(r) in 9. In the setting of 2, take re=l, d--2, o-( t) -= (0,1), r(t)=0, b(t)=0, T=I, B= W, and define the stopping time z&inf{t6 [0, 1]; t+B2(t) 1} and the process -2B(t) (13.1) q(t)& (l-t) 2 0; UTILITY MAXIMIZATION IN AN INCOMPLETE MARKET u(t)&(q(t), 0)*. For the utility function U(x)=2x/- 729 we have l(y)=y -2, and (7.6), (7.7) give (13.2) Z(t)=exp (13.3) (y)=y- exp q(s) dB [ q(s) ds (s)-- (s) dB (s)+ s) ds It is shown in Liptser and Shiryaev [15, p. 224] (see also Karatzas and Shreve [14, p. 201]) that the process Z of (13.2) is not a maingale; in fact, the construction (13.1) is made with this propey in mind. This implies, in paicular, that E[exp{f2(s) ds}]=; (13.4) for otherwise Z would be a maingale, by Novikov’s theorem (Karatzas and Shreve [14, p. 199]). According to Liptser and Shiryaev [14, p. 225]: Io (a(- Io (s Io [( -4 ( -3](a, whence ( e exp 2 [(1- t)-4+(1- t)-3]B(t) dt If this last expectation were finite, then so would be [exp { ] P(s) ds}] =[exp {2 (1-t)-4B(t) dt}, contradicting (13.4). eleget. We are grateful to an anonymous referee for carefully reading this paper and offering helpful advice on exposition. Nte e i rf. 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