OPTIMAL CONSUMPTION AND PORTFOLIO POLICIES WHEN ASSET PRICES FOLLOW A DIFFUSION PROCESS

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1 OPTIMAL CONSUMPTION AND PORTFOLIO POLICIES WHEN ASSET PRICES FOLLOW A DIFFUSION PROCESS John C. Cox and Chi-fu Huang Massachusetts Institute of Technology WP# May 1986 Revised, January 1987 I

2 Table of Contents 1 Introduction The General Case The Formulation Main Results Relation to dynamic Programming The Relationship Between the Constrained an(l the Ilnconstrained Solutions A Special Case Formulation Explicit Formulas for Optimal Consumption and Portfolio Policies Concluding Remarks i

3 1. Introduction Optimal intertemporal consumption and portfolio policies in continlous time under uncertainty have traditionally been characterized by stochastic dynalmic programming. Merton (1971) is the pioneering paper in this regard. To show the existence of a solution to the consnlption-portfolio problem using dynamic programming, there are two approaches. The first is through application of the existence theorems in the theory of stochastic control. These existence theorems often require an admissible control to take its values in a compact set. However, if we are modeling a frictionless financial market, any compactness assumption on the values of controls is arbitrary and unsatisfactory. Moreover, many of the results are limited to cases where the controls affect only the drift term of the controlled processes. This, unfortunately, rules ot the portfolio problem under consideration. The second approach is through construction: construct a control. usually by solving a nonlinear partial differential equation, and then use the verification tllorem in dynamic programming to verify that it indeed is a solution. Merton's paper uses this second app)roach. It is in general very difficult, however, to construct a solution. Moreover, when there are constraints on controls, such as the nonnegative constraints on consumption and on the, wealth, this approach becomes even more difficult. R.ecently, a martingale representation technology has been used in place of the theory of stochastic control to show the existence of optimal consumptionll anld ortfolio policies without the requirement of compactness of the values of admissible controls; see Cox and Huang (1986) and Pliska (1986). Notably, Cox and Huang show that, for a quite general class of utility fmctions, it suffices to check, for the existence of optimal controls, whether the sufficient. conditions for the existence and uniqueness of a system of stochastic differential e(uations. (lerived completely from the price system, are satisfied. The sufficient conditions for te,xistence and the uniqueness of a solution to a system of stochastic differential equations lhave been well studied. The focus of this paper is on explicit construction of optimal controls while taking into account the nonnegativity constraints on consumptionl and on finlal wealth by using a martingale technique. We provide two characterization theoremls of optimal pilicies (Theorems 2.1 and 2.2) and a verification theorem (Theorem 2.3), which is a counterpart of tlle verification theorem in dynamic programming. One advantage of our approach is that we need only to solve a linear partial differential equation in constructing solutions unlike a nonlinear p)artial differential equation in the case of dynamic programming. In many specific situations, optimal controls can even be directly computed without solving any partial differential equation. The rest of this paper is organized as follows. Section 2 contains our general theory. We formulate a dynamic consumnption-portfolio problem for an agent in continuous time with general diffusion price processes in Section 2.1. The agent's )prob)leml is to dynamically manage a portfolio 1 I

4 II of securities and withdraw fnds out of it in order to imaximize his e(xpected ultility of consumption over time and of the final wealth, while facing onnegativity constraints on consumption as well as on final wealth. Section 2.2 contains the main results of Section 2. In Theorems 2.1 and 2.2, we give characterizations of optimal consumption and portfolio policies. We show in Theorem 2.3 how candidates for optimal policies can be constructed by solving a linear partial differential equation. We also show ways to verify that a candidate is indeed optimal. The relationship between our approach and dynamic programming is discussed in Section 2.3. We also demonstrate the connection between a solution with nonnegativity constraints and a solution without the constraints. In Section 3, we specialize the general model of Section 2 to a model considered originally by Merton (1971). Risky securities price processes follow a geometric Brownian motion. In this case, optimal consumption and portfolio policies can be computed directly without solving any partial differential equation. Several examples of utility fnctions are considered. In particular, we solve the consumption and portfolio problem for the family of HARA utility functions. In the unconstrained case given in Merton (1971), the optimal policies for HARA utility functions are linear in wealth; when nonnegativity constraints are included, this is no longer true. We also obtain some characterization of optimal policies that are of independent interest. Section 4 contains the concluding remarks. 2

5 2. The General Case In this section, a model of securities markets in continuous time with diffusion price processes will be formulated. We will consider t.he optimal consumption-portfolio policies of an agent. The agent's problem is to dynamically manage a portfolio of securities and withdraw ftmds out of the portfolio in order to maximize his expected t.ility of consumption over time and of final wealth, while facing a nonlegativity constraint on consumption and on final wealth. The connection between our approach and dynamic programming will e demonstrated and the advantages of our approach will be pointed out. We will also discuss the relationship between a solution to the agent's problem with the constraint and a solution to his problem witholut the constraint The Formulation Taken as primitive is a complete probability space (1,. P) and a time span 10. T], where T is a strictly positive real number. Let there be an N dimensional stan(lard Brownian mlotion defined on the probability space, denoted by w = {w,, (t); t E [0. T]. n = N}. Let F = {; t E [0, T]} be the filtration generated by w. (A filtration is all increasing family of slub sigma-fields of jr) We assume that F is complete in that jr contains all the P null sets and that jrt = r. Since for an N dimensional standard Brownian motion. w(0) = 0( n.s.. r is almost trivial. We use to denote the F-optional sigmla-field and v, to denote the product measure on Q x [0, T] generated by P and the Lebesgue measure. (The F optional sigma--field is the sigmafield cn Q x [0, T] generated by F- adapted right-continuous processes: see. e.g., Chung and Williams (1983).) The consumption space for an agent is L 2 (v) X L 2 (P)- L 2 ( x [0, T. T 0.,,). x L 2 (Q. Y. P). where L 2 (z,) is the space of consumption rate processes and L 2 (P) is the space of final wealth. Note that all element of L 2 (/,) are F-adapted processes (see.... (Chu; and Williams (1983)). All the processes to appear will be adapted to F. Consider frictionless securities markets with N + I long-lived secllrities traded, indexed by n = 0, 1, 2,..., N. Security n g 0 is risky an(l pays dividendsll at time t. We will henceforth use S(t) to (lenote (Sl(t)... Sx, (t))t. at rate,,(t) and sells for S,,(t), Assume that t,,(t) can be written as,,(s(t),t) with L,,(, t): ' x [X T] - R Borel measural)le. Security 0 is (locally) riskless, pays no dividends, and sells for B(t) = exp{f r(s)dl.} at time t. r(t) = r(s(t), t) with r(x, t) : R N trace. x [0, T] --4 R+ continuous. Assume further that We shall use the following notation: If ct is a matrix. thell 1t 2 denotes tr(rat r ) and tr denotes The price process for risky securities S follows a /o /0 ' It? l)process satisfying S(t) +, ((s).), d = s(0)+ (S(.,),)d+ T (s(.), )d t,(). T], q, (2.1) 3

6 III where is an N-vector of,,s, (, t): RN x [0, T] - RN and (.. t): RN x [0, TJ] - :NxN are continuous in z and t, and a(s, t) is nonsingular for all z. and t. Defining S*(t) S(t)/B(t) and L*(t) t- (t)/b(t), Ito6s lemma implies that S*(t) +,l*(s)ds = S'(0) + 1 B(t ls(s),) - r(s()..)s()]ldt f1 a(s(.s), ) S*(O) + J *(S(s) B(.s). )d. (2.2) + B,'(S(.).B(.).)d(.,) Vt E [0.T1. a.s. Next we define and ( (t) S= (t) +,(t) = i,(. ),t, S*(t) + I' (.q),.l: the former is the N vector of gains proccsses and the latter is the N vector of gains processes in units of the 0-th security, both for the risky securities. A martingale measure is a probability measure Q equivalent to P such that E[(dQ/dP)2] < oo and under which G*(t) is a martingale. We assume that there exists a unique martingale measure, denoted by Q. The existence of a martingale measure ensules that there be no arbitrage opportunities for simple strategies; 1 see Harrison and Kreps (1979). It follows also from Harrison and Kreps (1979. Theorem 3) that where dq/dp = exp {j K(S(.), q)t d w () - 1 I.((.s)..)1d2 }.,(S(t), t) -- (S(t) t)- ( (s(t).t) - r (S(t). t)s(t)). (2.3) For future use, we define a square-integrable martingale mnider P 71(t) = E[dQ/dPI,].. = exp (S(,), )T w() -.1 'K(S() )I2.} (2.4) We will use E*[.] to denote the expectation under Q. The following lemma will be useful later. IA strategy is said to be simple if it. is bounded and changes its valuhes at a finite numli,er of nonstochastic time points. Formally, (o, O) is a simple trading strateg- if there exist. t.ime points n = n < fi <... < IN = 1 and hounded random variables 3', Yjn t1 = 0,... N -1 and j = 1,...,Af, suchll that,, and y,, are meauralde with respect. to,, and (l a() = r,, and tj(t) = Yjn if I E [f_,,,+ 1). 4

7 Lemma 2.1. Under Q, w'(t) - (t) - (S(.R)., )(., is a Standard Brownian motion and G*(t) = S'(O) + j q((.,),. )/B(.),,(.) a..., a square-integrable martingale. Proof. The first assertion follows from the (irsanov Theorem: see. e.g., Liptser and Shiryayev (1977, Chapter 6). The second assertion follows from substitution of wt into (2.2). I Remark 2.1. Since P and Q are equivalent and thus have th sae probal)ility zero sets, the almost srely statements above and henceforth will be with resp)ect to ither. I Remark 2.2. For sufficient conditions for the existence ad luniquenlless of a martingale measure see, for example, Proposition 3.1 and Theorem 3.1 of Cox and Hiuang (1986). A trading strategy is an N + -vector process (. 0) = {a(t). 0,,(t): n = 1, 2..., N}, where a(t) and,,(t) are the numlers of shares of the 0-th security and the n th seculrity, respectively, held at time t. A trading strategy (a, 0) is admissible if E [f O(t)Trr(t)r 7 (t)to(t)dt < o. (2.5) if the stochastic integral J (O(")Tdl( ) +(.)B(. is well-defined, and if there exists a consulmption plan (. W4) E L 2 () X L 2 (P) such that a(t)b(t) + O(t)T(t) + c((,)d. (2.6) *= (O)B() s ) ( + () ((,)(.) + (q)t(lr( )) Vt (L.A., and W = a(t)b(t) + O(T)T S(T)..q. (2.7) The consumption plan (c. W) of (2.6) and (2.7) is said to be finaznced ly (. f) and the qladruple (a.. c, W) is said to be a self-financing strategy. Let. H denote tll space of self financing strategies. By the linearity of the stochastic integral andl the Caiuchy Scllwartt inequllality, it is easily verified that H is a linear space. We record a well knownl fact about a self financilng strategy and a mathem at ical result: 5

8 III Lemma 2.2. Let (a,, c, W) E H. For all t E [0(), T]. E* [Ijc*(s)ds +WI] =a(o) + o(o)ts ' (o) + J (.) (T () =a(t) + (t)ts*(t) +, (.)(, (... where W* - W/B(T) and c*(t) - c(t)/b(t), the nornmalized final wealth and time t consumption. Hence the value of (c, W) at time t is a(t)b(t) + O(t)TS(t) = B(t)E*tW + / (.s)d.,],. Moreover, fo T on (,, Q). c* ()ds + W* is an element of L 2 (Q), the spare of s(lqare integrable random variables Proof. See, e.g., Cox and Huang (1986, Proposition 3.2). Lemma 2.3. Let g E L 2 (v). Then E* [ g(,)dslt] = E [/ g(.,)r()diy n(/t), i. Proof. See Dellacherie and Meyer (1982, VI.57). I Consider an agent with a time-additive utility fuinltion fr n + x [0, T] oonsumptio R U{-oo} and a utility fiunction V: R+! RU{-oo} for fial wealthll. His I)rol)lem is to choose a self-financing strategy to maximize his expected utility: Sil) E u((t). t),tt + V(i z )]. T T(2.8) (fio,;-h s.t. (0)B(0) + 0(0O) s(0) < W()). 6 > 0 v - a.e. and W > () a..r.. where W(O) > 0 is the agent's initial wealth. Note that the utility finrctions are allowed to take the value -oo. We assume that ul(y,t) and V(y) are continuous. increasing,. and strictly concave in y. For fiture reference, we cite several properties of a concave utility finction. At every interior point of the domain of a concave fiunction, the right- han(l derivative and the left ha(lnd derivative exist. At. G

9 the left boundary of its domain, thie right-hand derivative exists. The right -hand derivatives and the left-hand derivatives are decreasing functions and are equal to eachll other except possibly at most a countable number of points. That is, a concave fclotion is differentiable except possibly at most a countalble number of points and thus continuously (lifferentiable except Ipossibly at most a countable number of points. (Note that for strictly concave flnctions, the relation above such as decreasing becomes a strict relation.) The right--hand derivative is a right continuous function. Moreover, at every point, the left-hand derivative is greater than the right-hand derivative. Now let uy+(y, t), V(y) and u_(y, t), V' (y) denote tlle right hand derivatives and left-hand derivatives, respectively, for u(y,t) with respect to and( for V(y). Let y < y', then V'(y) > V_+(y) > V' (y'); and similarly for u(y, t). We assume that lim u+(y, t) = n and liin V+(y) = 0. y-oce Define inverse functions (z-l,t) = inf{y E + : uy+(yt) < -1 ) and V-l(-1) = inf{y E R+: V.1(y) < -')}. By the right-continuity.of the right hanld df erivatives. the infilia are equal to minima. Remark 2.3. Note that the assumption that. utility fnctions are increasing and strictly concave implies that they are strictly increasing. I Now we will state two assumptions that will b)e sed in te next subsection. First define a process Z(t) = Z(0) + j(r(s()),s) + jr(s(.), ))Z()d.- K(S(.),q)TZ(.()(,) n.. (2.9) for some constant Z(0) > 0. Using It6's lemmlla, it is easily verified that Z(t) = Z(O)B(t)/l(t) al... (2.10) Note that (log Z(T) - log Z(O))/T is thile realized continuously comnlp)olmled growth rate from time 0 to time T of the growth-optimal portfolio the )portfolio that maximizes the expected continuously compounded growth rate. We adopt the following notation: ad Y 9" l... ;1 1n = Ml N Y. yni"v 7

10 11 for positive integers ml, m2,.. r. m. If g RN X T -4 R has lpartial derivatives with respect to its first N arguments, the vector (g/ay,..., Ag/tyN9) T is denoted y Dyg or gy. Assumption 2.1. Write (2.1) and (2.9) compactly as follows: ( (t) = ((o) z(t) =k Z(O) )+ S(s), Z(s), ) ds + J (S(.) Z(s). q.)d)(,) Vt E [, T], a.s. Suppose that there exist strictly positive constants K 1. K 2. K 3. and - such that for all t V (z, t) < K 1 (1 + x[), t) < Kl(1 + Hz{) Vx E ~N+1 I'(x, t) -(y, t)l < K 2 - Yl, la(, t) - (y t)l < K2 :I- 1J V,,l e N + (. 1) that D" (y, t) and D'" (y, t) exist fo rn. = 1, 2. a(l ar rnfltitllllls i? anl t. n.n(l -- y Y, t) anay -- nd that ID''(y, t) + DW{a(y, t)i < K 3 (1 + {y{) (2.12) for all y E RN+1. Assumption 2.2. f(x,t) z-if(z- 1,t) and g(z) - -V (z - ') are sch that for m < 2, D"r'f(, t) and D"g(x) are continuous, and for some strictly positive constants K andl. ID?'f(x, t)l < K(1 + ' ), ID"'g(x)l < K(1 + xl'). The purpose of (2.11) is to guarantee the existence andl the niqullleness of a solution of (2.1) and (2.9). The purpose of (2.12) and Assumption 2.2 is to einsure that certain fiunctionals of S and Z have two continuous derivatives. To have a feel of the restrictiveness of Assumption 2.2, we note that if V(y) = Lz l - h, then b < 1/3. In manly specific situations, differentiability will obtain under much weaker conditions. For example, in Section 3 of this paper, we consider a special case of the general model developed here. All the HARA differentiability conditions Main Results tility functions give rise to the desired We will give explicit characterizations of an optimal consumption portfolio policy under the following assumption that the optimal consumption plan is also a sollution to a corresponding static 8

11 maximization problem. Assumption 2.3 is valid under quite nlil(l regilarity conditions, to which we refer readers to Sections 2-4 of Cox and Hlang [1986]. Assumption 2.3. There exists a solution to (2.8), dented y (. a solution to sup (,Ti)EL+(v)xL 2 (P) E [ u(, t),lt + V(W) s.t. E [f (t)(t)ib(t)dt + Wr(T)/IB(T)] < W(o), where L2(v,) denotes the positive orthant of L 2 (v) and likewise for L2 (P). 0, c. W). if and only if (c, W) is (2.13) Remark 2.4. The idea behind Assumption 2.3 is as follows. The assumnption that the martingale measure is unique together with the square- integrale restriction on the trading strategies of (2.5) implies that any element of L 2 (v') x L2(Q) is attainable 1)y all a(llnissil)le trading strategy, or is marketed. where v* is the product measure generated l)y Q and the Lebesgul measure. As long as the solution to (2.13) lies in L 2 (v ' ) x L2(Q), Assumption 2.3 will be valid. I this case, any consumption plan that is a candidate for the optimal solution for sone initial wealth is marketed and thus markets canll be said to l)e dynamically complete. It follows from the Lagrangian theory (see, e.g., Holmes (1975) and R.ockafellar (1975)) that if (c, W) is a solution to (2.13), there exists a strictly positive real number A such that J < A(w, t)/b(w, t) < u,_(c(w, t). t) for v a.e. (w, t) sch tllat c(w, t) > 0, (c(w t)t) < Ari(w, t)/b(w, t) for I, a.. (w. t) such tllat c(w, t) = 0; (TV < A (w, T)/B(w, T) < V' (W(w)) fo rp a.e w sch that W(w) > 0, ±+( )) '.< A (wt) /B(wT) fo r P a. W slch that W(w) = 0. Now let {Z(t); t E [, T]} be defined as in (2.9) the above first order conditions become with the initial condition Z(O) = 1/A. (2.14) Then J< Z(w,t)-' < U,_(C(wt), t).,+(c(wt),t) - < z(w t) - V+' ( M) < Z(w, T)- < V' (w)) v(w()) < Z(w, T)-' (()) where we have used (2.9). Thus we have for a.e. (w. t) sch that c(w, t) > 0, for vr a.e. (w. t) sllch that c(w, t) = 0; for Pr a. w siuch that W(w) > 0. for P a. w such that W(w) = 0. ' (2.15) c(t) = (z(t)-'. t), -.C. W = V+-'(Z(T)-' 1 ).,,. (2.16) Here is our first main result: 9

12 Theorem 2.1. Suppose that Assumptions 2.1 and 2.2 ar( satisfied and that there exists a solution to (2.13). Let Z(O) = 1/A and let the finction F he definedr by F(Z(t) S(t), t) = Z(t)E [ Z(.)- (Z(s)- s)ds + V+- 1 (Z(T)-)Z(T)-Z(t),S(t ), where we have fixed a right-continuous version of the onditional expectation. Then D' F(y, t) and F,(y, t) exist and are continuous for m < 2, and together with F satisfy the following linear partial differential equation with a boundary condition: F + if = FzZT -F rrr + rf - (2.17) F(Z, S,T)= V-'(Z-'). (2.18) where C is the differential generator of (Z, S). The otimal portfoli, p)tlicy is 0(t) = [Fs(Z(t), S(t), t) + ((S(t), t)'(s (t), t)t ) -1 ( (S(t),t)-r(S(t),)St)))Z(t)Fz(Zt)). (t),t)]. (2.19) a(t) =(F(Z(t), S(t), t) - (t) T S(t))/B(t) - a.r.. and the optimal consumption plan is specified in (2.16). Proof. By Lemma 2.2, the value of (c, W) at time t is B(t)E* [ f(z(s)-l,.)/b(,)ds + V'- (Z(T)-)/B(T) ] =t7(t)-b(t)e [f T f(z(, )r 1 (s)/b()d. + V+ '(Z(T)')n(T)/B(T)Y] =Z(t)E Z(s)' f(z(s)-l s)d + V 1-(Z(T)-')Z(T)- Z(t).S( )] where the first equality follows from Lemma 2.3 an tlh second e(iquality follows from (2.9) and the fact that under Assumption 2.1, (Z, S) is a diffusion process and thlllls )ossesses the Strong Markov property. Putting F(Z(t), S(t), t) E Z()- 1 f(z(s)-',.s)d.s + V-'(Z(Z(T))Z(T)- IZ(t), (t)] Asslmiption 2.2 and a multidimensional version of Remark 11.3 of (iihmlan and Skorohod (1972, p.77) shows that D' F(, t) and Ff(y, t) exist and continuous for m < 2, and F + P, + z-'f = o. 10

13 where is the differential generator of (Z, S). In addition. F(Z(T). S(T),T) = WZ(T) - ' a.s. This implies that Cl + F, = FZT.- rstk. + rf -. and F(z,, T) = V-'(Z-'). which are (2.17) and (2.18). It.'s formula implies that F(Z(t), S(t), t)/b(t) + = F(Z(0), S(0). 0) + j(cf(s) + F(.))/B(.)d. + I jz(r)-lld)/b(d)nff IF(Z(,). s).) (() )d() rs (Z (). (). ) T, (,5(\$), q)dw(,r) - FZ(Z(.,), S(.)..)Z(.)K.(Z(,,), S(,).,q)T/B(s)d(q) - / r -(.)F(Z(.). S(s)., )/B(.,q)d,. o'.{ = F(Z(O), S(O), O) + ' Fs(Z(.,), (.,),.)Td (. ) - Fz(Z(s),S(s),,)Z(,.),(Z(,). S(.). q)t(z(q) S(,q).,)-' (:'(.), (2.20) where the second equality follows from (2.17) and Lemmall 2.1. Evaluating (2.20) at t = T, we get W/B(T) + f(z( ()-', )/B(.s)dR = i(z(0). S (0). ()) + 1 (Fs(Z(s), S(q), q,)t + Fz(Z(s), s(),)z )( (.), (Z(. S ()..(. T (Z(s(Z(). (.),) - 1) dg*(s). From Lemma 2.2 we know W/B(T) + ft =F(Z(0), (0). 0) T (2.21) (2.22) Lemma 2.2 also says that the left-hand- side of (2.22) lies ill L 2 (Q). We can thus subtract (2.22) from (2.21), and then use Jacod (1979, Proposition 2.48) tlo prod(uct: E [/T -(F + E' (0i(t) - (s + (-rt)-i(t By the hypothesis, f ((.) - '. -,) / B(--) d. take expectation und(er Q of its inner - rs)zfz))t.*a*tr(o(t) - (Fs + (T )-( - rs)zfz))dt] = 0. *'a* T is positive definitive. Therefore. w must have '.t (t) =[Fs(Z(t), S(t), t) + (T(S(t).t)r(S(t). t)t) - ( (S(t),t)- r(s(t), t)s(t))z(t)fz(z(t). S(t). t)]. V - (t.e. 11

14 The rest of the assertion is easy. I Theorem 2.1 makes enough assumptions to ensure that F Ias desired derivatives and (2.17) and (2.18) are satisfied. Sometimes, we call explicitly compllte the function F and it may have the property that D"F(ly, t) and F exist and are continuous for m < 2. The theorem below gives a way to verify the existence of a solution and to compute optimal consmlinltion and )ortfolio policies. Theorem 2.2. Suppose that F defined in Theorem 2.1 is schll that D" F(y, t) and F(y, t) exist and are continuous for m < 2. Suppose further that there exists Z sch that F(Zo, S(0), O) = W(O) and that (2.16) lies in L 2 (v) x L 2 (P) when Z(O) = Zo. Then there exists a solution to (2.8) with optimal policies specified in (2.16) and (2.19). Moreover. F satisfies (2.17) and (2.18). Proof. Define Z by (2.9) with Z(0) = Z. Then F(Zo, S(0). 0) is the present value of (2.16) and by hypothesis is equal to W(O). Also, (2.16) lies in L 2 (i,) x L 2 (P). Thus the first order conditions for (2.13) are satisfied. Since (2.13) is a concave program., the first order conditions are sufficient. Hence there exists a solution to (2.13) and, by Assumnption 2.3. to (2.8). Since D"yF(y, t) and F(y, t) exist and are continuous for m. < 2. It6's lenlna implies that F(Z(t), S(t), t)/b(t) + j(z()', s)/b(s)d =F(Z(O), S(O), 0) + J Fs(Z(s) S(.), )Tq(S( 5 ) s)/b()uw(s) - ' Fz(Z(s), S(8), s)z(s)k (Z(S), S(s), q)t/b(,)du)*(q), + (CF(s) + F,(s) - Fz(\$)Z(.s)K(,) T K (.) + F (. )T()K(.q) - r(.)f(,q) + f(s)) /B(s)ds. By Lemma 2.2, the left-hand side of the above relatioi is a squiare integral)e martingale under Q. Now note that the integrantis of t.he two stochastic integrals on the right hand side are continuous functions of Z(t) and S(t) and thus are olunded on,bomlded sets. Let T, = inf{t E [0, T l Js(s)Trr(.q) - Fz()r.(.,)TI 2 / B 2 (s) > n}. It is clear that T, - T a.s. On the stochastic interval [0. T,,], the two stochastic integrals on the right-side of the above equation are square--integrable martingales under Q. Hence the integrand of the Lebesgue integral on the right-hand side must be zero on [0. T,,]. since any martingale having continuous and bounded variation paths must be zero or be a constant; cf. Fisk (1965). Since T, - T with probability one, we have (2.17). Finally. (2.18) follows directly from the definition of F. I 12

15 When F can not be explicitly computed, we call tilize the following theorem, which is a counterpart of the verification theorem in dynamiic programniling. Theorem 2.3. Let u(y, t) and V(y) be such that 1 -?f(x- 1,t) + -V4+-7( -')l < K(1 + IzI)~ for some K and y and let (2.11) be satisfied. Suppose F : R+1 x [0., T] - R with IF(y,t)l < K(1 + Iyl)- for some constants K and -y, is a solution to the partial differential qluation of (2.17) with a boundary condition (2.18). Suppose also that there exists Zo > () sulrlh that F(Zo. S(O), 0) = W(0) and that ({f(z(t)-'.t)}, Vl-'(Z(T)-')) E L 2 (V) x L 2 (P) for Z with Z(O) = Zo. Then there exists a solution to (2.8) with optimal policies descrihed in (2.16) and (2.19) an(l with Z(O) = Zo. Proof. First we note that (2.11) implies that for all positive integers m. there exist constants L,, such that E[Z(t)j 2 "'] < (1 + IZ(0) 2 '" ) exp{l,,,t): see, e.g., Theorem of Friedman (1975). Therefore. E [ Z(t)- 1 (Z(Zt)-.t)dt < oo. for every Z(O) > 0. This, (2.17), (2.18). and Theorem V.5.2 f Fleming and Rishel (1975) then imply that F(Z(O) S(0), O) = Z(O)-1E [f Z(t)-'f(Z(t)-. t)dt + -l,(z(t)-) ] E' [J f(z(t)-.t)/b(t)dt + V -(Z(T))/(T) In particular, we can take Z(O) = Zo in the above relation. This is simply tlhe value at time zero of ({f(z(t)-',t)}, V-1(Z(T)-)). which lies in L 2 (.) x L 2 (P) by hypothsis. Also by the hypothesis that there exists Zo such that F(Zo, S(O)O) ) = W(0). Hence ({f(z(t)-. t)}. V+-l(Z(T)- 1 )) satisfies the first order condition for an optimlmn for the program (2.13). Therefore. it is a solution to (2.13). The rest of the assertion then follows from Assimptio l 2.3 and Theorem 2.2. I Unlike the verification theorem in dynamic programming. the verification procedure in Theorem 2.3 involves a linear partial differential equation. 13

16 III For the rest of this section, we will assume that t.llere exists a solution to (2.13) and that Assumptions 2.1 and 2.2 are satisfied. We use {W(t); t C [0, T]} to denote the process of the optimally invested wealth: W(t) = F(Z(t), S(t). t). It is clear that W(T) = W a.s.. The following prol)ositioll shows that after the optimally invested wealth reaches zero, the optimal consumption and portfolio policies are zeros. The following notation will be needed. Define an optional time T = inf{t E [0, T) : W(t) < 0}, the first time the optimally invested wealth reaches zero. As a convention, wllell the infimnlul doles not exist. it is set to be T. Proposition 2.1. On the stochastic interval [T, T], O(Z(t), S(t), t) = 0,, -,.e. a(z(t), S(t). t) =,, - a.r. c(t) = I/ - a.e. W=0 a..s. Proof. From the definition of F, it is clear that it is equlal to zero at Tif and only if on [T, T] the optimal consumption and final wealth are zeros. Argmll1ents similar to the last half of the proof of Theorem 2.1 starting from (2.21) prove the rest of the assertion. I Note that if we consider the agent's probllem in the context of the theory of stochastic control, given the set up of the securities markets, we would like the optimal cntrols sch as (a, 0, c, W) to be feedback controls. That is, the optimal controls at eachll time t dependl(ll only nipon time t, the values of S(t), and the agent's optimally invested wealth at that timlle. In the above theorem, the optimal controls are functions of S(t), Z(t), and't. However, Z is (letermilled in I)art by the agent's initial wealth through the initial condition Z(0) = 1/A. The following proposition shows that given S(t) and t, the agent's optimally invested wealth at tilme t is all inverti)le filnction of Z if u(y, t) and V(y) are differentiable in. Hence, the optimal controls are indeed feedlack controls. Proposition 2.2. Fz >. Suppose that u(y, t) andl V(yI) are differentiable in. Then Fz > 0 if F > O. Thuls there exists a finction F-l(W(t). S(t). t) = Z(t) if W(t) > (). I addition. FlT TV, 14

17 Jl', F s Fsr, and F7' l exist and are ontinlouls. We ranl writ. _(Z(t), S(t), t) = O-(F (W(t), S(t), t). \$(t), t), -.e. 1.0~~~~~ if W(t) ~ifw(t) > 0 = 0: a(z(t), S(t),t)= O(F (W(t), (t), t).(t),t) r- a.c. if W(t) > 0: (t) f(l/f-(w (t) St), t), t) 0 W=W (T) a..., - t.. ifw(t) > :; if W(t) = O; Proof. It follows from Friedman (1975, Theorem 5.5.5) and thll( fact that az(.,)/iz(t) if s > t we have Z(.s)/Z(t) Fz(Z(t), (t), t) = -E f d + ( (T Z(t)() Z 2 (s) Z 2 (T) ( where f' denotes the derivative of f with respect to its first arpnient. andl where V+.-" denotes the derivative of V Thus Fz O0, since f(y, t) and V+-l(y) are d(creasing in y. Note that if u(y, t) is differentiable in y then is strictly decreasing wheln (y, t) > 0: anl similarly V - ' is strictly decreasing if V is differentiable and if V-l(y) >. If Fz =, it must be that F(Z(t), S(t), t) = and Proposition 2.1 gives the optimal consumption and p)ortfolio policies. If F(Z(t), S(t). t) > 0 then Fz(Z(t) S(t),t) > 0. Therefore, given S(t) and t. Z(t) is an invertiblle fiuction of W(t) if W(t) > 0. Let this function be denoted by F-I(W(t), S(t) t). The differential)ility of F-' follows fiom the implicit function theorem; see. e.g.. Hestenes (1975. follows from Theorem 2.1 and substitution..172). The rest of the assertion then Remark 2.5. For Fz > 0 when F > 0. it is certainly not ncessary that?l(y, t) and V(y) be differentiable in y. In the special case of our current general mlol(del to ) (l dealt with many in Section utility 3, functions that are concave and nonlinear yieldrs F > ) for F > 0. When utility functions have a finite marginal may involve zero consumption. optimal consumption is zero. I tility at zero. the otiallal consumption policy The following proposition irdentlifies the circumstances in which Proposition 2.3. Suppose that u+(o. t) < oo. C(onslim)tionl at tine t is zero only if W(t) < F(u,+(O, t), S(t), t). Sppose in addition that (y. t) anld V(y) are differentiable in y. optimal policy has the property that consumption will he zrr stochastic boundary F(u,(O, t)-', S(t). t). Then an if and only if wealth is less than a 15

18 11 Proof. From (2.15) we know that c(t) < 0 if and only if tu,(0. t) < Z(t)-'. It then follows from Proposition 2.2 that u,(o,t) < Z(t) - 1 only if F(uc(O,t)- 1,S(t),t) > F(Z(t), S(t) t) = W(t). This is the first assertion. Next suppose that both u(y, t) and V(y) are differentiable. We want to show that if W(t) < F(u,(, t)-, S(t), t), then c(t) = 0. We take two cases. Case 1: W(t) = 0. Then Proposition 2.1 shows that c(t) = 0. Case 2: W(t) > 0. Proposition 2.1 also shows that when W(t) > 0, Fz > 0. Thus u(o,t) < Z(t) - ' if and only if F(u,(O, t)-', S(t), t) > F(Z(t). S(t), t) = W(t). By inspection of (2.19) and (2.23), we easily see that, when Fz > 0, the feedback controls are differentiable functions of W(t) and S(t). In particular., the oltimal conlsmltion policy is twice continuously differentiable in W(t) and S(t), which follows directly from the assumption that f(y, t) is two times continuously differentiable with respect to (see Assumnltion 2.2). The following proposition gives a complete characterization of utility functions such that j(y, t) is twice contimnuously differentiable with respect to y, given that (y, t) is d(ifferentiable in y. Proposition 2.4. Sppose that u(y, t) is differential, e with respect to y. Df(y, t) exists and is continuous for m < 2 if and only if D.t u(y, t) exists and is continuous for m < 3, and for uy+(o, t) < 00oo, and Similar conclusions also hold for V(y). limra u(yt) _ 0: (2.24) hlm - Y.- = 0. (2.25) 0 Utlyyy(,t)(ly(t) )(_ (2.25) Proof. On the interval (0, uy(o, t)), uy is continuous and strictly ldecreasing. Hence D'"f(y, t) exists and is continuous for m < 2 if and only if D"'L(y, t) exists and is continlls for m < 3. When uy(0. t) < oo, on (uy(o, t), oo), D' j(, f t) is equal to zero for m < 2. This implies (2.24) and (2.25). The proof for V(./) is identical. I 2.3. Relation to dynamic Programming Traditionally, the agent's optimal consumption portfolio policy is compute(d by stochastic dynamic programming; see, e.g., Merton (1971). We will demonstr'ate the connection between our approach and the stochastic dynamic programming. 16

19 The usual formulation of the consumption--portfolio pirolblem uises a consumption policy and a vector of dollar amounts invested in risky assets to 1)e the controls. The former is denoted by c(w(t), S(t), t) and the latter will be called an investmeat policy and,)e denloted bly A(W(t), S(t), t). Given a pair of controls (c, A), dynamic behavior of the wealth is W(t) = W(0) + (W(.s)r(s) - c(.s) + A(.,)Is-,(t) ( (.) - r()s(.)))ds + A(s)Is- (s)r(i)dwi(s) Vt E [0. T where Is- (t) is a diagonal matrix with diagonal elements S,,(t)-'. Define J(W(t),S(t),t) = sup E u(c(.s)..)ds + V(W(T)))IW(t). S(t)] subject to the constraints that the wealth follows the aove dynamics, that consumption cannot be negative, and that.j(o, S(t), t) = J (0. )ds + v(0). (2.26) The last constraint is basically a nonnegative wealth constraint that rles out arbitrage opportunities. The existence of a pair of optimal controls is a nontrivial plroblem. We will refer readers to, for example, Krvlov (1980) for an extensive treatment using the theory of stochla.-tic controls. For a much easier approach specific to the consutmption--portfolio problem. we refer readers to Cox and Huang (1986) and the references given there. We assume that there exists a pair of optimal controls (c. A) and that.1 hlas two continuous derivatives with respect to its first two arguments ad a ontinuous derivative with respect to t. The Bellman equation is 0 = Ii).A {u((t), t) + L.(W(t). S(t).t) +.(W (t). (t).)}. (2.27) where is the differential generator of (W. S). The ol)timal (c(otruls satisfy the first or(ler necessary conditions:,(c (t),t) Lt) < u,_(c(t).t) if (t) > 0:28 <. -M if (t) = 0. V (W) { < JI (T) < V'(W) if > ( + <.IT (T) if W = O: At =s)- I(t) Js() ()+(tr(t)= + ((t)(t (t)- rt). ir(t) ) ) (2.29) Substituting (2.28) and (2.29) into (2.27), we have a nonlinear partial differential equation of.. To compute the optimal controls, we nee(l to solve this nonlinear partial dlifferential equation with 17

20 two boundary conditions: (2.26) and J(W, S, T) = V(W). Once we solve this plartial differential equation, the optimal controls can be gotten y simply sllstitnting the solution into (2.28) and (2.29). Note that in solving the nonlinear partial differential equation, the nonnegativity constraint on consumption usually makes this nontrivial problem even more lifficult; see, e.g., Karatzas, Lehoczky, Sethi, and Shreve (1986) in a special case of our general model. To see that dynamic programming is consistent with our approach, note that at each time t, dynamic strategy corresponds to the allocation that would 1e chosen in a newly initiated static problem of the form of (2.13) and that Z(t)-' is the marginal and tility of wealth. Hence t) < Z(w, t) - ' < u,_(c(w, t), t) for v-a.e. (w. t) such that (W, t) > 0, < Z(w, t)' for v-a.c. (w. t) sch that c(w, t) = 0; (2.30) V (W()) < Z(w, t)-' < V (W(w)) for P-a.e w such that W(w) > 0, < Z(w, t) - for P-a.e w such that W(w) = 0. Recall that JwTI(t) = Z(t) - = 1/F-'(W(t), S(t), t) I - rd.c. (2.31) JTIww(t) = -FT-(t)/(F(t)) 2 and.i1 1,s(t) = Fs'(t)/(F-'(t)) 2. (2.32) Z(t) = F- (W(t). S(t), t),v- a.r. Therefore, Fzr F = 1 (2.33) and FzFs -1 + Fs = o. (2.34) Relations (2.33) and (2.34) imply that and Hence, it follows from (2.29) that ',ll't'v = F-FT 1 = FzZ. A(t) = Is(t) (Fs(t) + ((t)(t) T )-l((t) - r(t)(t))fz(t)z(t)). - a.c. (2.35) Relations (2.30) and (2.35) are consistent with (2.15) and (2.19). Although our approach and stochastic dynamic programming are essentially consistent, there are several advantages to our approach. 18

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