Lifetime Consumption-Portfolio Choice underTrading Constraints, Recursive Preferences, and

Nontradeable Income

Mark Schroder�and Costis Skiadasy

April 2003; this revision: July 2004Working Paper No. 324

Abstract

We analyze the lifetime consumption-portfolio problem in a compet-itive securities market with continuous price dynamics, possibly non-tradeable income, and convex trading constraints. We de�ne a class of�translation-invariant� recursive preferences, which includes additive ex-ponential utility, but also non-additive recursive and multiple-prior for-mulations, and allows for �rst and second-order source-dependent riskaversion. For this class, we show that the solution reduces to a singleconstrained backward stochastic di¤erential equation (BSDE), which foran interesting class of incomplete-market problems simpli�es to a systemof ordinary di¤erential equations of the Riccati type.

1 Introduction

In this paper we analyze the lifetime consumption-portfolio problem for an agentwith a possibly nontradeable income stream who can trade in a competitivesecurities market with essentially arbitrary continuous price dynamics, under theconstraint that the vector of risky-asset position values must lie in a given convexset at all times. Starting with the general �rst-order conditions of optimality,we consider increasingly restrictive and increasingly tractable formulations.It is well-known in economics that an easily solvable class of problems with

nontradeable income arises with additive exponential utility and Gaussian dy-namics (see, for example, Svensson and Werner (1993)). Additivity of utility,however, is known to impose an ad-hoc relationship between intertemporal sub-stitution and risk aversion (see, for example, Epstein (1992)). In particular, the

�Eli Broad Graduate School of Management, Dept. of Finance, Michigan State University,323 Eppley Center, East Lansing, MI 48824. email: schroder@bus.msu.edu

yKellogg School of Management, Dept. of Finance, Northwestern University, 2001 SheridanRoad, Evanston, IL 60208. email: c-skiadas@kellogg.norhwestern.edu

1

risk aversion coe¢ cient with expected discounted exponential utility is entirelydetermined by an agent�s preferences over deterministic consumption plans.A main contribution of this paper is to de�ne translation-invariant recursive

utility, a generalization of additive exponential utility that retains the tractabil-ity advantages of the latter, but frees the speci�cation of absolute risk aversionfrom preferences over deterministic plans. Moreover, risk aversion can be di¤er-ent for every source of risk, for example, re�ecting the source�s ambiguity in thesense of Ellsberg (1961), and also risk aversion can be either �rst or second order,in a dynamic version of the distinction made by Segal and Spivak (1990). For aspecial class of translation-invariant recursive utilities and dynamic investmentopportunity sets, the Gaussian case included, the incomplete market problemreduces to a system of Riccati-type ordinary di¤erential equations. The resultis a modeling framework that, relative to the exponential-Gaussian benchmark,is signi�cantly more �exible, yet still highly tractable.This paper�s analysis is a natural continuation of that in Schroder and Ski-

adas (2003), hereafter abbreviated to SS. In the latter there is no endowedincome stream (or if there is one, it is traded), portfolio constraints are imposedin terms of proportions of wealth, and the focus is on simpli�cations achievedthrough the assumption of scale invariance (or homotheticity), which is inconsis-tent with a nontradeable income stream. In contrast, in this paper constraintsare imposed on dollar amounts, a nontradeable income is allowed, and the focusis on simpli�cations achieved by translation invariance (or quasilinearity) rela-tive to a �xed plan. In SS we de�ned source-dependent relative risk aversion,while here we de�ne source-dependent absolute risk aversion.The �rst-order conditions of optimality with recursive utility take the form

of a system of forward-backward stochastic di¤erential equations (FBSDE), cor-responding to a PDE system in a Markovian setting. The forward and back-ward components of the FBSDE system are shown to uncouple under a scale-invariance assumption in SS, and under a translation-invariance assumptionin this paper. Methodologically, both papers follow the approach originatingin Cox and Huang (1989) and Karatzas, Lehoczky, and Shreve (1987) for thecase of additive utility and complete markets, and extended in Skiadas (1992),Du¢ e and Skiadas (1994), Schroder and Skiadas (1999), and El Karoui, Peng,and Quenez (2001) to recursive utility settings. For a broader context of FB-SDE systems in control theory we refer to the books by Ma and Yong (1999)and Yong and Zhou (1999).A related literature on constrained portfolio selection with additive utilities

(He and Pearson (1991), Karatzas, Lehoczky, Shreve, and Xu (1991), Shreve andXu (1992), Cvitanic and Karatzas (1992), Kramkov and Schachermayer (1999),Hugonnier and Kramkov (2002)) focuses on dual formulations that involve theminimization of the value function over all possible completions of the market.Cuoco (1997) showed existence of an optimum under nontradeable income andadditive utility with primal methods. We do not discuss duality or existence inthis paper. Extensions of the analysis in SS and this paper to non-Browniansettings are developed in Schroder and Skiadas (2004).An alternative to the utility gradient approach, since Merton (1971), is dy-

2

namic programming. Appendix C gives a non-Markovian dynamic programmingoptimality veri�cation argument for translation-invariant recursive utility, anal-ogous to the one given in SS for the scale-invariant case. Related Markovian dy-namic programming treatments with nontradeable income, additive discountedpower utility, and i.i.d. instantaneous returns include Du¢ e, Fleming, Soner,and Zariphopoulou (1997) and Koo (1998). The recent additive-exponential-utility solution of Musiela and Zariphopoulou (2004) is nested in the setting ofthis paper, and is extended (in Example 25) by allowing recursive utility and amore general asset structure.The remainder of the paper is organized in seven sections and three ap-

pendices. In Section 2 the market and optimality are de�ned. In Section 3optimality is characterized in terms of the relationship between a utility (su-per)gradient density and a constrained notion of a state-price density. In Sec-tion 4 we apply the characterization of optimality to generalized recursive utility,followed by the translation-invariant case in Section 5. In Section 6 the impor-tant �quasi-quadratic� speci�cation is introduced, used in applications in the�nal two sections. Appendix A characterizes aggregators of translation-invariantgeneralized recursive utilities, Appendix B contains proofs, and Appendix C ison the dynamic programming approach.

2 Market and Optimality

Given is an underlying probability space (;F ; P ) supporting a d-dimensionalstandard Brownian motion B over the �nite time horizon [0; T ]. All processesappearing in this paper are assumed to be progressively measurable with respectto the augmented �ltration fFt : t 2 [0; T ]g generated by B. We also assumethat FT = F . The conditional expectation operator E [�jFt] will be abbreviatedto Et throughout. The quali�cation �almost surely� (or a.s.) will be oftenomitted where it is implied by the context.Given any subset S of a Euclidean space, L (S) denotes the set of all S-

valued (progressively measurable) processes, and Lp (S) denotes the set of allx 2 L (S) such that

R T0kxtkp dt < 1 a.s. (where k�k is Euclidean norm). We

will make frequent use of the spaces:

Sp =

(x 2 L (R) : E

"supt2[0;T ]

jxtjp#<1

);

H =

(x 2 L (R) : E

"Z T

0

x2tdt+ x2T

#<1

):

We consider H as a Hilbert space under the inner product

(xjy) = E"Z T

0

xtytdt+ xT yT

#:

As usual, we identify any elements x and ~x in H such that (x� ~xjx� ~x) = 0.

3

There is a securities market that allows instantaneous default-free borrowingand lending at a rate given by the stochastic process r 2 L1 (R). We refer totrading at this rate as the �money market.�The rest of the securities marketconsists of trading in m � d risky assets, whose instantaneous excess returns(relative to r) are represented by the m-dimensional Ito process R, with dynam-ics

dRt = �Rt dt+ �

R0t dBt;

where �R 2 L1 (Rm) and �R 2 L2�Rd�m

�. We assume throughout that �R is

everywhere full-rank (and therefore everywhere invertible if m = d).A trading plan is any process � 2 L (Rm) such thatZ t

0

����0s�Rs ��+ �0s�R0s �Rs �s� ds <1 a.s. for all t < T:

(All vectors are assumed to be column vectors, with a prime denoting trans-position.) We interpret the ith component of �t as the time-t market value ofthe agent�s investment in security i 2 f1; : : : ;mg, the remaining wealth beinginvested in the money market. A (�nancial) wealth process is any process Wsuch thatW� 2 S2, whereW�

t = max f0;�Wtg : (This restriction is imposed torule out doubling-type strategies.) A plan (c; �;W ) is a triple of a consumptionplan c, a trading plan �, and a wealth process W .We consider an agent characterized by the primitives (C; U0; w0; e;�). The

set C � H, whose elements are the consumption plans, is a convex set such thatc + b 2 C for any c 2 C and bounded b 2 L (R). Given any c 2 C, we thinkof ct, t < T , as the time-t consumption rate and cT as a terminal lump-sumconsumption or bequest. The function U0 : C ! R is a (strictly) increasingutility function over consumption plans. The positive scalar w0 > 0 is an initial�nancial wealth, while e 2 H is an endowed income stream, with et representinga time-t income rate, and eT representing a terminal lump-sum payment. Theset �, a nonempty convex closed subset of Rm; is the agent�s constraint set, inwhich the vector of risky-asset investment values is restricted to lie at all times.(The general analysis would also go through if � were made state and timedependent, subject to some technical measurability conditions.)

Example 1 Missing market i is modeled by requiring that �i = 0 for all � 2 �:A minimum investment constraint in asset i is modeled by requiring that �i � lfor all � 2 �; for some lower limit l: If l = 0; this becomes a short-sale constraint.

The plan (c; �;W ) is feasible if1

�t 2 �; t < T;

and it satis�es the budget equation:

W0 = w; dWt = (Wtrt + et � ct) dt+ �0tdRt; cT =WT + eT : (1)

1More precisely, this means that the process x de�ned by xt = 1 f�t =2 �g ; t < T; andxT = 0; is the zero process as an element of H: The analogous interpretation will always beimplied for statements of the form �xt 2 S�or �xt = yt:�

4

A consumption plan c is feasible if (c; �;W ) is feasible for some trading plan� and wealth process W . A consumption plan c is optimal if it is feasible andU (c) � U (~c) for any other feasible consumption plan ~c; while the plan (c; �;W )is optimal if it is feasible and c is optimal. A trading plan � is feasible (resp.optimal) if (c; �;W ) is feasible (resp. optimal) for some (c;W ).

3 Utility Gradient and State Pricing

Fixing a reference plan (c; �;W ) ; we formulate conditions for its optimalityin terms of a utility (super)gradient density and a constrained notion of statepricing. These conditions are applied in subsequent sections in increasinglyrestrictive settings.A process x 2 H is a feasible incremental cash �ow at c if c+ x is a feasible

consumption plan. A process � 2 H is a state-price density at c if (�jx) � 0for every feasible incremental cash �ow x at c: A process � 2 H is a utilitysupergradient density of U0 at c if for all x 2 H,

c+ x 2 C implies U0 (c+ x) � U0 (c) + (�jx) : (2)

A process � 2 H is a utility gradient density of U0 at c if for all x 2 H,

c+ �x 2 C for some � > 0 implies (�jx) = lim�#0

U0 (c+ �x)� U0 (c)�

:

If � is a supergradient density of U0 at c and the utility gradient of U0 at cexists, then the utility gradient density equals �.The supergradient and state-price properties combine to characterize opti-

mality in the following proposition, whose proof is essentially the same as thatof Proposition 3 in SS (short for Schroder and Skiadas (2003)).

Proposition 2 Suppose (c; �;W ) is a feasible plan. If � 2 H is both a super-gradient density of U0 at c and a state-price density at c, then the plan (c; �;W )is optimal. Conversely, if the plan (c; �;W ) is optimal and � 2 H is a utilitygradient density of U0 at c, then � is a state-price density at c.

To apply the last proposition at the reference plan (c; �;W ), we need tocompute the dynamics of the utility gradient (or supergradient) density at c,and to develop a criterion for recognizing these dynamics as the dynamics of astate-price density at c: Examples of utility (super)gradient calculations appearin Du¢ e and Skiadas (1994), Schroder and Skiadas (1999), Chen and Epstein(2002), El Karoui, Peng, and Quenez (2001), as well as in SS. The characteriza-tion of state-price dynamics in the current setting is provided in the followingresult, stated in terms of the support function �� : Rm ! (�1;1] de�ned by

�� (") = sup��0" : � 2 �

: (3)

5

Theorem 3 Suppose that (c; �;W ) is a feasible plan.(a) (Su¢ ciency) Suppose that � 2 S2, �W 2 S1; and for some � 2 L2

�Rd�,

d�t�t

= �rtdt� �0tdBt for t � T; and �0t"t = �� ("t) for t < T; (4)

where " = �R � �R0�:

Then � is a state-price density at c:

(b) (Necessity) Suppose (for simplicity) that r is bounded, and � 2 S1 is astrictly positive Ito process. If � is a state-price density at c, then (4) holds forsome � 2 L2

�Rd�:

By Proposition 2, if � is a utility (super)gradient, (4) states the �rst-orderconditions of optimality. In this case we can think of � as being the agent�ssubjective marginal pricing kernel at the optimum, which may not be consis-tent with the way the market prices assets, due to the agent�s constraints. Forexample, in equilibrium, an agent can regard a stock overvalued if the agentis constrained from short-selling the stock. If � follows the dynamics of condi-tion (4), the agent assesses a subjective market-price-of-risk process �. Givenrisky asset positions � 2 �, �0" can be thought of as the instantaneous expectedmispricing bene�t to the agent, which condition (4) states must be maximizedalong the optimal path.The market is complete if m = d and � = Rd; in which case there exists a

unique, up to positive scaling, state-price density, �; whose dynamics are givenin (4) with � =

��R0��1

�R, provided the process � so de�ned is in H. Inthe constrained case, the utility gradient density, �; at the optimum, assumingit exists and is a strictly positive Ito process, de�nes a market-price-of-riskprocess �; which can be interpreted in terms of a �ctitious complete market asfollows. (Note that the assumption m = d is without loss in generality, sincenontradeability of an asset can be modeled through �.)

Corollary 4 Suppose that (c; �;W ) is an optimal plan, m = d; r is bounded, thestrictly positive Ito process � 2 S2 is the gradient density of U0 at c; and �W 2S1: Then there exists � 2 L

�Rd�such that condition (4) holds. Moreover, the

plan (c; �;W ) is optimal for the same agent, but with endowed income e+�� (")instead of e; and no constraints, in a �ctitious complete market obtained from theoriginal market by assuming instantaneous expected excess returns �R = �R0�instead of �R:

Proof. The necessity part of Theorem 3 gives the �rst-order conditions ofoptimality in the original market. Feasibility of (c; �;W ) in the �ctitious marketis con�rmed by direct computation of the budget equation. Finally, optimalityin the �ctitious market follows by the su¢ ciency part of Theorem 3.

6

4 Recursive Utility

In this section we elaborate on the �rst-order conditions of optimality for thecase of generalized recursive utility, by which we mean a utility process that isde�ned as a solution to a backward stochastic di¤erential equation (BSDE), acontinuous-time version of a general backward recursion on an information tree.The speci�cation nests the continuous-time recursive utility of Du¢ e and Ep-stein (1992), time-additive utility being a special case, the multiple-prior formu-lation of Chen and Epstein (2002), as well as robust-control type criteria throughthe type of argument in Skiadas (2003). Further properties of generalized re-cursive utility are discussed in Lazrak and Quenez (2003), El Karoui, Peng, andQuenez (2001), as well as SS. In some cases,2 the equivalence in Schroder andSkiadas (2002) can be used to mechanically extend the analysis that follows toinclude linear habit formation.We de�ne the utility in terms of an aggregator function F : � [0; T ] �

R � R1+d ! R, and a set U � S2 of utility processes. The random variableF (T; c; U;�) does not depend on the arguments (U;�), which are thereforenotationally suppressed. We henceforth assume that, given any consumptionplan c, there is a unique

�U;�U

�2 U � L2

�Rd�that solves the BSDE

dUt = �F�t; ct; Ut;�

Ut

�dt+�U 0t dBt; UT = F (T; cT ) ; (5)

and we de�ne the utility process of c by letting U (c) = U: We also assume thatF (!; t; �) is a concave function for all (!; t), and the derivative, Fc (!; t; �; U;�) ;of F (!; t; �; U;�) exists and maps R onto (0;1).

Example 5 (Du¢ e-Epstein Utility) Du¢ e and Epstein (1992) analyze theaggregator form

F (t; c; U;�) = f (t; c; U)� A (t; U)2

�0�; (6)

for some functions f : � [0; T ]� (0;1)�R! R and A : � [0; T ]�R! R.The coe¢ cient A can always (under some technical regularity assumptions) beset to zero after a suitable choice of an ordinally equivalent utility. Additiveutility is obtained if f (t; c; U) = u (t; c)� �tU and A = 0:

The aggregator F need not be smooth in (U;�) ; a generality that is useful,for example, in incorporating the Chen and Epstein (2002) formulation. Thesuperdi¤erential of F with respect to the variables (U;�) at (!; t; c; U;�) isde�ned as the set (@U;�F ) (!; t; c; U;�) of all pairs (a; b) 2 R� Rd such that

F (!; t; c+ x; U + y;�+ z) � F (!; t; c; U;�) + Fc (!; t; c; U;�)x+ ay + b0z:2For example, this is the case under conical trading constraints if the short rate process and

the habit parameters are all deterministic. The equivalence applies generally in the complete-markets case.

7

We now �x a reference plan (c; �;W ) and characterize its optimality. Letting�U;�U

�be the solution to BSDE (5), we de�ne the process

�t = Fc�t; ct; Ut;�

Ut

�: (7)

In a time-t formulation of the agent�s problem, �t is the Lagrange multiplierfor the time-t wealth constraint, since it provides the �rst-order utility in-crement (per unit of wealth) as a result of slightly changing time-t wealth.Consumption can be expressed in terms of � by inverting the above equation:ct = I (t; �t; Ut;�t), where the function I : � [0; T ] � (0;1) � R1+d ! C isde�ned implicitly by

Fc (t; I (t; �; U;�) ; U;�) = �; � 2 (0;1) :

The utility (super)gradient density can be expressed as a stochastically dis-counted version of � as follows. Given any processes (a; b) 2 L1 (R)� L2

�Rd�;

the stochastic exponential process E (a; b) is de�ned by the SDEdEt (a; b)Et (a; b)

= atdt+ b0tdBt; E0 (a; b) = 1: (8)

Suppose that the process � 2 H is given by

�t = Et (FU ; F�)�t; t 2 [0; T ] ; (9)

for some (FU ; F�) 2 L1 (R)� L2�Rd�such that E (FU ; F�) 2 S2 and3

(FU (t) ; F� (t)) 2 (@U;�F )�t; ct; Ut;�

Ut

�; t 2 [0; T ] :

Then, by Proposition 13 of SS, � is a utility supergradient density of U0 at c.Applying Ito�s lemma to equation (9), condition (4), along with the wealth

dynamics and the utility speci�cation results in the following �rst-order condi-tions of optimality, to be solved jointly in

�U;�U ; �; ��;W

�:

dUt = �F�t; I

�t; �t; Ut;�

Ut

�; Ut;�

Ut

�dt+�U 0t dBt; UT = F (T;WT ) ;

d�t�t

= ��rt + FU (t) + �

�0t F� (t)

�dt+ ��0t dBt; �T = Fc (T;WT ) ;

dWt =�Wtrt + �

0t�Rt + et � I

�t; �t; Ut;�

Ut

��dt+ �0t�

R0t dBt; W0 = w0;

"t = �Rt + �

R0t

�F� (t) + �

�t

�; �t 2 �; �0t"t = �� ("t) ;

(FU (t) ; F� (t)) 2 (@U;�F )�t; I

�t; �t; Ut;�

Ut

�; Ut;�

Ut

�:

The above is a constrained forward-backward system of stochastic di¤er-ential equations (FBSDE). Using last section�s results the system is su¢ cientand necessary for optimality, under suitable regularity assumptions (as given inProposition 2 and Theorem 3). In a Markovian setting, the above system canbe formulated as a PDE system, following the Ma, Protter, and Yong (1994)approach. The basic idea is illustrated in SS, and can be easily adapted to thispaper�s setting.

3More precisely, we are assuming that the indicator function of (FU (t) ; F� (t)) 62�@U;�F

� �t; ct; Ut;�Ut

�is equal to zero as an element of H:

8

5 Translation-Invariant Formulation

Having characterized optimality for general recursive utility, in this section weintroduce translation invariance relative to a �xed cash �ow , a condition thatis this paper�s main focus. As noted in the Introduction, translation-invariantutility retains the tractability of additive exponential utility without imposingthe ad-hoc risk-aversion restrictions of additivity. Provided that the cash �ow is traded without constraints, we will see that translation invariance relative to uncouples the forward and backward components of the �rst-order conditions,thus reducing last section�s FBSDE system to a single BSDE.For each consumption plan c; let U (c) denote the corresponding utility

process solving BSDE (5) : We �x throughout a bounded process 2 L (R++),and we assume that the dynamic utility c 7! U (c) is translation-invariant rela-tive to ; meaning that for any c1; c2 2 C and t 2 [0; T ],

Ut�c1�= Ut

�c2�

=) Ut�c1 + k

�= Ut

�c2 + k

�for all k 2 R:

Appendix A shows that if we further assume that U is in certainty-equivalentform using as the numeraire, then U is quasilinear with respect to :

Ut (c+ k ) = Ut(c) + k for all k 2 R; c 2 C; t 2 [0; T ] : (10)

Appendix A also shows that quasilinearity with respect to is essentially char-acterized by the aggregator form in the following condition, which is assumedfor the remainder of this paper.

Condition 6 (Standing Assumption) The following restrictions hold for abounded process 2 L (R++) and a function G : � [0; T ]�R1+d ! R that wecall an absolute aggregator.

(a) For every c 2 C, U0 (c) = U0; where�U;�U

�solves, uniquely in U�L2

�Rd�;

the BSDE

dUt = �G�t;ct t� Ut;�Ut

�dt+�U 0t dBt; UT =

cT T: (11)

(b) There exists % 2 Rm and a strictly positive process � such that

d�t = (rt�t � t) dt+ %0dRt; �T = T ;

and v 2 � implies v + k% 2 � for all k 2 R:

The notion of an absolute aggregator in a translation-invariant setting isanalogous to that of a proportional aggregator in a scale-invariant setting (asexplained in Appendix A and SS). Part (b) of the above condition states thatthe agent can trade with no restrictions in a portfolio % that generates thedividend stream and has corresponding time-t value �t. Since trading in % isunconstrained, �� (") <1 implies %0" = 0: The �rst-order conditions thereforeimply that the agent correctly prices the portfolio %.

9

Example 7 If F is a Du¢ e and Epstein (1992) aggregator, then the absoluteaggregator G must take the functional form

G (!; t; x;�) = g (!; t; x)� A (!; t)2

�0�:

In the special case in which

A (!; t) = 1 and g (!; t; x) = � (!; t)� exp (�x) ;

under suitable integrability restrictions, the ordinally equivalent utility

Vt = � exp (�Ut)

is time-additive discounted exponential utility:

Vt = Et

"Z T

t

� exp��Z s

t

�udu�cs s

�ds� exp

�Z T

t

�udu�cT T

!#:

The key to simplifying the solution in the translation-invariant case is thefollowing informal argument. Given an optimal plan, suppose that, on sometime-t event, k units of account are added to the agent�s �nancial wealth. Be-cause of translation invariance with respect to , the agent will �nd it optimalto invest all additional k units of wealth in k=�t shares of the portfolio %, and toconsume all of the ensuing dividend stream fk s=�t; s > tg : Given the quasi-linearity of the utility function with respect to , this suggests the following keyrelationships at the optimum:

�t =1

�t; Ut =

1

�t(Yt +Wt) ; and �t = �

0t + Ut%; (12)

where�Y;�Y ; �0

�solves a constrained BSDE, given below, that is independent

of �nancial wealth. The second equation states that, taking the market valueof the fund % as the unit of account, the value function is the sum of �nancialwealth and the value of the endowed income stream.To state the relevant constrained BSDE, we introduce some notation. The

(partial) derivative of G (!; t; �;�) at x is denoted Gx (!; t; x;�), and the su-perdi¤erential of the function G with respect to � (de�ned analogously to@U;�F ) is denoted @�G. The functions X ; G� : � [0; T ]� (0;1)�Rd ! R arede�ned by

Gx (!; t;X (!; t; y;�) ;�) = y;

G�(!; t; y;�) = supx2R

fG (!; t; x;�)� yxg

= G(!; t;X (!; t; y;�) ;�)� yX (!; t; y;�) :

Given the conjectured conditions (12) ; the �rst-order conditions reduce toa constrained BSDE:

10

Condition 8�Y;�Y

�2 L (R)� L

�R1+d

�and �0 2 L (Rm) solve

dYt =

�Ytrt � �00t �Rt � et � �tG�

�t; t�t;�Yt + �

Rt �

0t

�t

�+��Rt %

�0��Yt + �Rt �0t�t

��dt

+�Y 0t dBt; YT = eT ;

"t = �Rt + �

R0t

�G�(t)�

1

�t�Rt %

�; �0t 2 �; �� ("t) = �

00t "t;

G� (t) 2 @�G�t;X

�t; t�t;�Yt + �

Rt �

0t

�t

�;�Yt + �

Rt �

0t

�t

�:

Given�Y;�Y ; �0

�and equations (12) ; the optimal consumption plan is

ct = t (Ut + xt) ; where xt = X�t; t�t;�Yt + �

Rt �

0t

�t

�: (13)

The wealth process dynamics are obtained by substituting the optimal con-sumption and trading plan expressions in the budget equation:

dWt =

�Wtrt +

Yt +Wt

�t

�%0�Rt � t

�+ �00t �

Rt + et � txt

�dt (14)

+

��0t +

Yt +Wt

�t%

�0�R0t dBt; W0 = w0:

The optimality characterization for the translation-invariant formulation iscompleted in the following result, proved in Appendix B.

Theorem 9 Suppose Condition 6 holds.(a) (Su¢ ciency) Suppose Condition 8 is satis�ed, the consumption plan c isde�ned by equation (13), the wealth process W solves SDE (14), and the utilityprocess U 2 U and trading plan � are given by equations (12). Finally, supposeE = E (� =�; G�) 2 S2; as well as E=� 2 S2 and EW=� 2 S1. Then the plan(c; �;W ) is optimal and U = U (c).

(b) (Necessity) Suppose the plan (c; �;W ) is optimal, G (!; t; x;�) is di¤er-entiable in (x;�), and r is bounded. Let U = U (c) ; x = c= � U , Gx (t) =Gx�t; xt;�

Ut

�, and G� (t) = G�

�t; xt;�

Ut

�, and suppose that E = E (�Gx; G�) 2

S2 and 1=� 2 S2. Then Condition 8 holds with Y = U��W and �0 = ��U%.

Remark 10 The �rst-order conditions for the case of no intermediate con-sumption or the case of no terminal consumption are obtained by omitting thecorresponding consumption argument in the utility and supergradient dynam-ics. The optimality veri�cation argument in those cases remains essentially thesame. In the translation-invariant formulation, if there is no intermediate con-sumption we omit the dependence of G (!; t; x;�) on x and let t = 0 for t < T ,and if there is no terminal consumption we let UT = T = 0.

11

6 Quasi-Quadratic Absolute Aggregator

We have established that translation invariance reduces the FBSDE of the �rst-order conditions to a single BSDE. To gain further insight into the relation-ship between trading strategies and risk aversion, in this section we impose a�quasi-quadratic�absolute aggregator functional form. The formulation is suf-�ciently �exible to model possibly source-dependent �rst or second-order riskaversion, and includes all translation-invariant Du¢ e-Epstein utilities, additivediscounted exponential utility being a special case. Unlike additive utility, theformulation allows the various types of risk aversion to be arbitrarily selectedgiven the agent�s preferences over deterministic plans. The �rst-order conditionsfor a quasi-quadratic absolute aggregator are derived generally in this section,while several more speci�c solutions are given in the �nal two sections.In the remainder of this paper we assume the following condition, using the

notation jxj0 = (jx1j ; : : : ; jxdj) ; x 2 Rd:

Condition 11 (Standing Assumption) The absolute aggregator G is quasi-quadratic, meaning that it takes the form

G(!; t; x;�) = g(!; t; x)� q(!; t)0�� �(!; t)0 j�j � 12�0Q(!; t)�; (15)

for some (progressively measurable) functions g : � [0; T ] � R ! R; q : �[0; T ]! Rd; � : �[0; T ]! Rd+; and Q : �[0; T ]! Rd�d, such that Q (!; t) issymmetric positive de�nite for all (!; t). The processes �; q; and Q are assumedbounded (for simplicity).

The interpretation of the various terms of equation (15) parallels that of theproportional quasi-quadratic representation in SS, whose Section 5.1 appliesin this context with obvious modi�cations. Assuming it is state independent,the function g entirely determines the agent�s preferences over deterministicconsumption plans. The linear term q0� can be thought of as re�ecting theagent�s beliefs, in the sense of a Girsanov change of measure (as explainedin SS). The remaining coe¢ cients, � and Q, measure, respectively, �rst andsecond-order absolute risk aversion that can be time and state dependent, aswell as dependent on the source of risk. The source-dependence of risk aversioncan be thought of as re�ecting the ambiguity of the risk source in the sense ofEllsberg (1961) (see Klibano¤, Marinacci, and Mukerji (2002) for related ideas).Alternatively, the �-term can arise from the Chen and Epstein (2002) notion of�-ignorance, while the Q-term is related to robust control type criteria throughthe type of argument given in Skiadas (2003).Under a quasi-quadratic absolute aggregator, we obtain the simpli�cations

X (!; t; y;�) = X g (!; t; y) ; and

G� (!; t; y;�) = g� (!; t; y)� q (!; t)0 �� � (!; t)0 j�j � 12�0Q (!; t) �;

12

where the functions X g; g� : � [0; T ]� (0;1)! R are de�ned analogously toX and G� by

gx (!; t;X g (!; t; y)) = y;

g�(!; t; y) = supx2R

(g (!; t; x)� yx)

= g(!; t;X g (!; t; y))� yX g (!; t; y) :

The superdi¤erential @�G is characterized as follows. For any � 2 Rd; let @ j�jdenote the set of all � 2 [�1; 1]d whose ith coordinate, �i; satis�es

�i = 1 if �i > 0; �i = �1 if �i < 0;

and �i 2 [�1; 1] if �i = 0:

Moreover, for every x; y 2 Rd, let x y 2 Rd denote element-by-element multi-plication, that is, (x y)i = xiyi for all i: One can easily show:

Lemma 12 G� (t) 2 (@�G) (t; xt;�t) if and only if there exists �t 2 @ j�tjsuch that G� (t) = � (qt + �t �t +Qt�t).

Direct computation using the above lemma gives the constrained BSDE ofthe �rst-order conditions as in the following proposition. The term �� in the�rst-order conditions (which vanishes if the aggregator is smooth) is computedexplicitly in the following section assuming that Q is diagonal.

Proposition 13 For the quasi-quadratic absolute aggregator G, the constrainedBSDE of Condition 8 is equivalent to

dYt =

�rtYt � et � �� ("t)� �tg�

�t; t�t

�+�Y 0t Qt�

Yt � �00t �R0t Qt�Rt �0t

2�t

�dt

+�Y 0t

�dBt +

�qt + �t �t +

�Rt %

�t

�dt

�; YT = eT ;

�0t = �t��R0t Qt�

Rt

��1��Rt � "t � �R0t

�qt + �t �t +

�Rt %+Qt�Yt

�t

��;

�0t 2 �; �� ("t) = �00t "t; �t 2 @

���Yt + �Rt �0t �� :Remark 14 (a) If � = Rm; corresponding to possibly incomplete markets(if m < d) but no further trading constraints, the above �rst-order conditionssimplify by letting " = 0 and �� (") = 0:

(b) If the processes r; ; �R; �R; �; q;Q; and e are all deterministic, and the func-tion g is state independent, then the �rst-order conditions simplify by letting�Y = 0 and % = 0:

One can now restate Theorem 9 for the quasi-quadratic case by replacingCondition 8 with the above constrained BSDE. Given a solution to the latter,the complete solution is recovered using equations (12), (13), and (14).

13

Example 15 Suppose that the quasi-quadratic absolute aggregator G is smooth(that is, � = 0). A simple expression for the optimal trading strategy is obtainedunder a single linear constraint. Suppose � = fv 2 Rm : l � p0v � ug ; wherep 2 Rm and �1 � l � u � 1. (For example, a short-selling constraint onasset i corresponds to l = 0, u =1, and p a vector with a one in the ith positionand zeros elsewhere.) De�ning

�0�t = At

��Rt � �R0t

�qt +

�Rt %+Qt�Yt

�t

��; At = �t

��R0t Qt�

Rt

��1;

the pair��0; "

�in the �rst-order conditions of Proposition 13 is given by

�0t = �0�t �At"t; "t = � (p0Atp)�1 p

�min

�max

�p0�0�t ; l

; u� p0�0�t

�:

Note that the above expression for �0� gives the optimal �0 as a function of �Y

in the unconstrained case. The above claim follows easily from the �rst-orderconditions. The details, as well as the extension to multiple linear constraints,are analogous to Example 34 in SS, and are therefore omitted.

The optimal consumption dynamics for a smooth quasi-quadratic absoluteaggregator are summarized in the following proposition, which is followed by anapplication to equilibrium asset pricing.

Proposition 16 Suppose that Condition 11 holds with � = 0; the partial gx isstate independent, the process follows the dynamics d = = � dt + � 0dB;and that the �rst-order conditions of optimality are satis�ed at the optimal plan(c; �;W ) ; with corresponding utility process U . De�ne

At = �gxx (t; xt)

gx (t; xt); St = �

gxt (t; xt)

gx (t; xt); Pt = �

gxxx (t; xt)

gxx (t; xt); xt =

ct t� Ut;

and � =�t = � t � ��1t �Rt %. Then dct = �ctdt+�

c0t dBt; where

�ct = t�t

��Yt + �

Rt �t +

�1

At� Ut

��Rt %

�+

�ct �

tAt

�� t ;

�ct = � tG(t; xt;�Ut ) + ct�� t � �

0t �

t

�+�c0t �

t +

t2

PtA2t� =�0t �

=�t

� tAt

�St + � t � rt +

t�t� 1

�t%0��Rt + �

R0t �

=�t

��:

Example 17 (Equilibrium Asset Pricing) In addition to the assumptionsof the last proposition, we assume that � = Rm; meaning that markets can beincomplete (m < d) ; but there are no further trading constraints. The optimalconsumption dynamics can be inverted to obtain the following equilibrium pricing

14

expressions (shown in Appendix B):

�Rt =1

t�R0t

�Qt�

ct +

� tAt� ct

�Qt�

t +

t�t

�I � 1

AtQt

��Rt %

�;

rt = � t + t�t� 1

�t%0��Rt + �

R0t �

=�t

�+ St �

Pt2At

� =�0t �

=�t

+At�G�t; xt;�

Ut

�+1

t

��ct + ct

�� 0t �

t � �

t

�� �c0t �

t

��:

For example, if g (!; t; x) = � (!; t)�exp (�ax), a 2 R++, the above expressionssimplify since At = Pt = a and St = 0: If markets are complete, or preferencesare Du¢ e-Epstein (that is, Q = bI, b 2 L (R++)), the risk-premium equilibriumexpression can be added-up across agents in an economy with agent-speci�c pa-rameters (a; b; �), to obtain an equilibrium expression for �R in terms of theaggregate endowment and its volatility.

7 Solutions under Quasi-Quadratic Aggregator

This section presents more detailed solutions under a quasi-quadratic absoluteaggregator, including an explicit expression for the term �� in the �rst-orderconditions given diagonal Q; clarifying the role of �rst-order risk aversion, aswell as some special solutions emphasizing the e¤ect of undiversi�able income.The following normalization of the return volatility process �R and the

second-order risk aversion parameter Q will be henceforth assumed. The re-striction is vacuous if m = d; which can be assumed without loss in generalityby modeling missing markets through �. In applications, however, it is oftenmore convenient to assume m < d:

Condition 18 (Standing Assumption) For some �RMM ; QMM 2 L (Rm�m),

�R =

��RMM

0

�and Q =

�QMM 00 QNN

�:

Remark 19 By Lemma 10 of SS, �R can always be put in the above formby passing to a new Brownian motion that generates the same �ltration as theoriginal one. Suppose Q = aI for some a 2 R (where I denotes the identitymatrix), as is the case with Du¢ e-Epstein utility, including time-additive utility.Then the change of Brownian motion does not a¤ect Q, and therefore imposingCondition 18 entails no loss in generality (even if m < d).

The notational scheme suggested in the above condition will be followedthroughout: M andN refer to the sets ofmarketed and non-marketed securities,respectively, and are also used as indices of corresponding matrix blocks. Inparticular, given any d-dimensional process or vector x, we write x0 = [x0M ; x

0N ],

where xM is valued in Rm and xN is valued in Rd�m. Given the new subscripts,we often omit dummy time indices to simplify the notation. For example, the

15

above condition implies that dR = �Rdt + �R0MMdBM : It is important to notethat, unless otherwise stated, the coe¢ cients �R and �RMM are not required tobe adapted to the �ltration generated by BM .The following processes will be of repeated use:

�M =��R0MM

��1 ��R � "

�and �M = �M � qM � 1

��RMM%: (16)

Moreover, in the �rst order conditions, we will often refer to the term

�M = �M � �M �M : (17)

For example, in this context, the optimal trading strategy is � = �0+U%; where

�0 =��RMM

��1 ��Q�1MM �M � �YM

�: (18)

Example 20 (Fictitious Completion) This example illustrates Corollary 4.Suppose that m < d and � = Rm. The �ctitious complete market of Corollary 4is obtained by introducing unrestricted trading in d�m �ctitious securities withprice dynamics speci�ed by �RN = qN + �N �N + ��1QNN�YN ; �RNN equalto the identity matrix, and �RMN = �R0NM = 0. The agent�s endowment andthe marketed asset price dynamics are the same in the original and �ctitiousmarkets. The �rst-order conditions in the �ctitious market imply that �N = 0;and therefore the impossibility of trading in �ctitious assets is a non-bindingconstraint. Direct computation shows that the BSDE for Y is the same in the�ctitious market and the original market.

The following result clari�es the role of � in the �rst-order conditions, givena diagonal process Q.

Proposition 21 Suppose Condition 18 holds, Q is diagonal, and the process�M is de�ned in (16) : Then, in the �rst-order conditions of Proposition 13, theterms � � and �0 can equivalently be speci�ed by

�M �M = max fmin f�M ; �Mg ;��Mg ; �N �N = �N sign(�YN );

and equation (18), where �M can be written (equivalently to (17)) as

�M = min fmax f0; �M � �Mg ; �M + �Mg = QMM�UM :

Given the assumption that Q is diagonal, the last equation implies that, forany i 2 M , �Ui = 0 whenever j�ij � �i; that is, the agent completely hedgesthe ithdimension of risk, as a consequence of �rst-order risk aversion. Analogousresults for the scale-invariant case are presented in SS (see also Segal and Spivak(1990), Chen and Epstein (2002), and Epstein and Miao (2000)).

Example 22 Suppose further that the parameters�r; ; �R; �R; �; q;Q; e

�are

all deterministic, and the function g is state independent. Then Y is also deter-ministic, and therefore �Y = 0 and �0 =

��RMM

��1��UM . In this case, �

Ui = 0

16

can imply complete nonparticipation in the market for asset i. For example,suppose that �RMM (as well as Q) is diagonal, and � = Rm. Suppose also thatthe portfolio % consists of a share in asset one, generating dividend stream .Then, for any i � 2,

����Ri =�Rii�� qi�� � �i implies �i = 0. That is, a su¢ cientlysmall belief-adjusted market price of the risk traded by asset i � 2 results in theagent�s nonparticipation in this market.

This section�s last main result focuses on the role of a nontradeable endow-ment income stream. We assume that � = Rm, which implies that markets canbe incomplete (m < d), but there are no further trading constraints. Assumingthat all parameters, except for the endowment e; depend only on uncertaintygenerated by BM , we derive a decomposition of Y into a subjective endowmentvalue component and a market component.

Proposition 23 Suppose that � = Rm, and the processes r; ; �R; �R; �M ;qM ; QMM ; and g� ( =�) are all adapted to the (augmented) �ltration

�FMt

generated by BM . Then, in the �rst-order conditions of Proposition 13, theprocess Y and its volatility can be decomposed as

Y = YM + Y e; �YM = �YM

M +�Ye

M ; and �YN = �Y e

N ;

where�YM ;�Y

M

M

�is adapted to

�FMt

and solves the BSDE:

dYM = ���g�

� �

�� rYM +

�

2�0MQ

�1MM �M

�dt

+�YM 0

M (dBM + �Mdt) ; YMT = 0;

while�Y e;�Y

e�is adapted to the �ner �ltration fFtg and solves the BSDE:

dY e = ��e� rY e � 1

2��Y

e0N QNN�

Y e

N

�dt

+�Ye0

M (dBM + �Mdt) + �Y e0N (dBN + (qN + �N �N ) dt) ; Y eT = eT :

To present integrated versions of the above BSDEs, we introduce some no-tation with regard to Girsanov changes of measure. Given any b 2 L

�Rd�, we

use the notation Bb and �b to denote the processes satisfying

dBbt = dBt + btdt; Bb0 = 0; andd�bt

�bt= �b0tdBt; �b0 = 1: (19)

If �b is a martingale (for example, if b satis�es the Novikov condition), then Bb

is Brownian motion under the probability P b with density dP b=dP = �bT . Theexpectation operator corresponding to the probability P b is denoted Eb.Consider now the processes

� =

��M

qN + �N �N

�and Bt;s = exp

��Z s

t

rudu

�;

17

and assume that �� is a martingale. The above BSDEs can then be integrated,provided the integrals below are well-de�ned, to obtain the following expressionsfor the two components of Y :

YMt = E�

"Z T

t

Bt;s�s�g��s; s�s

�+1

2�0M (s)Q

�1MM (s) �M (s)

�ds j FMt

#;

Y et = E�

"Z T

t

Bt;s�es �

1

2�s�e0N (s)QNN (s)�

eN (s)

�ds+ Bt;T eT j Ft

#:

Example 24 (Complete Markets) Suppose that markets are complete, thatis, m = d and � = Rm. Then � =

��R0��1

�R is the unique market price of

risk process, and Y et = E�t

hR TtBt;sesds+ Bt;T eT

iis the time-t market value of

the endowment.

Example 25 We consider two special cases of Proposition 23 in which the sub-jective value of the endowment, Y e, is given by an expression of the form

exp

�� a�tY et

�= Ebt

"exp

�Z T

t

a

�sesds�

a

�TeT

!#; (20)

for properly de�ned a and b, extending a more special result by Musiela and Za-riphopoulou (2004). The proofs of the following claims are given in Appendix B:In addition to the assumptions of Proposition 23, we assume that there is nointermediate consumption (meaning that for every c 2 C, ct = 0 for t < T ), andin particular t = 0 for all t < T (which presents no problems since the strictpositivity of T implies that of �). Even though we have not treated the caseof no-intermediate consumption separately, the results are essentially the same(see Remark 10). We also assume that r and T > 0 are deterministic, andthat �N = 0. Within this setting we consider two special cases:Case 1 (idiosyncratic endowment). Suppose that e is adapted to the �ltra-

tion generated by BN ; and QNN (t) = aI; where a > 0 and I is an identitymatrix. Then �Y

e

M = 0 and equation (20) holds for any b such that bN = qN ,provided �b is a martingale and the stated integral exists.Case 2 (non-idiosyncratic endowment). Suppose that QNN is constant over

time; �M and qN are deterministic; and the endowment process satis�es

det = �tdt+ �t�0dBt;

where � 2 Rd, and � and � are real-valued and adapted to the �ltration gener-ated by e. Then equation (20) holds with a = �0NQNN�N=�

0� and b0 = �0 =[�0M ; q

0N ] : The solution of Musiela and Zariphopoulou (2004) is the subcase ob-

tained by assuming Q = I (time-additive utility), q = 0, Markovian dynamicsfor e and R, and a single lognormally distributed risky asset.

18

8 Incomplete Markets and Quadratic BSDEs

In incomplete but otherwise unconstrained markets, and under a quasi-quadraticsmooth absolute aggregator, the BSDE of the �rst-order conditions character-izing Y takes a quadratic form. A quadratic BSDE is also obtained in SS ina scale-invariant setting with di¤erent preference restrictions. As in SS, wesummarize below a set of conditions under which the quadratic BSDE can bereduced to a system of ordinary di¤erential equations (ODEs), leading to somenew tractable applications with nontradeable income. The technique can be ap-plied to BSDEs as well as the corresponding PDEs in a Markovian setting. ThePDE version of the approach is familiar from the a¢ ne term-structure literature(see Du¢ e, Filipovic, and Schachermayer (2003) and Piazzesi (2005)), and hasbeen utilized in scale-invariant consumption/portfolio problems (subsumed bythe formulation of SS) by Kim and Omberg (1996), Chacko and Viceira (1999),Schroder and Skiadas (1999), and Liu (2001).The following restrictions are imposed throughout this subsection:

Condition 26 Markets can be incomplete but there are no other constraints(� = Rm) ; the absolute aggregator is smooth (� = 0), and q = 0:

The assumption q = 0 is without loss of generality (via a Girsanov change-of-measure argument). Under these assumptions, the BSDE characterizing Yin the �rst-order conditions of Proposition 13 becomes

dYt = ��et + pt � rtYt +�Y 0t ht +

1

2�Y 0t Ht�

Yt

�dt+�Y 0t dBt; YT = eT ; (21)

where

pt = �tg��t; t�t

�+�t2

��Rt �

1

�t�R0t �

Rt %

�0 ��R0t Qt�

Rt

��1��Rt �

1

�t�R0t �

Rt %

�;

ht = ��Rt %

�t�Qt�Rt

��R0t Qt�

Rt

��1��Rt �

1

�t�R0t �

Rt %

�; and

Ht =1

�t

�Qt�

R��R0t Qt�

Rt

��1�R0Qt �Qt

�:

We now formulate a set of conditions that reduce BSDE (21) to an ODEsystem. We introduce a state process Z 2 L

�Rk�with dynamics

dZt = �Zt dt+ �

Z0t dBt; where �Z 2 L1

�Rk�; �Z 2 L2

�Rd�k

�:

Moreover, we split these state variables into two blocks, of dimensionality aand b; respectively, where a + b = k: We treat a and b as integers denotingdimensionality, as well as indices of corresponding matrix blocks, writing

Z =

�Za

Zb

�and �Z =

��Za �Zb

�;

19

where Za 2 L (Ra) ; Zb 2 L�Rb�; �Za 2 L

�Rd�a

�; and �Zb 2 L

�Rd�b

�:

We seek a solution of the form

Yt = Y0 (t) + Y1 (t)0Zt +

1

2Za0t Y2 (t)Z

at ; (22)

for some deterministic processes Y0 2 L (R) ; Y1 = [Y a01 ; Y b01 ]0 2 L�Rk�; and

Y2 2 L (Ra�a). A su¢ cient set of conditions for this type of solution is statedbelow (omitting time indices). We assume, without loss of generality, that allmatrices appearing in quadratic forms, including Y2 (t) above, are symmetric.

Condition 27 (a) The processes r and �Za are deterministic.(b) For some deterministic processes J0;K0 2 L (R) ; J1;K1 2 L

�Rk�; and

J2;K2 2 L (Ra�a),

e = J0 + J01Z +

1

2Za0J2Z

a and p = K0 +K10Z +

1

2Za0K2Z

a:

(c) For some deterministic processes L0 = [La00 ; Lb00 ]0 2 L

�Rk�; La1 2 L (Ra�a) ;

Lb1 2 L�Rb�k

�, and Lb2[i] 2 L (Ra�a) ;

�Z + �Z0h = L0 +

�La1Z

a

Lb1Z +�Za0Lb2[i]Z

a�i=1;:::;b

�:

(d) For some deterministic processes Daa0 2 L (Ra�a) ; Dab

0 2 L�Ra�b

�;

Dbb0 2 L

�Rb�b

�; Dab

1 [i; j] 2 L (Ra), Dbb1 [i; j] 2 Rk and Dbb

2 [i; j] 2 L (Ra�a),

�Z0H�Z =

�Daa0 Dab

0

Dab00 Dbb

0

�+

0

�Za0Dab

1 [i; j]�i=1;:::;a;j=1;:::;b�

Za0Dab1 [i; j]

�0i=1;:::;a;j=1;:::;b

�Z 0Dbb

1 [i; j]�i;j=1;:::;b

!

+

�0 00�Za0Dbb

2 [i; j]Za�i;j=1;:::;b

�:

We let D0 2 L�Rk�k

�denote the �rst term of the right-hand side.

Suppose Conditions 26 and 27 hold, � 2 L (Ra�a) is de�ned to have ith row

�i� =bXj=1

Y b1jDab1 [i; j]

0;

and the deterministic processes (Y0; Y1; Y2) solve the following ODE system

20

(where the left-hand sides denote time-derivatives):

_Y0 = rY0 � J0 �K0 � Y 01L0 �1

2Y 01D0Y1 �

1

2trace

�Y2�

Za0�Za�;

_Y1 = rY1 � J1 �K1 � Lb01 Y b1 �1

2

bXi=1

bXj=1

Y b1iYb1jD

bb1 [i; j]

��La01 Y

a1 +

Pai=1

Pbj=1 Y

a1iY

b1jD

ab1 [i; j] + Y2

�Daa0 Dab

0

�Y1 + Y2L

a0

0

�;

_Y2 = rY2 � J2 �K2 � Y2La1 � La01 Y2 � Y2Daa0 Y2 � Y2�� �0Y2

�bXi=1

bXj=1

Y b1iYb1jD

bb2 [i; j]� 2

bXi=1

Y b1iLb2[i];

with terminal conditions Yi (T ) = Ji (T ) ; i = 0; 1; 2:Direct computation using Ito�s lemma shows that then equation (22) de�nes

a solution to BSDE (21) (provided that the drift and di¤usion terms of (21) aresuitably integrable so that the respective integrals are well-de�ned).

Example 28 We assume that Condition 26 is satis�ed, that the processes Q;r; and are deterministic, that g� is state independent, and (without loss ofgenerality) that dR = �Rdt+ �R0MMdBM . These assumptions imply that % = 0;and that �t and g�(t; t=�t) are also deterministic. We also recall the price ofmarketed risk expression �M =

��R0MM

��1�R: The following two problem classes

satisfy Condition 27.

Class 1. Z = Za (a = k, b = 0), and

dZt = (�� �Zt) dt+�0dBt;

�M = � + V Zt; et = J0 + J1Zt +1

2Z 0tJ2Zt;

where � 2 Ra, � 2 Rd�a, � 2 Ra�a, � 2 Rm, V 2 Rm�a, J0 2 R, J1 2 Ra, andJ2 2 Ra�a.Class 2. Z = Zb (a = 0, b = k), Q satis�es Condition 18 with QNN = Ifor some positive constant , and

dZt = (�� �Zt) dt+�0diag�p

� + V Zt

�dBt;

�M = diag�p

�M + VM�Zt

�'; et = J0 + J

01Zt;

where diag(x) denotes the diagonal matrix with x on the diagonal,px de-

notes the vector with ith elementpxi; and � 2 Rb, � 2 Rd�b, � 2 Rb�b,

� = [�0M ; �0N ]0 2 Rd, V = [V 0M�; V

0N�]

0 2 Rd�b, ' 2 Rm, J0 2 R, J1 2 Rb.In this case, Y2 = 0, and only the �rst two Riccati equation are needed, thesecond unlinked to the �rst. This formulation includes the square-root volatilityspeci�cation of Heston (1993).

21

A Appendix: Translation-Invariant RecursiveUtility Representation

In this appendix we argue the necessity of the functional form of Condition 6for translation invariance relative to : We consider the dynamic utility thatassigns to each consumption plan c a utility process U (c) that is the uniqueprocess in U solving the BSDE (5) : We �x a bounded process 2 L (R++)throughout, and we assume that for every U 2 U and time t; there exists aconsumption plan c such that cs = Ut s for all s � t: We also recall that thedynamic utility c 7! U (c) is translation-invariant relative to ; or -TI forshort, if for any k 2 R, c1; c2 2 C; and t 2 [0; T ] ; Ut

�c1�= Ut

�c2�implies

Ut�c1 + k

�= Ut

�c2 + k

�:

The dynamic utility c 7! U (c) is in certainty equivalent form relative to ,or -CE for short, if at any time-t information node, Ut (c) represents the unitsof the cash �ow that result in the same time-t utility as the consumption planc. More formally, given any time t; let the set C t consist of all consumptionplans c such that cs= s = ct= t for all s � t: The dynamic utility U is de�nedto be -CE if for any time t and any c 2 C t ; Ut (c) t = ct: To interpret this,consider any c 2 C: By our earlier assumption, there exists some �c 2 C suchthat �cs = Ut (c) s for all s � t; and therefore �c 2 C t : If U is -CE, thenUt (�c) t = �ct = Ut (c) t; and therefore Ut (�c) = Ut (c) : In this sense, the cash�ow is the yardstick relative to which utility is cardinalized.Suppose now that c 7! U (c) is both -CE and -TI. Then the quasi-linearity

restriction (10) is satis�ed. To see this, �x any c 2 C, any time t; and any scalark: Select a �c 2 C such that �cs = Ut (c) s for s � t: Arguing as above, �c 2 C

t and

Ut (�c) = Ut (c) : Since U is -TI, it follows that Ut (�c+ k ) = Ut (c+ k ) : SinceU is -CE and �c+ k 2 C t ; Ut (�c+ k ) t = �ct + k t: Therefore, Ut (c+ k ) =Ut (�c+ k ) = �ct= t + k = Ut (c) + k; proving equation (10) :Let Uk = U (c+ k ) and U = U0 = U (c). Since Uk = U + k, the volatility

process of Uk is �U for any scalar k: The de�nition of U (c+ k ) implies

dUkt = �F�t; ct + k t; U

kt ;�

Ut

�dt+�U 0t dBt:

On the other hand, since Uk = U + k, the de�nition of U (c) gives

dUkt = �F�t; ct; Ut;�

Ut

�dt+�U 0t dBt;

implying F�t; ct + k t; Ut + k;�

Ut

�= F

�t; ct; Ut;�

Ut

�. The last equality holds

almost everywhere for any given value of k: Ignoring some uninteresting techni-calities regarding null events, we formally set k = �Ut and de�ne the functionG (!; t; x; y; z) = F (!; t; (!; t)x; 0; z) to �nd

F�t; ct; Ut;�

Ut

�= G

�t;ct t� Ut;�Ut

�:

Noting that UT = cT = T (U being -CE), we obtain the BSDE of Condition 6:

dUt = �G�t;ct t� Ut;�Ut

�dt+�U 0t dBt; UT =

cT T:

22

The above conclusions are isomorphic to corresponding conclusions in SSwith regard to homotheticity, via the transformations ~ct = exp (�ct= t) and~Ut (~c) = exp (�Ut(c)) ; in which case U is -TI if and only if ~U is homothetic,~U is 1-CE if and only if U is -CE, and ~U is homogeneous of degree one if andonly if U satis�es the quasi-linearity property (10). The dynamics

d ~Ut= ~Ut = � ~G�t; ~ct= ~Ut; ~�t

�dt+ ~�U 0t dBt; ~Ut = ~cT ;

in SS can be mapped to the above dynamics for U by letting G (!; t; x; y) =� ~G (!; t; exp (�x) ;�y)� y0y=2; and applying Ito�s lemma.

B Appendix: Proofs

B.1 Proof of Theorem 3

(a) Su¢ ciency: Given any feasible plan�~c; ~�; ~W

�and stopping time � , in-

tegration by parts, the dynamics of �, and the budget equation imply

�� ~W� � �0w0 =Z �

0

�t~�0t"tdt�

Z �

0

�t (~ct � et) dt+Z �

0

�t

��Rt~�t � ~Wt�t

�0dBt;

(23)and therefore

�� ~W� � �0w0 �Z �

0

�t�� ("t) dt�Z �

0

�t (~ct � et) dt+M1t ;

for a local martingale M1. Given condition (4), the same argument applied at(c; �;W ) gives

��W� � �0w0 =Z �

0

�t�� ("t) dt�Z �

0

�t (ct � et) dt+M2t ;

for another local martingale M2. Letting x = ~c � c, and M = M1 �M2, weconclude that

�� ~W� � ��W� � �Z �

0

�txtdt+M� :

Let f�n : n = 1; 2; : : : g be an increasing sequence of stopping times that con-verges to T and such thatM stopped at �n is a martingale. Taking expectationsin the last inequality, we �nd

0 � E�Z �n

0

�txtdt+ ��n~W�n � ��nW�n

�: (24)

The idea is to let n ! 1 to conclude that (�jx) � 0. There remains to justifythe interchange of limits and integration. For the integral of �x, we apply thedominated convergence theorem, using the assumptions �; x 2 H. For the term,

23

involving � ~W , we use the assumptions � 2 S2 and ~W� 2 S2, and Fatou�s lemmato conclude that

lim inft!1

Eh��n

~W�n

i� E

h�T ~WT

i:

Finally, the assumption �W 2 S1 implies that E [��nW�n ] converges to E [�TWT ].Taking the limit inferior on both sides of inequality (24), we therefore reach theconclusion (�jx) � 0.

(b) Necessity: By taking the accumulated value of a unit invested at timezero in the money market as the numeraire, we assume without loss in generality,that r = 0.We begin by showing that � is a martingale. Consider any time t < T

and any event F 2 Ft. Given any h 2 (t; T � t), consider the feasible plans(c+ x; �;W + z) and (c� x; �;W � z), where x is equal to the indicator functionof F � [t; t+ h], while z is given by dzs = �xsdt; z0 = 0: Since � is a state-pricedensity, it follows that (�jx) = 0, and therefore

E

"1F

1

h

Z t+h

t

�sds� �T

!#= 0:

Since � is continuous and � 2 S1; taking the limit implies that E [1F�t] =E [1F�T ]. Applying this condition over all t and F 2 Ft, implies that � is amartingale.Since � 2 S1 is a strictly positive Ito process and a martingale, there exists

� 2 L2�Rd�such that

d�t�t

= ��0tdBt:

De�ning "t = �Rt ��R0t �t; we complete the proof by showing that �0t"t = �� ("t) :We �rst consider a lemma suggestive of the result:

Lemma 29 Suppose that (c; �;W ) and (c+ x; �+ y;W + z) are feasible plans,yT = 0; y

0" 2 H and �z 2 S1. Then (�jx) = (�jy0") :

Proof. As in the su¢ ciency proof, for any feasible plan�~c; ~�; ~W

�and

stopping time � , equation (23) holds. Taking the di¤erence of the equationapplied at (c+ x; �+ y;W + z), and the equation applied at (c; �;W ), we �ndthat Z �

0

�txtdt+ ��z� =

Z �

0

�ty0t"tdt+M� ;

for some local martingale M . Consider now an increasing sequence of stoppingtimes f�n : n = 1; 2; : : : g that converges to T , such that M stopped at �n is amartingale. It then follows that

E

�Z �n

0

�txtdt+ ��nz�n

�= E

�Z �n

0

�ty0t"tdt

�; n = 1; 2 : : :

24

Feasibility of (c+ x; �+ y;W + z) implies that xT = zT , and therefore the left-hand side converges to (�jx), given the assumptions that �z 2 S1 and �; x 2 H.The right-hand side converges to (�jy0") since �; y0" 2 H.

Finally, we show that, for every process �+y valued in �; y0t"t � 0: Supposeinstead that there exists some process y such that ~� = �+ y is valued in � but

EhR T01 fy0t"t > 0g dt

i> 0. For each positive integer N , de�ne

yNt = yt1 f0 � y0t"t � N; kytk < Ng :

Since 1�yN 0t "t > 0

monotonically converges almost surely to 1 fy0t"t > 0g, it

follows that EhR T01�yN 0t "t > 0

dti> 0 for someN:We therefore assume, with-

out loss in generality, that y0" is non-negative and bounded, and y is bounded,which in turn implies that ~� is a trading plan. Similarly, for every positiveinteger N , de�ne the stopping time

�N = inf

�t :

Z t

0

y0sdRs > N or t = T

�;

and the process yNt = yt1 ft < �Ng : The same convergence argument used aboveshows that E

hR T01�yN 0t "t > 0

dti> 0 for some N . We therefore also assume,

without loss in generality, that the process z, de�ned by dzt = y0tdRt; z0 = 0,is bounded, and therefore W + z is a wealth process. Letting xt = 0 for t < T ,and xT = zT , it follows that (c+ x; �+ y;W + z) is a feasible plan. ApplyingLemma 29, it follows that (�jx) = (�jy0") > 0, implying that � cannot be astate-price density.

B.2 Proof of Theorem 9

(a) Su¢ ciency: We begin by con�rming the budget equation and that U =U (c) : Given the construction of U and � in (12), and equation (13) ; the wealthdynamics (14) are easily con�rmed to be equivalent to the budget equation (1).If dUt = : : : dt+�U 0t dBt, integration by parts applied to Y = U��W implies

�Ut =�Yt + �

Rt �

0t

�tand xt =

ct t� Ut = X

�t; t�t;�Ut

�: (25)

To verify that U = U (c) ; we use the above expression for �U ; and the fact that

G��t; t�t;�

�= G(t; xt;�)�

t�txt;

to rewrite the dynamics of Y as

dYt =�Ytrt � �00t �Rt � et � �tG

�t; xt;�

Ut

�+ txt +

��Rt %

�0�Ut

�dt+�Y 0t dBt:

25

Integration by parts applied to Y = U� �W , simpli�ed using the dynamicsof Y;�; and W; and the identities (12) and (13) ; results in the utility dynam-ics (11) ; con�rming that U = U (c) :Recalling that � = �0 + U% 2 � if and only if �0 2 �; we therefore have a

feasible plan (c; �;W ) ; with U = U (c) :We prove optimality of this plan by ap-plying Proposition 2, showing that the process � = E=� is both a supergradientdensity of U0 at c and a state-price density at c:Let �t = Fc

�t; ct; Ut;�

Ut

�: By Proposition 13 of Schroder and Skiadas

(2003), we know that if (FU ; F�) 2 L1 (R)�L2�Rd�is such that E (FU ; F�) 2 S2

and (FU (t) ; F� (t)) 2 (@U;�F )�t; ct; Ut;�

Ut

�for all t, and � = E (FU ; F�)� de-

�nes an element of H; then � is a supergradient density of U0 at c: The ag-gregator form implies �t =

�1t Gx

�t; xt;�

Ut

�; and therefore, by (25) ; � = 1=�:

The aggregator form also implies that (FU (t) ; F� (t)) 2 (@U;�F )�t; ct; Ut;�

Ut

�if

and only if FU (t) = �Gx�t; xt;�

Ut

�(= � t=�t) and F� (t) 2 @�G

�t; xt;�

Ut

�:

Putting the above observations together, con�rms that � is a supergradientdensity of U0 at c:Finally, we use part (a) of Theorem 3 to con�rm that � is a state-price

density at c: If d�t=�t = ��t dt+ ��0t dBt, then, using the fact that � = 1=�;

��t = �rt + t�t� %

0�Rt�t

+ ��0t ��t and ��t = �

�Rt %

�t:

Since trading in % is unrestricted, �� (") < 1 implies %0" = 0; and therefore%0�R=�� ��0�� = ��0G�; resulting in �� = �r + =�� ��0G�: Using this factand applying integration by parts to � = E�, we obtain condition (4) ; where� = �

�G� � ��1�R%

�:We have therefore veri�ed that � is a state-price density

at c; completing the optimality veri�cation.

(b) Necessity: We begin by proving that � = 1=�. Since G is smooth, byProposition 13 of Schroder and Skiadas (2003), � 2 E�; where �t = Gx (t; xt;�t) = t,de�nes a utility gradient density. By Proposition 2, � is also a state-price densityat c. Since trading the portfolio % is unrestricted,

�t =1

�tEt

"Z T

t

�s sds+ �T T

#:

This follows by a standard argument that we now outline. Suppose, withoutloss of generality (by taking one unit invested initially in the money marketas the numeraire), that r = 0. We have shown in the proof of part (b) ofTheorem 3 that � is a martingale. Suppose now that at time t and event F 2 Ftthe agent borrows from the money market to buy portfolio %, consuming theensuing dividend stream , and paying back the loan �t by reducing terminalconsumption. The resulting feasible incremental cash �ow x is given by xs = 0for s < t, xs = 1F s for s 2 (t; T ), and xT = 1F ( T � �t). Since �x is alsofeasible, we have

0 = (�jx) = E"1F

Z T

t

�s sds+ �T ( T � �t)!#

:

26

Since F 2 Ft is arbitrary, it follows that EthR Tt�s sds+ �T T

i= �tEt [�T ] =

�t�t, as claimed.Taking the derivative of the function f(k) = U(c+ k ) at zero, we obtain

Etf 0t(0) = Et

"Z T

t

�s sds+ �s T

#= �t�t:

Since U(c + k ) = k + U(c), it also the case that f 0(0) = 1, and therefore� = E=� = 1=�:Having shown the key fact that � = 1=�, we de�ne the process Y = U��W ,

and apply Ito�s lemma, using the dynamics of U and �, the de�nition of G� andX , as well as the budget equation, to obtain

dYt = Utd�t + �tdUt +�U 0 ��R%� dt� dWt

=

�Ytrt � �0t�Rt +�U 0t �Rt %� �tG�

�t; t�t;�Ut

�� et

�dt

+��t�

Ut � �R�0t

�0dBt

The di¤usion term con�rms the expression for �U in (25), which when substi-tuted into the drift term results in the dynamics of Y in Condition 8. The restof the condition follows easily from the necessary conditions of Theorem 3 for �to be a state-price density at c:

B.3 Proof of Proposition 13

Substituting G� = ��q + � � +Q�U

�; � 2 @

���U �� ; into the equation for "in Condition 8 gives

" = �R � �R0�q + � � + �

R%

�+Q�U

�:

The formula for �0 is obtained after substituting �U = ��1��Y + �R�0

�. Com-

puting the inner product of " and �0 results in

�00�R = �� (") + �00�R0

�q + � � + ��1�R%+Q�U

�:

Letting dY = �Y dt+�Y 0dB, we substitute for G� in Condition 8 to �nd

�Y = Y r � �00�R � e� �g� +��R%

�0�U + �

�q0�U + �0

���U ��+ 12�U 0Q�U

�:

Using the fact that �0���U �� = �U 0 (� �) and the above expression for �00�R,

the drift term of Y becomes �Y = Y r � e� �g� � �� (") +Q; where

Q =���U 0 � �00�R0

� �q + � � + ��1�R%

�+1

2��U 0Q�U � �00�R0Q�U

= �Y 0�q + � � + ��1�R%

�+1

2��1

��Y 0Q�Y � �00�R0Q�R�0

�:

27

B.4 Proof of Proposition 21

The expression for �0 in (18) implies

��1Qii��Y + �R�0

�i= �i � �i�i; i 2M:

Since �i 2 @���Y + �R�0��, there are three possibilities: (1) ��Y + �R�0�

i> 0,

�i = 1, �i > �i; (2)��Y + �R�0

�i= 0, �i = �i�i; and (3)

��Y + �R�0

�i< 0,

�i = �1, �i < ��i. In all three cases, �M �M = max fmin f�M ; �Mg ;��Mg.Since

��Y + �R�0

�N= �YN , we have �N 2 @

���YN �� and therefore �Y 0N (�N �N ) =�Y 0N

��N sign(�YN )

�. Finally, from �U = Y + W; we get ��UM = �YM +

�RMM�0 = � (QMM )

�1�M .

B.5 Proof of Proposition 23

Substituting the expression for �0 in (18) into the equation for Y in Proposition13, we �nd

dY =

�Y r � e� �g�

� �

�� �2�0Q�1MM �+

1

2��Y 0N QNN�

YN

�dt

+�Y 0�dB +

��M

qN + �N �N

�dt

�:

Suppose Y et satis�es the stated BSDE and �MN = 0. Then direct computation

shows that YMt = Y � Y et satis�es the claimed BSDE.

B.6 Proof of Example 25

Only the proof Case 2 is presented. The proof of Case 1 is similar. The assump-tions of the example imply that e is the only source of uncertainty driving Y et ,and therefore �Y

e

t = �t� for some scalar-valued process �. Therefore,

dY et = ��et � Y et rt �

a

2�t�2t�

0�

�dt+ �t�

0dB�t :

De�ning zt = exp (�aY et =�t), we have

dztzt

= � a�tdY et +

1

2

�a

�t

�2�2t�

0�dt� Y et d�a

�t

�=

a

�t(et � Y et rt) dt�

a

�t�t�

0dB�t � Y et d�a

�t

�= �aY et

�d

�1

�t

�+rt�tdt

�+a

�tetdt+ �

z0t dB

�t ;

where �zt = ���t�e=�t. Since r and are deterministic,

d

�1

�t

�= �

�rt � t=�t

�t

�dt;

28

and therefore z solves the BSDE:

dztzt=1

�t(aet � t log(zt)) dt+ �z0t dB�t ; zT = exp

�� a

�TeT

�:

Under the assumption that t = 0; t < T; the equation for z in (20) is easilycon�rmed.

B.7 Proof of Proposition 16 and Example 17

We derive the consumption dynamics under the more general absolute aggrega-tor form

G (!; t; x;�) = g(!; t; x) +H (!; t;�) ;

where gx(t; x) is assumed state independent. Letting xt = ct= t � Ut; andapplying Ito�s lemma to gx (t; xt) = t=�t; gives

gxt (t; xt) dt+ gxx (t; xt) dxt +1

2gxxx (t; xt) (dxt)

2= d

� t�t

�: (26)

Squaring this expression results in

(dxt)2=

�d ( t=�t)

gxx (t; xt)

�2=

1

A2t� =�0t �

=�t :

Matching the di¤usion terms in (26) and rearranging, we �nd that

1

t�ct =

1

At�Rt %

�t+

�ct t� 1

At

�� t +�

Ut : (27)

Substituting �U = ��1��Y + �R�� �R%U

�; and rearranging gives the formula

for �c.The formula for �ct is obtained similarly by matching the drift terms in

equation (26), and the equilibrium short rate is obtained by inverting and solvingfor r as a function of �ct . The risk premium expression is derived by solving (27)for �U ; and substituting into

�Rt = ��R0t�H�

�t;�U

�� �

Rt %

�t

�;

which follows from the �rst-order conditions with " = 0: The quasi-quadraticcase corresponds to H�

�t;�U

�= �Qt�U .

29

C Appendix: Dynamic Programming Approach

In this appendix we provide an alternative dynamic programming veri�cationargument in the translation-invariant setting, complementing the veri�cationargument given in Schroder and Skiadas (2003) for the scale-invariant case. Asin the utility gradient approach, a Markovian structure is not required. Theappendix e¤ectively provides an alternative proof of part (a) of Theorem 9.Neither proof is simpler than the other, their main steps can be matched stepfor step.The translation-invariance Condition 6 is assumed through the appendix.

To informally motivate the argument, we let Jt (w) be the agent�s time-t opti-mal utility value given time-t �nancial wealth w. We conjecture that Jt (w) =(Yt + w) =�t, where � is de�ned in Condition 6, and Y is an Ito process thatdoes not depend on wealth, and whose dynamics are denoted

dYt = �Yt dt+�

Y 0t dBt; YT = eT :

Using hats to denote (candidate) optimal quantities, suppose that�c; �; W

�is

an optimal plan, with corresponding utility process U = U (c) ; and de�ne

xt =ct t� Yt + Wt

�tand �

0= �� Yt + Wt

�t%: (28)

Optimality requires that Ut = Jt�Wt

�=�Yt + Wt

�=�t. Applying Ito�s lemma,

using the budget equation and the dynamics for �, we conclude that

�tdUt = Dt�xt; �

0

t

�dt+

��Yt + �

Rt �

0

t

�0dBt;

where

Dt�xt; �

0t

�= �Yt � rtYt + �00t �Rt + et �

��Rt %

�0��Yt + �Rt �0t�t

�� txt:

Comparing to the dynamics of U ; we conclude that at the optimum

Dt�xt; �

0

t

�+ �tG

t; xt;

�Yt + �Rt �

0

t

�t

!= 0: (29)

Given any other feasible plan (c; �;W ), and letting

xt =ct t� Yt +Wt

�tand �0 = �� Yt +Wt

�t%; (30)

the usual dynamic programming argument suggests the inequality

Dt�xt; �

0t

�+ �tG

�t; xt;

�Yt + �Rt �

0t

�t

�� 0: (31)

The theorem below veri�es the above conjectures, using the following technicalregularity condition:

30

Condition 30 Given any feasible plan (c; �;W ), the process J � (Y +W ) =�is in S2, and there exist processes (FU ; F�) such that

(FU (t) ; F� (t)) 2 (@U;�F )�ct; Jt;�

Jt

�and E (FU ; F�) 2 S2:

Theorem 31 Suppose Conditions 6 and 30 hold. If the feasible plan�c; �; W

�satis�es equation (29), where

�x;�

0�is de�ned in (28), while any other feasible

plan (c; �;W ) satis�es inequality (31), where�x; �0

�is de�ned in (30), then�

c; �; W�is optimal.

Proof. Consider any feasible plan (c; �;W ), and let U = U (c) 2 U � S2and J = (Y +W ) =� 2 S2. We will show that U0 � J0, with equality holding if(c; �;W ) =

�c; �; W

�.

Applying Ito�s lemma to J and using the budget equation (14) and thedynamics of �, we �nd

dJt =1

�t

��Yt � rtYt + �00t �Rt + et �

��Rt %

�0�Jt � txt

�dt+�J0t dBt

�Jt =1

�t

��Yt + �

Rt �

0t

�:

Let p1 be the nonnegative (by inequality (31)) process such that

0 =1

�t

��Yt � rtYt + �00t �Rt + et �

��Rt %

�0�Jt � txt

�+G

�t; xt;�

Jt

�+ p1t :

Then the above dynamics for J can be written as

dJt = ��F�t; ct; Jt;�

Jt

�+ p1t

�dt+�J0dBt:

On the other hand, dUt = �F�t; ct; Ut;�

Ut

�dt + �U 0t dBt: Letting y = U � J;

and z = �U � �J ; we therefore have

dyt = ��F�t; ct; Ut;�

Ut

�� F

�t; ct; Jt;�

Jt

�� p1t

�dt+ z0tdBt

Selecting (FU ; F�) 2 (@U;�F )�c; J;�J

�as in Condition 30,

F�t; ct; Ut;�

Ut

�� F

�t; ct; J;�

J�= FU (t) yt + F� (t) zt � p2t ;

for some nonnegative process p2. Therefore, with p = p1 + p2;

dyt = ��FU (t) yt + F� (t)

0zt � pt

�dt+ z0tdBt:

Letting E = E (FU ; F�) 2 S2, integration by parts gives

d (Etyt) = Etdyt + ytdEt + dytdEt = Etptdt+ dMt;

31

for a local martingaleM . Let f�Ng be a corresponding localizing stopping timesequence converging to T almost surely. Then

y0 = E

��Z �N

0

Espsds+ E�N y�N�� E [E�N y�N ] :

The last term converges since Ey 2 S1, and therefore y0 � E [yTET ] = 0 (sinceyT = 0). We have proved that U0 � J0.Repeating the argument for

�c; �0;W

�=�c; �

0; W�and p1 = 0, gives U0 =

J0, proving optimality of�c; �; W

�, and completing the proof.

Conditions (29) and (31) are clearly implied by the Bellman equation

0 = maxx2R; �02 �

Dt�x; �0

�+ �tG

�t; x;

�Yt + �Rt �

0

�t

�; (32)

with the optimal xt and �0

t providing the maximizing values. We concludeby relating the Bellman equation to the �rst-order conditions of optimality asstated in Condition 8. For the sake of brevity, we assume that G (!; t; �) issmooth. (The analogous argument applies using the supergradient of G.)Maximization with respect to x is equivalent to

xt = X t; t�t;�Yt + �

Rt �

0

t

�t

!:

Substituting this expression into equation (29) shows that �Yt matches the driftof Y in Condition 8.The gradient of the function maximized in (32) with respect to �0 at �

0is

"t = �Rt + �

R0t

G�

t; xt;

�Yt + �Rt �

0

t

�t

!� 1

�t�Rt %

!:

Maximization over �0 2 � implies that��0 � �

0�0" � 0 for every �0 2 �, a

condition that is equivalent to �00" = �� (").

Based on the above observations, one can now easily con�rm that the �rst-order conditions of Condition 8 imply the Bellman equation (32), and thereforethe above veri�cation argument applies.

32

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