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http://www.econometricsociety.org/ Econometrica, Vol. 76, No. 4 (July, 2008), 841–907 EQUILIBRIUM IN CONTINUOUS-TIME FINANCIAL MARKETS: ENDOGENOUSLY DYNAMICALLY COMPLETE MARKETS ROBERT M. ANDERSON University of California at Berkeley, Berkeley, CA 94720-3880, U.S.A. ROBERTO C. RAIMONDO University of Melbourne, Victoria 3010, Australia The copyright to this Article is held by the Econometric Society. It may be downloaded, printed and reproduced only for educational or research purposes, including use in course packs. No downloading or copying may be done for any commercial purpose without the explicit permission of the Econometric Society. For such commercial purposes contact the Ofﬁce of the Econometric Society (contact information may be found at the website http://www.econometricsociety.org or in the back cover of Econometrica). This statement must the included on all copies of this Article that are made available electronically or in any other format.

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Econometrica, Vol. 76, No. 4 (July, 2008), 841–907 EQUILIBRIUM IN CONTINUOUS-TIME FINANCIAL MARKETS: ENDOGENOUSLY DYNAMICALLY COMPLETE MARKETS 1 BY ROBERT M. ANDERSON AND ROBERTO C. RAIMONDO We prove existence of equilibrium in a continuous-time securities market in which the securities are potentially dynamically complete: the number of securities is at least one more than the number of independent sources of uncertainty. We prove that dy- namic completeness of the candidate equilibrium price process follows from mild ex- ogenous assumptions on the economic primitives of the model. Our result is universal, rather than generic: dynamic completeness of the candidate equilibrium price process and existence of equilibrium follow from the way information is revealed in a Brownian ﬁltration, and from a mild exogenous nondegeneracy condition on the terminal security dividends. The nondegeneracy condition, which requires that ﬁnding one point at which a determinant of a Jacobian matrix of dividends is nonzero, is very easy to check. We ﬁnd that the equilibrium prices, consumptions, and trading strategies are well-behaved functions of the stochastic process describing the evolution of information. We prove that equilibria of discrete approximations converge to equilibria of the continuous-time economy. KEYWORDS: Dynamic completeness, convergence of discrete-time ﬁnance models, continuous-time ﬁnance, general equilibrium theory. 1. INTRODUCTION IN AN ARROW–DEBREU MARKET, agents are allowed to shift consumption across states and times by trading a complete set of Arrow–Debreu contingent claims. Mas-Colell and Richard (1991), Dana (1993), and Bank and Riedel (2001) proved existence of equilibrium in a continuous-time Arrow–Debreu market. By contrast, in a securities market, agents are restricted to trading a prespec- iﬁed set of securities. The securities market is said to be dynamically complete 1 This paper is dedicated to the memory of Gerard Debreu and Shizuo Kakutani. We are very grateful to Jonathan Berk, Theo Diasakos, Darrell Dufﬁe, Steve Evans, Chiaki Hara, Ken Judd, Felix Kubler, John Quah, Paul Ruud, Jacob Sagi, Larry Samuelson, Karl Schmedders, Chris Shannon, Bill Zame, and four anonymous referees for very helpful discussions and comments. Versions of this paper have been presented, starting in 2005, at the National University of Sin- gapore, the Conference in Memory of Gerard Debreu at Berkeley, Arizona State University, the 21 COE Conference at Keio University, the International Congress on Nonstandard Methods in Pisa, the Australasian Meetings of the Econometric Society, the Finance Lunch at the Haas School of Business, the Berkeley–Stanford Theory Workshop, Kyoto University, Seoul National University, and UC Davis; we are grateful to all participants for their comments. The work of both authors was supported by Grant SES-9710424, and Anderson’s work was supported Grant SES-0214164, from the U.S. National Science Foundation. Raimondo’s work was supported by Grant DP0558187 from the Australian Research Council. Anderson’s work was also generously supported by the Coleman Fung Chair in Risk Management at UC Berkeley. Anderson gratefully acknowledges the support and hospitality of the University of Melbourne and the hospitality of the Korea Securities Research Institute. Both authors gratefully acknowledge the support and hospitality of the Institute for Mathematical Sciences, National University of Singapore. 841

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842 R. M. ANDERSON AND R. C. RAIMONDO if agents can, by rapidly trading the given set of securities, achieve all the con- sumption allocations they could achieve in an Arrow–Debreu market. When the uncertainty is driven by a Brownian motion, markets are said to be poten- tially dynamically complete if the number of securities is at least one more than 2 the number of independent sources of uncertainty. A securities market which is potentially dynamically complete may or may not be dynamically complete. Existence of equilibrium in continuous-time ﬁnance models with a single agent has been established in a number of papers (Bick (1990), He and Leland (1995), Cox, Ingersoll and Ross (1985), Dufﬁe and Skiadas (1994), Raimondo 3 (2002, 2005) ). Surprisingly, the existing literature does not contain a satisfactory theory of the existence of equilibrium in continuous-time securities markets with more than one agent. With dynamic incompleteness, essentially nothing is known. While we hope that some of the techniques developed in this paper will help in addressing the dynamically incomplete case, we do not here present a result in that case. This paper and the existing literature all deal with the case in which markets are potentially dynamically complete. Every previous paper assumes, in one form or another, that the candidate equilibrium price process is dynamically complete and shows that this implies that the candidate equilibrium is, in fact, an equilibrium; see Basak and Cuoco (1998), Cuoco (1997), Dana and Jean- blanc (2002), Detemple and Karatzas (2003), Dufﬁe (1986, 1991, 1995, 1996), Dufﬁe and Huang (1985), Dufﬁe and Zame (1989), Karatzas, Lehoczky, and Shreve (1990), and Riedel (2001). The form of the assumption varies. Some assume directly that the dispersion matrix of the candidate equilibrium price process is almost surely nonsingular, which is well known to be equivalent to dynamic completeness. Others assume that the dispersion matrix of the can- didate equilibrium price process is uniformly positive deﬁnite; this is stronger than dynamic completeness and it fails for an open set of primitives in our model. Still others assume that the candidate equilibrium price of consump- tion is uniformly bounded above and uniformly bounded away from zero, but this is inconsistent with unbounded consumption (found in the Black–Scholes and most other continuous-time ﬁnance models) and standard utility functions. Finally, some construct securities which sufﬁce to complete the markets, rather than taking the securities’ dividends as given by the model. But the candidate equilibrium price process is determined from the eco- nomic primitives of the model by a ﬁxed point argument, which makes it im- possible, except in knife-edge special cases, to determine from the primitives whether or not the dynamic completeness assumption is satisﬁed. Thus, if we 2 With non-Brownian processes, such as Lévy processes that allow jumps, the number of secu- rities needed for potential dynamic completeness may be larger. 3 All of these papers except Raimondo (2002, 2005) require one or more endogenous assump- tion(s) that is (are) not expressed solely in terms of the primitives of the model.

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EQUILIBRIUM IN FINANCIAL MARKETS 843 consider one of these models and choose speciﬁc utility functions, endowment, and dividend processes, except in the rare cases where we can solve explicitly for the candidate equilibrium, we cannot apply any of the previous theorems to determine whether or not an equilibrium exists. Indeed, the previous results do not rule out the possibility that equilibrium generically fails to exist; see the example in Section 3. While dynamic completeness plays a role in proving the existence of equi- librium, its main application in ﬁnance is to derivative pricing. Given a dynam- ically complete securities price process, options and other derivatives can be uniquely priced by arbitrage arguments and can be replicated by trading the underlying securities. With dynamic incompleteness, arbitrage considerations do not determine a unique option price and replication is not possible. The previous results provide no guarantee that equilibrium prices will support the standard theory of pricing and replicating options, which depends on dynamic completeness. In this paper, we prove that the candidate equilibrium price process is, in fact, dynamically complete and that the candidate equilibrium is, in fact, an equilib- 4 rium. Dynamic completeness of the candidate equilibrium price process and the existence of equilibrium follow from the way information is revealed by a Brownian motion and from a mild exogenous nondegeneracy condition on the terminal security dividends. To motivate our nondegeneracy condition, suppose we are given a mar- ket with K independent sources of uncertainty and K + 1 securities labeled j = 0K , so the market is potentially dynamically complete. Now suppose that two of the securities are perfect clones of each other, in that they pay exactly the same dividends at all nodes. Clearly, this is the same as a market with K sources of uncertainty and K securities, so it cannot possibly be dy- namically complete. Similarly, if the dividend processes of the securities were linearly dependent, we could not possibly have dynamic completeness. Thus, we need to assume that the dividend processes are not linearly dependent. In our model, the dividend of security j at the terminal date T is given as a func- tion Gj(β(T ω)) of the terminal realization β(T ω) of the Brownian motion in state ω. In the important special case in which security 0 is a zero-coupon 1 bond, our condition requires that G1G K be C functions on some open K set V ⊂ R and that the Jacobian determinant of (G1G K) be nonzero at one point x ∈ V . In particular, our assumption depends only on the securities 4 More speciﬁcally, we show that any consumption which is adapted to the Brownian ﬁltration and has ﬁnite value at the candidate equilibrium price process can be replicated by an admissible self-ﬁnancing trading strategy. We allow the ﬁltration in the original continuous-time economy to be larger than the Brownian ﬁltration, and the ﬁltration in the Loeb-measure economy we construct is always larger than the Brownian ﬁltration. This does not pose a problem for the existence of equilibrium because any consumption adapted to the larger ﬁltration is a mean- preserving spread of a consumption adapted to the Brownian ﬁltration, so agents’ demands are always adapted to the Brownian ﬁltration.

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844 R. M. ANDERSON AND R. C. RAIMONDO dividends at the terminal date, not on the other economic primitives; changing the utility function or endowment of an agent or the initial ownership of the securities has no effect on the existence of equilibrium. Clearly, the nondegen- eracy condition is generically satisﬁed. Moreover, one can easily tell whether the condition is satisﬁed for any particular value of the economic primitives simply by checking whether a determinant is nonzero at least at one point; this contrasts with the situation in most generic results, in which one knows that the result holds except on a small set of primitives, but it is hard to tell whether the result holds for any speciﬁc value of the primitives. If there are just enough securities for potential dynamic completeness, then some form of linear independence of the securities dividends is a necessary condition for dynamic completeness of the Arrow–Debreu securities prices. Thus, some form of linear independence of the dividends is implicitly assumed in all the previous papers. We chose to place our nondegeneracy assumption on the lump terminal dividends because it is convenient to do so. Not all of the previous papers have lump terminal dividends; we believe it should be possible to place the assumption instead on the intermediate ﬂow dividends, although the statement of the assumption would be more complex. We obtain explicit formulas for the equilibrium price process and its disper- sion matrix, each trader’s equilibrium securities wealth, and the dispersion ma- trix for each traders’ equilibrium trading strategy in terms of the equilibrium consumptions; each trader’s equilibrium trading strategy can then be calcu- lated using linear algebra. These formulas are expressed in terms of the equi- librium consumptions, which are not known a priori. However, since the equi- librium is Pareto optimal, there is a vector of Negishi (1960) utility weights λ such that, at each node, the equilibrium consumptions maximize the weighted sum of the utilities of the agents. Thus, the key features of the equilibrium can be calculated explicitly from knowledge of the primitives of the model (endow- ments and utility functions of the individuals, and the dividends of the secu- rities) and the equilibrium utility weights. Moreover, even if the equilibrium utility weights are not known, the explicit nature of the formulas can poten- tially be used to establish general properties of equilibria, such as comparative statics results. We prove that all key elements of equilibria of discrete approximations converge to the corresponding elements of equilibria of the continuous-time 5 model. This is important for two reasons: First, many people regard the dis- crete models as the appropriate models for “real” economies. In particular, ﬁnancial transaction data are very high frequency, but nonetheless discrete. 5 When the continuous-time model has multiple equilibria, it is possible that the equilibria of the discrete approximations will be near one continuous-time equilibrium for some approxima- tions and near a different continuous-time equilibrium for other approximations, so the sequence of equilibria of the discrete approximations converges to the set of equilibria of the continuous- time economy, rather than to a single equilibrium of the continuous-time economy.

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EQUILIBRIUM IN FINANCIAL MARKETS 845 The theoretical literature has focused on continuous time because the formu- las are much simpler in continuous time than in discrete time. Cox, Ross, and Rubinstein (1979) showed that one can compute the prices of options when the price of a stock is given by a geometric binomial random walk and showed that the discrete option prices converge to those given by the Black–Scholes formula. Our convergence theorems tell us that the equilibrium pricing for- mulas from the continuous-time model apply asymptotically to equilibria of the discrete models as the discretization gets ﬁner. This shows that the for- mulas are applicable to discrete models. It also shows that the formulas can be used in the econometric analysis of data which are high frequency but nonethe- 6 less discrete. Second, since we show that the equilibria of the discretizations are close to equilibria of the continuous-time economy, algorithms to compute 7 equilibria for discrete economies provide a means to compute equilibria of the continuous-time economy. Our starting point is a continuous-time model, on which we state our exis- tence theorem, Theorem 2.1, and our characterization of the elements of the equilibrium, Proposition 2.2. A function is said to be real analytic if, at every point in its domain, there is a power series which converges to the function on an open set containing the point. We assume that the primitives of the economy are given by real analytic functions of time and the current value of the Brown- 8 ian motion for times t ∈ (0T) , where T is the terminal date. The assumption that utility functions are analytic is not problematic; most of the utility func- tions commonly studied are in fact analytic. Option payoffs are not analytic because of the kink when the stock price equals the exercise price of the op- tion. However, options can be handled under certain conditions; see the two paragraphs preceding Equation (1) in Section 2. The assumption that the div- idends at time t are a function of t and the value of the Brownian motion at time t is discussed in Section 6. 6 As noted above, if the continuous-time economy has multiple equilibria, equilibria of the discrete approximation economies converge to the set of continuous-time equilibria, rather than to a single equilibrium. Nonetheless, this tells us that the formulas given in Proposition 2.2 apply approximately to the equilibria of the discretizations. For example, Equation (6) gives a formula for the dispersion matrix of the securities prices in terms of the Brownian motion, the future dividends, and the endogenously determined prices of consumption. The convergence results in Theorem 4.2 imply that Equation (6) holds asymptotically over any sequence of equilibria of the discrete approximations, using the prices of consumption in the given discrete approximations. 7 See, for example, Brown, DeMarzo, and Eaves (1996) and Judd, Kubler, and Schmedders (2000, 2002, 2003). 8 We also require that the utility functions satisfy Inada conditions (ruling out Constant Ab- solute Risk Aversion (CARA) utility) and be differentiably strictly concave (ruling out risk neu- trality). It is important to emphasize that the primitives of the model, and as we will ﬁnd, the equilibrium, are analytic functions of time and the Brownian motion, not analytic functions of time alone. Brownian motion is almost surely nowhere differentiable and almost surely of un- bounded variation on every interval of time. Consequently, the equilibrium prices and trading strategies are nowhere differentiable, and of unbounded variation, as functions of time.

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846 R. M. ANDERSON AND R. C. RAIMONDO We discretize the continuous-time model to construct a sequence of mod- ˆ 9 els; in each, we replace the Brownian motion β by a random walk β. A naive discretization would approximate the K-dimensional Brownian motion by a K K-dimensional random walk in which each node has 2 successors, and the random walk moves independently in each direction at each node. However, K in a discrete random walk in which each node has 2 successors, one needs K K at least 2 securities to obtain dynamic completeness; for K > 1, K + 1 < 2 and the discrete model cannot possibly be dynamically complete. So that the discrete approximations correspond closely to the continuous-time model it is critical that the discrete model have the same number of securities as the 10 continuous-time model and that it be dynamically complete. Thus, we con- ˆ struct a K-dimensional random walk β in which each nonterminal node has exactly K + 1 successors. Endowments, dividends, and utility functions are induced on the discrete economies from the speciﬁcation of the continuous- time economy. These discrete economies are general equilibrium incomplete markets (GEI) models; the book by Magill and Quinzii (1996) is an excellent reference on GEI models. Endowments and dividends are perturbed as neces- sary to ensure the existence of a dynamically complete equilibrium under the Dufﬁe–Shafer (1985, 1986) or Magill–Shafer (1990) theorems. We then prove our convergence theorem, Theorem 4.2, that equilibria of the discrete approx- imations converge to equilibria of the continuous-time economy. Since equilibrium prices are arbitrage-free, the equilibrium securities prices at the terminal date in the discrete model must be given by the exogenously speciﬁed dividends at the terminal date. It is well known that the equilib- rium securities price process at time t is the conditional expectation of fu- ture dividends, valued at the Arrow–Debreu equilibrium prices of consump- tion; this comes from the ﬁrst-order conditions for utility maximization (see, for example, Magill and Quinzii (1996) in the discrete case). We show that because of the smoothness properties of the Gaussian distribution, such con- ˆ ditional expectations are given by real analytic functions of (t β(t ω)) ∈ K (0T) × R , so the equilibrium securities price process is a real analytic func- ˆ tion of (t β(t ω)) . Moreover, the dispersion matrix of the securities prices ˆ and its associated determinant is a real analytic function of (t β(t ω)) ; this is a property of the way information is revealed by a Brownian motion and does not depend on any speciﬁc assumptions on the functional form of the primi- tives. 9 Zame (2001) considered a model in which time is discretized but the probability space is not, so each step of the random walk is normally distributed. In that setting, he found that the discrete-time approximation does not converge to the continuous-time limit. Raimondo (2001) studied the existence of equilibrium in models with discrete time and a continuum of states. 10 If one were to extend these methods to the analysis of dynamically incomplete continuous- time models, it would be critical to ensure that the discrete approximation has the same degree of incompleteness as the continuous-time model, so the random walk would have to be chosen carefully in that context also.

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EQUILIBRIUM IN FINANCIAL MARKETS 847 We use nonstandard analysis to project the discrete model back into the continuous-time model and ﬁnd that the dispersion matrix of the continuous- time securities prices is the projection of the dispersion matrix in the discrete model, and hence is real analytic. A real analytic function cannot be zero on a set of positive measure unless it is identically zero. Since the determinant as- sociated with the dispersion matrix is nonzero on a set of positive measure, it must be nonzero except on a set of measure zero. This implies that the disper- sion matrix has rank K except on a set of measure zero, but it is well known that this condition is equivalent to dynamic completeness; thus, we have shown that the candidate equilibrium prices in the continuous-time model are dynam- ically complete. We verify that the projected prices are equilibrium prices in the continuous-time model. Our proof depends heavily on nonstandard analysis and, in particular, on the nonstandard theory of stochastic processes. Nonetheless, the statements of our main results, Theorems 2.1 and 4.2 and Proposition 2.2, can be understood without any knowledge of nonstandard analysis. Nonstandard analysis provides powerful tools to move from discrete to con- tinuous time, and from discrete distributions like the binomial to continuous distributions like the normal; in particular, it provides the ability to transfer computations back and forth between the discrete and continuous settings. Our sequence of discrete approximations extends to a hyperﬁnite approxima- tion, one which is inﬁnite but has all the formal properties of ﬁnite approxi- mations. In particular, the hyperﬁnite approximation has a GEI equilibrium which is dynamically complete in the hyperﬁnite model. We then use non- standard analysis to produce a candidate equilibrium in the continuous-time model, show that the equilibrium in the hyperﬁnite model is inﬁnitely close to the candidate equilibrium in the continuous-time model, verify that the candi- date prices are dynamically complete, and are in fact equilibrium prices. Anderson (1976) provided a construction for Brownian motion and Brown- ian stochastic integration using Loeb (1975) measure—a measure in the usual standard sense produced by a nonstandard construction. Anderson’s Brown- ian motion is a hyperﬁnite random walk which can simultaneously be viewed as being a standard Brownian motion in the usual sense of probability theory. While the standard stochastic integral is motivated by the idea of a Stieltjes integral, the actual standard deﬁnition of the stochastic integral is of neces- sity rather indirect because almost every path of Brownian motion is of un- bounded variation, and Stieltjes integrals are only deﬁned with respect to paths of bounded variation. However, a hyperﬁnite random walk is of hyperﬁnite variation, and hence a Stieltjes integral with respect to it makes perfect sense. Anderson showed that the standard stochastic integral can be obtained readily from this hyperﬁnite Stieltjes integral. We modify Anderson’s construction of the hyperﬁnite random walk to a ran- dom walk with branching number equal to K + 1 and extend the results on stochastic integration to that random walk. We show that equilibrium con- sumptions are nonzero at all times and states. Consequently, we can use the

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848 R. M. ANDERSON AND R. C. RAIMONDO ﬁrst-order conditions to characterize the equilibrium prices. Then we use the Loeb measure construction to produce a candidate equilibrium of the original continuous-time model. The central limit theorem then allows us to explicitly describe the candidate equilibrium prices as integrals with respect to a normal distribution; however, with more than one agent, the prices depend on the ter- minal distributions of wealth, which are not described in closed form. We show that the hyperﬁnite equilibrium is inﬁnitely close to the candidate equilibrium, which implies that the equilibria of the discrete approximations converge to candidate equilibria of the continuous-time model. Finally, in a process analo- gous to that ﬁrst used in Brown and Robinson (1975), we show that the candi- 11 date equilibrium is an equilibrium of the continuous-time economy. 2. THE MODEL In this section we deﬁne the continuous-time model. There is a single consumption good. Trade and consumption occur over a compact time interval [0T ], endowed with a measure ν which agrees with Lebesgue measure on [0T) and such that ν({T }) = 1. Consumption and div- idends on [0T) are ﬂows; consumption at the terminal date T is a lump. We choose this formulation to give a ﬁnite-horizon model in which the securities will have positive value at the terminal date T ; if securities paid only a ﬂow dividend on [0T ], they would expire worthless at time T and it would not be possible to formulate our nondegeneracy condition on their dividends. An al- ternative would be to take an inﬁnite-horizon model, but this would require addressing certain additional technical problems. We think of our ﬁnite-time horizon T as a truncation of an inﬁnite-horizon model. The lump of consump- tion at time T aggregates the ﬂow consumption on the interval (T ∞) in the inﬁnite-horizon model, conditional on the information available at time T . Our nondegeneracy assumption will be imposed on the lump dividend at the termi- nal date T . The uncertainty in the model is described by a standard K-dimensional Brownian motion β on a probability space (Ω Fμ) ; the components of β are independent of each other and the variance of βk(t ·) equals t. We de- ﬁne I(t ω) = (t β(t ω)) . The primitives of the economy—dividends, endow- ments and utility functions—will be described as functions of I(t ω) . We are given a right-continuous ﬁltration {Ft : t ∈ [0T ]} on (Ω Fμ) such that F0 contains all null sets and β is adapted to {Ft}, that is, for all t, β(t ·) is measurable with respect to Ft. Let J = K. There are J + 1 securities, indexed by j = 0J ; security j is in net supply ηj ∈ {0 1}. Security j pays dividends (measured in consumption units) at a ﬂow rate Aj(t ω) = gj(I(t ω)) at times t ∈ [0T) , and a lump 11 The argument is more complicated here because our continuous-time economy is more com- plicated than the economy in Brown and Robinson (1975).

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EQUILIBRIUM IN FINANCIAL MARKETS 849 K dividend Aj(T ω) =Gj(I(T ω)) at time T . We assume that g : [0T ]×R → K K R + is real analytic on (0T) ×R and that Gj is continuous almost everywhere K on {T } × R . For example, A0 could be a zero-coupon bond (g0(t ω) = 0 for σj ·β(tω) t ∈ [0T) , g0(T ω) is constant) or Aj(t ω) = e , where σj is the jth row of a J × J matrix σ, for j = 1J . There are I agents i = 1I . Agent i has a ﬂow rate of endowment ei(t ω) = fi(I(t ω)) at times t ∈ [0T) , and a lump endowment ei(T ω) = K Fi(I(T ω)) , where fi is analytic on (0T) × R and Fi is continuous almost ∑ K I everywhere on {T } × R . Let e(t ω) = i=1 ei(t ω) denote the aggregate en- dowment. The utility functions are von Neumann–Morgenstern utility functions, ex- pectations of functions of the consumption and the process I which are ana- K lytic on (0T) ×R . More formally, given a measurable consumption function ci : [0T ] ×Ω →R++, the utility function of the agent is [∫ ] T Ui(c) = Eμ hi(ci(t ·) I(t ·))dt +Hi(ci(T ·) I(T ·)) 0 K where the functions hi :R+ × ([0T) ×R ) →R ∪ {−∞} and Hi :R+ × ({T } × K K 2 R ) →R∪{−∞} are analytic on R++ × ((0T) ×R ) and C on R++ × ({T }× K R ), respectively, and satisfy ∂hi K lim = ∞ uniformly over ([0T ] ×R ) c→0+ ∂c ∂Hi K lim = ∞ uniformly over {T } ×R c→0+ ∂c ∂hi K lim = 0 uniformly over ([0T ] ×R ) c→∞ ∂c ∂Hi K lim = 0 uniformly over {T } ×R c→∞ ∂c K lim hi(c (t x)) = hi(0(tx)) uniformly over ([0T ] ×R ) c→0+ K lim Hi(c (T x)) =Hi(0(Tx)) uniformly over {T } ×R c→0+ ∂hi K > 0 on R++ × ([0T ] ×R ) ∂c ∂Hi K > 0 on R++ × ({T } ×R ) ∂c 2 ∂ hi K < 0 on R++ × ([0T ] ×R ) 2 ∂c 2 ∂ Hi K < 0 on R++ × ({T } ×R ) 2 ∂c

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