Dynamically complete markets under Brownian motion

This paper investigates how continuous-time trading renders complete a financial market in which the underlying risk process is a Brownian motion. A sufficient condition, that the instantaneous dispersion matrix of the relative dividends is non-degenerate, has been established in the literature for single-commodity, pure-exchange economies with many heterogenous agents where the securities’ dividends as well as the agents’ utilities and endowments include flows during the trading horizon which are analytic functions. In sharp contrast, the present analysis is based upon a different mathematical argument that assumes neither analyticity nor a particular underlying economic environment. The novelty of our approach lies in deriving closed-form expressions for the dispersion coefficients of the securities’ prices. To this end, we assume only that the pricing kernels and dividends satisfy standard growth and smoothness restrictions (mild enough to allow even for options). In this sense, our sufficiency conditions apply irrespectively of preferences, endowments or other structural elements (for instance, whether or not the budget constraints include only pure exchange).

completeness, however, this is no longer the case: no-arbitrage restrictions do not suffice to guarantee unique option prices while replication may not be possible. It is crucial therefore to be able to associate dynamic completeness with the economic primitives of a given financial environment-in a manner that remains unambiguously verifiable and holds at least generically across the space of these primitives. The present paper provides sufficient conditions for continuous-time trading to render an asset market dynamically complete when the underlying risk process is a Brownian motion.
The typical generic result for dynamic completeness in the continuous-time literature establishes the validity of the corresponding sufficient condition except for a small subset of the domain space of the primitive parameters. Yet it remains difficult, if not impossible in some cases, to establish whether the condition is valid for particular values of these parameters. Notable exceptions are the sufficiency results in Anderson and Raimondo [2], Hugonnier et al. [17], Riedel and Herzberg [28] (see also Riedel and Herzberg [27]) as well as Kramkov [19]. These studies refer to financial markets in single-commodity, pure-exchange economies with many heterogenous agents, and where all intermediate flows of utilities and endowments are analytic functions. In sharp contrast, the current exposition is based upon a different mathematical argument that assumes neither analyticity nor a particular underlying economic environment.
The novelty of our approach lies in deriving closed-form expressions for the dispersion coefficients of the relative asset prices. The respective formulae allow us in turn to establish sufficient conditions for the asset market to be dynamically complete that apply under general specifications for the pricing kernel and the securities' dividends, as long as both are continuous and satisfy standard in the literature smoothness and growth conditions. As to be expected, our sufficient conditions are non-degeneracy ones on the instantaneous dispersion matrix of primitive parameters. They do coincide with those in the aforementioned papers under analyticity. In general, however, they are stronger. This renders our analysis applicable on more general settings regarding the underlying structure for economic activity or the agents' preferences and endowments. It also sheds light on how the relation between dynamic completeness and the non-degeneracy conditions in question extends in directions that are important for applications.
The relevance of our study becomes evident when viewed in the context of general equilibrium analysis. The typical method in the literature for obtaining financial equilibria in continuous time has been to compute an Arrow-Debreu equilibrium and use the associated consumption process as pricing kernel in order to construct equilibrium prices for the traded securities. 1 To ensure, however, that the starting Arrow-Debreu allocation is implementable by trading the given set of securities, their market needs to be dynamically complete-and, thus, to permit the construction of the equilibrium pricing process via a representative agent. Yet, the equilibrium pricing process is determined endogenously (via fixed-point arguments) from the model's primitives (the utility functions of the agents, their endowments and the dividend processes of the securities) and are expressed as expectations of properly discounted future payoffs. As a result, especially in economies with many heterogenous agents and apart from the extremely special cases where one can obtain sufficiently straightforward closedform solutions, verifying from the primitives that the equilibrium pricing process is indeed dynamically complete is a highly non-trivial problem, known as "endogenous completeness." Essential progress in this problem was achieved by the aforementioned papers. The fundamental insight is that, for the asset market to be dynamically complete, it suffices that the instantaneous dispersion matrix of the relative dividends is non-degenerate (i.e., nonsingular) at some point in the space. The crucial underlying assumption is that the securities' dividends as well as the agents' utilities and endowments include flows during the trading horizon which are analytic functions (at least in time). By relaxing this restriction, the present study complements these papers shedding light on the intuition behind their fundamental underpinnings.
For example, when the securities' dividends comprise only lump sums at the terminal date of the trading horizon, our sufficiency condition is exactly the same as above-even though we do not assume analyticity. This remains the case when the dividends include flows during the trading horizon, as long as a collection of simple options is available for trading. More tellingly perhaps, when the securities' dividends comprise only flows during the trading horizon (a common setup especially in applied finance models), our sufficiency condition requires that the instantaneous dispersion matrix of the relative nominal dividends is nondegenerate everywhere in the space. This highlights the implications of assuming analyticity. Without it we end up with a sufficiency condition that is, on the one hand, more burdensome to verify. On the other hand, it does ensure that the instantaneous dispersion matrix of the relative securities' prices is non-degenerate everywhere, not almost everywhere, in the space.
The balance of the paper is organized as follows. The next section introduces the theoretical structure under investigation. Section 3 presents our results and Sect. 4 interprets them in the context of the pertinent literature. Section 5 concludes while the "Appendix" presents the supporting technical material and results.

Theoretical framework
Consider a financial market where the trading horizon is T = [0, T ] for some T ∈ R ++ or T = R + while the informational structure is given by a K -dimensional (K ∈ N\ {0}) standard Brownian process, defined on a complete probability space ( , F , π) and denoted by β : × T → R K (or β k : × T → R with k ∈ K := {1, . . . , K } for the typical dimension). As usual, the process is meant to fully describe the exogenous financial risk in the sense that the collection of the sample paths {β (ω, t) : t ∈ T } ω∈ specifies all the distinguishable events.
The underlying risk process being a Brownian motion, a necessary condition for any securities market to be dynamically complete is that the number of securities exceeds that of independent Brownian motions by at least one (i.e., that the market is potentially complete). In what follows the trading structure will consist of K +1 dividend-paying securities, indexed by j ∈ K ∪ {0} and traded continuously over T . These are real assets in the sense that their dividends are in units of an underlying physical commodity. The dividends will take one or both of two forms, a flow and a lump sum. Specifically, letting I : × T → T × R K denote the process {t, β (ω, t)} (ω,t)∈ ×T , we consider the functions G j , g j : T ×R K → R + and that the typical security pays the dividend flow g j (I (ω, ·)) along the Brownian path {β (ω, t) : t ∈ T } and/or the lump sum G j (I (ω, T )) on the terminal date (if the time horizon is finite: T = [0, T ] for some T ∈ R ++ ). The supports S G j := β ∈ R K : G j (T , β) > 0 and S g j := (t, β) ∈ T × R K : g j (t, β) > 0 will be taken as open and connected. Letting moreover D k := ∂/∂β k denote the instantaneous dispersion operator, we will also assume that, for any k ∈ K, the functions D k G j (T , ·) and D k g j (·) are well defined and continuous on S G j and S g j , respectively. Given that trading occurs over a time interval while the informational structure is driven by a Brownian motion, well-known no-arbitrage conditions ensure that the securities' prices are the current expectations of their future dividends valued at some pricing kernel, a strictly positive one-dimensional Itô process. In what follows, we take the pricing kernel to be given by the functions M, m : T ×R K → R ++ which are continuously differentiable on int (T )×R K . In addition, we assume that the functions M j (·) := MG j (·) and m j (·) := mg j (·) satisfy appropriate growth conditions (see assumptions A1-A2 below) for the price of the typical security to be written as follows All prices above being strictly positive, any one can be used as deflator. Without loss of generality, therefore, we may deflate all prices by that of the zeroth security and restrict attention to the relative pricing process p j (·) := P j (·) /P 0 (·), j ∈ K. This normalization renders the zeroth security instantaneously risk free (for its price remains constant at 1) so that the strategy of buying and holding it becomes a (trivial) money-market account. As a result, the financial market under consideration is dynamically complete if and only if the (Jacobian) matrix of the instantaneous dispersions of the relative prices, J p (·) := [D k p n (·)] (n,k)∈K×K , is non-singular almost everywhere on int (T ) × R K . 2 In what follows, we seek to establish sufficient conditions for the Jacobian matrix to have the desired property. To this end we observe that, since p n (·) = (m P n ) (·) / (m P 0 ) (·) using the nominal prices m P j (·) := m (·) P j (·) for j ∈ K ∪ {0}, an equivalent statement of our aim is to ensure that (n,k)∈K×K remains non-singular almost everywhere on int (T ) × R K . As it turns out, the latter matrix is well defined on int (T ) × R K under the following conditions. 3 2 The claims in this paragraph are based on well-known concepts and results. See, for example, Chapters 5A-C, 6I, and 6K in Duffie [11] as well as Sections 4.1-4.4 and Theorem 5.6 in Nielsen [24]. Observe also that, as usual, β t denotes a (random) value of the mapping t → {β (ω, t)} ω∈ . 3 As usual, | · | denotes the Euclidean norm of a vector or the determinant of a (square) matrix. Notice also that the conditions A1-A2 are trivially satisfied on {T } × int R K \S G j and int T × R K \S g j when these are non-empty (for instance, when the security is a European option-see Claim 3.2 below). Indeed, we have G j (T , ·) = 0 = D k G j (T , ·) on the former and g j (·) = 0 = D k g j (·) on the latter.

Analysis
Starting with the first term in (1), and letting T := [0, T ] for some T ∈ R ++ , recall that the increment β (·, T ) − β (·, t) is independent of F t and distributed N 0 K , (T − t) I K over . Hence, the term in question can be written as where φ (·) denotes the K -dimensional standard-normal pdf. Under Assumption A1(i), the pricing function in (2) above is well defined. And so is the dispersion with respect to the typical Brownian dimension of its nominal version. Specifically, we have For t → T , moreover, the continuity in t commutes inside the expectation operator above. As a result, letting p 1n (·) := P 1n (·) /P 10 (·) and G n/0 (·) := 1 S G 0 × G n (·) /G 0 (·) denote, respectively, the relative prices and terminal dividends, the (Jacobian) dispersion matrix of the relative prices, J p 1 (I (ω, t)), approaches that of the relative terminal dividends The latter observation leads to a sufficient condition for the financial market to be dynamically complete in the case where the time horizon is finite and the securities pay only lump-sum dividends on the terminal date.
Theorem 3.1 Let T = [0, T ] for some T ∈ R ++ and suppose that the price process is given by (2) with A1(i) satisfied. Then J p 1 (·) has full rank almost everywhere on (0, T ) × R K if J G (·) is non-singular at some β T ∈ S G 0 .
Proof Our argument will be based upon the fact that, even though we have not required analyticity anywhere above, the entries of the dispersion matrix J p (·) are analytic on (0, T ) × R K under Assumption A1(i). This is because, for any ( j, k) ∈ K ∪ {0} × K, the functions m P j (·) and D k m P j (·) are analytic on (0, T ) × R K . For the former function, its analyticity follows immediately from Lemma A.7 in the "Appendix". For the latter function, since |x k | ≤ |x| < e |x| < 1 + e |x| everywhere on R K , by Lemma A.1 in the "Appendix", there exist constants (C, r ) ∈ R ++ × 0, 1 2T − r j such that |x k | ≤ Ce r |x| 2 for any x ∈ R K . Under Assumption A1(i), therefore, we have x k M j (T , x) ≤ C 0 Ce (r 0 +r )|x| 2 anywhere on R K , with r 0 + r ∈ 0, 1 2T . By Lemma A.7 in the "Appendix" then The claims in the main text above are based on supporting results in the "Appendix". Specifically, that m P 1 j (·) and D k m P 1 j (·) are both well defined with the latter given by (3) follow from Lemma A.5. Moreover, that continuity with respect to time commutes inside the expectations in (2) and (3) follows from Lemmas A.6-A. 3. Finally, that J p 1 (I (ω, t)) approaches J G (I (ω, T )) is due to Proposition A.1.
is an analytic function of (t, β) on (0, T ) × R K . That D k m P j (·) itself is analytic on (0, T ) × R K follows now from the fact that the sum, product and ratio of two real analytic functions are also analytic (see Propositions 1.1.7 and 1.1.12 in Krantz and Parks [20]). Suppose now that J G T , β 0 = 0 for some β 0 ∈ S G 0 , and take ω 0 ∈ such that β 0 = β (ω 0 , T ). By continuity, there exists δ 0 > 0 such that |J G (T , β)| = 0 for any β ∈ B β 0 (δ 0 ). Take also ω 1 ∈ such that lim t→T J p (t, β (ω 1 , t)) = J G T , β 1 where β 1 := β (ω 1 , T ) ∈ B β 0 (δ 0 ). By Proposition A.1 in the "Appendix", and as the determinant of a matrix is a continuous operator, we have As established above, though, the entries of J p (·) are analytic on (0, T ) × R K . The determinant of a matrix involving nothing but the operations of product and sum on its entries, J p (·) is also analytic on (0, T ) × R K . But then (4) necessitates that J p (·) = 0 almost everywhere on (0, T ) × R K . 5 Our argument for establishing Theorem 3.1 hinges crucially upon the fact that the functions m P j (·) and D k m P j (·) are both analytic on (0, T ) × R K , even though M j (·) itself is not assumed to be analytic. Unfortunately, this fact does not obtain for the respective functions when it comes to the second term in (1). The analysis of the pricing process for the flow divedends requires its own approach.
Under Assumption A1(ii), the pricing function but also the dispersion with respect to the typical Brownian dimension of its nominal version are both well defined. Specifically, we have 6 And for the case T = R + , note that the respective right-hand sides of (6)-(7) above remain well defined at all T ∈ R ++ under condition A2. Hence, their limiting versions are given by Letting then p 2n (·) := P 2n (·) /P 20 (·), the above findings can be summarized as follows.
Claim 3.1 Let the price process be given by (5) with T = [0, T ] for some T ∈ R ++ or T = R ++ and A1(ii) or A2, respectively, satisfied. Then The preceding relations shed light as to why our proof for Theorem 3.1 does not extend to the case of flow dividends. For it would establish here that the intergrands in (6)-(7) are analytic in (t, β) for each given s. However, unless one assumes in addition that these integrands are also analytic in s, it does not follow that m P 2 j (·) and D k m P 2 j (·) are necessarily analytic on int (T ) × R K .
This notwithstanding, the explicit expressions for the diffusion coefficients in Claim 3.1 allow a more direct approach. To this end, we define the collection m * and a stronger than the standard notion of matrix non-singularity.
We will say that { f k } k∈K satisfies strong non-singularity on X := k∈K X k if there exists no x 1 , . . . , x K ∈ X rendering the matrix This notion of matrix non-singularity leads to a sufficient condition for the financial market to be dynamically complete in the case where the securities' dividends comprise only flows during the time horizon. Specifically, letting J p 2 (·) := [D k p 2n (·)] (n,k)∈K×K , we have the following.
Theorem 3.2 Let the price process be given by (5) with T = [0, T ] for some T ∈ R ++ or T = R ++ and A1(ii) or A2, respectively, satisfied. Then J p 2 (·) has full rank everywhere on int (T )×R K if the collection m * n/0 n∈K satisfies strong non-singularity on n∈K S g n × S g 0 .
Proof Take arbitrary (t, β) ∈ int (T ) × R K and v ∈ R K \ 0 K . By Claim 3.1, the typical entry in the matrix (m P 0 ) (t, β) 2 J p (t, β) is given as Suppose now that the collection m * n/0 n∈K satisfies strong non-singularity on n∈K S g n × S g 0 . There exists then no s 1 , β s 1 , . . . , s K , β s K ∈ n∈K S g n and no τ 1 , β τ 1 , . . . , It is trivial to check that this property requires in turn the existence of some n ∈ K such that As S g n × S g 0 though is connected, by continuity this can be only if m * n/0 (·) maintains the same sign everywhere on S g n × S g 0 . Without loss of generality, therefore, we may let m * n/0 (·) > 0 everywhere on S g n × S g 0 . But this implies in turn that That is, J p (t, β) v = 0 K and the claim follows.
Turning finally to the case when both terms on the right-hand side of (1) apply, the pricing process P j (·) = P 1 j (·) + P 2 j (·) produces even more complex dynamics for the relative prices. Nonetheless, the preceding analysis provides again a sufficient condition for the financial market to be dynamically complete when the securities pay both dividend flows and lump sums-as long as the market includes a zero-coupon bond maturing at T as well as, for each j ∈ K, a European call option maturing at T along with its equivalent put.

Claim 3.2 Let
for some T ∈ R ++ and the price process be given by (1) with A1(i) satisfied. Suppose also that the zeroth security is a zero-coupon bond maturing at T (i.e., G 0 (T , ·) = 1 and g 0 (·) = 0) while, for each j ∈ K, a European call maturing at T (with strike price some X j > 0) along with its equivalent put are available for trading. Then J p (·) has full rank almost everywhere on Proof Observe first that, for any given β ∈ R K , we have that lim t→T m P 2 j (t, β) = 0. As a result, we have that lim t→T m P j (I (ω, t)) = lim t→T m P 1 j (I (ω, t)) = M j (I (ω, T )) almost everywhere on . Clearly, at any (t, β) ∈ [0, T ) × R K , the nominal price of a European call on the jth security with maturity date T and strike price X j and that of its equivalent put are given, respectively, by This implies of course the put-call parity is the current price of the zero-coupon bond. For each j ∈ K consider the portfolio that consists of being long one unit on the corresponding call, short one unit on the equivalent put, and long K j units on the zero-coupon bond. The value of this portfolio being P 1 j (·), the Jacobian of the relative prices for the overall K such portfolios is given by J G (·). The claim now follows from Theorem 3.1.

Dynamics with money-market accounts
The preceding analysis applies also when the zeroth security is a money-market account. To see this, let {r t : t ∈ T } be an instantaneously riskless rate process and P 0t := P 0 exp t 0 r s ds for some initial value P 0 > 0. 7 As D k P 0t = 0 for any k ∈ K, we now have J p (·) = P −1 0t [D k P n (·)] (n,k)∈K×K . Hence, the dispersion matrix of the relative prices is non-singular if and only if so is the dispersion matrix of the absolute prices.
With respect to the case where the securities' dividends comprise only flows during the time horizon, the latter equivalence means that the argument establishing Theorem 3.2 above remains valid. For the equivalence ensures that, as far as dynamic completeness is concerned, there is no loss of generality if we consider the (counterfactual) market in which the securities' prices are normalized using an annuity (i.e., a security with dividend G 0 (·) = 0 and g 0 (·) = 1) instead of the money-market account as the numeraire. 8 Corollary 3.1 Let the price process be given by (5) with T = [0, T ] for some T ∈ R ++ or T = R ++ and A1(ii) or A2, respectively, satisfied. Then [D k P 2n (·)] (n,k)∈K×K has full rank everywhere on int (T )×R K if the collection m * n n∈K of the functions m * n : S g n ×R K → R K given by satisfies strong non-singularity on n∈K S g n × R K .
Proof Take arbitrary (t, β) ∈ int (T ) × R K and (n, k) ∈ K × K. By rearranging terms in the first expression in Claim 3.1, we get that The remainder of the argument is the same as in the latter part of the proof for Theorem 3.2.
Regarding the cases investigated in Theorem 3.1 and Claim 3.2 above, our analysis remains valid when the price process {P n } n∈K derives from the risk-neutral valuation method. More precisely, when P n (·) = 1 T =R + × P 1n (·) + P 2n (·) is given by with β : × T → R K being another K -dimensional Brownian motion on ( , F , π ), and π a martingale measure equivalent to π. 9 Comparing these expressions with those in (2) and (5), it follows immediately that the dynamics with respect to the typical Brownian dimension are given here by (3) and the first expression in Claim 3.1 for G 0 (T , ·) = 1 = g 0 (·) and M (t, ·) = exp − t 0 r s ds = m (t, ·). Theorem 3.1, Claim 3.2, and Corollary 3.1 can be stated, respectively, as follows.
for some T ∈ R ++ and suppose that the price process is given Suppose also that the price process P n (·) = P 1n (·) + P 2n (·) is given by (9) Suppose also that, for each j ∈ K, a European call maturing at T (with strike price some X j > 0) along with its equivalent put are available for trading. Then [D k P n (·)] (n,k)∈K×K has full rank almost everywhere on [0, T ) × R K if [D k G n (T , ·)] (n,k)∈K×K is non-singular at some β ∈ R K . Corollary 3.4 Let the price process be given by (10) with A1(ii) and A2 satisfied for m (t, ·) := exp − t 0 r s ds , respectively, when T = [0, T ] for some T ∈ R ++ and T = R ++ . Then [D k P 2n (·)] (n,k)∈K×K has full rank everywhere on int (T ) × R K if the collection {g n (·)} n∈K satisfies strong non-singularity on n∈K S g n .

Discussion and related literature
As pointed out already, the issue under investigation here has been analyzed also in Anderson and Raimondo [2], Riedel and Herzberg [28] (see also Riedel and Herzberg [27]) as well as Kramkov [19]. Anderson and Raimondo [2] considers the pricing process in (1) (c (t, β t ) , t, β t ), which satisfy in addition the Inada conditions uniformly. In Riedel and Herzberg [28], this setting is extended by allowing the underlying risk process to be a time-homogenous diffusion. Attention is restricted though to the case in which the dividends and the individual endowments are time homogenous, one of the securities is a zero-coupon bond maturing at the terminal date, the aggregate endowment is bounded, while the agents' utilities are state independent and depend on time only via an "impatience" rate. In Kramkov [19], on the other hand, the setting in Anderson and Raimondo [2] is extended to allow for general diffusions . Yet this is done at the expense of restricting the agents' relative risk aversion coefficients over terminal lump sums to be bounded while their utilities over flows and their endowment flows are time homogenous.
With respect to growth conditions, Anderson and Raimondo [2] imposes one that is stronger than that in our Assumption A1 on the dividends, the agents' marginal utilities and endowments (both in terms of flows and lump sums) as well as on the instantaneous dispersion of the dividend and endowment flows. In Riedel and Herzberg [28] these quantities are all bounded while in Kramkov [19] the same condition as in Anderson and Raimondo [2] applies on the agents' utilities, marginal utilities and endowments (both in terms of flows and lump sums) as well as on the instantaneous dispersion of the terminal lump-sum dividends.
The above features not withstanding, in all of the aforementioned studies, the sufficient condition for the respective financial market to be dynamically complete is the same as that in our Theorem 3.1. And despite the presence of intermediate dividend flows, the intuition for the sufficiency of the non-degeneracy condition in question is also the same as that in our proof for Theorem 3.1. Specifically, even though P j (·) = P 1 j (·) + P 2 j (·), as there is no value left to a security on the terminal date other than its lump-sum dividend, we do have lim t→T P j (t, ·) = lim t→T P 1 j (t, ·) and, thus, lim t→T J p (I (ω, t)) = D k G n/0 (I (ω, T )) (n,k)∈K×K almost everywhere on . Moreover, since all intermediate dividend and marginal utility flows are analytic on [0, T ) × R K , so is the integral P 2 j (·). As a result, P j (·) and, thus, also J p (·) are analytic on [0, T ) × R K . Clearly, the latter part of our argument in the proof for Theo-rem 3.1 remains valid when P 1 j (·) [resp. p 1 j (·)] is replaced by P j (·) [resp. p j (·)] under analyticity. 10 This line of reasoning depicts also the intuition behind the non-degeneracy condition in Hugonnier et al. [17]. Compared to the aforementioned papers, this study extends the setting in Riedel and Herzberg [27] by allowing the underlying risk process to follow a general diffusion and the time horizon to be infinite (albeit, in the latter case, under the restriction that the agents' relative risk aversion coefficients are bounded). The main analysis in Hugonnier et al. [17] is conducted under a finite time horizon in the presence of a money-market account and only dividend flows. The sufficient condition for the financial market to be dynamically complete applies on the dispersion matrix of the flow-dividends: [D k g n (·)] (n,k)∈K×K is taken to be non-degenerate at some (t, β) ∈ (0, T ) × R K with t being arbitrarily close to T . The underlying intuition emerges now in light of our expessions in Claim 3.1.
Recall in particular the first expression in Claim 3.1. Letting t be arbitrarily close to T , the dispersion of the typical security price can be approximated by the relation where the first and last equalities above follow by the fact that the time integrals vanish as we approach the terminal date:

β) (T − t)+o (T − t), and thus also P 2n (t, β) = −g n (t, β) (T − t)+ o (T − t). Clearly, if [D k g n (t, β)] (n,k)∈K×K
is non-singular then so will be J P 2 (t, β) -the remainder of the argument being the same as above but for replacing P n (·) with P 2n (·). 11 It is noteworthy of course that, even though our Corollary 3.1 refers to a financial setting that is consistent with that underpinning the main analysis in Hugonnier et al. [17], the respective non-degeneracy conditions coincide only locally. To see this, observe that [D k g n (t, β)] (n,k)∈K×K is non-singular only if the matrix m * 1 ((t, β) , (t, β)) · · · m * K ((t, β) , (t, β)) is also non-singular. Yet the latter property necessitates in turn the existence of some neighbourhood V of (t, β) in (0, T ) × R K such that m * 1 s 1 , β s 1 , τ 1 , β τ 1 · · · m * K s K , β s K , τ K , β τ K remains non-singular for any collection s k , β s k , τ k , β τ k k∈K from V (see Claim C.1 in the "Appendix"). Hence, the requirement that [D k g n (t, β)] (n,k)∈K×K is non-singular guarantees in fact that m * n k∈K satisfies strong non-singularity on V ; equivalently, that [D k P 2n (·)] (n,k)∈K×K has full rank everywhere on V (recall our Corollary 3.1). Similarly, even though Claim 3.2 refers to a financial setting that is consistent with those in Anderson and Raimondo [2], Riedel and Herzberg [28] or Kramkov [19], we had to assume the availability of options in order to establish the same sufficiency result. Either of these observations attests to the mileage one gets out of analyticity. Equally importantly perhaps, the obvious reason why our argument in the proof of Theorem 3.1 cannot apply also in the settings of Theorem 3.2 or Claim 3.2 offers an intuitive understanding of the claim that one may do away with assuming analyticity in the space variables, yet analyticity in time remains sine qua non. 12 Yet another insight born out of the relation our analysis bears to the aforementioned papers is the indication that the sufficiency of the respective non-degeneracy conditions extends beyond pure-exchange economies. In the context of general equilibrium analysis, the pricing kernel cannot be but a weighted average of the agents' equilibrium marginal utilities. This notwithstanding, the asset-pricing equation in (1) allows also for non-financial wealth (and, thus, production). In this sense, the essential premise is that the time-and state-dependency of the primitive variables (utilities, dividends, endowments, and other non-financial wealth) obtains as a function of (t, β t ). As an approach towards equilibrium asset-pricing theory this has been the building block for much of the seminal literature.
The starting point was to assume that the agents have identical preferences. This has been the launching pad of two related strands of the literature. The first restricts attention to what is essentially the continuous-time analogue of the static (one-period) model: the time horizon is finite and securities pay only lump-sum dividends on the terminal date. 13 The second approach has been to allow for securities that pay also dividend flows during the time interval while the time horizon may be infinite. 14 The next step in the literature was to study agents with heterogenous preferences. Even in this case, however, the pricing kernel remains a linear function of the equilibrium marginal utilities (the Negishi weights are constant) if the equilibrium allocation is Pareto-optimal. 15 As a result, even in this case the pricing formula retains the same basic form as that under the present investigation. 16 Clearly, in the context of general equilibrium, as long as we maintain Pareto optimality as a desideratum, the scope of our analysis is large. By considering general formulations for the pricing kernel and the securities' dividends, our analysis applies irrespectively of preferences, endowments, and other structural elements (such as whether the agents' budget constraints include only pure exchange). And this is important in its own right; it has become by now clear in the literature that the choice of pricing kernel can play a crucial role for the model's results. 17 Equally importantly, our analysis remains valid even in the presence of multiple underlying consumption commodities. With more than one consumption commodities, the dividend specifications in (1) would be given as G j (·) = Q j G j (·) and g j (·) = q j g j (·) with 12 For enlightening discussions on this point, see Section A in the Supplementary Appendix to Hugonnier et al. [17] or Remark 3.4 in Kramkov [19]. 13 Our pricing Eq. (2) can be found, for instance, in Bick [4,5], He and Leland [16], Raimondo [25] or Anderson and Raimondo [1]. 14 Perhaps the most well-known paper in this strand is Cox et al. [8], the continuous-time analogue of the famous model in Lucas [21] enhanced to include production. Our pricing Eq. (5) is consistent with the analysis in Cox et al. [8]; it can be found also in Cochrane et al. [7], Martin [22], Farhi and Panageas [15], Merton [23] (see  or Wang [30]. 15 When the equilibrium allocation is not Pareto-optimal, the representative agent's utility function will be state-dependent even if all individuals have state-independent preferences and homogenous beliefs (see, for example, Cuoco and He [9]). In fact, the Negishi weights in the construction of the representative agent may even play the role of endogenous state variables which cannot be recovered as functions of the exogenous ones. 16 Our pricing formula (1) can be found in Basak and Cuoco [3], Duffie and Zaime [12] (see Theorem 1 and the subsequent discussion in Sect. 5), Dumas [13], Karatzas et al. [18] (see Corollary 10.4) or Riedel [26] (Theorem 2.1). 17 See footnote 2 in Riedel and Herzberg [28] for relevant references. Q j (·) and q j (·) being, respectively, the terminal and intermediate values for the relative price (with respect to that of the numeraire commodity) of the commodity in whose units the dividend of the jth security is paid. 18 Obviously, our results remain valid under these dividend specifications. Yet the latter entail also endogenously determined quantities; the commodity prices Q j (·) and q j (·) are determined endogenously by the equilibrium itself. As a result, apart from very special cases, the specifications of the securities' dividends become conditional on the choice of numeraire. Needless to say, the latter qualification applies also for the sufficiency of our non-degeneracy conditions; they guarantee that the financial market will be dynamically complete as long as the normalization remains with respect to the given numeraire.

Concluding remarks
In an Arrow-Debreu economy, the agents may shift consumption or income across states and time by trading a complete set of contingent claims, once and for all at the beginning of time. When they are instead constrained to trade a given set of securities, the market is said to be dynamically complete if repeated trading of the securities can still deliver any allocation that would be feasible under a complete set of contingent claims. Under continuous-time trading, this may be possible by trading a finite set of securities rapidly enough, even though the information about the state of the world is revealed through a stochastic process. In particular, when the underlying uncertainty is driven by Brownian motions, this can happen if the securities' market is potentially dynamically complete (i.e., the number of securities exceeds that of independent Brownian motions by at least one). 19 However, potential dynamic completeness does not suffice by itself: some form of independence amongst the securities' payoffs must obtain in addition. In general, once the securities' prices are appropriately deflated, this refers to the non-degeneracy of their instantaneous dispersion with respect to the underlying stochastic process. The present paper establishes sufficient conditions for such non-degeneracy when the pricing process is given by (1)-as long as some standard in the literature growth conditions obtain. 20 Of course, our attention was restricted to the exogenous risk process being a Brownian motion while the time-and state-dependency of the primitive variables obtain as functions of (t, β t ). And even though this combination has been used extensively in the literature, it does mean that our results do not extend to a larger class of stochastic models (such as Ornstein-Uhlenbeck or more generally affine processes) that is becoming increasingly the forefront of the research in financial economics. Nevertheless, the setting of the present study has always been an important theoretical benchmark in the quest for fundamental equilibrium insight. The significance of being able to determine explicitly if and when dynamic completeness obtains in this setting is obvious.
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A Supporting results
Taking (δ, x) ∈ R ++ × R K , B x (δ) denotes in what follows the open ball in R K of radius δ that is centered at x.
Lemma A.2 Let (τ, β) ∈ R ++ × R K be a parameter vector. Suppose also that f : Proof Observe first that, given any α > 0, we have This implies in turn that Choosing, therefore, α ∈ 0, τ −2r 0 2τ establishes that To complete the argument let δ := |β|/α. (X , A, μ) and a non-empty, non-degenerate open interval (a, b) ⊆ R, let f : (a, b) × X → R be a function satisfying 21 x) is continuous for every fixed x ∈ X , and (iii) there exists a function h :

Lemma A.3 Given a measure space
Then the function v : (a, b) → R given by is continuous.
Lemma A.4 Let (τ, β) ∈ R ++ × R K be a parameter vector. Suppose also that the continuous function f : for some constants (r , C) ∈ (0, τ/2) × R ++ . Letting then m = (m 1 , . . . , m K ) ∈ N K , the k∈K ∂β m k k dx is well defined and C 1 , with its partial derivatives given by Fix an arbitrary β ∈ R K and an arbitrary dimension k ∈ K. Our argument-which will focus upon the interval (β k − k , β k + k ) for some k ∈ (0, 1)-will be presented in steps.
Step 1(i). If m = 0 K , that F (m) (·) is well defined follows immediately from Lemma A.2 above. 22 For the case m = 0 K , notice first that a well-known property of the pdf of a scalar standard normal variable is that its nth derivative is given by with H n (·) being the Hermite polynomial of order n. It follows therefore that, writing H n (x k ) = n i=0 c i x i k for some constants c 0 , . . . , c n ∈ R, we have for any α > 0, all x k ∈ R\ (−1, 1) : |x k | ≥ 2n/α and where C n = max {|c i | : i ∈ {0, . . . , n}}.
It is trivial to check moreover that, letting g n √ τ And regarding the domain-restrictions above, we have that for any z k ∈ (− k , k ) and any x k ∈ R : |x k | ≥ 1 + 2n Step 1(ii). It follows from the preceding step that for all x ∈ R K and all m ∈ N K , while for any α > 0, any z k ∈ (− k , k ), and any x k ∈ R\ (−δ k , δ k ) where δ k = 1 + 2m k α √ τ + |β k |. Set now z k = 0 for all k ∈ K, and let m 0 = max {m k : k ∈ K} and β 0 = max {|β k | : k ∈ K}. Then where Step 1(iii). We are now in position to show that F (m) (·) is well defined. Taking α ∈ (0, 1), (14) above implies that By hypothesis, moreover, there are r ∈ (0, τ/2) and C > 0 such that | f (x)| ≤ Ce r |x| 2 on R K . Choosing, therefore, α ∈ 0, τ −2r τ so that r < τ (1−α) 2 , Lemma A.2 above ensures the existence of r , δ 0 > 0 such that And as the functions f , φ, and H m k are all continuous with respect to x, the claim follows.
Step 2(i). We will show next that the partial derivatives exist and are given by the expression in the statement. Observe that, by (12) above, we have By the mean value theorem, however, there exist γ, ρ ∈ (0, 1) such that Step 2(ii). Recall now (13) above. For any x k ∈ R\ (−δ k , δ k ) we have Clearly, the first two terms of the integrand in (15) give By hypothesis, however, there exist r ∈ (0, τ/2) and C > 0 such that | f (x)| ≤ Ce r |x| 2 on R K . Moreover, by Lemma A.1 above, for any r k > 0, we can find C k > 0 such that e τ (1−α)|x k | < C k e r k x 2 k . Choosing, thus, r k ∈ (0, τ/2 − r ) and α ∈ 0, τ −2(r +r k ) τ Lemma A.2 ensures the existence of r , δ 0 > 0 such that We have just established that, viewed as a function of z k , the integrand in (15) is bounded above on R K \B 0 max δ k , δ 0 by a function that is independent of z k and integrable on R K \B 0 max δ k , δ 0 . Regarding the interior of the left-out neighbourhood, observe that and, thus, where C 0 = e (m k +2)(|β k |+1) m k +2 i=0 |c i |τ i/2 . Clearly, also on B 0 max δ k , δ 0 , as a function of z k , the integrand in (15) is bounded above by an integrable function that is independent of z k .
Given these obsevations, and as g m k +2 √ τ (x k − β k − γρz k ) is continuous in z k for any x k ∈ R, Lemma A.3 above ensures that the continuity in z k on (− k , k ) is preserved under the integral sign in (15). That is, Step 2(iii). To establish now the required differentiability, it suffices to show that the integral on the right-hand side of (17) above is well defined. To this end, recall once more (13). Letting for any x ∈ R K \B 0 (δ 0 ) and where C 0 = C m k +2 l∈K\{k} C m l . Choosing though α as in Step 1(iii) we have that Clearly, the integrand on the right-hand side of (17) is bounded above on R K \B 0 max δ 0 , δ 0 by the function CC 0 φ √ r x ; hence, its integral is well defined. To complete the argument, let k → 0 in (15).
Step 3. To establish finally continuous differentiability, it suffices to show that the partial derivatives are continuous. And as m above was taken arbitrarily, it suffices in turn to show that F m (·) itself is continuous. Taking ∈ (0, 1), consider then the neighborouhood B β ( ). For any z ∈ B β ( ), using again (13) above and the mean-value theorem, we have where γ k ∈ (0, 1) for k ∈ K. The remainder of the argument is trivially similar to that in steps 2(ii)-(iii) above.
Lemma A.5 Let (τ, β) ∈ R ++ × R K be a parameter vector. Suppose also that the continuously-differentiable function f : is well defined and C 1 , with its partial derivatives given by By changing the variables of integration, we have That F (·) is well defined and C 1 follows from Lemma A.4. Moreover, its partial derivatives are given by To establish the expession in the claim, fix first an arbitrary z −k ∈ R K −1 . By Lemma A.2 above, we can find r 0 , δ 0 > 0 such that Proof Under the given growth condition, for any (t, s, Recall now Lemma A.1 above. We can find constants C 1 ∈ R ++ and r 1 ∈ 0, 1 2 − r 0 T such that The claim follows by setting r = 1 − 2 (r 0 T − r 1 ) and C = C 0 C 1 e r 0 |β| 2 / √ 2π .
Proposition A.1 Let T = [0, T ] for some T ∈ R ++ and suppose that the price process is given by (2) in the main text with condition A.1(i) satisfied. Then, for any ( j, n, k) ∈ K ∪ {0} × K × K, the following hold (i) lim t→T m P j (I (ω, t)) = M j (I (ω, T )) almost everywhere (a.e.) on .
Moreover, M j (T , ·) being continuous so is the function t → M j T , β + √ T − tx everywhere on (0, T ) for any given (β, x) ∈ R K × R K . Furthermore, by Lemma A.6, there exist constants r , C > 0 such that By Lemma A.3 then the preceding observations mean that, for any given β ∈ R K , m P 1 j (t, β) is continuous in t everywhere on (0, T ). Taking now any ∈ 0, 1 2r − T , the argument above remains valid if T is replaced by T + . That is, the function t → m P 1 j (t, β) is continuous also on (0, T + ). It is therefore leftcontinuous at T in the original problem and thus (ii) Replacing the function M j (T , ·) with D k M j (T , ·) in the preceding argument, we obtain also that lim t→T D k m P j (t, β) = D k M j (T , β).
Lemma A.7 Suppose f : R K → R is measurable on R K and satisfies 23 Let also β be a standard K-dimensional Brownian motion and Then F (·) is an analytic function of (t, β) on (0, T ) × R K .
Proof The proof proceeds in exactly the same way as in that of Theorem B.4 in Anderson and Raimondo [2], but for a trivial adaptation of the argument that supports equations (23)- (27) in that paper. Specifically, fixing t < T , Eq. (21) in Anderson and Raimondo [2] gives Under the present growth condition, however, Eqs. (22)- (27) in Anderson and Raimondo [2] can be replaced now by the following steps 1 (2π (T − t)) K /2 R K y k 1 1 · · · y k K K k 1 ! · · · k K ! f (y) e − |y| 2 2(T −t) dy ≤ 1 (2π (T − t)) K /2 R K |y 1 | k 1 · · · |y K | k K k 1 ! · · · k K ! | f (y) |e − |y| 2 2(T −t) dy ≤ C (2π (T − t)) K /2 R K |y 1 | k 1 · · · |y K | k K k 1 ! · · · k K ! e − (1−2r(T −t))|y| 2 where the second inequality above follows from the growth condition itself and the third from the formula for the k i th moment of the absolute value of a normal random variable. Clearly, the last inequality above provides an upper bound for the absolute value of the coefficient of β k 1 1 · · · β k K K in the power series of equation (27) in Anderson and Raimondo [2]. Hence, the power series in question converges also here absolutely within a positive radius of convergence (see Proposition 2.2.10 in Krantz and Parks [20]). The remainder of the proof is identical to that in Anderson and Raimondo [2]. Proof Since r 0 < 1 2T < 1 2(T −t) < 1 2(s−t) ∀s ∈ (t, T ), it follows from Lemma A.4 (for m = 0 K ) that is well defined on (t, T ). By Lemma A.6, moreover, there exist a constant r > 0 and a continuous function C : The claim follows.
The preceding discussion establishes that the expressions in (5)-(6) in the main text follow from (19)- (20), and are indeed well defined.
With respect now to (7) in the main text, by Lemma A.5, at any s ∈ (t, T ] the integrand in (6) has a well-defined partial derivative: D k E x m j s, β + √ s − tx = E x D k m j s, β + √ s − tx . Given then Assumption A1(ii), Lemma A.4 ensures that the partial derivative is continuous in β for any given (t, s) ∈ (0.T ) × (t.T ). Moreover, we can apply again here the argument in the proof of Proposition A.1(i) to establish that, for any given (t, β) ∈ [0.T ) × R K , the partial derivative is also continuous in s everywhere on (t, T ). And as these observations about continuity apply also for the quantity E x m j s, β + √ s − tx itself, that the dispersion operator D k commutes under the time-integral in (6) follows by Leibniz's rule.

C Strong non-singularity
Recall the definition of strong non-singularity in the main text.
where e n denotes the K −dimensional column vector with one as its nth entry and zeroes everywhere else.