The Role of (Quasi) Analyticity in Establishing Completeness of Financial Markets Equilibria

Abstract

Existence of equilibrium in a continuous-time securities market in which the securities are endogenously dynamically complete has been proved only recently. Given the fundamental nature of this result various extensions have been proposed. In the present paper we prove all results under optimal condition. Namely, we only assume quasi-analyticity rather than analyticity of the basic economic ingredients, and we prove everything based solely on this hypothesis.

Appendices

Appendix 1: Quasi-Analytic Functions of Several Variables

In this Appendix, we summarize the results on quasi-analytic functions of several variables used in our proofs.

Let the sequence of positive numbers $\left \{ M_{k}\right \} _{k=0}^{\infty }$ satisfy the following conditions (see Tanabe, 1979): there exist positive numbers d ₀, d ₁, and d ₂ such that

Let Ω be an open subset of $\mathbb {R} ^{n}$. We denote by $\mathcal {D}(\Omega ,\left \{ M_{k}\right \} )$ the set of all infinitely differentiable functions u defined on Ω such that for each compact set $\mathfrak {K}\subset \Omega$ there exist positive constants C ₀ and C so that for every multi-index α we have the inequality

$$\displaystyle \begin{aligned} \max_{x\in \mathfrak{K}}\left\vert D^{\alpha }u(x)\right\vert \leq C_{0}C^{\left\vert \alpha \right\vert }M_{\left\vert \alpha \right\vert }. {} \end{aligned} $$

(12)

The class $\mathcal {D}(\Omega ,\left \{ k!\right \} )$ coincides with the class of real analytic functions on Ω. A class $\mathcal {D}$ of functions defined on Ω is said to be quasi-analytic if the only function u in the class $\mathcal {D}$ such that D ^α u(x ₀) = 0 for all α for a fixed x ₀ ∈ Ω is the identically zero function, u ≡ 0. It is well-known (Carleman-Denjoy Theorem) (see Hörmander, 1990 and Tanabe, 1979) that the class $\mathcal {D}(\Omega ,\left \{ M_{k}\right \} )$ is quasi-analytic if and only if $\sum _{k=0}^{\infty }\left ( M_{k}\right ) ^{-\frac {1}{k}}=\infty .$ The sequence $M_{k}=\left ( k\log k\right ) ^{k}$ determines a non-real analytic, but a quasi-analytic, class and satisfies Q1 − Q5. The following class of functions which are nowhere real analytic in the real line but are nevertheless determined uniquely by the values of all the derivatives at one point was given by Borel (1901):

$$\displaystyle \begin{aligned} g(x)=\sum_{q=1}^{\infty }\sum_{p=-\infty }^{\infty }\sum_{p^{\prime }=-\infty }^{\infty }\frac{\varphi (p,p^{\prime },q^{\prime })}{x+i\sqrt{2}- \frac{p+ip^{\prime }}{q}} \end{aligned} $$

with

$$\displaystyle \begin{aligned} \left\vert \varphi (p,p^{\prime },q^{\prime })\right\vert <e^{-e^{p^{8}+p^{\prime 8}+q^{8}}} \end{aligned} $$

and the coefficients φ(p, p ^′, q ^′)^′s are nowhere zero (for real functions consider the class $f(x)=\left \vert g(x)\right \vert ^{2}).$

Note the well-known facts that if $f\in \mathcal {D}(\Omega ,\left \{ M_{k}\right \} )$ then $D^{\alpha }f\in \mathcal {D}(\Omega ,\left \{ M_{k}\right \} )$ for any α, and if $f,g\in \mathcal {D}(\Omega ,\left \{ M_{k}\right \} )$ then the product $f\cdot g\in \mathcal {D}(\Omega ,\left \{ M_{k}\right \} ).$ We are going to make a constant use of these facts.

Quasi-analytic functions of several variables have been studied extensively in Bierstone and Milman (2004). They prove an implicit function theorem for quasi-analytic functions.

Theorem 5 (The Quasi-Analytic Implicit Function Theorem)

Suppose that U is open in R ⁿ ×R ^p (with product coordinates (x, y) = (x ₁, …, x _n, y ₁, …, y _p)). Suppose that f ₁, …, f _p are quasi-analytic, (a, b) ∈ U, f(a, b) = 0 and

$$\displaystyle \begin{aligned} (\partial f/\partial y)(a,b) \end{aligned}$$

is invertible, where f = (f1, …, fp). Then there is a product neighborhood V × W of (a, b) in U and a quasi–analytic mapping g : V → W such that g(a) = b and

$$\displaystyle \begin{aligned} f\left( x,g(x)\right) =0\mathit{\ }\forall x\in V. \end{aligned}$$

The main subject of Bierstone and Milman (2004) is proving the possibility of resolution of singularities for quasi-analytic functions. As observed in Bierstone and Milman (2004, Corollary 5.13), if U is an open set in R ⁿ and f : U →R is quasi-analytic, then for every x ₀ ∈ U, there exists a neighborhood $V_{x_{0}}$ of x ₀ such that either F(x) = 0 for all $x\in V_{x_{0}}$ or $\{x\in \mathcal {V}_{x_{0}}:F(x)=0\}$ is a finite union of quasi-analytic varieties of dimension < n. From this follows the following

Corollary 6

Let $\mathcal {O}\subset {\mathbf {R}}^{n}$ be open and convex, $f:\mathcal {O}\rightarrow \mathbf {R}$ is quasi-analytic. If $\{x\in \mathcal {O}:f(x)=0\}$ has positive Lebesgue measure, then f is identically zero on $\mathcal {O}$.

Proof

If f(x) = 0 for all $x\in \mathcal {V}_{x_{0}}$ and $y\in \mathcal {O}$, there is a ray that passes through $\mathcal {V}_{x_{0}}$ then the function f has to vanish identically on the connected set $\mathcal {O}$. On the other hand, if $\{x\in \mathcal {V}_{x_{0}}:f(x)=0\}$ is a finite union of quasi-analytic varieties of dimension < n, $\{x\in \mathcal {V}_{x_{0}}:F(x)=0\}$ has Lebesgue measure zero. There is a countable collection {x _n : n ∈N} such that $\cup _{n\in \mathbf {N}}\mathcal {V}_{x_{n}}\supset U$, so {x ∈ U : f(x) = 0} has Lebesgue measure zero. $\blacksquare$

Quasi-analyticity of solutions of parabolic partial differential equations such as those that are satisfied by securities prices has been obtained recently in Kannai and Raimondo (2013) in a form sufficient for establishing dynamic completeness. The relevant definitions and assumptions are stated explicitly in the cited paper.

Theorem 7

Let f(t, x) ∈ $\mathcal {D}( \mathbb {R} ^{n\mathit{}}\times \left ( 0,T\right ) ,\left \{ M_{k}\right \} )$ , $u_{0}(x)\in L_{\mathit{\text{loc}}}^{1}( \mathbb {R} ^{n})$ be such that for every δ > 0

$$\displaystyle \begin{aligned} \int \left\vert u_{0}(x)\right\vert \exp \left( -\delta \sum_{i=1}^{n}\left\vert x_{i}\right\vert ^{\frac{m}{m-1}}\right) dx<\infty , {} \end{aligned} $$

(13)

and there exist constants C ₀, C such that

$$\displaystyle \begin{aligned} \int \left\vert \left( \frac{\partial }{\partial t}\right) ^{k}f(t,x)\right\vert \exp \left( -\delta \sum_{i=1}^{n}\left\vert x_{i}\right\vert ^{\frac{m}{m-1}}\right) dx<C_{0}C^{k}M_{k} {} \end{aligned} $$

(14)

uniformly in $\left [ 0,T\right ] ,$ for every non-negative integer k. Let A satisfy the assumptions of Lemma 3 of Kannai and Raimondo ( 2013 ). Then the solution u(t, x) of the differential equation

$$\displaystyle \begin{aligned} \frac{\partial u}{\partial t}+Au=f \end{aligned} $$

(15)

with the initial condition

$$\displaystyle \begin{aligned} u(x,0)=u_{0}\end{aligned} $$

(16)

given by the formula

$$\displaystyle \begin{aligned} u(x,t)=\int_{ \mathbb{R} ^{n\mathit{}}}U(x,t,y,0)u_{0}(y)dy+\int_{0}^{t}\int_{ \mathbb{R} ^{n\mathit{}}}U(x,t,y,s)f(s,y)dyds\end{aligned} $$

(17)

is quasi-analytic in $\mathbb {R} ^{n}\times \left ( 0,T\right )$.

Observe that in our case m = 2.

Appendix 2: Sketch of Proof of Main Theorem

Proof

Here we only sketch the proof, as it follows the one in Anderson and Raimondo (2008), emphasizing only the points where we differ. In order to prove existence we observe that (see Dana, 1993) the consumption is given by the solution of the social planner’s problem

$$\displaystyle \begin{aligned} \max_{\sum_{i=1,..,I}c_{i}\leq e}\sum_{i=1}^{I}\lambda _{i}U_{i}(c_{i}), \end{aligned}$$

where the parameters λ _i are Negishi weights. Since we have von Neumann-Morgenstern utility functions the solution is equivalent to the solutions of the problem state-by-state. Therefore the solution is given by

$$\displaystyle \begin{aligned} u(\lambda _{1},\ldots,\lambda _{I},t,x)=\max_{\sum_{i=1,..,I}x_{i}\leq x}\sum_{i=1}^{I}\lambda _{i}h_{i}(t,x_{i})\end{aligned} $$

and this is equivalent to the solution of the following (Lagrange multipliers) system

$$\displaystyle \begin{aligned} \left\{ \begin{array}{cc} \lambda _{1}\frac{\partial h_{1}}{\partial c} & =\widetilde{\mu } \\ \lambda _{2}\frac{\partial h_{2}}{\partial c} & =\widetilde{\mu } \\ \vdots & \\ \lambda _{I}\frac{\partial h_{I}}{\partial c} & =\widetilde{\mu } \\ \sum_{i=1,..,I}x_{i} & =x \end{array} \right. \end{aligned}$$

The Implicit Function Theorem of Bierstone and Milman (2004, see theorem 5 in Appendix 1) implies that there exist quasi-analytic functions $x_{1},\ldots ,x_{I},\widetilde {\mu }$ which are solutions of the system and this implies, by a well-known result (Dana, 1993), that

$$\displaystyle \begin{aligned} \left( c_{1},\ldots,c_{I},p_{C}\right) =\left( x_{1}(t,e),\ldots,c_{I}(t,e), \widetilde{\mu }(t,e)\right) \end{aligned}$$

is a contingent Arrow Debreu equilibrium with quasi-analytic data, in particular the relative price of consumption at equilibrium p _C(t, x) is quasi-analytic. It follows that the securities prices $p_{A}(t,X)=\left ( p_{A_{0}}(t,X),\ldots ,p_{A_{K}}(t,X)\right )$ are given by the expectations (8) and satisfy the partial differential equation (2) with the final condition

$$\displaystyle \begin{aligned} p_{Aj}(T,x)=D_{j}(T,X(T))p_{C}(T,x) {} \end{aligned} $$

(18)

and Theorem 7 of Appendix 1 applies. Hence the security prices are quasi-analytic.

Finally quasi-analyticity and non-degeneracy of the dispersion of the final lump dividend (7), together with corollary (6), imply the non-degeneracy of the price dispersion matrix. Hence (see Karatzas and Shreve, 1998) the equilibrium is dynamically complete. $\blacksquare$