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 dynamic completeness of the candidate equilibrium price process follows from mild exogenous 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 of 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.

A number of authors have attempted to apply Nonstandard Analysis to Probability Theory. Unfortunately, the nonstandard reformulations heretofore proposed have retained most of the essential difficulties inherent in the standard formulations. As a result, the application of nonstandard techniques has met with limited success. Hersh [4] produced a nonstandard analogue of Wiener measure. His "measure", however, is not countably additive; moreover, it is supported on a countable subset of C([0, 1]). Using a different approach, Hersh and Greenwood [5] established some interesting results about nonstandard increments in Brownian motion and other stochastic processes, but failed to produce a successful formulation of the Itô integral or a proof of Itô's Lemma.In a recent paper [9], Peter A. Loeb introduced a new technique for formulating probabilistic problems in nonstandard terms. He showed that any nonstandard measure space within a denumerably comprehensive enlargement could be converted into a standard measure space which inherited important structural properties from the nonstandard space. Loeb gave applications to coin tossing and the Poisson process. The present paper outlines how Brownian motion and Itô integration can be successfully treated using Loeb's technique; the details will be presented in a subsequent article.Let 7? be an infinite natural number, in the sense of nonstandard analysis. Define an internal measure space (£2, 21, v) by £2 = {-1, l} 7 *, 31 = {internal subsets of 12}, v(A) = \A \l2 n for A G 21. Thus, v is counting measure. Loeb's results show that the standard part of v is necessarily countably additive; hence, by the Carathéodory Extension Theorem, it has a unique extension to the a-field generated by 21. We shall call the completion of this extension the Loeb space corresponding to (12, 21, v), and denote it by (12,1(21), L(y)). AMS (MOS) subject classifications (1970). Primary 60H05, 60J65, 02H25.

A number of authors have attempted to apply Nonstandard Analysis to Probability Theory. Unfortunately, the nonstandard reformulations heretofore proposed have retained most of the essential difficulties inherent in the standard formulations. As a result, the application of nonstandard techniques has met with limited success. Hersh [4] produced a nonstandard analogue of Wiener measure. His "measure", however, is not countably additive; moreover, it is supported on a countable subset of C([0, 1]). Using a different approach, Hersh and Greenwood [5] established some interesting results about nonstandard increments in Brownian motion and other stochastic processes, but failed to produce a successful formulation of the Itô integral or a proof of Itô's Lemma.In a recent paper [9], Peter A. Loeb introduced a new technique for formulating probabilistic problems in nonstandard terms. He showed that any nonstandard measure space within a denumerably comprehensive enlargement could be converted into a standard measure space which inherited important structural properties from the nonstandard space. Loeb gave applications to coin tossing and the Poisson process. The present paper outlines how Brownian motion and Itô integration can be successfully treated using Loeb's technique; the details will be presented in a subsequent article.Let 7? be an infinite natural number, in the sense of nonstandard analysis. Define an internal measure space (£2, 21, v) by £2 = {-1, l} 7 *, 31 = {internal subsets of 12}, v(A) = \A \l2 n for A G 21. Thus, v is counting measure. Loeb's results show that the standard part of v is necessarily countably additive; hence, by the Carathéodory Extension Theorem, it has a unique extension to the a-field generated by 21. We shall call the completion of this extension the Loeb space corresponding to (12, 21, v), and denote it by (12,1(21), L(y)).Define a random walk x-*[0, 1] x 12 -> *R by Xfcco)-Z ^ + (i,r-[ l ,f])îi2ttl±l k<[nt] \nVf?and define 0: [0, 1] x 12 -• R by 0(f, co) = °x(t, co).

Genericity analysis is widely used to show that desirable properties that fail in certain "knife-edge" economic situations nonetheless obtain in "typical" situations. For finite-dimensional spaces of parameters, the usual notion of genericity is full Lebesgue measure. For infinite dimensional spaces of parameters (for instance, the space of preferences on a finite-dimensional commodity space, no analogue of Lebesgue measure is available; the lack of such an analogue has prompted the use of less compelling topological notions of genericity. Christensen (1974) and Hunt, Sauer and Yorke (1992) have proposed a measure-theoretic notion of genericity, which Hunt, Sauer and Yorke call prevalence, which coincides with full Lebesgue measure in Euclidean space and which extends to infinite-dimensional vector spaces. This notion is not directly applicable in most economic settings because the relevant parameter sets are small subsets of vector spaces -- especially cones or order intervals -- not vector spaces themselves. We adapt the notion to economically relevant environments by defining two notions of prevalence relative to a convex set in a topological vector space. The first notion is very easy to understand and apply, and has all of the properties one would desire except that it is not closed under countable unions; the second notion contains the first and has all the good properties of the first notion except simplicity; it is closed under countable unions. We provide four economic applications: 1) generic existence of equilibrium in financial models, 2) generic finiteness of the number of pure strategy Nash equilibria and Pareto inefficiency of "non-vertex" Nash equilibria for games with a continuum of actions and smooth payoffs, 3) generic regularity of exchange economies when some agents are constrained to have 0 endowment of some goods, 4) generic single-valuedness of the core of transferable utility games.