Model Theory at the University of Munich: Non-standard Analysis
Using model theory it is possible to construct a new model of mathematics, having the same properties as the standard model of mathematics, such that each set is k-compact, where k is an arbitrarily given cardinality. We obtain a new notion of finiteness: in the new model there exist infinite sets, which can be treated as though they were finite sets. For example, we may sum up the elements in a "finite infinite" additive semigroup, without having any topology. This is quite surprising even unbelievable; the idea for the proof of the existence of such a model is similar to the proof that any field has an algebraically closed extension. The extension of the standard notion of finiteness makes it possible to approximate infinite mathematical entities by finite infinite objects from above with an infinitesimal error, that is, the error is smaller than any positive real number. This theory is often called "Nonstandard analysis", which is not at all a good phrase, because Nonstandard Analysis is not apart from but a part of standard Analysis. Both theories are living under the common roof of Zermelo-Fraenkel set theory with the Axiom of Choice. In my research these new notion of finiteness is applied to probability theory and stochastics:
- It is well known that in the new model there exist measure spaces for which the product is richer than the usual standard product. In a joint paper with Y. Sun (Singapore), J.L. Wu (Swansea), J. Berger (Muenchen) we have studied this new product systematically: in contrast to the usual product there exist a sequence along the real numbers of independent events which is product measurable. This fact may have applications to mathematical economy.
- It can be seen that the continuous time line for stochastic processes is almost indistinguishable from the -in the new sense- finite time line: using this fact, it is possible to construct pathwise a continuous Brownian motion with values in any abstract Wiener space; this Brownian motion is infinitely close to a finite dimensional Brownian motion which belongs to the new model.
- In a similar way it is possible to construct a continuous infinite dimensional Ornstein Uhlenbeck process, by solving pathwise an infinite dimensional stochastic Langevin differential equation (joint work with J.L. Wu), which is -up to an infinitesimal error- a finite difference equation.
- The constructed Brownian motion is used to obtain Malliavin calculus on the space of continuous functions defined on the real time line with values in any abstract Wiener space. Taking constant functions we obtain the established Malliavin calculus on abstract Wiener spaces. Since, contrary to my approach, in this theory there does not exist a real time line, Uestuenel and Zakai recently introduced time by means of a resolution of the identity on the underlying Hilbert space. Applications of my approach are Girsanov theorems for time anticipating shifts of the Banach space valued Brownian motion by the Bochner integral of square integrable stochastic processes with values in the underlying Hilbert space.