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This work is based on Nelson’s paper [], where the central question was: under suitable regularity conditions, what is the form of the infinitesimal generator of a Markov semigroup?

In the elementary approach using IST []. the idea is to replace the continuous state space, such as ℝ with a finite state space possibly containing an unlimited number of points. The topology on arises naturally from the probability theory. For ε , let be the set of all ∈ vanishing at where is the multiplier algebra of the domain of the infinitesimal generator. To describe the structure of the semigroup generator , we want to split ()=∑() () so that the contribution of the external set of the points far from appears separately. A definition of the quantity ()=∑() () is given using the least upper bound of the sums on all internal sets included in the external set . This leads to the characterization of the global part of the infinitesimal generator.

Suitable notions of “unfairness” that measure how far an empirical discounted asset price process is from being a martingale are introduced for complete and incomplete-market settings. Several limit processes are involved each time, prompting a nonstandard approach to the analysis of this concept. This leads to an existence result for a “fairest price measure” (rather than a martingale measure) for an asset that is simultaneously traded on several stock exchanges. This approach also proves useful when describing the impact of a currency transaction tax.

Based on a *-algcbraic approach to quantum probability theory we construct basic discrete internal quantum stochastic processes with independent increments. We obtain a one-parameter family of (classical) Bernoulli experiments as linear combinations of these basic processes.

Then we use the nonstandard hull of the internal GNS-Hilbert space corresponding to the chosen state (the underlying quantum probability measure) in order to derive nonstandard hulls of our internal processes. Finally continuity requirements lead to the specification of a certain subspace of to which the nonstandard hulls of our internal processes can be restricted and which turns out to be isomorphic to the Loeb-Guichardet space introduced by Leitz-Martini []. A subspace of then is shown to be isomorphic to the symmetric Fock space (([0,1], λ)) and our basic processes agree with the processes of Hudson and Parthasarathy on this subspace.

We review some recent work by Yeneng Sun and the author. Sun’s work shows that there are results, some used for decades without a rigourous foundation, that arc only true for spaces with the rich structure of Loeb measure spaces. His joint work with the author uses that structure to extend an important result on the purification of measure valued maps.

In their important (but often overlooked) paper []. C. Ward Henson and Frank Wallenberg introduced the notion of . and showed that S-measurable functions are “approximately standard” (in a sense made precise in the next section).

The conclusion of a Radon-Nikodým theorem is that a measure can be represented as an integral with respect to a reference measure such that for all measurable sets () = ∫() with a (Bochner or Lebesgue) integrable derivative or density . The measure is usually a countably additive -finite measure on the given measure space and the measure is absolutely continuous with respect to . Different theorems have different range spaces for . which could be the real numbers, or Banach spaces with or without the Radon-Nikodým property. In this paper we generalize to derivatives of vector valued measures with respect a vector-valued reference measure. We present a Radon-Nikodým theorem for vector measures of bounded variation that are absolutely continuous with respect to another vector measure of bounded variation. While it is easy in settings such as << , where is Lebesgue measure on the interval [0,1] and is vector-valued to write down a nonstandard Radon-Nikodým derivative of the form : *[0,1] → fin(*) by a vector valued reference measure does not allow this approach, as the quotient of two vectors in different Banach spaces is undefined. Furthermore, generalizing to a vector valued control measure necessitates the use of a generalization of the Bartle integral, a bilinear vector integral.

We introduce a general definition of -differentiability of an internal measure and compare different special cases. It will be shown how -differentiability of an internal measure yields differentiability of the associated Loeb measure. We give some examples.

The search for useful non standard minimization conditions on functionals defined on Banach spaces lead us to a very simple argument which shows that if a function : → between Banach spaces is actually Gâteaux differentiable on finite points along finite vectors, then it is uniformly continuous on bounded sets if and only if it is lipschitzian on bounded sets. The following is a development of these ideas starting from locally convex spaces.

We present nonstandard versions of the Palais-Smale condition (PS) below, some of them generalizations, but still sufficient to prove Mountain Pass Theorems, which are quite important in Critical Point Theory.

A nonstandard approach to averaging theory for ordinary differential equations and functional differential equations is developed. We define a notion of perturbation and we obtain averaging results under weaker conditions than the results in the literature. The classical averaging theorems approximate the solutions of the system by the solutions of the averaged system, for Lipschitz continuous vector fields, and when the solutions exist on the same interval as the solutions of the averaged system. We extend these results to perturbations of vector fields which are uniformly continuous in the spatial variable with respect to the time variable and without any restriction on the interval of existence of the solution.