Department Mathematik
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Inhaltsbereich

"Infinitesimal" Group

within the Working Group Mathematical Logic of the Mathematical Institute at the Ludwig Maximilians Universität München:

Eva Aigner, Josef Berger, Horst Osswald

It follows from results in model theory, a field within mathematical logic, that there exists a model of mathematics, in which, on the one side, we are able to make the same mathematics as in the usual standard model. On the other side each (internal) set can be handled as though it were k-kompakt (k-saturated), where k is an arbitrarily fixed sufficiently large cardinality. Then any infinite set in the usual model of mathematics can be extended to a finite set in the sense of this new model. This fact makes it possible to approximate infinite entities of standard mathematics by finite objects in the new model from above. The difference between the standard entity and this finite approximating object is in general smaller than any positive real number.

In our group we apply these results to Measure Theory, Functional Analysis and Stochastic Analysis. Here are publications on this subject and related topics:

  • C. Hort, H. Osswald, On nonstandard model in higher order logic, J. Symb. Logic 49, (1984) 204-219
  • H. Osswald, A convergence theorem for the Lewis integral, Tagungsberichte der Bergischen Universität Wuppertal (1988) 19-29
  • H. Osswald, On vector valued measures in Nonstandard Analysis and the stochastic integral, Tagungsberichte der Bergischen Universität Wuppertal (1989) 121-140
  • H. Osswald, Y. Sun, On the extensions of vector-valued Loeb measures, Proc. Am. Math. Soc. 111, (1991) 663-675
  • H. Osswald, Vector valued Loeb measures and the Lewis integral, Math. Scand. 68 (1991) 247-268
  • H. Osswald, A note on liftings of linear continuous functionals, Proc. Am. Math. Soc. 120 (1994) 253-256
  • H. Osswald, A nonstandard approach to the Pettis integral, Proc. of the int. Conf. at the H. Fabri Institut, Kluver Ac. Publishers, Math. Appl. Dordr. 314 (1995) 75-90
  • P. A. Loeb, H. Osswald, Nonstandard Integration Theory in Topological Vector Lattices, Monatshefte für Mathematik 124 (1997) 53-82
  • P. A. Loeb, H. Osswald, Refining the uniform convergence topology, Classical and Modern Potential Theory and Appl. ed K. Gowribankavan et al. Kluver Acad. Press (1994)
  • H. Osswald, Infinitesimals in abstract Wiener spaces, in Stochastic processes, physics and geometry: new interplays, II (Leipzig, 1999) (A volume in honor of Sergio Albeverio, eds. F. Gesztesy et al), CMS Conf. Proc 29, Amer. Math. Soc., Providence, RI, 2000, 539-546.
  • H. Osswald, The Existence of poly-staturated models, Nonstandard-Analysis for the working mathematician, P. Loeb, M. Wolff eds., Kluwer Scientific, Dordrecht (2000), 57-72.
  • H. Osswald, Y. Sun, Measure and Probability Theory and Applications, Nonstandard-Analysis for the working mathematician, P. Loeb, M. Wolff eds., Kluwer Scientific, Dordrecht, (2000), 137-256.
  • J. Berger, H. Osswald, Y. Sun, J. L Wu, On nonstandard product measure spaces, Illinois Journal of Mathematics 46 (2002) 319-330.
  • P. A. Loeb, H. Osswald, Y. Sun, Z. Zhang, Uncorrelatedness and Orthogonality for Vector-Valued Processes, Trans. Amer. Math. Soc., 356, (8), (2004) 3209-3225.
  • H. Osswald, J.-L. Wu, On infinite-dimensional continuous Ornstein-Uhlenbeck processes, Acta Appl. Math., 83 (2004) 289-312.
  • H. Osswald, Malliavin Calculus in Abstract Wiener Spaces using infinitesimals, Advances in Mathematics 176 (2003) 1-37.
  • H. Osswald, On the Clark-Ocone formula for the abstract Wiener space, Advances in Mathematics 176 (2003) 38-52.
  • P. Schuster, U. Berger, H. Osswald eds. Reuniting the Antipodes, Proc. of the Venedig Conference, Kluwer Scientific (2001).
  • H. Osswald, Malliavin calculus on product measures of $% \mathbb{R}^{\mathbb{N}}$ based on chaos, Stochastics 77 (6) (2005), 501-514.
  • H. Osswald, A smooth approach to Malliavin calculus for Lévy processes, Journal of Theoretical Probability, 22 (2009), 441-473.
  • H. Osswald, Malliavin calculus on extensions of abstract Wiener spaces, Journal of Mathematics of the Kyoto University, 8} (2) (2008), 239-262.
  • H. Osswald, On Anicipative Girsanov Transformations, Journal of Theoretical Probability, 22 (2009), 474-481.
  • H. Osswald, Computation of te kernels of Lévy functionals and applications, to appear in Illinois Journal of Mathematics.
  • H. Osswald, Ist Zeit kontinuierlich oder diskret?, Mathe-LMu.de, Vol 26, (2012), 27-34.
  • H. Osswald, Malliavin calculus for Lévy processes and infinite-dimensional Brownian motion, Cambridge Tracts in Mathematics, Vol 191, Cambridge University Press, (2012)

    Cambridge University Press writes about this book

    Assuming only basic knowledge of probability theory and functional analysis, this book provides a self-contained introduction to Malliavin calculus and infinite-dimensional Brownian motion. In an effort to demystify a subject thought to be difficult, it exploits the framework of nonstandard analysis, which allows infinite-dimensional problems to be treated as finite-dimensional. The result is an intuitive, indeed enjoyable, development of both Malliavin calculus and nonstandard analysis.

    The main aspects of stochastic analysis and Malliavin calculus are incorporated into this simplifying framework. Topics covered include Brownian motion, Ornstein-Uhlenbeck processes both with values in abstract Wiener spaces, Lévy processes, multiple stochastic integrals, chaos decomposition, Malliavin derivative, Clark-Ocone formula. Skorohod integral/processes and Girsanov transformations. The exposition, which is neither too abstract nor too theoretical, makes this book accessible to graduate students as well as to researchers interested in the techniques.

  • H. Osswald, Existence of Polysaturated Nonstandard Models, Nonstandard-Analysis for the working mathematician, P. Loeb, M. Wolff eds., second edition, Springer, to appear 2015.
  • H. Osswald, Measure Theory and Integration, Nonstandard-Analysis for the working mathematician, P. Loeb, M. Wolff eds., second edition, Springer, to appear 2015.
  • H. Osswald, Stochastic Analysis, Nonstandard-Analysis for the working mathematician, P. Loeb, M. Wolff eds., second edition, Springer, to appear 2015.