Abstract
Differential algebras of generalized stochastic processes are constructed, which contain irregular processes as e. g. white noise. Solutions to nonlinear stochastic differential equations are obtained in these algebras. The methods extend the nonlinear theories of generalized functions as developed by J. F. Colombeau, Yu. V. Egorov, E. E. Rosinger to the stochastic setting. Both an approach based on regularized sample paths as well as on sequences of smooth L 0-valued functions are presented.
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References
S. Albeverio, Z. Haba, F. Russo, Trivial solutions for a non-linear two-space dimensional wave equation perturbed by space-time white noise, Prépublication 93 17, Université de Provence, Marseille, 1993.
H. Bunke, Gewöhnliche Differentialgleichungen mit zufälligen Parametern, Akademie-Verlag, Berlin, 1972.
J. F. Colombeau, New Generalized Functions and Multiplication of Distributions, North-Holland, Amsterdam, 1984.
J. F. Colombeau, Elementary Introduction to New Generalized Functions, North-Holland, Amsterdam, 1985.
J. F. Colombeau, Multiplication of distributions, Lecture Notes Math. 1532, Springer-Verlag, Berlin, 1992.
J. F. Colombeau, Multiplication of distributions, Bull. Am. Math. Soc. 23 (1990), 251 - 268.
Yu. V. Egorov, A contribution to the theory of generalized functions, Russian Math. Surveys 45:5 (1990), 1-49, Translated from: Uspekhi Mat. Nauk 45:5 (1990), 3 - 40.
I. M. Gel’fand, N. Ya. Vilenkin, Generalized Functions, Vol. 4: Applications of Harmonic Analysis, Academic Press, New York, 1964.
H. Holden, T. Lindstrom, B. Oksendal, J. Uboe, T.-S. Zhang, The Burgers equation with a noisy force and the stochastic heat equation, Comm. Part. Diff. Eqs. 19 (1994), 119 - 141.
P. E. Kloeden, E. Platen, Numerical Solution of Stochastic Differential Equations, Springer-Verlag, Berlin, 1992.
D. Laugwitz, Anwendung unendlich kleiner Zahlen. I. Zur Theorie der Distributionen, J. reine angew. Math. 207 (1961), 53 - 60.
T. Lindstrom, B. Oksendal, J. Uboe, Wick multiplication and Itô-Skorohod stochastic differential equations, In: Ideas and Methods in Mathematical Analysis, Stochastics, and Applications (S. Albeverio, J. E. Fenstad, H. Holden, T. Lindstrom, eds.), vol. I, Cambridge University Press, 1992.
M. Loève, Probability Theory, vol. I and II. Fourth Ed., Springer-Verlag, New York, 1977/ 1978.
E. Nelson, Radically Elementary Probability Theory, Annals Math. Studies, vol. 117, Princeton University Press, 1987.
M. Oberguggenberger, Multiplication of Distributions and Applications to Partial Differential Equations, Pitman Research Notes Math., vol. 259, Longman, Harlow, 1993.
M. Oberguggenberger, Nonlinear theories of generalized functions, In: Advances in Analysis, Probability, and Mathematical Physics - Contribution from Nonstandard Analysis ( S. Albeverio, W. A. J. Luxemburg, M.P. H. Wolff, eds.), Kluwer, Dordrecht, 1994.
E. E. Rosinger, Distributions and Nonlinear Partial Differential Equations, Lecture Notes Math. 684, Springer-Verlag, Berlin, 1978.
E. E. Rosinger, Nonlinear Partial Differential Equations. Sequential and Weak Solutions, North Holland, Amsterdam, 1980.
E. E. Rosinger, Generalized Solutions of Nonlinear Partial Differential Equations, North-Holland, Amsterdam, 1987.
E. E. Rosinger, Non-linear Partial Differential Equations. An Algebraic View of Generalized Solutions, North Holland, Amsterdam, 1990.
E. Wong, M. Zakai, On the convergence of ordinary integrals to stochastic integrals, Ann. M.th. Statistics 36 (1965), 1560 - 1564.
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Oberguggenberger, M. (1995). Generalized Functions and Stochastic Processes. In: Bolthausen, E., Dozzi, M., Russo, F. (eds) Seminar on Stochastic Analysis, Random Fields and Applications. Progress in Probability, vol 36. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-7026-9_16
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DOI: https://doi.org/10.1007/978-3-0348-7026-9_16
Publisher Name: Birkhäuser, Basel
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