Nonstandard construction of the stochastic integral and applications to stochastic differential equations. I

Hoover, Douglas N.; Perkins, Edwin

doi:10.1090/S0002-9947-1983-99928-9

Nonstandard construction of the stochastic integral and applications to stochastic differential equations. I
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by Douglas N. Hoover and Edwin Perkins PDF

Trans. Amer. Math. Soc. 275 (1983), 1-36 Request permission

Abstract:

R. M. Anderson has developed a nonstandard approach to Itô integration in which the Itô integral is interpreted as an internal Riemann-Stieltjes sum. In this paper we extend this approach to integration with respect to semimartingales. Lifting and pushing down theorems are proved for local martingales, semimartingales and other right-continuous processes on a Loeb space.

References

Star-finite probability theory

Robert M. Anderson, A non-standard representation for Brownian motion and Itô integration, Israel J. Math. 25 (1976), no. 1-2, 15–46. MR 464380, DOI 10.1007/BF02756559
Robert M. Anderson, Star-finite representations of measure spaces, Trans. Amer. Math. Soc. 271 (1982), no. 2, 667–687. MR 654856, DOI 10.1090/S0002-9947-1982-0654856-1
Patrick Billingsley, Convergence of probability measures, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR 0233396
D. L. Burkholder, B. J. Davis, and R. F. Gundy, Integral inequalities for convex functions of operators on martingales, Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971) Univ. California Press, Berkeley, Calif., 1972, pp. 223–240. MR 0400380
Claude Dellacherie and Paul-André Meyer, Probabilités et potentiel, Publications de l’Institut de Mathématique de l’Université de Strasbourg, No. XV, Hermann, Paris, 1975 (French). Chapitres I à IV; Édition entièrement refondue. MR 0488194
Catherine Doléans, Variation quadratique des martingales continues à droite, Ann. Math. Statist. 40 (1969), 284–289 (French). MR 236982, DOI 10.1214/aoms/1177697823
J. L. Doob, Stochastic processes, John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London, 1953. MR 0058896

A notion of equivalence for adapted processes

Jean Jacod, Intégrales stochastiques par rapport à une semimartingale vectorielle et changements de filtration, Seminar on Probability, XIV (Paris, 1978/1979) Lecture Notes in Math., vol. 784, Springer, Berlin, 1980, pp. 161–172 (French). MR 580121
Jean Jacod and Jean Mémin, Existence of weak solutions for stochastic differential equations with driving semimartingales, Stochastics 4 (1980/81), no. 4, 317–337. MR 609691, DOI 10.1080/17442508108833169
H. Jerome Keisler, An infinitesimal approach to stochastic analysis, Mem. Amer. Math. Soc. 48 (1984), no. 297, x+184. MR 732752, DOI 10.1090/memo/0297
Tom L. Lindstrøm, Hyperfinite stochastic integration. I. The nonstandard theory, Math. Scand. 46 (1980), no. 2, 265–292. MR 591606, DOI 10.7146/math.scand.a-11868
Tom L. Lindstrøm, Hyperfinite stochastic integration. I. The nonstandard theory, Math. Scand. 46 (1980), no. 2, 265–292. MR 591606, DOI 10.7146/math.scand.a-11868
Tom L. Lindstrøm, Hyperfinite stochastic integration. I. The nonstandard theory, Math. Scand. 46 (1980), no. 2, 265–292. MR 591606, DOI 10.7146/math.scand.a-11868
Peter A. Loeb, Conversion from nonstandard to standard measure spaces and applications in probability theory, Trans. Amer. Math. Soc. 211 (1975), 113–122. MR 390154, DOI 10.1090/S0002-9947-1975-0390154-8
Peter A. Loeb, Weak limits of measures and the standard part map, Proc. Amer. Math. Soc. 77 (1979), no. 1, 128–135. MR 539645, DOI 10.1090/S0002-9939-1979-0539645-6
Peter A. Loeb, An introduction to nonstandard analysis and hyperfinite probability theory, Probabilistic analysis and related topics, Vol. 2, Academic Press, New York-London, 1979, pp. 105–142. MR 556680
Michel Métivier and Jean Pellaumail, Stochastic integration, Probability and Mathematical Statistics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London-Toronto, Ont., 1980. MR 578177
P. A. Meyer, Un cours sur les intégrales stochastiques, Séminaire de Probabilités, X (Seconde partie: Théorie des intégrales stochastiques, Univ. Strasbourg, Strasbourg, année universitaire 1974/1975), Lecture Notes in Math., Vol. 511, Springer, Berlin, 1976, pp. 245–400 (French). MR 0501332

Hyperreal probability spaces: some applications of the Loeb construction

A non-standard approach to Brownian local time

Edwin Perkins, On the construction and distribution of a local martingale with a given absolute value, Trans. Amer. Math. Soc. 271 (1982), no. 1, 261–281. MR 648092, DOI 10.1090/S0002-9947-1982-0648092-2

Limit theorems for stochastic processes

1

Charles Stone, Weak convergence of stochastic processes defined on semi-infinite time intervals, Proc. Amer. Math. Soc. 14 (1963), 694–696. MR 153046, DOI 10.1090/S0002-9939-1963-0153046-2
K. D. Stroyan and W. A. J. Luxemburg, Introduction to the theory of infinitesimals, Pure and Applied Mathematics, No. 72, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1976. MR 0491163

Introduction to infinitesimal stochastic analysis