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

Authors: Douglas N. Hoover and Edwin Perkins
Journal: Trans. Amer. Math. Soc. 275 (1983), 1-36
MSC: Primary 60H10; Secondary 03H05
Part II: Trans. Amer. Math. Soc. (1) (1983), 37-58
MathSciNet review: 678335
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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.

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  • [1] R. M. Anderson, Star-finite probability theory, Ph.D. thesis, Yale University, 1977.
  • [2] -, A nonstandard representation for Brownian motion and Itô integration, Israel J. Math. 25 (1976), 15-46. MR 0464380 (57:4311)
  • [3] -, Star-finite representations of measure spaces, Trans. Amer. Math. Soc. (to appear). MR 654856 (83m:03077)
  • [4] P. Billingsley, Convergence of probability measures, Wiley, New York, 1968. MR 0233396 (38:1718)
  • [5] D. L. Burkholder, B. J. Davis and R. F. Gundy, Integral inequalities for convex functions of operators on martingales, Proc. 6th Berkeley Sympos., vol. 2, University of California Press, Berkeley, 1972, pp. 223-240. MR 0400380 (53:4214)
  • [6] C. Dellacherie and P. A. Meyer, Probabilités et potentiel, Hermann, Paris, 1975. MR 0488194 (58:7757)
  • [7] C. Doléans-Dade, Variation quadratique des martingales continues à droite, Ann. Math. Statist. 40 (1969), 284-289. MR 0236982 (38:5275)
  • [8] J. L. Doob, Stochastic processes, Wiley, New York, 1953. MR 0058896 (15:445b)
  • [9] D. N. Hoover and H. J. Keisler, A notion of equivalence for adapted processes (preprint).
  • [10] J. Jacod, Intégrales stochastiques par rapport à une semi-martingale vectorielle et changements de filtration, Séminaire de Probabilités, Exposé XIV, Lecture Notes in Math., vol. 784, Springer-Verlag, Berlin, 1980. MR 580121 (82g:60073)
  • [11] J. Jacod and J. Memin, Existence of weak solutions for stochastic differential equations with driving semimartingales, Stochastics (to appear). MR 609691 (82i:60107)
  • [12] H. J. Keisler, An infinitesimal approach to stochastic analysis, Mem. Amer. Math. Soc. (to appear). MR 732752 (86c:60086)
  • [13] T. L. Lindstrøm, Hyperfinite stochastic integration. I: The nonstandard theory, Math. Scand. 46 (1980), 265-292. MR 591606 (83a:60091a)
  • [14] -, Hyperfinite stochastic integration. II: Comparison with the standard theory, Math. Scand. 46 (1980), 293-314. MR 591607 (83a:60091b)
  • [15] -, Hyperfinite stochastic integration. III: Hyperfinite representations of standard martingales, Math. Scand. 46 (1980), 315-331. MR 591608 (83a:60091c)
  • [16] P. A. Loeb, Conversion from nonstandard to standard measure spaces and applications in probability theory, Trans. Amer. Math. Soc. 211 (1975), 113-122. MR 0390154 (52:10980)
  • [17] -, Weak limits and the standard part map, Proc. Amer. Math. Soc. 77 (1979), 128-135. MR 539645 (80i:28020)
  • [18] -, An introduction to nonstandard analysis and hyperfinite probability theory, Probabilistic analysis and related topics, vol. 2, edited by A. Barucha-Reid, Academic Press, New York, 1979. MR 556680 (80m:60005)
  • [19] M. Metivier and J. Pellaumail, Stochastic integration, Academic Press, New York, 1980. MR 578177 (82b:60060)
  • [20] P. A. Meyer, Un cours sur les intégrales stochastiques, Séminaire de Probabilités, Exposé X, Lecture Notes in Math., vol. 511, Springer-Verlag, Berlin, 1976. MR 0501332 (58:18721)
  • [21] L. Panetta, Hyperreal probability spaces: some applications of the Loeb construction, Ph.D. thesis, University of Wisconsin, 1978.
  • [22] E. Perkins, A non-standard approach to Brownian local time, Ph.D. thesis. University of Illinois, 1979.
  • [23] -, On the construction and distribution of a local martingale with a given absolute value, Trans. Amer. Math. Soc. (to appear). MR 648092 (83h:60044)
  • [24] A. V. Skorokhod, Limit theorems for stochastic processes, Theory Probab. Appl. 1 (1956), 261-290.
  • [25] C. J. Stone, Weak convergence of stochastic processes defined on semi-infinite time intervals, Proc. Amer. Math. Soc. 14 (1963), 694-696. MR 0153046 (27:3015)
  • [26] K. D. Stroyan and W. A. J. Luxemburg, Introduction to the theory of infinitesimals, Academic Press, New York, 1976. MR 0491163 (58:10429)
  • [27] K. D. Stroyan and J. M. Bayod, Introduction to infinitesimal stochastic analysis (preprint), 1980.

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Keywords: Stochastic integration, local martingale, semimartingale, quadratic variation, Skorokhod topology, nonstandard analysis
Article copyright: © Copyright 1983 American Mathematical Society

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