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Book Information

Author(s): A. E. Hurd and P. A. Loeb
Title: An introduction to nonstandard real analysis
Additional book information: Pure and Applied Mathematics, vol. 118, Academic Press, 1985, xii + 232 pp., $35.00. ISBN 0-12-362440-1

1.: S. Albeverio, J. E. Fenstad, R. Heogh-Krohn, and T. Lindstrom, Nonstandard methods in stochastic analysis and mathematical physics, Academic Press, New York (forthcoming).
2.: Robert M. Anderson, A nonstandard representation for Brownian motion and Itô integration, Israel J. Math. 25 (1976), 15-46. MR 464380
3.: Robert M. Anderson, An elementary core equivalence theorem, Econometrica 46 (1978), 1483-1487. MR 513701
4.: Robert M. Anderson, Core theory with strongly convex preferences, Econometrica 49 (1981), 1457-1468. MR 636163
5.: Robert M. Anderson, Star-finite representations of measure spaces, Trans. Amer. Math. Soc. 271 (1982), 667-687. MR 654856
6.: Robert M. Anderson, Strong core theorems with nonconvex preferences, Econometrica 53 (1985), 1283-1294. MR 809911
7.: Robert M. Anderson, Notions of core convergence, Contributions to Mathematical Economics in Honor of Gerard Debreu (Werner Hildenbrand and Andreu Mas-Colell eds.), North-Holland Publishing Company, Amsterdam, 1986, pp. 25-46. MR 902875
8.: Robert M. Anderson, The second welfare theorem with nonconvex preferences, Working Papers in Economic Theory and Econometrics #IP-327, Center for Research in Management, Univ. of Calif., Berkeley (January 1986).
9.: Leif Arkeryd, A nonstandard approach to the Boltzmann equation, Arch. Rational Mech. Anal. 77 (1981), 1-10. MR 630118
10.: Leif Arkeryd, Intermodular forces of infinite range and the Boltzmann equation, Arch. Rational Mech. Anal. 77 (1981), 11-21. MR 630119
11.: Leif Arkeryd, A time-wise approximated Boltzmann equation, IMA J. Appl. Math. 27 (1981), 373-383. MR 633809
12.: Leif Arkeryd, Asymptotic behaviour of the Boltzmann equation with infinite range force, Comm. Math. Phys. 86 (1982), 475-484. MR 679196
13.: Leif Arkeryd, Loeb solutions of the Boltzmann equation, Arch. Rational Mech. Anal. 86 (1984), 85-97. MR 748925
14.: Robert J. Aumann, Markets with a continuum of traders, Econometrica 32 (1964), 39-50. MR 172689
15.: Allen R. Bernstein and Abraham Robinson, Solution of an invariant subspace problem of K. T. Smith and P. R. Halmos, Pacific J. Math. 16 (1966), 421-431. MR 193504
16.: Donald J. Brown and Abraham Robinson, The cores of large standard exchange economies, J. Econom. Theory 9 (1974), 245-254. MR 475731
17.: Donald J. Brown and Abraham Robinson, Nonstandard exchange economies, Econometrica 43 (1975), 41-55. MR 443867
18.: Nigel J. Cutland, Nonstandard measure theory and its applications, Bull. London Math. Soc. 15 (1983), 529-589. MR 720746
19.: Martin Davis, Applied nonstandard analysis, Wiley, New York, 1977. MR 505473
20.: A. Dvoretzky, P. Erdős and S. Kakutani, Double points of paths of Brownian motion in n-space, Acta Sci. Math. 12B (1950), 75-81. MR 34972
21.: C. Ward Henson and Lang Moore, Jr., Nonstandard analysis and the theory of Banach spaces, Nonstandard Analysis—Recent Developments (Albert E. Hurd, ed.), Lecture Notes in Math., vol. 983, Springer-Verlag, Berlin and New York, 1983, pp. 27-112.
22.: Werner Hildenbrand, Core and equilibria of a large economy, Princeton Univ. Press, Princeton, N. J., 1974. MR 389160
23.: Douglas N. Hoover and Edwin A. Perkins, Nonstandard construction of the stochastic integral and applications to stochastic differential equations. I, Trans. Amer. Math. Soc. 275 (1983), 1-36. MR 678335
24.: Douglas N. Hoover and Edwin A. Perkins, Nonstandard construction of the stochastic integral and applications to stochastic differential equations. II, Trans. Amer. Math. Soc. 275 (1983), 37-58. MR 678335
25.: J. Jacod and J. Memin, Existence of weak solutions for stochastic differential equations with driving semimartingales, Stochastics 4 (1981), 317-337. MR 609691
26.: H. Jerome Keisler, Elementary calculus, Prindle, Weber and Schmidt, Boston, 1976.
27.: H. Jerome Keisler, Foundations of infinitesimal calculus, Prindle, Weber and Schmidt, Boston, 1976.
28.: H. Jerome Keisler, An infinitesimal approach to stochastic analysis, Mem. Amer. Math. Soc. no. 297 (1984). MR 732752
29.: M. Ali Khan, Some equivalence theorems, Review of Economic Studies 41 (1974), 549-565.
30.: M. Ali Khan, Oligopoly in markets with a continuum of traders: An asymptotic approach, J. Econom. Theory 12 (1976), 273-297. MR 411572
31.: M. Ali Khan and Salim Rashid, Limit theorems on cores with costs of coalition formation, preprint, Johns Hopkins Univ., 1976.
32.: Gregory F. Lawler, A self-avoiding random walk, Duke Math. J. 47 (1980), 655-693. MR 587173
33.: Tom L. Lindstrom, Hyperfinite stochastic integration. I, The nonstandard theory, Math. Scand. 46 (1980), 265-292. MR 591606
34.: Tom L. Lindstrom, Hyperfinite stochastic integration. II, Comparison with the standard theory, Math. Scand. 46 (1980), 293-314. MR 591607
35.: Tom L. Lindstrom, Hyperfinite stochastic integration. III, Hyperfinite representations of standard martingales, Math. Scand. 46 (1980), 315-331. MR 591608
36.: Peter A. Loeb, Conversion from nonstandard to standard measure spaces with applications in probability theory, Trans. Amer. Math. Soc. 211 (1975), 113-122. MR 390154
37.: Peter A. Loeb, Applications of nonstandard analysis to ideal boundaries in potential theory, Israel J. Math. 25 (1976), 154-187. MR 457757
38.: Robert Lutz and Michel Goze, Nonstandard analysis: A practical guide with applications, Lecture Notes in Math., vol. 881, Springer-Verlag, Berlin and New York, 1981. MR 643624
39.: Edward Nelson, Internal set theory: A new approach to nonstandard analysis, Bull. Amer. Math. Soc. 83 (1977), 1165-1198. MR 469763
40.: Edwin Perkins, A global intrinsic characterization of Brownian local time, Ann. Probability 9 (1981), 800-817. MR 628874
41.: Edwin Perkins, The exact Hausdorff measure of the level sets of Brownian motion, Z. Wahrsch. Verw. Gebiete 58 (1981), 373-388. MR 639146
42.: Edwin Perkins, Weak invariance principles for local time, Z. Wahrsch. Verw. Gebiete 60 (1982), 437-451. MR 665738
43.: Edwin Perkins, On the construction and distribution of a local martingale with a given absolute value, Trans. Amer. Math. Soc. 271 (1982), 261-281. MR 648092
44.: Edwin Perkins, Stochastic processes and nonstandard analysis, Nonstandard Analysis—Recent Developments (Albert E. Hurd, ed.), Lecture Notes in Math., vol. 983, Springer-Verlag, Berlin and New York, 1983, pp. 162-185.
45.: Edwin Perkins, Work on measure-valued diffusions (in preparation).
46.: Salim Rashid, Economies with many agents: A nonstandard approach, Johns Hopkins, Baltimore (forthcoming). MR 871874
47.: Abraham Robinson, Non-standard analysis, North-Holland Publishing Company, Amsterdam, 1970. MR 205854
48.: H. L. Royden, Real analysis, MacMillan Publishing Co., New York, 1968. MR 151555
49.: Andreas Stoll, A nonstandard construction of Levy Brownian motion with applications to invariance principles, Probability Theory and Related Fields 71 (1986), 321-334. MR 824706
50.: Andreas Stoll, Self-repellent random walks and polymer measures in two dimensions, Dissertation, Ruhr-Universität Bochum, 1985.
51.: K. D. Stroyan and W. A. J. Luxemburg, Introduction to the theory of infinitesimals, Academic Press, New York, 1976. MR 491163
52.: K. D. Stroyan and J. M. Bayod, Introduction to infinitesimal stochastic analysis, North-Holland, Amsterdam, 1986. MR 849100

Review Information:
Journal: Bull. Amer. Math. Soc. 16 (1987), 298-306.
DOI: 10.1090/S0273-0979-1987-15523-6
PII: S 0273-0979(1987)15523-6

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ISSN 1088-9485(e) ISSN 0273-0979(p)

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