Bulletin of the American Mathematical Society

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Book Information:

Authors: 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).

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, An elementary core equivalence theorem, Econometrica 46 (1978), no. 6, 1483–1487. MR 513701, DOI 10.2307/1913840

Robert M. Anderson, Core theory with strongly convex preferences, Econometrica 49 (1981), no. 6, 1457–1468. MR 636163, DOI 10.2307/1911411

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

Robert M. Anderson, Strong core theorems with nonconvex preferences, Econometrica 53 (1985), no. 6, 1283–1294. MR 809911, DOI 10.2307/1913208

Robert M. Anderson, Notions of core convergence, Contributions to mathematical economics, North-Holland, 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).

Leif Arkeryd, A nonstandard approach to the Boltzmann equation, Arch. Rational Mech. Anal. 77 (1981), no. 1, 1–10. MR 630118, DOI 10.1007/BF00280402

Leif Arkeryd, Intermolecular forces of infinite range and the Boltzmann equation, Arch. Rational Mech. Anal. 77 (1981), no. 1, 11–21. MR 630119, DOI 10.1007/BF00280403

L. Arkeryd, A time-wise approximated Boltzmann equation, IMA J. Appl. Math. 27 (1981), no. 3, 373–383. MR 633809, DOI 10.1093/imamat/27.3.373

Leif Arkeryd, Asymptotic behaviour of the Boltzmann equation with infinite range forces, Comm. Math. Phys. 86 (1982), no. 4, 475–484. MR 679196

Leif Arkeryd, Loeb solutions of the Boltzmann equation, Arch. Rational Mech. Anal. 86 (1984), no. 1, 85–97. MR 748925, DOI 10.1007/BF00280649

Robert J. Aumann, Markets with a continuum of traders, Econometrica 32 (1964), 39–50. MR 172689, DOI 10.2307/1913732

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

Donald J. Brown and Abraham Robinson, The cores of large standard exchange economies, J. Econom. Theory 9 (1974), no. 3, 245–254. MR 475731, DOI 10.1016/0022-0531(74)90050-7

Donald J. Brown and Abraham Robinson, Nonstandard exchange economies, Econometrica 43 (1975), 41–55. MR 443867, DOI 10.2307/1913412

Nigel J. Cutland, Nonstandard measure theory and its applications, Bull. London Math. Soc. 15 (1983), no. 6, 529–589. MR 720746, DOI 10.1112/blms/15.6.529

Martin Davis, Applied nonstandard analysis, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1977. MR 0505473

A. Dvoretzky, P. Erdös, and S. Kakutani, Double points of paths of Brownian motion in $n$-space, Acta Sci. Math. (Szeged) 12 (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.

Werner Hildenbrand, Core and equilibria of a large economy, Princeton Studies in Mathematical Economics, No. 5, Princeton University Press, Princeton, N.J., 1974. With an appendix to Chapter 2 by K. Hildenbrand. MR 0389160

Douglas N. Hoover and Edwin Perkins, Nonstandard construction of the stochastic integral and applications to stochastic differential equations. I, II, Trans. Amer. Math. Soc. 275 (1983), no. 1, 1–36, 37–58. MR 678335, DOI 10.1090/S0002-9947-1983-0678335-1

Douglas N. Hoover and Edwin Perkins, Nonstandard construction of the stochastic integral and applications to stochastic differential equations. I, II, Trans. Amer. Math. Soc. 275 (1983), no. 1, 1–36, 37–58. MR 678335, DOI 10.1090/S0002-9947-1983-0678335-1

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

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.

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

29.

M. Ali Khan, Some equivalence theorems, Review of Economic Studies 41 (1974), 549-565.

M. Ali Khan, Oligopoly in markets with a continuum of traders: an asymptotic interpretation, J. Econom. Theory 12 (1976), no. 2, 273–297. MR 411572, DOI 10.1016/0022-0531(76)90078-8

31.

M. Ali Khan and Salim Rashid, Limit theorems on cores with costs of coalition formation, preprint, Johns Hopkins Univ., 1976.

Gregory F. Lawler, A self-avoiding random walk, Duke Math. J. 47 (1980), no. 3, 655–693. MR 587173

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, Applications of nonstandard analysis to ideal boundaries in potential theory, Israel J. Math. 25 (1976), no. 1-2, 154–187. MR 457757, DOI 10.1007/BF02756567

Robert Lutz and Michel Goze, Nonstandard analysis, Lecture Notes in Mathematics, vol. 881, Springer-Verlag, Berlin-New York, 1981. A practical guide with applications; With a foreword by Georges H. Reeb. MR 643624

Edward Nelson, Internal set theory: a new approach to nonstandard analysis, Bull. Amer. Math. Soc. 83 (1977), no. 6, 1165–1198. MR 469763, DOI 10.1090/S0002-9904-1977-14398-X

Edwin Perkins, A global intrinsic characterization of Brownian local time, Ann. Probab. 9 (1981), no. 5, 800–817. MR 628874

Edwin Perkins, The exact Hausdorff measure of the level sets of Brownian motion, Z. Wahrsch. Verw. Gebiete 58 (1981), no. 3, 373–388. MR 639146, DOI 10.1007/BF00542642

Edwin Perkins, Weak invariance principles for local time, Z. Wahrsch. Verw. Gebiete 60 (1982), no. 4, 437–451. MR 665738, DOI 10.1007/BF00535709

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

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).

Salim Rashid, Economies with many agents, Johns Hopkins University Press, Baltimore, MD, 1987. An approach using nonstandard analysis. MR 871874

Abraham Robinson, Non-standard analysis, North-Holland Publishing Co., Amsterdam, 1966. MR 0205854

H. L. Royden, Real analysis, The Macmillan Company, New York; Collier Macmillan Ltd., London, 1963. MR 0151555

Andreas Stoll, A nonstandard construction of Lévy Brownian motion, Probab. Theory Relat. Fields 71 (1986), no. 3, 321–334. MR 824706, DOI 10.1007/BF01000208

50.

Andreas Stoll, Self-repellent random walks and polymer measures in two dimensions, Dissertation, Ruhr-Universität Bochum, 1985.

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

K. D. Stroyan and José Manuel Bayod, Foundations of infinitesimal stochastic analysis, Studies in Logic and the Foundations of Mathematics, vol. 119, North-Holland Publishing Co., Amsterdam, 1986. MR 849100

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 0464380

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