Book Review
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MathSciNet review:
1567097
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Book Information:
Author:
A. U. Kussmaul
Title:
Stochastic integration and generalized martingales
Additional book information:
Pitman Publishing, London, San Francisco, Melbourne, 1977, 163 pp., $14.00.
Ludwig Arnold, Stochastic differential equations: theory and applications, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. Translated from the German. MR 0443083
L. Bachelier, Théorie de la spéculation, Ann. Sci. École Norm. Sup. (3) 17 (1900), 21–86 (French). MR 1508978
Claude Dellacherie, Capacités et processus stochastiques, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 67, Springer-Verlag, Berlin-New York, 1972 (French). MR 0448504
C. Doléans-Dade, Quelques applications de la formule de changement de variables pour les semimartingales, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 16 (1970), 181–194 (French). MR 283883, DOI 10.1007/BF00534595
C. Doléans-Dade and P.-A. Meyer, Intégrales stochastiques par rapport aux martingales locales, Séminaire de Probabilités, IV (Univ. Strasbourg, 1968/69) Lecture Notes in Mathematics, Vol. 124, Springer, Berlin, 1970, pp. 77–107 (French). MR 0270425
J. L. Doob, Stochastic processes, John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London, 1953. MR 0058896
7. A. Einstein, On the movement of small particles suspended in a stationary liquid demanded by the molecular-kinetic theory of heat, Ann. Physik 17 (1905), 549-560; English Transl., Investigations on the theory of the Brownian movement, Methuen, London, 1926.
L. I. Gal′čuk, The structure of certain martingales, Proceedings of the School and Seminar on the Theory of Random Processes (Druskininkai, 1974) Inst. Fiz. i Mat. Akad. Nauk Litovsk. SSR, Vilnius, 1975, pp. 9–33 (Russian). MR 0410905
Kiyosi Itô, Stochastic integral, Proc. Imp. Acad. Tokyo 20 (1944), 519–524. MR 14633
Hiroshi Kunita, Stochastic integrals based on martingales taking values in Hilbert space, Nagoya Math. J. 38 (1970), 41–52. MR 264754
Hiroshi Kunita and Shinzo Watanabe, On square integrable martingales, Nagoya Math. J. 30 (1967), 209–245. MR 217856
E. J. McShane, Stochastic calculus and stochastic models, Probability and Mathematical Statistics, Vol. 25, Academic Press, New York-London, 1974. MR 0443084
E. J. McShane, Stochastic differential equations, J. Multivariate Anal. 5 (1975), 121–177. MR 373006, DOI 10.1016/0047-259X(75)90036-6
14. M. Metivier, The stochastic integral with respect to processes with values in a reflexive Banach space, Theor. Probability 19 (1974), 758-787.
Michel Métivier and Giovanni Pistone, Une formule d’isométrie pour l’intégrale stochastique hilbertienne et équations d’évolution linéaires stochastiques, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 33 (1975/76), no. 1, 1–18 (French). MR 383527, DOI 10.1007/BF00539856
P. A. Meyer, A decomposition theorem for supermartingales, Illinois J. Math. 6 (1962), 193–205. MR 159359
P.-A. Meyer, Intégrales stochastiques. I, II, III, IV, Séminaire de Probabilités (Univ. Strasbourg, Strasbourg, 1966/67) Springer, Berlin, 1967, pp. 72–94, 95–117, 118–141, 142–162 (French). MR 0231445
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
P. A. Meyer, Notes sur les intégrales stochastiques. I. Intégrales hilbertiennes, Séminaire de Probabilités, XI (Univ. Strasbourg, Strasbourg, 1975/1976) Lecture Notes in Math., Vol. 581, Springer, Berlin, 1977, pp. 446–462 (French). MR 0501333
R. E. A. C. Paley, N. Wiener, and A. Zygmund, Notes on random functions, Math. Z. 37 (1933), no. 1, 647–668. MR 1545426, DOI 10.1007/BF01474606
Jean Pellaumail, Sur l’intégrale stochastique et la décomposition de Doob-Meyer, Astérisque, No. 9, Société Mathématique de France, Paris, 1973 (French). MR 0339333
Philip Protter, Markov solutions of stochastic differential equations, Z. Wahrsch. Verw. Gebiete 41 (1977/78), no. 1, 39–58. MR 477420, DOI 10.1007/BF00535013
23. P. E. Protter, A comparison of stochastic integrals, Ann. Probability (to appear).
R. L. Stratonovich, A new representation for stochastic integrals and equations, SIAM J. Control 4 (1966), 362–371. MR 0196814
25. N. Wiener, Differential-space, J. Math. and Physics 2 (1923), 131-174.
- 1.
- L. Arnold, Stochastic differential equations: theory and applications, Wiley, New York, 1974. MR 0443083
- 2.
- L. Bachelier, Théorie de la spéculation, Ann. Sci. École Norm. Sup. 17 (1900), 21-86. MR 1508978
- 3.
- C. Dellacherie, Capacités et processus stochastiques, Springer-Verlag, Berlin and New York, 1972. MR 448504
- 4.
- C. Doléans-Dade, Quelques applications de la formule de changement de variables pour les semimartingales, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 16 (1970), 181-194. MR 283883
- 5.
- C. Doléans-Dade and P. A. Meyer, Intégrales stochastiques par rapport aux martingales locales, Lecture Notes in Math., vol. 124, Springer-Verlag, Berlin and New York, 1970, pp. 77-107. MR 270425
- 6.
- J. L. Doob, Stochastic processes, Wiley, New York, 1953. MR 58896
- 7.
- A. Einstein, On the movement of small particles suspended in a stationary liquid demanded by the molecular-kinetic theory of heat, Ann. Physik 17 (1905), 549-560; English Transl., Investigations on the theory of the Brownian movement, Methuen, London, 1926.
- 8.
- L. I. Galčuk, The structure of a class of martingales, Proceedings of the School-Seminar on the Theory of Random Processes, (Druskininkai), Lithuanian SSR 1 (1974), 9-33. MR 410905
- 9.
- K. Itô, Stochastic integral, Proc. Japan Acad. 20 (1944), 519-524. MR 14633
- 10.
- H. Kunita, Stochastic integrals based on martingales taking values in Hilbert space, Nagoya Math. J. 38 (1970), 41-52. MR 264754
- 11.
- H. Kunita and S. Watanabe, On square integrable martingales, Nagoya Math. J. 30 (1967), 209-245. MR 217856
- 12.
- E. J. McShane, Stochastic calculus and stochastic models, Academic Press, New York, 1977. MR 443084
- 13.
- E. J. McShane, Stochastic differential equations, J. Multivariate Anal. 5 (1975), 121-177. MR 373006
- 14.
- M. Metivier, The stochastic integral with respect to processes with values in a reflexive Banach space, Theor. Probability 19 (1974), 758-787.
- 15.
- M. Metivier and G. Pistone, Une formule d'isométrie pour l'intégrale stochastique Hilbertienne et équations d'évolution linéaires stochastiques, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 33 (1975), 1-18. MR 383527
- 16.
- P. A. Meyer, A decomposition theorem for supermartingales, Illinois J. Math. 6 (1962), 193-205. MR 159359
- 17.
- P. A. Meyer, Intégrales stochastiques. IV, Lecture Notes in Math., vol. 39, Springer-Verlag, Berlin and New York, 1967, pp. 142-162. MR 231445
- 18.
- P. A. Meyer, Un cours sur les intégrales stochastiques, Lecture Notes in Math., vol. 511, Springer-Verlag, Berlin and New York, 1976, pp. 245-400. MR 501332
- 19.
- P. A. Meyer, Intégrales Hilbertiennes, Lecture Notes in Math., vol. 581, Springer-Verlag, Berlin and New York, 1977, pp. 446-461. MR 501333
- 20.
- R. E. A. C. Paley, N. Wiener and A. Zygmund, Notes on random functions, Math. Z. 37 (1933), 647-668. MR 1545426
- 21.
- J. Pellaumail, Sur l'intégrale stochastique et la décomposition de Doob-Meyer, Asterique 9 (1973), 1-125. MR 339333
- 22.
- P. E. Protter, Markov solutions of stochastic differential equations, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 41 (1977), 39-58. MR 477420
- 23.
- P. E. Protter, A comparison of stochastic integrals, Ann. Probability (to appear).
- 24.
- R. L. Stratonovich, A new representation for stochastic integrals and equations, SIAM. J. Control 4 (1966), 362-371. MR 196814
- 25.
- N. Wiener, Differential-space, J. Math. and Physics 2 (1923), 131-174.
Review Information:
Reviewer:
Philip Protter
Journal:
Bull. Amer. Math. Soc.
84 (1978), 1346-1351
DOI:
https://doi.org/10.1090/S0002-9904-1978-14572-8
