Book Review
The AMS does not provide abstracts of book reviews.
You may download the entire review from the links below.
Full text of review:
PDF
This review is available free of charge.
Book Information:
Author:
Marta Sanz-Solé
Title:
Malliavin calculus with applications to stochastic partial differential equations
Additional book information:
Fundamental Sciences,
EPFL Press, Lausanne, distributed by CRC Press, Boca Raton, FL,
2005,
viii+162 pp.,
ISBN 978-0849340307,
US$84.95$
Vlad Bally and Etienne Pardoux, Malliavin calculus for white noise driven parabolic SPDEs, Potential Anal. 9 (1998), no. 1, 27–64. MR 1644120, DOI 10.1023/A:1008686922032
Denis R. Bell, The Malliavin calculus, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 34, Longman Scientific & Technical, Harlow; John Wiley & Sons, Inc., New York, 1987. MR 902583
Jean-Michel Bismut, Large deviations and the Malliavin calculus, Progress in Mathematics, vol. 45, Birkhäuser Boston, Inc., Boston, MA, 1984. MR 755001
R. H. Cameron and W. T. Martin, The transformation of Wiener integrals by nonlinear transformations, Trans. Amer. Math. Soc. 66 (1949), 253–283. MR 31196, DOI 10.1090/S0002-9947-1949-0031196-6
Rene A. Carmona and Boris Rozovskii (eds.), Stochastic partial differential equations: six perspectives, Mathematical Surveys and Monographs, vol. 64, American Mathematical Society, Providence, RI, 1999. MR 1661761, DOI 10.1090/surv/064
J. M. C. Clark, The representation of functionals of Brownian motion by stochastic integrals, Ann. Math. Statist. 41 (1970), 1282–1295. MR 270448, DOI 10.1214/aoms/1177696903
Robert C. Dalang, Extending the martingale measure stochastic integral with applications to spatially homogeneous s.p.d.e.’s, Electron. J. Probab. 4 (1999), no. 6, 29. MR 1684157, DOI 10.1214/EJP.v4-43
Giuseppe Da Prato and Jerzy Zabczyk, Stochastic equations in infinite dimensions, Encyclopedia of Mathematics and its Applications, vol. 44, Cambridge University Press, Cambridge, 1992. MR 1207136, DOI 10.1017/CBO9780511666223
Donald A. Dawson, Measure-valued Markov processes, École d’Été de Probabilités de Saint-Flour XXI—1991, Lecture Notes in Math., vol. 1541, Springer, Berlin, 1993, pp. 1–260. MR 1242575, DOI 10.1007/BFb0084190
Helge Holden, Bernt Øksendal, Jan Ubøe, and Tusheng Zhang, Stochastic partial differential equations, Probability and its Applications, Birkhäuser Boston, Inc., Boston, MA, 1996. A modeling, white noise functional approach. MR 1408433, DOI 10.1007/978-1-4684-9215-6
Kiyosi Ito, On stochastic differential equations, Mem. Amer. Math. Soc. 4 (1951), 51. MR 40618
Kiyosi Itô, Multiple Wiener integral, J. Math. Soc. Japan 3 (1951), 157–169. MR 44064, DOI 10.2969/jmsj/00310157
Hui-Hsiung Kuo, A quarter century of white noise theory, Quantum information, IV (Nagoya, 2001) World Sci. Publ., River Edge, NJ, 2002, pp. 1–37. MR 1947556, DOI 10.1142/9789812777324_{0}001
Shigeo Kusuoka and Daniel Stroock, Applications of the Malliavin calculus. I, Stochastic analysis (Katata/Kyoto, 1982) North-Holland Math. Library, vol. 32, North-Holland, Amsterdam, 1984, pp. 271–306. MR 780762, DOI 10.1016/S0924-6509(08)70397-0
Shigeo Kusuoka and Daniel Stroock, Applications of the Malliavin calculus. I, Stochastic analysis (Katata/Kyoto, 1982) North-Holland Math. Library, vol. 32, North-Holland, Amsterdam, 1984, pp. 271–306. MR 780762, DOI 10.1016/S0924-6509(08)70397-0
S. Kusuoka and D. Stroock, Applications of the Malliavin calculus. III, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 34 (1987), no. 2, 391–442. MR 914028
Paul Malliavin, Stochastic calculus of variation and hypoelliptic operators, Proceedings of the International Symposium on Stochastic Differential Equations (Res. Inst. Math. Sci., Kyoto Univ., Kyoto, 1976) Wiley, New York-Chichester-Brisbane, 1978, pp. 195–263. MR 536013
Paul Malliavin, Stochastic analysis, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 313, Springer-Verlag, Berlin, 1997. MR 1450093, DOI 10.1007/978-3-642-15074-6
Paul Malliavin and Anton Thalmaier, Stochastic calculus of variations in mathematical finance, Springer Finance, Springer-Verlag, Berlin, 2006. MR 2189710
M. Métivier, Stochastic partial differential equations in infinite-dimensional spaces, Scuola Normale Superiore di Pisa. Quaderni. [Publications of the Scuola Normale Superiore of Pisa], Scuola Normale Superiore, Pisa, 1988. With a preface by G. Da Prato. MR 982268
Annie Millet and Marta Sanz-Solé, A stochastic wave equation in two space dimension: smoothness of the law, Ann. Probab. 27 (1999), no. 2, 803–844. MR 1698971, DOI 10.1214/aop/1022677387
David Nualart, The Malliavin calculus and related topics, 2nd ed., Probability and its Applications (New York), Springer-Verlag, Berlin, 2006. MR 2200233
Daniel Ocone, Stochastic calculus of variations for stochastic partial differential equations, J. Funct. Anal. 79 (1988), no. 2, 288–331. MR 953905, DOI 10.1016/0022-1236(88)90015-8
Szymon Peszat and Jerzy Zabczyk, Nonlinear stochastic wave and heat equations, Probab. Theory Related Fields 116 (2000), no. 3, 421–443. MR 1749283, DOI 10.1007/s004400050257
Daniel W. Stroock, The Malliavin calculus, a functional analytic approach, J. Functional Analysis 44 (1981), no. 2, 212–257. MR 642917, DOI 10.1016/0022-1236(81)90011-2
John B. Walsh, An introduction to stochastic partial differential equations, École d’été de probabilités de Saint-Flour, XIV—1984, Lecture Notes in Math., vol. 1180, Springer, Berlin, 1986, pp. 265–439. MR 876085, DOI 10.1007/BFb0074920
S. Watanabe, Lectures on stochastic differential equations and Malliavin calculus, Tata Institute of Fundamental Research Lectures on Mathematics and Physics, vol. 73, Published for the Tata Institute of Fundamental Research, Bombay; by Springer-Verlag, Berlin, 1984. Notes by M. Gopalan Nair and B. Rajeev. MR 742628
28.N. Wiener. Differential space, J. Math. Phys. 2(1923), 131-174.
Norbert Wiener, The Homogeneous Chaos, Amer. J. Math. 60 (1938), no. 4, 897–936. MR 1507356, DOI 10.2307/2371268
- 1.
- V. Bally and E. Pardoux. Malliavin calculus for white noise driven parabolic SPDEs, Potential Anal. 9(1998), 27-64. MR 1644120
- 2.
- D. Bell. The Malliavin Calculus, Pitman Monograph 34, Wiley, 1987. MR 0902583
- 3.
- J.M. Bismut. Large Deviations and the Malliavin Calculus, Prog. Math. 45, Birkhäuser, 1984. MR 0755001
- 4.
- R.H. Cameron and W.T. Martin. Transformations of Wiener integrals by nonlinear transformations, Trans. Amer. Math. Soc. 66(1949), 253-283. MR 0031196
- 5.
- R.A. Carmona and B. Rozovskii. Stochastic Partial Differential Equations: Six Perspectives, Math. Surveys and Monographs 44, AMS, 1999. MR 1661761
- 6.
- J.M.C. Clark. The representation of functionals of Brownian motion by stochastic integrals, Ann. Math. Statist. 41 (1970), 1282-1295. MR 0270448
- 7.
- R.C. Dalang. Extending martingale measure stochastic integral with applications to spatially homogeneous s.p.d.e.'s, Elect. J. Probab. 4(1999), paper 6, 1-29. MR 1684157
- 8.
- G. Da Prato and J. Zabczyk. Stochastic Equations in Infinite Dimensions, Cambridge University Press, second ed. (1998). MR 1207136
- 9.
- D.A. Dawson. Measure-valued Markov processes, in Ecole d'été de probabilités de Saint Flour XXI, Lecture Notes in Math. 1541, Springer-Verlag, 1993. MR 1242575
- 10.
- H. Holden, B. Øksendal, J. Ubøe and T. Zhang. Stochastic Partial Differential Equations, Birkhäuser, 1996. MR 1408433
- 11.
- K. Itô. On stochastic differential equations, Mem. Amer. Math. Soc. 4(1951). MR 0040618
- 12.
- K. Itô. Multiple Wiener integral, J. Math. Soc. Japan 3(1951), 157-169. MR 0044064
- 13.
- H.H. Kuo. A quarter century of white noise theory, in Quantum Information IV, T. Hida and K. Saitô, eds., World Scientific (2002), pp. 1-37. MR 1947556
- 14.
- S. Kusuoka and D.W. Stroock. Application of the Malliavin calculus, I, in Stochastic Analysis, Proc. Taniguchi Inter. Symp. on Stochastic Analysis, Katata and Kyoto, 1982, ed. K. Itô, Kinokuniya/North-Holland, Tokyo, 1984, pp. 271-306. MR 0780762
- 15.
- S. Kusuoka and D.W. Stroock. Application of the Malliavin calculus, II, J. Fac. Sci. Univ. Tokyo IA Math. 32(1985), 1-76. MR 0780762
- 16.
- S. Kusuoka and D.W. Stroock. Application of the Malliavin calculus, III, J. Fac. Sci. Univ. Tokyo IA Math. 34(1987), 391-442. MR 0914028
- 17.
- P. Malliavin. Stochastic calculus of variations and hypoelliptic operators, in Proc. Inter. Symp. on Stoch. Diff. Equations, Kyoto (1976), Wiley, 1978, pp. 195-263. MR 0536013
- 18.
- P. Malliavin. Stochastic Analysis, Grundlehren der Mathematischen Wissenschaften 313, Springer-Verlag, 1997. MR 1450093
- 19.
- Paul Malliavin and Anton Thalmaier. Stochastic Calculus of Variations in Mathematical Finance, Springer, 2006. MR 2189710
- 20.
- M. Métivier. Stochastic partial differential equations in infinite-dimensional spaces, with a preface by G. Da Prato. Scuola Normale Superiore, Pisa, 1988. MR 0982268
- 21.
- A. Millet and M. Sanz-Solé. A stochastic wave equation in two space dimension: smoothness of the law, Ann. Probab. 27 (1999), 803-844. MR 1698971
- 22.
- D. Nualart. The Malliavin Calculus and Related Topics, 2nd ed., Springer-Verlag, 2006. MR 2200233
- 23.
- D. Ocone. Stochastic calculus of variations for stochastic partial differential equations, J. Funct. Anal. 79(1988), 231-288. MR 0953905
- 24.
- S. Peszat and J. Zabczyk. Nonlinear stochastic wave and heat equations, Probab. Theory Relat. Fields 116(2000), 421-443. MR 1749283
- 25.
- D.W. Stroock. The Malliavin calculus, a functional analytic approach, J. Functional Anal. 44(1981), 14-171. MR 0642917
- 26.
- J.B. Walsh. An introduction to Stochastic Partial Differential Equations, in Ecole d'été de probabilités de Saint Flour XIV, Lecture Notes in Math. 1180, Springer-Verlag, 1986. MR 0876085
- 27.
- S. Watanabe. Lectures on Stochastic Differential Equations and Malliavin Calculus, Tata Institute of Fundamental Research, Springer-Verlag, 1984. MR 0742628
- 28.
- N. Wiener. Differential space, J. Math. Phys. 2(1923), 131-174.
- 29.
- N. Wiener. The homogeneous chaos, Amer. J. Math. 60(1938), 879-936. MR 1507356
Review Information:
Reviewer:
Donald Dawson
Affiliation:
Carleton University
Email:
ddawson@math.carleton.ca
Journal:
Bull. Amer. Math. Soc.
44 (2007), 497-504
Published electronically:
April 11, 2007
Review copyright:
© Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.