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Elliott, Robert J.; Kohlmann, Michael

Martingale representation and the Malliavin calculus. (English) Zbl 0685.60043

Appl. Math. Optimization 20, No. 1, 105-112 (1989).
A new approach to Malliavin calculus for stochastic differential equations is presented based on the idea of martingale representation of stochastic integrals. The presentation is very interesting although the reader must know some results about partial SDEs. The main idea is not very far from J.-M. Bismut’s one [Z. Wahrscheinlichkeitstheor. Verw. Geb. 56, 469-505 (1981; Zbl 0445.60049)] but gives another point of view on the integration by parts formula.
Reviewer: A.Yu.Veretennikov

Cited in 2 Documents

MSC:

60G35 Signal detection and filtering (aspects of stochastic processes)
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60G44 Martingales with continuous parameter

Keywords:

Malliavin calculus; martingale representation of stochastic integrals; integration by parts formula

Citations:

Zbl 0445.60049
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Full Text: DOI

References:

[1] D. Bell. The Malliavin Calculus. Longman, London, 1987. · Zbl 0678.60042
[2] K. Bichteler and D. Fonken. A Simple Version of the Malliavin Calculus in Dimension One. Lecture Notes in Mathematics, vol. 939. Springer-Verlag, Berlin, 1982. · Zbl 0483.60077
[3] J. M. Bismut. Martingales, the Malliavin calculus and hypoellipticity under general H?rmander’s conditions. Z. Wahrsch. Verw. Gebiete, 56 (1981), 469-505. · Zbl 0445.60049 · doi:10.1007/BF00531428
[4] R. J. Elliott and M. Kohlmann. Integration by parts, homogeneous chaos expansions and smooth densities. Ann. Probab. To appear. · Zbl 0671.60050
[5] N. Ikeda and S. Watanabe. Stochastic Differential Equations and Diffusion Processes. North Holland, Amsterdam, 1981. · Zbl 0495.60005
[6] H. Kunita. Stochastic Partial Differential Equations Connected with Non-Linear Filtering. Lecture Notes in Mathematics, vol. 972. Springer-Verlag, Berlin, 1982. · Zbl 0527.60067
[7] P. Malliavin. Stochastic calculus of variations and hypoelliptic operators. Proceedings of the International Conference, Kyoto, 1976, Wiley, New York, 1978, pp. 195-263.
[8] J. Norris. Simplified Malliavin Calculus. Lecture Notes in Mathematics, vol. 1204. Springer-Verlag, Berlin, 1986. · Zbl 0609.60066
[9] D. Stroock. The Malliavin calculus, a functional analytic approach. J. Funct. Anal., 44 (1981), 212-257. · Zbl 0475.60060 · doi:10.1016/0022-1236(81)90011-2
[10] M. Zakai. The Malliavin calculus. Acta Appl. Math., 3 (1985), 175-207. · Zbl 0553.60053 · doi:10.1007/BF00580703
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