Stochastic integrals: A combinatorial approach

Annals of Probability

Volumn 25, Issue 3, 1997, Pages 1257-1283

  Rota, Gian Carlo ^a   Wallstrom, Timothy C ^b  

^a MASSACHUSETTS INSTITUTE OF TECHNOLOGY   (United States)

^b LOS ALAMOS NATIONAL LABORATORY   (United States)

Author keywords

Discrete and homogeneous chaos; Kailath Segall formula; Multiple stochastic integrals; Orthogonal polynomials; Partitions of sets; Symmetric functions

Indexed keywords

EID: 0031526259     PISSN: 00911798     EISSN: None     Source Type: Journal    
DOI: 10.1214/aop/1024404513     Document Type: Article

References (40)

1. BJORKEN, J. D. and DRELL, S. D. (1964). Relativistic Quantum Mechanics. McGraw-Hill, New York.
2. [21 BOGOLIUBOV, N. N. and SHIRKOV, D. V. (1980). Introduction to the Theory of Quantized Fields. Wiley, New York.
3. CAMERON, R. H. and MARTIN, W. T. (1947). The orthogonal development of nonlinear functionals in series of Fourier-Hermite functionals. Ann. Math. 48 385-392.
4. p estimates on iterated stochastic integrals. Ann. Probab. 19 354-368.
5. CHIHARA, T. S. (1978). An Introduction to Orthogonal Polynomials. Gordon and Breach, New York.
6. COX, D. R. and ISHAM, V. (1980). Point Processes. Chapman and Hall, London.
7. [71 DOLÉANS-DADE, C. (1970). Quelques applications de la formule de changement de variables pour les semimartingales. Z. Wahrsch. Verw. Gebiete 16 181-194.
8. DOUBILET, P. (1972). On the foundations of combinatorial theory. VII: symmetric functions through the theory of distribution and occupancy. Stud. Appl. Math. 51 377-396.
9. ENGEL, D. D. (1982). The multiple stochastic integral. Mem. Am. Math. Soc. 38 1-82.
10. [101 FLOREANINI, R., LETOURNEUX, J. and VINET, L. (1993). Quantum mechanics and polynomials of a discrete variable. Ann. Physics 226 331-349.
11. HIDA, T. (1970). Stationary Stochastic Processes. Princeton Univ. Press.
12. HIDA, T. and IKEDA, N. (1965). Analysis on Hilbert space with reproducing kernel arising from multiple Wiener integral. Proc. Fifth Berkeley Symp. Math. Statist. Probab. 2 117-143. Univ. California Press, Berkeley.
13. ITÔ, K. (1951). Multiple Wiener integral. J. Math. Soc. Japan 3 157-169.
14. ITÔ, K. (1956). Spectral type of the shift transformation of differential processes with stationary increments. Trans. Amer. Math. Soc. 81 253-263.
15. KAKUTANI, S. (1950). Determination of the spectrum of the flow of Brownian motion. Proc. Nat. Acad. Sci. U.S.A. 36 319-323.
16. KAKUTANI, S. (1961). Spectral analysis of stationary Gaussian processes. Proc. Fourth Berkeley Symp. Math. Statist. Probab. 2 239-247. Univ. California Press, Berkeley.
17. KALLENBERG, O. (1983). Random Measures. Academic Press, New York.
18. KALLENBERG, O. and SZULGA, J. (1989). Multiple integration with respect to Poisson and Lévy processes. Probab. Theory Related Fields 83 101-134.
19. LEBEDEV, N. N. (1972). Special Functions and Their Applications. Dover, New York.
20. MAJOR, P. (1981). Multiple Wiener-Itô Integrals. Lecture Notes in Math. 849. Springer, New York.
21. MCKEAN, JR., H. P. (1955). Sample functions of stable processes. Ann. Math. 61 564-579.
22. MCKEAN, JR., H. P. (1969). Stochastic Integrals. Academic Press, New York.
23. METIVIER, M. and PELLAUMAIL, J. (1980). Stochastic Integration. Academic Press, New York.
24. MEYER, P.-A. (1976). Un cours sur les integrales stochastiques. Seminaire de Probabilités X: Université de Strasbourg. Lecture Notes in Math. 511 245-400. Springer, Berlin.
25. MEYER, P.-A. (1993). Quantum Probability for Probabilists. Lecture Notes in Math. 1538. Springer, New York.
26. NIKIFOROV, A. F., SUSLOV, S. K. and UVAROV, V. B. (1991). Classical Orthogonal Polynomials of a Discrete Variable. Springer, Berlin.
27. OGURA, H. (1972). Orthogonal functionals of the Poisson process. IEEE Trans. Inform. Theory IT-18 473-481.
28. PROTTER, P. (1990). Stochastic Integration and Differential Equations: A New Approach. Springer, Berlin.
29. ROTA, G.-C. (1964). The number of partitions of a set. Amer. Math. Monthly 71 498-504.
30. ROTA, G.-C. (1964). On the foundations of combinatorial theory I. Theory of Möbius functions. Z. Wahrsch. Verw. Gebiete 2 340-368.
31. ROTA, G.-C. (1975). Finite Operator Calculus. Academic Press, New York.
32. ROTA, G.-C., KAHANER, D. and ODLYZKO, A. (1973). On the foundations of combinatorial theory. VIII. Finite operator calculus. J. Math. Anal. Appl. 42 684-760.
33. ROTA, G.-C. and TAYLOR, B. D. (1994). The classical umbral calculus. SIAM J. Math. Anal. 25 694-711.
34. SEGAL, I. E. (1956). Tensor algebras over Hilbert spaces, I. Trans. Amer. Math. Soc. 81 106-134.
35. SEGALL, A. and KAILATH, T. (1976). Orthogonal functionals of independent increment processes. IEEE Trans. Inform. Theory IT-22 287-298.
36. STREATER, R. F. and WIGHTMAN, A. S. (1964). PCT, Spin and Statistics, and All That. Benjamin/Cummings, Reading, MA.
37. WICK, G.-C. (1950). Phys. Rev. 80 268-272.
38. WIENER, N. (1938). The homogeneous chaos. Amer. J. Math. 55 897-936.
39. WIENER, N. and WINTNER, A. (1941). The discrete chaos. Amer. J. Math. 65 279-298.
40. ZINN-JUSTIN, J. (1989). Quantum Field Theory and Critical Phenomena. Clarendon Press, Oxford.