Chain rules and p-variation

Studia Mathematica

Volumn 149, Issue 3, 2002, Pages 197-238

  Norvaisa R ^a,b  

^a Institute of Mathematics and Informatics ^*

^b VILNIUS UNIVERSITY   (Lithuania)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 26844471222     PISSN: 00393223     EISSN: None     Source Type: Journal    
DOI: 10.4064/sm149-3-1     Document Type: Article

References (47)

1. L. Ambrosio and G. Dal Maso, A general chain rule for distributional derivatives, Proc. Amer. Math. Soc. 108 (1990), 691-702.
2. V. I. Averbukh and O. G. Smolyanov, The various definitions of the derivative in linear topological spaces, Math. Surveys 23 (1968), 67-113
3. Uspekhi Mat. Nauk 23 (1968), 67-116 (in Russian).
4. Z. Ciesielski, G. Kerkyacharian et B. Roynette, Quelques espaces fonctionnels associés à des processus gaussiens, Studia Math. 107 (1993), 171-204.
5. R. M. Dudley, Sample functions of the Gaussian process, Ann. Probab. 1 (1973), 66-103.
6. _, Fréchet differentiability, p-variation and uniform Donsker classes, Ann. Probab. ibid. 20 (1992), 1968-1982.
7. R. M. Dudley and R. Norvaiša, Differentiability of Six Operators on Nonsmooth Functions and p-Variation, Lecture Notes in Math. 1703, Springer, Berlin, 1999.
8. _, _, An Introduction to p-Variation and Young Integrals, Lecture Notes, No. 1, MaPhySto, Univ. of Aarhus, 1998.
9. _, _, Concrete Functional Calculus, book in preparation.
10. A. M. D'yačkov, Conditions for the existence of Stieltjes integral of bounded generalized variation, Anal. Math. (Budapest) 14 (1988), 295-313.
11. H. Föllmer, Calcul d'Itô sans probabilités, in: Séminaire de Probabilités XV, 1979/80, J. Azéma and M. Yor (eds.), Lecture Notes in Math. 850, Springer, Berlin, 1981, 143-150.
12. H. Föllmer, P. Protter and A. N. Shiryayev, Quadratic covariation and an extension of Itô's formula, Bernoulli 1(1/2) (1995), 149-169.
13. R. Henstock, Theory of Integration, Butterworths, London, 1963.
14. E. W. Hobson, The Theory of Functions of a Real Variable and the Theory of Fourier's Series, Vol. 1, Dover, New York, 1957.
15. T. H. Hildebrandt, Introduction to the Theory of Integration, Academic Press, New York, 1963.
16. K. Itô, Stochastic differentials, Appl. Math. Optim. 1 (1975), 374-381.
17. Reprinted in: Kiyosi Itô Selected Papers, D. W. Stroock and S. R. S. Varadhan (eds.), Springer, New York, 1987, 558-565.
18. A. N. Kolmogorov, Untersuchungen über Integralbegriff, Math. Ann. 103 (1930), 654-696.
19. English transl, in: Selected Works of A. N. Kolmogorov, Vol. I, Mathematics and Mechanics, V. M. Tikhomirov (ed.), Kluwer, Dordrecht, 1991, 100-143.
20. J. Kurzweil, Generalized ordinary differential equations and continuous dependence on a parameter, Czechoslovak Math. J. 7 (1957), 568-583 (in Russian).
21. _, On integration by parts, Czechoslovak Math. J. ibid. 8 (1958), 356-359.
22. S. Leader, The Kurzweil-Henstock Integral and its Differentials, Dekker, New York, 2001.
23. R. Leśniewicz and W. Orlicz, On generalized variations (II), Studia Math. 45 (1973), 71-109.
24. P. Lévy, Le mouvement brownien plan, Amer. J. Math. 62 (1940), 487-550.
25. _, Intégrales de Stieltjes généralisées, Ann. Soc. Polon. Math. 25 (1952), 17-26.
26. Reprinted in: Oeuvres de Paul Lévy, Vol. II, Gauthier-Villars, Ecole Polytechnique, Paris, 1974, 473-482.
27. T. Lyons, On the non-existence of path integrals, Proc. Roy. Soc. London Ser. A 432 (1991), 281-290.
28. T. Lyons, Differential equations driven by rough signals, Rev. Mat. Iberoamericana 14 (1998), 215-310.
29. R. M. McLeod, The Generalized Riemann Integral, Carus Math. Monographs 20, Math. Assoc. Amer., 1980.
30. E. J. McShane, A theory of limits, in: Studies in Modern Analysis, Vol. 1, R. C. Buk (ed.), Math. Assoc. Amer., 1962, 7-29.
31. E. J. McShane and T. A. Botts, Real Analysis, Van Nostrand, Princeton, 1959.
32. J. J. Moreau and M. Valadier, A chain rule involving vector functions of bounded variation, J. Funct. Anal. 74 (1987), 333-345.
33. R. Norvaiša, p-variation and integration of sample functions of stochastic processes, in: Probab. Theory and Math. Statist. (Vilnius, 1998), B. Grigelionis et al. (eds.), TEV, 1999, 521-540.
34. _, Quadratic variation, p-variation and integration with applications to stock price modelling, preprint, The Fields Institute, Toronto, 2001.
35. E. R. Phillips, An Introduction to Analysis and Integration Theory, with a new historical introduction and notes by the author, Dover, New York, 1984.
36. 1 -functions of semimartingales, Probab. Theory Related Fields 104 (1996), 27-41.
37. S. Saks, Theory of the Integral, transl, by L. C. Young, Hafner, New York, 1937.
38. J. Serrin and D. E. Varberg, A general chain rule for derivatives and the change of variables formula for the Lebesgue integral, Amer. Math. Monthly 76 (1969), 514-520.
39. H. L. Smith, On the existence of the Stieltjes integral, Trans. Amer. Math. Soc. 27 (1925), 491-515.
40. B. S. Thomson, Some remarks on differential equivalence, Real Anal. Exchange 12 (1986-87), 294-312.
41. A. I. Vol'pert, The spaces BV and quasi-linear equations, Mat. Sb. 73 (115) (1967), 255-302 (in Russian).
42. A. J. Ward, The Perron-Stieltjes integral, Math. Z. 41 (1936), 578-604.
43. L. C. Young, An inequality of the Hölder type, connected with Stieltjes integration, Acta Math. 67 (1936), 251-282.
44. _, General inequalities for Stieltjes integrals and the convergence of Fourier series, Math. Ann. 115 (1938), 581-612.
45. _, A converse to some inequalities and approximations in the theory of Stieltjes and stochastic integrals, and for nth derivatives, Studia Math. 55 (1976), 215-223.
46. W. H. Young, On integration with respect to a function of bounded variation, Proc. London Math. Soc. (2) 13 (1914), 109-150.
47. M. Zähle, Integration with respect to fractal functions and stochastic calculus. I, Probab. Theory Related Fields 111 (1998), 333-374.