# Product integrals, young integrals and p-variation

• R. M. Dudley
• R. Norvaiša
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 1703)

## Abstract

Let $$\mathbb{B}$$be a Banach algebra with identity $$\mathbb{I}$$. Consider a function f defined on an interval [a, b] and with values in $$\mathbb{B}$$. Let {a=x0<...<b=x n } be a partition of [a, b]. Then the product integral with respect to f over [a, b] is defined as the limit of the product from i=1 to n of $$\mathbb{I}$$+f(xi-1), if it exists, where the limit is taken under refinements of partitions. It is proved that the product integral with respect to f over [a, b] exists if fW p ([a,b];)$$\mathbb{B}$$, 0<p<2, i.e., if f has bounded p-variation for some 0<p<2, as shown for f continuous by M. A. Freedman, Trans. Amer. Math. Soc. 279 (1983), 95–112. A necessary and sufficient condition for the existence of the product integral is given when $$\mathbb{B} = \mathbb{R}$$. An operator $$\mathcal{P}_a$$from $$\mathcal{W}_p ([a,b];\mathbb{B})$$into itself is induced by an indefinite product integral. The main result says that $$\mathcal{P}_a$$is Fréchet differentiable. R. D. Gill and S. Johansen (Ann. Statist. 18, 1990, 1501–1555) had shown compact differentiability in the supremum norm, on sets uniformly bounded in 1-variation norm. The present paper shows that when restricted to rightor left-continuous elements of $$\mathcal{W}_p ([a,b];\mathbb{B})$$, $$\mathcal{P}_a$$is analytic. To prove these results a generalized Stieltjes integral due to L. C. Young is developed, as are variants of it called left Young (LY) and right Young (RY) integrals, and the Duhamel formula is extended to (LY) and (RY) integrals. Also using Young integrals a logarithm operator $$\mathcal{L}_a$$is defined so that $$\mathcal{L}_a (f)$$exists for each $$f \in \mathcal{W}_p ([a,b];\mathbb{B})$$, 0<p<2, such that the function xf(x)−1 is in $$\ell _\infty ([a,b];\mathbb{B})$$. The operators $$\mathcal{P}_a$$and $$\mathcal{L}_a$$are shown to be inverses of each other. This allows one to determine a unique solution of a linear integral equation and to solve an evolution representation problem whenever the evolution has bounded p-variation for some 0<p<2.

## Keywords

Banach Algebra Product Integral Left Endpoint Left Limit Linear Integral Equation
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. Averbukh, V. I. and Smolyanov, O. G. (1967). The theory of differentiation in linear topological spaces. Uspekhi Mat. Nauk, 22(6), 201–260 (in Russian). Translated in Russian Math. Surveys, 22(6), 201–258.
2. Birkhoff, Garrett (1937). On product integration. J. Math. and Phys. (MIT, Cambridge, Mass), 16, 104–132.
3. Bonsall, F. F. and Duncan, J. (1973). Complete normed algebras. Springer-Verlag, Berlin.
4. Bruneau, M. (1974). Variation totale d'une fonction. Lect. Notes in Math. (Springer-Verlag), 721.Google Scholar
5. Chae, S. B. (1985). Holomorphy and calculus in normed spaces. Dekker, New York.
6. Dobrushin, R. L. (1953). Generalization of Kolmogorov's equations for Markov processes with a finite number of possible states. Mat. Sb., 33 (3), 567–596 (in Russian).Google Scholar
7. Doléans-Dade, C. (1970). Quelques applications de la formule de changement de variables pour les semimartingales. Z. Wahrsch. Verw. Gebiete, 16, 181–194.
8. Dollard, J. D. and Friedman, C. N. (1979). Product integration with applications to differential equations. Encyclopedia of Math. and its Appl., Vol. 10, Addison-Wesley, Reading, Massachusetts.Google Scholar
9. Doran, R. S. and Belfi, V. A. (1986). Characterizations of C *-algebras: the Gelfand-Naimark theorems. Dekker, New York.
10. Dudley, R. M. (1973). Sample functions of the Gaussian process. Ann. Probab., 1, 66–103.
11. Dudley, R. M. (1992). Fréchet differentiability, p-variation and uniform Donsker classes. Ann. Probab., 20, 1968–1982.
12. Dudley, R. M. (1997). Empirical processes and p-variation. In Festschrift for Lucien Le Cam, Eds. D. Pollard, E. Torgersen, G. L. Yang. Springer-Verlag, New York, pp. 219–233.
13. Dvoretzky, A. and Rogers, C. A. (1950). Absolute and unconditional convergence in normed linear spaces. Proc. Nat. Acad. Sci. U.S.A., 36, 192–197.
14. Emery, M. (1978). Stabilité des solutions des équations différentielles stochastiques application aux intégrales multiplicatives stochastiques. Z. Wahrsch. Verw. Gebiete, 41, 241–262.
15. Fernique, X. (1964). Continuité des processus gaussiens. C. R. Acad. Sci. Paris, 258, 6058–6060.
16. Fernique, X. (1964). Régularité de processus gaussiens. Invent. Math., 12, 304–320.
17. Freedman, M. A. (1983a). Operators of p-variation and the evolution representation problem. Trans. Amer. Math. Soc., 279, 95–112.
18. Freedman, M. A. (1983b). Necessary and sufficient conditions for discontinuous evolutions with applications to Stieltjes integral equations. J. Integral Equations, 5, 237–270.
19. Fristedt, B. and Taylor, S. J. (1973). Strong variation for the sample functions of a stable process. Duke Math. J., 40, 259–278.
20. Gal'chuk, L. I. (1984). Stochastic integrals with respect to optional semimartingales and random measures. Theory Probab. Appl., 29, 93–108.
21. Gantmacher, F. R. (1960). The Theory of Matrices. Vol. 2. Chelsea, New York.
22. Gill, R. D. (1994). Lectures on survival analysis. In: Ecole d'Eté de Probabilités de Saint Flour XXII, (ed. P. Bernard). Lect. Notes in Math. (Springer-Verlag), 1581, 115–241.Google Scholar
23. Gill, R. D. and Johansen, S. (1990). A survey on product-integration with a view toward application in survival analysis. Ann. Statist., 18, 1501–1555.
24. Gochman, E. Ch. (1958). The Stieltjes integral and its applications. Moscow, GIFML (in Russian).Google Scholar
25. Hildebrandt, T. H. (1959). On systems of linear differentio-Stieltjes-integral equation. Illinois J. Math., 3, 352–373.
26. Hildebrandt, T. H. (1963). Introduction to the theory of integration. Academic Press, New York.
27. Hinton, D. B. (1966). A Stieltjes-Volterra integral equation theory. Canad. J. Math., 18, 314–331.
28. Hönig, C. S. (1975). Volterra Stieltjes-Integral Equations. Mathematics Studies 16. North-Holland, Amsterdam.
29. Hönig, C. S. (1980). Volterra Stieltjes-Integral Equations. In: Functional Differential Equations and Bifurcation, (ed. A. F. Izé). Lect. Notes in Math. (Springer-Verlag), 799, 173–216.Google Scholar
30. Jarník, J. and Kurzweil, J. (1987). A general form of the product integral and linear ordinary differential equations. Czechoslovak Math. J., 37, 642–659.
31. Kawada, T. and Kôno, N. (1973). On the variation of Gaussian processes. In: Proc. of the Second Japan-USSR Symposium on Probability Theory, Lect. Notes in Math. (Springer-Verlag), 330, 176–192.Google Scholar
32. Köthe, G. (1969). Topological Vector Spaces I. Translated by D. J. H. Garling. Springer, New York.
33. Krabbe, G. L. (1961a). Integration with respect to operator-valued functions. Bull. Amer. Math. Soc., 67, 214–218.
34. Krabbe, G. L. (1961b). Integration with respect to operator-valued functions. Acta Sci. Math. (Szeged), 22, 301–319.
35. Lebesgue, H. (1973). Leçons sur l'intégration et la recherche des fonctions primitives. 3-rd Edition. Chelsea, New York.
36. Leśniewicz, R. and Orlicz, W. (1973). On generalized variations (II). Studia Math., 45, 71–109.
37. Love, E. R. (1951). A generalization of absolute continuity. J. London Math. Soc., 26, 1–13.
38. Love, E. R. (1993). The refinement-Ross-Riemann-Stieltjes (R3S) integral. In Analysis, geometry and groups: a Riemann legacy volume, Eds. H. M. Srivastava, Th. M. Rassias, Part I. Hadronic Press, Palm Harbor, FL, pp. 289–312.Google Scholar
39. Love, E. R. and Young, L. C. (1938). On fractional integration by parts. Proc. London Math. Soc., 44, 1–35.
40. Lyons, Terry (1994). Differential equations driven by rough signals (I): an extension of an inequality of L. C. Young. Math. Res. Lett., 1, 451–464.
41. MacNerney, P. R. (1955). Stieltjes integrals in linear spaces. Ann. of Math., 61, 354–367.
42. Maisonneuve, B. (1968). Quelques martingales remarquables associées à une martingale continue. Publ. Inst. Statist. Univ. Paris, 17, no. 3, 13–27.
43. Manturov, O. V. (1990). A multiplicative integral. In: Itogi Nauki i Tekhniki, Ser. Probl. Geom., Vol. 22, R. V. Gamkrelidze ed., VINITI Moscow, pp. 167–215 (in Russian). Translated in J. Soviet Math. 55 (1991), no. 5, 2042–2076.Google Scholar
44. Marcus, M. B. and Shepp, L. A. (1971). Sample behavior of Gaussian processes. Proc. Sixth Berkeley Symp. math. Statist. Prob. (1970) 2, 423–442. Univ. of Calif. Press, Berkeley and Los Angeles.Google Scholar
45. Masani, P. R. (1947). Multiplicative Riemann integration in normed rings. Trans. Amer. Math. Soc., 61, 147–192.
46. Moore, E. H. (1915). Definition of limit in general integral analysis. Proc. Nat. Acad. Sci. U.S.A., 1, 628.
47. Nachbin, L. (1973). Recent developments in infinite dimensional holomorphy. Bull. Amer. Math. Soc., 79, 625–640.
48. Nashed, M. Z. (1971). Differentiability and related properties of non-linear operators: some aspects of the role of differentials in nonlinear analysis. In: Nonlinear Functional Analysis and Applications, L. B. Rall (ed.), Academic Press, New York, 103–309.
49. Peano, G. (1886–7). Integrazione per serie delle equazioni differenziali lineari. Atti Accad. sci. Torino, 22, 437–466. Engl. transl. in “Selected works of Giuseppe Peano”, ed. H. C. Kennedy, Univ. Toronto Press, 1973, pp. 58–66.
50. Pollard, S. (1923). The Stieltjes' integral and its generalizations. Quart. J. of Pure and Appl. Math., 49, 73–138.
51. Rolewicz, S. (1985). Metric Linear Spaces. Reidel, Dordrecht; PWN, Warsaw.
52. Schlesinger, L. (1931). Neue Grundlagen für einen Infinitesimalkalkul der Matrizen. Math. Z., 33, 33–61.
53. Schlesinger, L. (1932). Weitere Beiträge zum Infinitesimalkalkul der Matrizen. Math. Z., 35, 485–501.
54. Schmidt, G. (1971). On multiplicative Lebesgue integration and families of evolution operators. Math. Scand., 29, 113–133.
55. Schwabik, Š. (1990). The Perron product integral and generalized linear differential equations. Časopis Pěst. Mat., 115, 368–404.
56. Schwabik, Š. (1994). Bochner product integration. Math. Bohem., 119, 305–335.
57. Segal, I. E. and Kunze, R. A. (1968). Integrals and Operators. McGraw-Hill, New York.
58. Taylor, S. J. (1972). Exact asymptotic estimates of Brownian path variation. Duke Math. J., 39, 219–241.
59. Volterra, V. (1887). Sui fondamenti della teoria delle equazioni differenziali lineari. Memorie della Società Italiana delle Scienze (detta dei XL), serie III, vol. VI, n. 8. In: Opere Matematiche I, 1954, pp. 209–289. Accad. Nazionale dei Lincei, Roma.Google Scholar
60. Volterra, V. and Hostinsky, B. (1938). Opérations infinitesimales, linéaires. Gauthier-Villars, Paris.
61. Wiener, N. (1924). The quadratic variation of a function and its Fourier series. J. Math. and Phys. (MIT, Cambridge, Mass), 3, 73–94.Google Scholar
62. Young, L. C. (1936). An inequality of the Hölder type, connected with Stieltjes integration. Acta Math., 67, 251–282.
63. Young, L. C. (1938). General inequalities for Stieltjes integrals and the convergence of Fourier series. Math. Ann., 115, 581–612.