The Integral, a Vision of its Evolution Through Time IV. Harmonic Analysis, Stochastic Processes. Stochastic Integration

Authors

DOI:

https://doi.org/10.17268/sel.mat.2024.01.09

Keywords:

BMO, Brownian motion, random game, martingale, stochastic integral

Abstract

The objective of this article is to give an overview of the relationship between harmonic analysis and stochastic processes, a relationship that has motivated many investigations and applications to specific problems. We will also emphasize stochastic integration.

References

Petersen KE. Brownian motion, Hardy spaces and bounded mean oscillation: Cambridge University Press. Cambridge; 1977.

Fefferman C. Characterizations of bounded mean oscillation, American Mathematical Society.1971; 77(4): 587-588.

Fefferman Ch, Stein EM. Hp spaces of several variables. Acta Math; 1972. 129: 137-192.

Fefferman C. Harmonic analysis and Hp spaces, Studies in Math, Math. Assoc. of Amer. 1976; 13.

Calderón AP, Zygmund A. On the existence of certain singular integrals. Acta Math. 1952; 88: 85-139.

Burkholder DL. Martingale transforms, Ann. Math. Stat. 1966; 37: 1494-1504.

Gundy RF. On the class LlogL, martingales and singular integrals, Studia Mathematica. 1969; T. XXXIII.

Doob JL. Stochastic Processes, John Wiley and Sons, Inc; New York. 1953.

Rincón L. Introducción a los procesos estocásticos, Dep. Mat. UNAM; 2012.

Burkholder DL, Gundy RF, Silverstein ML. A maximal function characterization of the class Hp, Trans. Amer. Math. Soc. 1971; 157.

John F, Nirenberg L. On functions of bounded mean oscillation. Comm. Pure and Applied Math. 1961; 14.

Neri U. Fractional integration on the space Hp and its dual, Studia Math. 1975.

Hardy GH, Littlewood JE. A maximal theorem with function-theoretic applications, Acta Math. 1930; 54.

Lusin N. Sur une propri´et´e des fonctions ´a carr´e sommable, Bull. Calcutta Math. Soc. 1930; 20.

Marcinkiewicz J, Zygmund A. A theorem of Lusin, Duke math. J. 1938; 4.

Spencer DC. A function-theoretic identity, Amer. J. Math. 1943; 65.

Littlewood JE, Paley RE. Theorems on Fourier series and power series, I. J. London Math. Soc. 1931; 6.

Gasper GJr. On the Littlewood-Paley g-function and the Lusin S-function, Trans. Amer. Math. Soc. 1968; 134.

Kolmogorov AN. Sur les fonctions harmoniques conjugées et les séries de Fourier, Fund. Math. 1925; 7.

Davis B. On the weak type (1, 1) inequality for conjugate functions, Proc. Amer. Math. Soc. 1974; 44.

Taylor AE. Weak convergence in the spaces Hp, Duke math. J. 1950; 17.

Davis B. On the integrability of the martingale square function, Israel J. math. 1970; 8.

Burkholder DL, Gundy RF. Extrapolation and interpolation of quasi-linear operators on martingales, Acta Math. 1970; 124.

Roussanov N. Martingale Hardy spaces and bounded mean oscillation. University of Pennsylvania, Dpt. Math. (Internet).

Milman M. Interpolation of martingale spaces and application, 11 Seminario Brasileiro de An´alise. Pre-Public.

Stein EM, Weiss G. Introduction to Fourier analysis on euclidean spaces, Princeton University Press. Princeton. N. J. 1971.

Garsia AM. The Burges Davis inequalities via Fefferman’s Inequality, Ark. Mat. 1973; 11.

Herz C. Bounded mean oscillation and regulated martingales, Trans. of the American Mathematical Society. 1974; 193.

Herz C. Hp-spaces of martingales, 0 < p < 1. Z. Wahrscheinlichkeitstheorie verw Geb. 1974; 28.

Burkholder DL, Davis BJ, Gundy RF. Integral inequalities for convex functions of operators on martingales, Proceed. 6 Berkeley Sym. Math. Stat. Prob. 1970.

Burkholder DL. Inequalities for operators on martingales, Actes, Congr´es Inter. Math. 1970; 2.

Burkholder DL, Gundy RF. Boundary behaviour of harmonic functions in a half-space and brownian motion, Ann. Inst. Fourier. U. Grenoble. 1973; T. XXlll (4).

Calderón A. On the behaviour of harmonic functions at the boundary, Trans. Amer. Math. Soc. 1950; 68.

Carleson L. On the existence of boundary values for harmonic functions in several variables, Arkiv f¨or Mathematik. 1961; 4.

Calderón A. On a theorem of Marcinkiewicz and Zygmund. Trans. Americ. Math. Soc. 1950; 68.

Spencer D. A function-theoretic identity, Amer. J. Math. 1943; 65.

Privalov I. Integral Cauchy, Saratov. 1919.

Stein EM. On the theory of harmonic functions of several variables II, Behaviour near the boundary. Acta Math. 1961; 106.

Burkholder DL. Martingale theory and harmonic analysis in Euclidean spaces, Proceeding of Symp. in Pure Math. 1979; XXXV(2).

León JA. Integración estocástica con respecto al movimiento browniano, Matemáticas: Enseñanza Universitaria. 2006; XIV (2).

Burkholder DL. One-sided maximal functions and Hp, Journal of Functional Analysis. 1975; 18.

Burkholder DL. Distribution function inequalities for martingales, The Annals of Probability. 1973; 1(1).

Burkholder DL. Hp spaces and exit times of brownian motion. Bull, AMS. 1975; 81(3).

Ortiz A. La integral, una visión de su evolución a través del tiempo. I, II, III, UNT. Selecciones Matemáticas. 2023; I, 10(1); II y III, 10(2).

Liu J. Approximative theorem of incomplete Riemann-Stieltjes sum of stochastic integral, arXiv. 2018; 17:1803.05182v2 [MATH.pr].

Garro M. Notas breves de integración estocástica, Internet. Octubre. 2016.

Le Gall J-F. Brownian motion, martingales and stochastic calculus, Graduate Text in Mathematics. Springer. 2013.

Bojdecki T. Teoría general de procesos e integración estocástica, Texto 6. Aportaciones Matemáticas. S. M. Mex. 1995.

Bastons G, Joan C. The Ito integral and anticipating generalizations, Univ. Aut. de Madrid. Madrid. 2017.

Buldygin V, et al. Anatolii Skorohod(1930-2011), Newsletter of the EMS.

López R. Métodos numéricos para la solución de ecuaciones diferenciales estocásticas, Univ. Aut. De Puebla. 2014.

Ayed W, Kuo H-H. An extension of the Ito integral, Communications on Stochastic Analysis. 2008; 2(3).

AyedW, Kuo H-H. An extension of the Ito integral: toward a general theory of stochastic integration, Theory of Stochastic Processes. 2010; 16(32)(1).

Kuo H-H: The Ito calculus and white noise theory: a brief survey toward general stochastic integration. Communications on Stochastic Analysis. 2014; 8(1).

Kuo H-H. Introduction to stochastic integration, Universitext (UTX). Springer. 2006.

Kuo H, Tang AS, Szozda B. The Itˆo formula for a new stochastic integral. Thiele Centre. 2012; 4(I).

Hwang Ch, Kuo H, Saito K, Zhai J. A general Ito formula for adapted and instantly independent stochastic processes. Commun. on Stochastic Analysis. 2016; 10(3).

Parczewski P. Extensions of the Hitsuda-Skorokhod integral. Communications on Stochastic Analysis. 2017; 11(4).

Jarrow R, Protter Ph. A short history of Stochastic integration and mathematical finance: The early years, 1880-1970, A Festschrift for Herman Rubin. Inst. of Mathem. Statistics. Lect. Notes. 2004; 4.

Ito K. Stochastic integral, Proc. Imp. Acad. Tokyo. 1944; 20 (8).

Kuo H-H. Integration theory on infinite-dimensional manifolds. Transac. of the American Mathem. Society. 1971; 159.

Gross L. AbstractWiener Spaces, Proc. Fifth Berkeley Symp. Math.Statist. and Probability. Berkeley. 1965/66.

Stein EM. Topics in harmonic analysis. Related to the Littlewood-Paley theory: Princeton University Press; N. J. 1970.

Lé K. Quantitative John-Nirenberg inequality for stochastic processes of bounded mean oscillation, arXiv. 2022.

Kazamaki N. The martingale version of a theorem of Sarason on the class VMO, Math. Rep. Toyama Univ. 1979; 2.

Kazamaki N. Transformation of Hp-martingales by a change of law. Z, Wahrscheinlichkeistheorieverw. Gebiet. 1979; 46.

Kohatsu A. Cálculo estocástico y una aplicación a la estadística, Pro Mathemática. PUCP. 1988; II (4).

Ito K. Differential equations determining Markov processes, Zenkoku Shijo S. D. 1942; 244(1077).

Published

2024-07-29

How to Cite

Ortiz Fernández, A. . (2024). The Integral, a Vision of its Evolution Through Time IV. Harmonic Analysis, Stochastic Processes. Stochastic Integration. Selecciones Matemáticas, 11(01), 117 - 152. https://doi.org/10.17268/sel.mat.2024.01.09

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