Skip to main content
SpringerLink
Log in

Stochastic Processes in the Decades after 1950

  • Chapter
  • First Online:
The Splendors and Miseries of Martingales

Part of the book series: Trends in the History of Science ((TRENDSHISTORYSCIENCE))

  • 966 Accesses

Abstract

Paul-André Meyer (1934–2003), founder and leader of the Strasbourg school of probability, worked from the 1960s into the 1990s on the theory of stochastic processes, emphasizing processes in continuous time. In this sweeping review, which first appeared in French in 2000, Meyer emphasizes the founding role of Doob’s Stochastic Processes, published in 1953, which presented tools and topics that fueled probabilistic research for the rest of the century: filtrations, stopping times, martingales, Markov processes, diffusions, Itô’s stochastic integral, and stochastic differential equations. The review then concentrates on two periods of research. The period from 1950 to 1965 was dominated by the study of Markov processes and their connections with potential theory and martingales. In the period from 1965 to 1980, martingales became more prominent, along with the stochastic integral, excursions, the general theory of processes, and stochastic mechanics. The review extends into the 1980s, discussing the Malliavin calculus and noncommutative probability theory. The great contributions by Doob and Itô are given their due.

Paul-André Meyer (1934–2003) worked at the University of Strasbourg. This chapter was first published in French in 2000 [172]. Glenn Shafer, Rutgers University. Translation by Jeanine Sedjro.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    Which allows the distribution of the maximum of a Brownian motion to be calculated.

  2. 2.

    Weak convergence is associated with the integration of bounded continuous functions [14].

  3. 3.

    Brownian motion happens to be simultaneously a martingale and a Markov process, but these two notions are not related.

  4. 4.

    Of which the core is the elementary solution of the heat equation, that is, the Brownian transition function itself.

  5. 5.

    Technical definition weakening the integrability condition for martingales.

References

  1. Accardi, L., Frigerio, A., Lewis, J.T.: Quantum stochastic processes. Publications of the Research Institute for Mathematical Sciences, Kyoto 18(1), 97–133 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  2. Aldous, D.: Stopping times and tightness. The Annals of Probability 6(2), 335–340 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  3. Austin, D.G.: A sample function property of martingales. Annals of Mathematical Statistics 37(5), 1396–1397 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  4. Azéma, J., Yor, M.: Temps locaux. No. 52–53 in Astérisque. Société mathématique de France (1978)

    Google Scholar 

  5. Bachelier, L.: Théorie de la spéculation. Annales de l’École Normale Supérieure 17, 21–86 (1900)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bakry, D.: L’hypercontractivité et son utilisation en théorie des semigroupes. In: Lectures on probability theory, LN 1581, pp. 1–114. Springer (1994)

    Google Scholar 

  7. Barlow, M.T.: Study of a filtration expanded to include an honest time. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 44(4), 307–323 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  8. Barnett, C., Streater, R., Wilde, I.: The Itô-Clifford integral. Journal of Functional Analysis 48(2), 172–212 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  9. Benveniste, A., Jacod, J.: Systèmes de Lévy des processus de Markov. Inventiones Mathematicae 21(3), 183–198 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bernstein, S.: Sur l’extension du théorème limite du calcul des probabilités aux sommes de quantités dépendantes. Mathematische Annalen 97, 1–59 (1927)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bernstein, S.: Equations differentielles stochastiques. Act. Sci. Ind. 738, 5–31 (1938). Conf. Intern. Sci. Math. Univ. Genève, Hermann

    Google Scholar 

  12. Beurling, A., Deny, J.: Dirichlet spaces. Proceedings of the National Academy of Sciences 45(2), 208–215 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  13. Bichteler, K.: Stochastic integration and \(L^p\)-theory of semimartingales. The Annals of Probability 9(1), 49–89 (1981)

    Google Scholar 

  14. Billingsley, P.: Convergence of Probability Measures. Wiley, New York (1968)

    MATH  Google Scholar 

  15. Bismut, J.M.: Martingales, the Malliavin calculus and hypoellipticity under general Hörmander’s conditions. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 56(4), 469–505 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  16. Bismut, J.M.: The Atiyah-Singer theorems: A probabilistic approach. I. The index theorem. Journal of Functional Analysis 57(1), 56–99 (1984)

    Google Scholar 

  17. Bismut, J.M.: Large Deviations and the Malliavin Calculus. Birkhäuser (1984)

    Google Scholar 

  18. Bismut, J.M.: Last exit decompositions and regularity at the boundary of transition probabilities. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 69(1), 65–98 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  19. Blagoveshchenskii, Y.N., Freidlin, M.I.: Some properties of diffusion processes depending on a parameter. Doklady Akademii Nauk 138(3), 508–511 (1961)

    MathSciNet  MATH  Google Scholar 

  20. Bliedtner, J., Hansen, W.: Potential Theory: An Analytic and Probabilistic Approach to Balayage. Springer (1986)

    Google Scholar 

  21. Blumenthal, R.M.: An extended Markov property. Transactions of the American Mathematical Society 85(1), 52–72 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  22. Blumenthal, R.M., Getoor, R.K.: Markov processes and potential theory. Academic Press (1968)

    Google Scholar 

  23. Bochner, S.: Diffusion equation and stochastic processes. Proceedings of the National Academy of Sciences 35(7), 368–370 (1949)

    Article  MathSciNet  MATH  Google Scholar 

  24. Bochner, S.: Harmonic analysis and the theory of probability. University of California (1955)

    Google Scholar 

  25. Brelot, M.: Sur le principe des singularités positives et la topologie de R. S. Martin. Annales de l’université de Grenoble. Nouvelle série. Section sciences mathématiques et physiques 23, 113–138 (1948)

    Google Scholar 

  26. Brelot, M.: Le problème de Dirichlet, axiomatique et frontière de Martin. Journal de Mathématiques Pures et Appliquées 35, 297–335 (1956)

    MathSciNet  MATH  Google Scholar 

  27. Burkholder, D.L.: Maximal inequalities as necessary conditions for almost everywhere convergence. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 3(1), 75–88 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  28. Burkholder, D.L.: Martingale transforms. The Annals of Mathematical Statistics 37(6), 1494–1504 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  29. Burkholder, D.L.: Distribution function inequalities for martingales. The Annals of Probability 1(1), 19–42 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  30. Burkholder, D.L., Davis, B.J., Gundy, R.F.: Integral inequalities for convex functions of operators on martingales. In: Sixth Berkeley Symposium on Mathematical Statistics and Probability, vol. 2, pp. 223–240. University of California (1971)

    Google Scholar 

  31. Burkholder, D.L., Gundy, R.F.: Extrapolation and interpolation of quasi-linear operators on martingales. Acta Mathematica 124, 249–304 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  32. Burkholder, D.L., Gundy, R.F.: Distribution function inequalities for the area integral. Studia Mathematica 44(6), 527–544 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  33. Cairoli, R.: Une inégalité pour martingales à indices multiples et ses applications. Séminaire de probabilités de Strasbourg 4, 1–27 (1970)

    MATH  Google Scholar 

  34. Cairoli, R., Walsh, J.B.: Stochastic integrals in the plane. Acta Mathematica 134, 111–183 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  35. Cameron, R.H., Martin, W.T.: Transformations of Weiner integrals under translations. Annals of Mathematics pp. 386–396 (1944)

    Google Scholar 

  36. Cameron, R.H., Martin, W.T.: Fourier-Wiener transforms of analytic functionals. Duke Mathematical Journal 12(3), 489–507 (1945)

    MathSciNet  MATH  Google Scholar 

  37. Cameron, R.H., Martin, W.T.: Fourier-Wiener transforms of functionals belonging to \(L_2\) over the space \(C\). Duke Mathematical Journal 14(1), 99–107 (1947)

    Google Scholar 

  38. Cameron, R.H., Martin, W.T.: The orthogonal development of non-linear functionals in series of Fourier-Hermite functionals. Annals of Mathematics 48(2), 385–392 (1947)

    Article  MathSciNet  MATH  Google Scholar 

  39. Carlen, E.A.: Conservative diffusions. Communications in Mathematical Physics 94(3), 293–315 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  40. Cartan, H.: Théorie du potentiel newtonien: énergie, capacité, suites de potentiels. Bulletin de la Société Mathématique de France 73, 74–106 (1945)

    Article  MathSciNet  MATH  Google Scholar 

  41. Cartan, H.: Théorie générale du balayage en potentiel newtonien. Annales de l’université de Grenoble. Nouvelle série. Section sciences mathématiques et physiques 22, 221–280 (1946)

    Google Scholar 

  42. Choquet, G.: Theory of capacities. Annales de l’Institut Fourier 5, 131–295 (1954)

    Article  MathSciNet  MATH  Google Scholar 

  43. Chung, K.L.: Markov Chains with Stationary Transition Probabilities. Springer (1960)

    Google Scholar 

  44. Chung, K.L.: Excursions in Brownian motion. Arkiv för Matematik 14(1), 155–177 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  45. Chung, K.L., Doob, J.L.: Fields, optionality and measurability. American Journal of Mathematics 87(2), 397–424 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  46. Chung, K.L., Feller, W.: On fluctuations in coin-tossing. Proceedings of the National Academy of Sciences 35, 605–608 (1949)

    Article  MathSciNet  MATH  Google Scholar 

  47. Chung, K.L., Walsh, J.B.: To reverse a Markov process. Acta Mathematica 123, 225–251 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  48. Ciesielski, Z., Taylor, S.J.: First passage times and sojourn times for Brownian motion in space and the exact Hausdorff measure of the sample path. Transactions of the American Mathematical Society 103(3), 434–450 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  49. Clark, J.M.C.: The representation of functionals of Brownian motion by stochastic integrals. The Annals of Mathematical Statistics 41(4) (1970)

    Google Scholar 

  50. Coifman, R.R.: A real variable characterization of \({H}^p\). Studia Mathematica 51(3), 269–274 (1974)

    Google Scholar 

  51. Cushen, C.D., Hudson, R.L.: A quantum-mechanical central limit theorem. Journal of Applied Probability 8(3), 454–469 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  52. Davies, E.B.: Quantum Theory of Open Systems. Academic Press (1976)

    Google Scholar 

  53. Dellacherie, C.: Capacités et Processus Stochastiques. Springer (1972)

    Google Scholar 

  54. Dellacherie, C., Meyer, P.A.: Probabilités et potentiel, Chapitres I à IV. Hermann (1975)

    MATH  Google Scholar 

  55. Deny, J.: Les potentiels d’énergie finie. Acta Mathematica 82, 107–183 (1950)

    Article  MathSciNet  MATH  Google Scholar 

  56. Doléans-Dade, C.: On the existence and unicity of solutions of stochastic integral equations. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 36(2), 93–101 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  57. Doléans-Dade, C., Meyer, P.A.: Intégrales stochastiques par rapport aux martingales locales. Séminaire de probabilités de Strasbourg 4, 77–107 (1970)

    MATH  Google Scholar 

  58. Doléans-Dade, C.: Quelques applications de la formule de changement de variables pour les semimartingales. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 16(3), 181–194 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  59. Donsker, M.D., Varadhan, S.R.S.: Asymptotic evaluation of certain Markov process expectations for large time. Communications on Pure and Applied Mathematics 28, 1–47, 279–301, 389–461 (1976)

    Google Scholar 

  60. Doob, J.L.: Stochastic processes depending on a continuous parameter. Transactions of the American Mathematical Society 42(1), 107–140 (1937)

    Article  MathSciNet  MATH  Google Scholar 

  61. Doob, J.L.: Regularity properties of certain families of chance variables. Transactions of the American Mathematical Society 47(3), 455–486 (1940)

    Article  MathSciNet  MATH  Google Scholar 

  62. Doob, J.L.: Markoff chains—denumerable case. Transactions of the American Mathematical Society 58(3), 455–473 (1945)

    Article  MathSciNet  MATH  Google Scholar 

  63. Doob, J.L.: Continuous parameter martingales. In: Second Berkeley Symposium on Mathematical Statistics and Probability, pp. 269–177. University of California (1951)

    Google Scholar 

  64. Doob, J.L.: Stochastic Processes. Wiley, New York (1953)

    MATH  Google Scholar 

  65. Doob, J.L.: Semimartingales and subharmonic functions. Transactions of the American Mathematical Society 77(1), 86–121 (1954)

    Article  MathSciNet  MATH  Google Scholar 

  66. Doob, J.L.: A probability approach to the heat equation. Transactions of the American Mathematical Society 80(1), 216–280 (1955)

    Article  MathSciNet  MATH  Google Scholar 

  67. Doob, J.L.: Probability methods applied to the first boundary value problem. In: Third Berkeley Symposium on Mathematical Statistics and Probability, vol. 2, pp. 49–80. University of California (1956)

    Google Scholar 

  68. Doob, J.L.: Brownian motion on a Green space. Theory of Probability & Its Applications 2(1), 1–30 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  69. Doob, J.L.: Conditional Brownian motion and the boundary limits of harmonic functions. Bulletin de la Société Mathématique de France 85, 431–458 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  70. Doob, J.L.: Discrete potential theory and boundaries. Journal of Mathematics and Mechanics 8(3), 433–458 (1959)

    MathSciNet  MATH  Google Scholar 

  71. Doob, J.L.: Boundary properties of functions with finite Dirichlet integrals. Annales de l’Institut Fourier 12, 573–621 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  72. Doob, J.L.: Classical Potential Theory and its Probabilistic Counterpart. Springer (1984)

    Google Scholar 

  73. Dvoretsky, A., Erdős, P., Kakutani, S.: Nonincrease everywhere of the Brownian motion process. In: Fourth Berkeley Symposium on Mathematical Statistics and Probability, vol. 4, pp. 103–116. University of California (1961)

    Google Scholar 

  74. Dvoretzky, A., Erdős, P., Kakutani, S.: Double points of paths of Brownian motion in \(n\)-space. Acta Sci. Math. Szeged 12, 75–81 (1950)

    Google Scholar 

  75. Dvoretzky, A., Erdős, P., Kakutani, S.: Multiple points of paths of Brownian motion in the plane. Bulletin of the Research Council of Israel 3, 364–371 (1954)

    MathSciNet  Google Scholar 

  76. Dvoretzky, A., Erdős, P., Kakutani, S., Taylor, S.J.: Triple points of Brownian paths in 3-space. Mathematical Proceedings of the Cambridge Philosophical Society 53(4), 856–862 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  77. Dynkin, E.B.: Criteria of continuity and of absence of discontinuities of the second kind for trajectories of a Markov random process. Izv. Akad. Nauk Ser. Mat. 16(6), 563–572 (1952)

    MathSciNet  MATH  Google Scholar 

  78. Dynkin, E.B.: Foundations of the theory of Markov processes. Izd. Fiz.-Mat., Moscow (1959)

    Google Scholar 

  79. Dynkin, E.B.: Natural topology and excessive functions connected with a Markov process. Doklady Akademii Nauk 127(1), 17–19 (1959)

    MathSciNet  MATH  Google Scholar 

  80. Dynkin, E.B.: Markov Processes (in Russian). Izdat. Fiz.-Mat., Moscow (1963)

    MATH  Google Scholar 

  81. Dynkin, E.B.: Diffusion of tensors. Doklady Akademii Nauk 179(6), 1264–1267 (1968)

    MathSciNet  Google Scholar 

  82. Dynkin, E.B.: Wanderings of a Markov process. Theory of Probability & Its Applications 16(3), 401–428 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  83. Dynkin, E.B.: Regular Markov processes. Russian Mathematical Surveys 28(2), 33–64 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  84. Dynkin, E.B., Yushkevich, A.A.: Markov Processes: Theorems and Problems. Nauka, Moscow (1967)

    Google Scholar 

  85. Edgar, G.A., Sucheston, L.: Amarts: a class of asymptotic martingales. Journal of Multivariate Analysis 6, 193–221, 572–591 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  86. Edgar, G.A., Sucheston, L.: Stopping times and directed processes. Cambridge (1992)

    Google Scholar 

  87. El Karoui, N., Reinhard, H.: Compactification et balayage de processus droits. No. 21 in Astérisque. Société mathématique de France (1975)

    Google Scholar 

  88. Émery, M.: Stabilité des solutions des équations différentielles stochastiques application aux intégrales multiplicatives stochastiques. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 41(3), 241–262 (1978)

    Article  MATH  Google Scholar 

  89. Émery, M.: Stochastic calculus in manifolds. Springer (1989)

    Google Scholar 

  90. Fefferman, C.: Characterizations of bounded mean oscillation. Bulletin of the American Mathematical Society 77(4), 587–588 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  91. Fefferman, C., Stein, E.M.: \({H}^p\) spaces of several variables. Acta Mathematica 129, 137–193 (1972)

    Google Scholar 

  92. Feller, W.: An Introduction to Probability Theory and its Applications. Wiley (1950)

    Google Scholar 

  93. Feller, W.: Diffusion processes in one dimension. Transactions of the American Mathematical Society 77(1), 1–31 (1954)

    Article  MathSciNet  MATH  Google Scholar 

  94. Feller, W.: The general diffusion operator and positivity preserving semi-groups in one dimension. Annals of Mathematics 60(3), 417–436 (1954)

    Article  MathSciNet  MATH  Google Scholar 

  95. Feller, W.: An Introduction to Probability Theory and its Applications II. Wiley (1966)

    Google Scholar 

  96. Fisk, D.L.: Quasi-martingales. Transactions of the American Mathematical Society 120(3), 369–389 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  97. Fitzsimmons, P.J.: On a connection between Kuznetsov processes and quasi-processes. In: Stochastic Processes, 1987 PPS 15, pp. 123–133. Birkhäuser (1988)

    Google Scholar 

  98. Fitzsimmons, P.J., Maisonneuve, B.: Excessive measures and Markov processes with random birth and death. Probability Theory and Related Fields 72(3), 319–336 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  99. Fukushima, M.: Dirichlet Forms and Markov Processes. North Holland (1980)

    Google Scholar 

  100. Garsia, A.M.: Martingale Inequalities. Seminar Notes on Recent Progress. Benjamin, Reading, Massachusetts (1973)

    MATH  Google Scholar 

  101. Gelfand, I.M.: Generalized random processes. Dokl. Akad. Nauk 100(5), 853–856 (1955). Translated into English on pp. 529-533 of Gel’fand’s Collected Papers, Volume III, Springer 1989

    Google Scholar 

  102. Getoor, R.K.: Markov Processes, Ray Processes and Right Processes, LN 440. Springer (1975)

    Google Scholar 

  103. Getoor, R.K.: Excessive Measures. Birkhäuser (1990)

    Google Scholar 

  104. Getoor, R.K., Sharpe, M.J.: Last exit decompositions and distributions. Indiana University Mathematics Journal 23(5), 377–404 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  105. Getoor, R.K., Sharpe, M.J.: Last exit times and additive functionals. The Annals of Probability 1(4), 550–569 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  106. Getoor, R.K., Sharpe, M.J.: Balayage and multiplicative functionals. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 28(2), 139–164 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  107. Girsanov, I.V.: On transforming a certain class of stochastic processes by absolutely continuous substitution of measures. Theory of Probability & Its Applications 5(3), 285–301 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  108. Glimm, J., Jaffe, A.: Quantum Physics: A Functional Integral Point of View. Springer (1981)

    Google Scholar 

  109. Gross, L.: Logarithmic Sobolev inequalities. American Journal of Mathematics 97(4), 1061–1083 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  110. Halmos, P.R.: Measure theory. Van Nostrand (1950)

    Google Scholar 

  111. Harris, T.E.: The theory of branching processes. Springer, Berlin (1963)

    Book  MATH  Google Scholar 

  112. Herz, C.: Bounded mean oscillation and regulated martingales. Transactions of the American Mathematical Society 193, 199–215 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  113. Herz, C.: \(H_p\)-spaces of martingales, \(0 {<} p \le 1 \). Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 28(3), 189–205 (1974)

    Google Scholar 

  114. Hida, T.: Finite dimensional approximations to white noise and Brownian motion. Journal of Mathematics and Mechanics 16(8), 859–866 (1967)

    MathSciNet  MATH  Google Scholar 

  115. Hida, T.: Brownian Motion. Springer (1980)

    Google Scholar 

  116. Hille, E.: Functional Analysis and Semi-groups. American Mathematical Society (1948)

    Google Scholar 

  117. Hoffmann-Jørgensen, J.: Markov sets. Mathematica Scandinavica 24(2), 145–166 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  118. Hudson, R.L., Parthasarathy, K.R.: Quantum Itô’s formula and stochastic evolutions. Communications in mathematical physics 93(3), 301–323 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  119. Hunt, G.A.: Some theorems concerning Brownian motion. Transactions of the American Mathematical Society 81(2), 294–319 (1956)

    Article  MathSciNet  MATH  Google Scholar 

  120. Hunt, G.A.: Markoff processes and potentials I, II, III. Illinois Journal of Mathematics 1(1), 44–93 (1957). Part II 1(3):316–369; Part III 2(2):151–213

    Google Scholar 

  121. Hunt, G.A.: Markoff chains and Martin boundaries. Illinois Journal of Mathematics 4(3), 313–340 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  122. Ikeda, N., Watanabe, S.: Stochastic Differential Equations and Diffusion Processes. North-Holland (1988)

    Google Scholar 

  123. Itô, K.: Stochastic integral. Proceedings of the Imperial Academy 20(8), 519–524 (1944)

    MathSciNet  MATH  Google Scholar 

  124. Itô, K.: On a stochastic integral equation. Proceedings of the Japan Academy 22(1-4), 32–35 (1946)

    Article  MathSciNet  MATH  Google Scholar 

  125. Itô, K.: Stochastic differential equations in a differentiable manifold. Nagoya Mathematical Journal 1, 35–47 (1950)

    Article  MathSciNet  MATH  Google Scholar 

  126. Itô, K.: Multiple Wiener integral. Journal of the Mathematical Society of Japan 3(1), 157–169 (1951)

    Article  MathSciNet  MATH  Google Scholar 

  127. Itô, K.: On a formula concerning stochastic differentials. Nagoya Mathematical Journal 3, 55–65 (1951)

    Article  MathSciNet  MATH  Google Scholar 

  128. Itô, K.: On stochastic differential equations. Memoirs of the American Mathematical Society 4, 51 pp. (1951)

    Google Scholar 

  129. Itô, K.: Lectures on Stochastic Processes. Tata Institute, Bombay (1961)

    Google Scholar 

  130. Itô, K.: Poisson point processes attached to Markov processes. In: Sixth Berkeley Symposium on Mathematical Statistics and Probability, vol. 3, pp. 225–239 (1972)

    Google Scholar 

  131. Itô, K., McKean, H.P.: Diffusion processes and their sample paths. Springer (1965)

    Google Scholar 

  132. Itô, K., Watanabe, S.: Transformation of Markov processes by multiplicative functionals. Annales de l’Institut Fourier 15(1), 13–30 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  133. Jacod, J.: Calcul stochastique et problemes de martingales, LN 714. Springer (1979)

    Google Scholar 

  134. Jacod, J., Mémin, J.: Caractéristiques locales et conditions de continuité absolue pour les semi-martingales. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 35(1), 1–37 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  135. Jacod, J., Shiryaev, A.: Limit theorems for stochastic processes, vol. 288. Springer (1987)

    Google Scholar 

  136. Jacod, J., Yor, M.: Étude des solutions extrémales et représentation intégrale des solutions pour certains problèmes de martingales. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 38(2), 83–125 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  137. Jeulin, T.: Semi-martingales et grossissement d’une filtration, LN 833. Springer (1980)

    Google Scholar 

  138. John, F., Nirenberg, L.: On functions of bounded mean oscillation. Communications on Pure and Applied Mathematics 14(3), 415–426 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  139. Kac, M.: On distributions of certain Wiener functionals. Transactions of the American Mathematical Society 65(1), 1–13 (1949)

    Article  MathSciNet  MATH  Google Scholar 

  140. Kakutani, S.: On Brownian motions in \(n\)-space. Proceedings of the Imperial Academy 20(9), 648–652 (1944)

    Google Scholar 

  141. Kakutani, S.: Two-dimensional Brownian motion and harmonic functions. Proceedings of the Imperial Academy 20(10), 706–714 (1944)

    MathSciNet  MATH  Google Scholar 

  142. Kazamaki, N.: On a stochastic integral equation with respect to a weak martingale. Tôhoku Mathematical Journal 26(1), 53–63 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  143. Knight, F.: Note on regularization of Markov processes. Illinois Journal of Mathematics 9(3), 548–552 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  144. Knight, F.B.: Prediction processes and an autonomous germ-Markov property. The Annals of Probability 7(3), 385–405 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  145. Knight, F.B.: Foundations of the prediction process, vol. 1. Oxford (1992)

    Google Scholar 

  146. Kolmogorov, A.N.: Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung. Mathematische Annalen 104, 415–458 (1931)

    Article  MathSciNet  MATH  Google Scholar 

  147. Kolmogorov, A.N.: Grundbegriffe der Wahrscheinlichkeitsrechnung. Springer, Berlin (1933)

    Book  MATH  Google Scholar 

  148. Krylov, N.V.: Some estimates in the theory of stochastic integrals. Theory of Probability & Its Applications 18(1), 54–63 (1973)

    Article  MATH  Google Scholar 

  149. Krylov, N.V., Yushkevich, A.A.: Markov random sets. Tr. Mosk. Mat. Obs. 13, 114–135 (1965)

    MathSciNet  MATH  Google Scholar 

  150. Kunita, H., Watanabe, S.: On square integrable martingales. Nagoya Journal of Mathematics 30, 209–245 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  151. Kunita, H., Watanabe, T.: Markov processes and Martin boundaries Part I. Illinois Journal of Mathematics 9(3), 485–526 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  152. Kuznetsov, S.E.: Construction of Markov processes with random times of birth and death. Theory of Probability & Its Applications 18(3), 571–575 (1974)

    Article  MATH  Google Scholar 

  153. Lenglart, É., Lépingle, D., Pratelli, M.: Présentation unifiée de certaines inégalités de la théorie des martingales. Séminaire de probabilités de Strasbourg 14, 26–48 (1980)

    MATH  Google Scholar 

  154. Letta, G.: Martingales et Intégration Stochastique. Edizioni della Normale, Pisa (1984)

    MATH  Google Scholar 

  155. Lévy, P.: Théorie de l’addition des variables aléatoires. Gauthier-Villars, Paris (1937). Second edition 1954.

    Google Scholar 

  156. Lévy, P.: Processus Stochastiques et Mouvement Brownien. Gauthier-Villars, Paris (1948)

    MATH  Google Scholar 

  157. Liptser, R.S., Shiryaev, A.N.: Statistics of Random Processes. Springer (1977). Two volumes

    Google Scholar 

  158. Loève, M.: Probability Theory. Van Nostrand (1960)

    Google Scholar 

  159. Maisonneuve, B.: Systèmes régénératifs. No. 15 in Astérisque. Société mathématique de France (1974)

    Google Scholar 

  160. Malliavin, P.: Géométrie différentielle stochastique. University of Montréal (1977)

    Google Scholar 

  161. Malliavin, P.: Stochastic calculus of variation and hypoelliptic operators. In: K. Itô (ed.) Proc. Internat. Symposium on Stochastic Differential Equations, Kyoto, 1976, pp. 195–263 (1978)

    Google Scholar 

  162. Markov, A.A.: Extension of the law of large numbers to dependent quantities (in Russian). Izvestiia Fiz.-Matem. Obsch. Kazan Univ., (2nd Ser.) 15, 135–156 (1906)

    Google Scholar 

  163. Martin, R.S.: Minimal positive harmonic functions. Transactions of the American Mathematical Society 49(1), 137–172 (1941)

    Article  MathSciNet  MATH  Google Scholar 

  164. McShane, E.J.: Stochastic Calculus and Stochastic Models. Academic Press (1974)

    Google Scholar 

  165. Métivier, M., Pellaumail, J.: Mesures stochastiques à valeurs dans des espaces \({L}_0\). Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 40(2), 101–114 (1977)

    Google Scholar 

  166. Métivier, M., Pellaumail, J.: Stochastic integration. Academic Press (1980)

    Google Scholar 

  167. Meyer, P.A.: A decomposition theorem for supermartingales. Illinois Journal of Mathematics 6(2), 193–205 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  168. Meyer, P.A.: Decomposition of supermartingales: the uniqueness theorem. Illinois Journal of Mathematics 7(1), 1–17 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  169. Meyer, P.A.: Probability and Potentials. Blaisdell (1966)

    Google Scholar 

  170. Meyer, P.A.: Intégrales stochastiques I–IV. Séminaire de probabilités de Strasbourg 1, 72–162 (1967)

    MATH  Google Scholar 

  171. Meyer, P. A. Démonstration probabiliste de certaines inégalités de Littlewood-Paley, Séminaire de probabilités de Strasbourg, X, LN511. Springer (1976)

    Google Scholar 

  172. Meyer, P.A.: Les processus stochastiques de 1950 à nos jours. In: J.P. Pier (ed.) Development of Mathematics 1950–2000, pp. 813–848. Birkhäuser (2000)

    Google Scholar 

  173. Millar, P.W.: Martingale integrals. Transactions of the American Mathematical Society 133(1), 145–166 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  174. Minlos, R.A.: Generalized random processes and their extension in measure. Trudy Moskovskogo Matematicheskogo Obshchestva 8, 497–518 (1959)

    MathSciNet  Google Scholar 

  175. Mitro, J.B.: Dual Markov processes: construction of a useful auxiliary process. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 47(2), 139–156 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  176. Motoo, M.: The sweeping-out of additive functionals and processes on the boundary. Annals of the Institute of Statistical Mathematics 16(1), 317–345 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  177. Motoo, M., Watanabe, S.: On a class of additive functionals of Markov processes. Journal of Mathematics of Kyoto University 4(3), 429–469 (1965)

    MathSciNet  MATH  Google Scholar 

  178. Nagasawa, M.: Time reversions of Markov processes. Nagoya Mathematical Journal 24, 177–204 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  179. Nagasawa, M.: Segregation of a population in an environment. Journal of Mathematical Biology 9(3), 213–235 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  180. Nelson, E.: Dynamical Theories of Brownian Motion. Princeton (1967)

    Google Scholar 

  181. Nelson, E.: Construction of quantum fields from Markoff fields. Journal of Functional Analysis 12(1), 97–112 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  182. Nelson, E.: The free Markoff field. Journal of Functional Analysis 12(2), 211–227 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  183. Nelson, E.: Notes on non-commutative integration. Journal of Functional Analysis 15(2), 103–116 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  184. Nelson, E.: Quantum Fluctuations. Princeton (1985)

    Google Scholar 

  185. Neveu, J.: Entrance, exit and fictitious states for Markov chains. In: Proc. Aarhus Colloq. Combinatorial Probability, pp. 64–68 (1962)

    Google Scholar 

  186. Neveu, J.: Sur les états d’entrée et les états fictifs d’un processus de Markov. Annales de l’Institut Henri Poincaré 17(4), 323–337 (1962)

    MathSciNet  MATH  Google Scholar 

  187. Orey, S.: F-processes. In: Fifth Berkeley Symposium on Mathematical Statistics and Probability, vol. 2, Pt. 1, pp. 301–313. University of California (1967)

    Google Scholar 

  188. Parthasarathy, K.R.: An introduction to Quantum Stochastic Calculus. Birkhäuser (1992)

    Google Scholar 

  189. Prokhorov, Y.V.: Convergence of random processes and limit theorems in probability theory. Theory of Probability & Its Applications 1(2), 157–214 (1956)

    Article  MathSciNet  Google Scholar 

  190. Protter, P.: \({\cal{H}} ^p\) stability of solutions of stochastic differential equations. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 44(4), 337–352 (1978)

    Google Scholar 

  191. Protter, P.: Stochastic Integration and Differential Equations: A New Approach. Springer (1990)

    Google Scholar 

  192. Protter, P.E.: On the existence, uniqueness, convergence and explosions of solutions of systems of stochastic integral equations. The Annals of Probability 5(2), 243–261 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  193. Ray, D.: Resolvents, transition functions, and strongly markovian processes. Annals of Mathematics pp. 43–72 (1959)

    Google Scholar 

  194. Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion. Springer, Berlin (1991)

    Book  MATH  Google Scholar 

  195. Schilder, M.: Some asymptotic formulas for Wiener integrals. Transactions of the American Mathematical Society 125(1), 63–85 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  196. Schrödinger, E.: Über die Umkehrung der Naturgesetze. de Gruyter, Berlin (1931)

    MATH  Google Scholar 

  197. Schwartz, L.: Semi-martingales sur des variétés, et martingales conformes sur des variétés analytiques complexes, LN 780. Springer (1980)

    Google Scholar 

  198. Segal, I.E.: A non-commutative extension of abstract integration. Annals of Mathematics 57(3), 401–457 (1953)

    Article  MathSciNet  MATH  Google Scholar 

  199. Segal, I.E.: Tensor algebras over Hilbert spaces. I. Transactions of the American Mathematical Society 81(1), 106–134 (1956)

    Google Scholar 

  200. Sharpe, M.: General Theory of Markov Processes. Academic Press (1988)

    Google Scholar 

  201. Shih, C.T.: On extending potential theory to all strong Markov processes. Annales de l’Institut Fourier 20(1), 303–315 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  202. Shur, M.G.: Continuous additive functionals of Markov processes and excessive functions. Doklady Akademii Nauk 137(4), 800–803 (1961)

    MathSciNet  MATH  Google Scholar 

  203. Silverstein, M.L.: Symmetric Markov Processes. LN 426. Springer (1974)

    Google Scholar 

  204. Simon, B.: The \(\text{P}(\Phi )_{2}\) Euclidean (Quantum) Field Theory. Princeton (1974)

    Google Scholar 

  205. Simon, B.: Functional Integration and Quantum Physics. Academic Press (1979)

    Google Scholar 

  206. Skorokhod, A.V.: Limit theorems for stochastic processes. Theory of Probability & Its Applications 1(3), 261–290 (1956)

    Article  MathSciNet  MATH  Google Scholar 

  207. Skorokhod, A.V.: Investigations in the theory of random processes. University Press Kiev (1961)

    Google Scholar 

  208. Snell, J.L.: Applications of martingale system theorems. Transactions of the American Mathematical Society 73(2), 293–312 (1952)

    Article  MathSciNet  MATH  Google Scholar 

  209. Spitzer, F.: Recurrent random walk and logarithmic potential. In: Fourth Berkeley Symposium on Mathematical Statistics and Probability, vol. 2, pp. 515–534. University of California (1961)

    Google Scholar 

  210. Spitzer, F.: Principles of random walk. Van Nostrand (1964)

    Google Scholar 

  211. Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton (1970)

    Google Scholar 

  212. Stein, E.M.: Topics in Harmonic Analysis Related to the Littlewod-Paley Theory. Princeton (1970)

    Google Scholar 

  213. Stratonovich, R.L.: A new representation for stochastic integrals and equations. SIAM Journal on Control 4(2), 362–371 (1966). Translated from Vestnik Moscow Univ. Ser. I Mat. Mech. 1,3–12 (1964).

    Google Scholar 

  214. Stroock, D.W.: The Malliavin calculus and its application to second order parabolic differential equations: Part I. Mathematical Systems Theory 14(1), 25–65 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  215. Stroock, D.W., Varadhan, S.R.S.: Diffusion processes with continuous coefficients. Communications on Pure and Applied Mathematics 22(3,4), 3:345–400, 4:479–530 (1969)

    Google Scholar 

  216. Stroock, D.W., Varadhan, S.R.S.: Diffusion processes with continuous coefficients. Communications on Pure and Applied Mathematics 24(2), 147–225 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  217. Stroock, D.W., Varadhan, S.R.S.: On the support of diffusion processes with applications to the strong maximum principle. In: Sixth Berkeley Symposium on Mathematical Statistics and Probability, vol. 3, pp. 333–359. University of California (1972)

    Google Scholar 

  218. Stroock, D.W., Varadhan, S.R.S.: Multidimensional Diffusion Processes. Springer (1979)

    Google Scholar 

  219. Tanaka, H.: Note on continuous additive functionals of the 1-dimensional Brownian path. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 1, 251–257 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  220. Taylor, S.J.: The \(\alpha \)-dimensional measure of the graph and set of zeros of a Brownian path. Mathematical Proceedings of the Cambridge Philosophical Society 51(2), 265–274 (1955)

    Google Scholar 

  221. Trotter, H.F.: A property of Brownian motion paths. Illinois Journal of Mathematics 2(3), 425–433 (1958)

    Article  MathSciNet  MATH  Google Scholar 

  222. Varopoulos, N.T.: Aspects of probabilistic Littlewood-Paley theory. Journal of Functional Analysis 38(1), 25–60 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  223. Ville, J.: Étude critique de la notion de collectif. Gauthier-Villars, Paris (1939)

    MATH  Google Scholar 

  224. Volkonskii, V.: Additive functionals of Markov processes. Trudy Moskovskogo Matematicheskogo Obshchestva 9, 143–189 (1960)

    MathSciNet  MATH  Google Scholar 

  225. Watanabe, S.: On discontinuous additive functionals and Lévy measures of a Markov process. Japanese Journal of Mathematics 34, 53–70 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  226. Wiener, N.: Differential space. Journal of Mathematics and Physics 2, 131 (1923)

    Article  MathSciNet  MATH  Google Scholar 

  227. Wiener, N.: The homogeneous chaos. American Journal of Mathematics 60(4), 897–936 (1938)

    Article  MathSciNet  MATH  Google Scholar 

  228. Williams, D.: Diffusions, Markov Processes, and Martingales. I: Foundations. Wiley, Chichester (1979)

    Google Scholar 

  229. Williams, D. (ed.): Stochastic Integrals, Durham, UK, LN 851. Springer (1981)

    Google Scholar 

  230. Yamada, T., Watanabe, S.: On the uniqueness of solutions of stochastic differential equations. Journal of Mathematics of Kyoto University 11(1,3), 1:155–167, 3:553–563 (1971)

    Google Scholar 

  231. Yor, M., de Sam Lazaro, J.: Sous-espaces denses dans \({L}^1\) ou \({H}^1\) et représentation des martingales. Séminaire de probabilités de Strasbourg 12, 265–309 (1978)

    Google Scholar 

  232. Yosida, K.: On the differentiability and the representation of one-parameter semi-group of linear operators. Journal of the Mathematical Society of Japan 1(1), 15–21 (1948)

    Article  MathSciNet  MATH  Google Scholar 

  233. Yosida, K.: Functional Analysis. Iwanami (1951)

    Google Scholar 

  234. Zheng, W.A.: Tightness results for laws of diffusion processes application to stochastic mechanics. Annales de l’Institut Henri Poincaré Probabilités et statistiques 21(2), 103–124 (1985)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Rutgers University, 377 South Harrison Street, Apt. 20B, East Orange, New Jersey, 07018, USA

    Paul-André Meyer & Glenn Shafer

Authors
  1. Paul-André Meyer
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Glenn Shafer
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Glenn Shafer .

Editor information

Editors and Affiliations

  1. LPSM, Sorbonne Université, Paris, France

    Laurent Mazliak

  2. Rutgers Business School, Newark, NJ, USA

    Glenn Shafer

Rights and permissions

Reprints and permissions

Copyright information

© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Meyer, PA., Shafer, G. (2022). Stochastic Processes in the Decades after 1950. In: Mazliak, L., Shafer, G. (eds) The Splendors and Miseries of Martingales. Trends in the History of Science. Birkhäuser, Cham. https://doi.org/10.1007/978-3-031-05988-9_8

Download citation

Publish with us

Policies and ethics

Search

Navigation