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## A Short History of Stochastic Integration and Mathematical Finance The early years, 1880 – 1970 (2003)

Citations: | 11 - 1 self |

### Citations

749 | F.,1998, “Market efficiency, long term returns, and behavioral finance
- Fama
- 2000
(Show Context)
Citation Context ...s “the efficient market hypothesis.” The efficient market hypothesis caused a revolution in empirical finance; the debate and empirical investigation of this hypothesis is still continuing today (see =-=[23]-=-). Two other profound insights can be found in this early paper that subsequently, but only in a modified form, became the mainstay of option pricing theory. The first idea is the belief (postulate) t... |

659 | The valuation of options for alternative stochastic processes
- Cox, Ross
- 1976
(Show Context)
Citation Context ...uivalent martingale” probabilities. It is interesting to note that, contrary to common belief, this use of “equivalent martingale probabilities” under another guise predated the paper by Cox and Ross =-=[12]-=- by nearly 10 years. In fact, Merton (footnote 5 page 218, [48]) points out that Samuelson knew this fact as early as 1953! Again, by not invoking the no arbitrage principle, this paper just missed ob... |

147 |
Théorie de la spéculation, Annales scientifiques de l’École normale supérieure 17
- Bachelier
- 1900
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Citation Context ...tudying time series in 1880 [77]. 2 ; the second was that of L. Bachelier of Paris, who created a model of Brownian motion while deriving the dynamic behavior of the Paris stock market, in 1900 (see, =-=[1]-=-, [2], [11]); and the third was that of A. Einstein, who proposed a model of the motion of small particles suspended in a liquid, in an attempt to convince other physicists of the molecular nature of ... |

140 |
Stochastic integrals
- McKean
- 1969
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Citation Context ...pating processes. 11 9Here Itô cites the work of S. Bernstein [5] as well as that of Kolmogorov [39] and W. Feller [24] as antecedents for his work. 10The book by H. P. McKean, Jr., published in 1969 =-=[45]-=-, had a great influence in popularizing the Itô integral, as it was the first explanation of Itô’s and others’ related work in book form. But McKean referred to Itô’s formula as Itô’s lemma, a nomencl... |

135 |
Continuous time finance
- Merton
- 1990
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Citation Context ...hat, contrary to common belief, this use of “equivalent martingale probabilities” under another guise predated the paper by Cox and Ross [12] by nearly 10 years. In fact, Merton (footnote 5 page 218, =-=[48]-=-) points out that Samuelson knew this fact as early as 1953! Again, by not invoking the no arbitrage principle, this paper just missed obtaining the famous Black Scholes formula. The first use of the ... |

129 |
Multiple Wiener integral
- Itô
- 1951
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Citation Context ...ot what is known today as the “Multiple Wiener Integral”: indeed, it was K. Itô, in 1951, when trying to understand Wiener’s papers (not an easy task), who refined and greatly improved Wiener’s ideas =-=[34]-=-. The next step in the groundwork for stochastic integration lay with A. N. Kolmogorov. The beginnings of the theory of stochastic integration, from the non-finance perspective, were motivated and int... |

96 |
Introduction to Stochastic Integration
- Williams
- 1990
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Citation Context ...gebra, and the predictably measurable processes are a strict subset of jointly measurable, non-anticipating processes. This point is clarified in (for example) the book of K. L. Chung and R. Williams =-=[9]-=-, p. 63. 6sof Doob. Ornstein had shown that there were submartingales not satisfying the Class (D) property 12 , and G. Johnson and L. L. Helms [38] quickly provided an example in print in 1963, using... |

94 |
Capital Ideas: The Improbable Origins of Modern Wall
- Bernstein
- 1993
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Citation Context ...ontinue our discussion of the evolution of the theory of stochastic integration, however, let us digress to discuss the developments in economics. It is curious that Peter Bernstein, in his 1992 book =-=[4]-=-, states “Despite its importance, Bachelier’s thesis was lost until it was rediscovered quite by accident in the 1950’s by Jimmie Savage, a mathematical statistician at Chicago.” He goes on a little l... |

93 |
On the movement of small particles suspended in stationary liquids required by the molecular-kinetic theory of heat. Annalen der Physik
- Einstein
- 1905
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Citation Context ...d the third was that of A. Einstein, who proposed a model of the motion of small particles suspended in a liquid, in an attempt to convince other physicists of the molecular nature of matter, in 1905 =-=[19]-=-(See [61] for a discussion of Einstein’s model and his motivations.) Of these three models, those of Thiele and Bachelier had little impact for a long time, while that of Einstein was immediately infl... |

73 |
On transforming a certain class of stochastic processes by absolutely continuous substitution of measures
- Girsanov
- 1960
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Citation Context ...once on October 18 and 19, 1996 in East Lansing, Michigan, with Dynkin, Skorohod, Wentzell, Freidlin, Krylov, etc.), and Girsanov’s work on transformations of Brownian motion date to 1960 and earlier =-=[27]-=-. 21 Stratonovich developed a version of the Itô integral which obeys the usual Riemann-Steiltjes change of variables formula, but sacrifices the martingale property as well as much of the generality ... |

47 |
On continuous martingales
- Dubins, Schwarz
- 1965
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Citation Context ...erizing Brownian motion among continuous martingales via its quadratic variation process, as well as an extension from one to N dimensions of the spectacular 1965 theorem of L. Dubins and G. Schwartz =-=[16]-=- that a large class of continuous martingales can be represented as time changes of Brownian motion. This remarkable paper of Kunita and Watanabe was quickly appreciated by P.A. Meyer, now in Strasbou... |

27 |
Theorie de la Speculation, Gauthier-Villars
- Bachelier
- 1900
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Citation Context ...ng time series in 1880 [77]. 2 ; the second was that of L. Bachelier of Paris, who created a model of Brownian motion while deriving the dynamic behavior of the Paris stock market, in 1900 (see, [1], =-=[2]-=-, [11]); and the third was that of A. Einstein, who proposed a model of the motion of small particles suspended in a liquid, in an attempt to convince other physicists of the molecular nature of matte... |

27 |
Transformation of Markov processes by multiplicative functionals
- Itô, Watanabe
- 1965
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Citation Context ...to the full development of the theory of stochastic integration. Two years later, in 1965, Itô and S. Watanabe, while studying multiplicative functionals of Markov processes, define local martingales =-=[37]-=-. This turns out to be the key object needed for Doob’s original conjecture to hold. That is, any submartingale X, whether it is of Class (D) or not, has a unique decomposition Xt = Mt + At, where M i... |

19 |
A decomposition theorem for supermartingales
- Meyer
- 1962
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Citation Context ...mportance of probabilistic potential theory in the development of the stochastic integral, Meyer’s first paper, establishing the existence of the Doob decomposition for continuous time submartingales =-=[49]-=-, is written in the language of potential theory. Meyer showed that the theorem is false in general, but true if and only if one assumes that the submartingale has a uniform integrability property whe... |

17 |
Markoff processes and potentials
- Hunt
- 1958
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Citation Context ...e development of probabilistic potential theory, which began to parallel the development of axiomatic potential theory, especially with the publication of G. A. Hunt’s seminal papers in 1957 and 1958 =-=[29]-=-, [30], [31]. It took perhaps a decade for these papers to be fully appreciated, but in the late 1960’s and early 1970’s they led to even greater interest in Itô’s treatment of Markov processes as sol... |

17 |
Decompositions of supermartingales. The uniqueness theorem
- Meyer
- 1963
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Citation Context ...d G. Johnson and L. L. Helms [38] quickly provided an example in print in 1963, using three dimensional Brownian motion. Also in 1963, P. A. Meyer established the uniqueness of the Doob decomposition =-=[50]-=-, which today is known as the Doob-Meyer decomposition theorem. In addition, in this second paper Meyer provides an analysis of the structure of L 2 martingales, which later will prove essential to th... |

12 |
T.: The transformation f Wiener integrals by nonlinear transfor- mations
- Cameron, Martin
- 1949
(Show Context)
Citation Context ...ial geometry (see, e.g., [20]). The primary works of interest in the Soviet Union were the series of articles of Skorohod. 21 Girsanov’s work extends the much earlier work first of Cameron and Martin =-=[8]-=-, who in 1949 transformed Brownian paths for both deterministic translations and also some random translations, keeping the old and new distributions of the processes equivalent (in the sense of havin... |

12 |
Louis Bachelier on the centenary of théorie de la spéculation
- Courtault, Kabanov, et al.
- 2000
(Show Context)
Citation Context ...me series in 1880 [77]. 2 ; the second was that of L. Bachelier of Paris, who created a model of Brownian motion while deriving the dynamic behavior of the Paris stock market, in 1900 (see, [1], [2], =-=[11]-=-); and the third was that of A. Einstein, who proposed a model of the motion of small particles suspended in a liquid, in an attempt to convince other physicists of the molecular nature of matter, in ... |

11 |
Transformation de martingales locales par changement absolue continu de probabilits. Zeitschrift fr Wahrscheinlichkeit 39
- Lenglart
- 1977
(Show Context)
Citation Context ...Maruyama [46] in 1954, and then by Girsanov in 1960. It was not until 1974 that Van Schupen and Wong [78] extended these ideas to martingales, followed in 1976 by P.A. Meyer [55] and in 1977 Lenglart =-=[41]-=- for the current modern versions. See also (for example) pages 132–136 of [64] for an exposition of the modern results. 22 Indeed, the Stratonovich integral was not met with much excitement. In a book... |

10 |
Stochastic calculus in manifolds. With an appendix by
- Émery
- 1989
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Citation Context ... of Brownian motion with differentiable curves, the resultant integrals converge to the Stratonovich integral; this led to it being an intrinsic object in stochastic differential geometry (see, e.g., =-=[20]-=-). The primary works of interest in the Soviet Union were the series of articles of Skorohod. 21 Girsanov’s work extends the much earlier work first of Cameron and Martin [8], who in 1949 transformed ... |

8 |
Integrales stochastiques par rapport aux martingales locales
- Doleans-Dade, Meyer
- 1970
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Citation Context ...nce. After the paper of Kunita and Watanabe, and after P.A. Meyer’s four papers extending their results, there was a hiatus of three years before the paper of C. Doléans-Dade and P. A. Meyer appeared =-=[13]-=-. Prior to this paper the development of stochastic integration had been tied rather intimately to Markov processes, and was perhaps seen as a tool with which one could more effectively address certai... |

8 |
Thieles’s contributions to statistics
- Hald
- 1981
(Show Context)
Citation Context ...d in part by NSF grant DMS-0202958 and NSA grant MDA-904-03-1-0092 2 This was called to our attention by Ragnar Norberg, whom we thank, and the contributions of Thiele are detailed in a paper of Hald =-=[28]-=- 1sLet us now turn to Einstein’s model. In modern terms, Einstein assumed that Brownian motion was a stochastic process with continuous paths, independent increments, and stationary Gaussian increment... |

6 |
Quasi-martingales and stochastic integrals
- Fisk
(Show Context)
Citation Context ...date for the class of processes, but I did not see a clear proof. I gave the problem to Fisk to work on for a Ph.D. thesis, and he did come up with what was needed.”[66] Indeed, in D.L. Fisk’s thesis =-=[25]-=-, written under Rubin when he was at Michigan State University, Fisk developed what is now known as the Stratonovich integral, and he also coined the phrase and developed the modern theory of quasimar... |

6 |
On the ananitical methods in the probability theory
- Kolmogorov
(Show Context)
Citation Context ...s before his famous book establishing a rigorous mathematical basis for Probability Theory using measure theory, Kolmogorov refers to and briefly explains Bachelier’s construction of Brownian motion (=-=[39]-=-, pages 64, 102–103). It is this paper too in which he develops a large part of his theory of Markov processes. Most significantly, in this paper Kolmogorov showed that continuous Markov processes (di... |

5 |
Comments on the life and mathematical legacy of Wolfgang Doeblin. Finance and Stochastics
- BRU, YOR
- 2002
(Show Context)
Citation Context ...e second author is grateful to Marc Yor for having sent to him his beautiful article, written together with Bernard Bru [6]. This article, together with the companion (and much more detailed) article =-=[7]-=-, are the sources for this discussion of Doeblin. In addition, the story of Doeblin has recently been turned into a book in biographical form [62]. 3sreproduce the beautiful summary of his work and co... |

5 |
Quasi-martingales
- Fisk
- 1965
(Show Context)
Citation Context ...tion, again according to the recollections of Herman Rubin [66]. So Fisk dropped that part of the thesis and did not pursue it, publishing instead only the part on quasimartingales, which appeared as =-=[26]-=-. Returning now to the historical development of stochastic integration, we mention that P. A. Meyer’s development of the stochastic integral in [49] is skeletal at best, and a more systematic develop... |

5 |
On the transition probability functions of Markov processes
- MARUYAMA
- 1954
(Show Context)
Citation Context ...slations, keeping the old and new distributions of the processes equivalent (in the sense of having the same sets of probability zero); these ideas were extended to Markov processes first by Maruyama =-=[46]-=- in 1954, and then by Girsanov in 1960. It was not until 1974 that Van Schupen and Wong [78] extended these ideas to martingales, followed in 1976 by P.A. Meyer [55] and in 1977 Lenglart [41] for the ... |

4 |
Intégrales stochastiques et martingales de carré intégrable, Séminaire Brélot-Choquet-Dény (Théorie du Potentiel), 7e année
- Courrège
- 1963
(Show Context)
Citation Context ...astic integration, we mention that P. A. Meyer’s development of the stochastic integral in [49] is skeletal at best, and a more systematic development is next put forward by Philippe Courrège in 1963 =-=[10]-=-. The motivation clearly arises from potential theory, and the paper of Courrège is published not in a journal, but in the (at the time) widely circulated Séminaire Brélot-ChoquetDény (Théorie du Pote... |

4 |
Markov Processes (in two volumes
- Dynkin
- 1965
(Show Context)
Citation Context ...nd France. But important parallel developments were occurring in the Soviet Union. The books of Dynkin on Markov processes appeared early, in 1960 [17] and in English as Springer Verlag books in 1965 =-=[18]-=-. The famed Moscow seminar (reconstituted at least once on October 18 and 19, 1996 in East Lansing, Michigan, with Dynkin, Skorohod, Wentzell, Freidlin, Krylov, etc.), and Girsanov’s work on transform... |

3 |
Equations différentielles stochastiques
- Bernstein
(Show Context)
Citation Context ... 0 2 s ds). Once the isometry is established for continuous non-anticipating processes H, it then extends to jointly measurable non-anticipating processes. 11 9Here Itô cites the work of S. Bernstein =-=[5]-=- as well as that of Kolmogorov [39] and W. Feller [24] as antecedents for his work. 10The book by H. P. McKean, Jr., published in 1969 [45], had a great influence in popularizing the Itô integral, as ... |

3 |
The development of rigor
- DOOB
- 1996
(Show Context)
Citation Context ...olmogorov did not have the Itô integral available, and thus he relied on an analysis of the semigroup and its infinitesimal generator, and the resulting partial differential equations. 3 3 J. L. Doob =-=[15]-=- has complained that the PDE methods of Kolmogorov and Feller used to study 2sAfter Kolmogorov we turn to the fascinating and tragic story of Vincent Doeblin (born Wolfgang Döblin) the son of the auth... |

3 |
Ito Collected Papers
- ITO
- 1987
(Show Context)
Citation Context ...tant, and arrived at the notion of a stochastic differential equation governing the paths of a Markov process that could be formulated in terms of the differentials of a single differential process.” =-=[35]-=-. 8 Itô’s first paper on stochastic integration was published in 1944 ([32]), the same year that Kakutani published two brief notes connecting Brownian motion and harmonic functions. Meanwhile through... |

3 |
A random walk down wall street (7th edition
- Malkiel
- 2000
(Show Context)
Citation Context ...t discounted futures prices follow a martingale 15 . From this postulate, Samuelson proved that changes in futures prices were uncorrelated across time, a generalization of the random walk model (see =-=[44]-=-). The second insight is that this proposition can be extended to arbitrary functions of the spot price, and although he did not state it explicitly herein, this forebodes an immediate application to ... |

2 |
Systèmes de Lévy des processus de
- Benveniste, Jacod
- 1973
(Show Context)
Citation Context ...cesses, a structure which describes the jump behavior of a Hunt process, had only been developed a few years earlier in 1964 by S. Watanabe [81], and extended much later by A. Benveniste and J. Jacod =-=[3]-=-). This “retreat” must have seemed natural at the time, since stochastic integrals were, as noted previously, seen as intimately intertwined with Markov processes. And also, as an application of their... |

2 |
La vie de W. Doeblin et le Pli cacheté 11 668, La Lettre de L’Académie des Sciences
- Bru, Yor
- 2001
(Show Context)
Citation Context ..., have died so young and left behind a body of work so important.’ See [42] 5 The second author is grateful to Marc Yor for having sent to him his beautiful article, written together with Bernard Bru =-=[6]-=-. This article, together with the companion (and much more detailed) article [7], are the sources for this discussion of Doeblin. In addition, the story of Doeblin has recently been turned into a book... |

2 |
Zur Theorie der Stochastichen Prozesse (existenz-und Eindeutigkeitssatze
- Feller
- 1936
(Show Context)
Citation Context ...tructed a stochastic differential equation of the form: dXt = σ(Xt)dWt + µ(Xt)dt, 6 The interested reader can also consult [63]. 7 Here Itô is referring to the papers of Kolmogorov [39] and of Feller =-=[24]-=-. 8 Note that while Itô never mentions the work of Bachelier in his foreword, citing instead Kolmogorov, Lévy, and Doob as his main influences, it is reasonable to think he was aware of the work of Ba... |

2 |
W. Döblin (V. Doeblin) (1915-1940) Revue d’Histoire des Sciences et de leurs Applications 8
- Lévy
- 1955
(Show Context)
Citation Context ...to give an idea of Doeblin’s stature, that one can count on the fingers of one hand the mathematicians who, since Abel and Galois, have died so young and left behind a body of work so important.’ See =-=[42]-=- 5 The second author is grateful to Marc Yor for having sent to him his beautiful article, written together with Bernard Bru [6]. This article, together with the companion (and much more detailed) art... |

2 | Continuous time processes and stochastic equations - Maruyama - 1955 |

1 |
editors; 2002 Séminaire de Probabilités 1967–1980: A
- Emery, Yor
(Show Context)
Citation Context ...tochastic integration were published in the Séminaire de Probabilités series, and these papers have been recently reprinted in a new volume of the Séminaire, with a small amount of commentary as well =-=[21]-=-. 12sWe should further note at this point that Meyer was able to discard the Markov process framework used by Kunita and Watanabe in the first two of the four papers, and he established the general ch... |

1 |
The Behavior of Stock
- Fama
- 1965
(Show Context)
Citation Context ...965, two papers of ground breaking work. In his paper [69] he gives his economics arguments that prices must fluctuate randomly, 65 years after Bachelier had assumed it! This paper, along with Fama’s =-=[22]-=- work on the same topic, form the basis of what has come to be known as “the efficient market hypothesis.” The efficient market hypothesis caused a revolution in empirical finance; the debate and empi... |

1 |
On a formula concerning sotchastic differentials
- Itô
- 1951
(Show Context)
Citation Context ...ate the paths of X to the transition function of the diffusion. This amounted to showing that the distribution of X solves Kolmogorov’s forward equation. This effort resulted in his spectacular paper =-=[33]-=- in 1951, where he stated and proved what is now known as Itô’s formula: f(Xt) = f ′ (Xt)dXt + 1 2 f ′′ (Xt)d[X, X]t. Here the function f is of course assumed to be C 2 , and we are using modern notat... |