**Louis Jean-Baptiste Alphonse Bachelier** (French: [baʃəlje]; March 11, 1870 – April 28, 1946)^{[1]} was a French mathematician at the turn of the 20th century. He is credited with being the first person to model the stochastic process now called Brownian motion, as part of his PhD thesis *The Theory of Speculation* (*Théorie de la spéculation*, published 1900).

Bachelier’s Doctoral thesis, which introduced for the first time a mathematical model of Brownian motion and its use for valuing stock options, is historically the first paper to use advanced mathematics in the study of finance. Thus, Bachelier is considered as the forefather of mathematical finance and a pioneer in the study of stochastic processes.

Louis Bachelier | |
---|---|

Louis Bachelier, aged 15 | |

Born | March 11, 1870 |

Died | April 28, 1946 (aged 76) |

Nationality | French |

Alma mater | University of Paris |

Known for | Pioneer in mathematical finance |

Scientific career | |

Fields | Mathematics |

Institutions | University of Paris Université de Franche-Comté (Besançon) Université de Dijon University of Rennes |

Doctoral advisor | Henri Poincaré |

Influenced | Benoit Mandelbrot, Paul Samuelson, Fischer Black, Myron Scholes |

Bachelier was born in Le Havre. His father was a wine merchant and amateur scientist, and the vice-consul of Venezuela at Le Havre. His mother was the daughter of an important banker (who was also a writer of poetry books). Both of Louis' parents died just after he completed his high school diploma ("baccalauréat" in French), forcing him to take care of his sister and three-year-old brother and to assume the family business, which effectively put his graduate studies on hold. During this time Bachelier gained a practical acquaintance with the financial markets. His studies were further delayed by military service. Bachelier arrived in Paris in 1892 to study at the Sorbonne, where his grades were less than ideal.

Defended on March 29, 1900 at the University of Paris,^{[2]} Bachelier's thesis was not well received because it attempted to apply mathematics to an unfamiliar area for mathematicians.^{[3]} However, his instructor, Henri Poincaré, is recorded as having given some positive feedback (though socially insufficient for finding an immediate teaching position in France at that time). For example, Poincaré called his approach to deriving Gauss' law of errors

very original, and all the more interesting in that Fourier's reasoning can be extended with a few changes to the theory of errors. ... It is regrettable that M. Bachelier did not develop this part of his thesis further.

The thesis received a grade of *honorable,* and was accepted for publication in the prestigious *Annales Scientifiques de l’École Normale Supérieure*. While it did not receive a mark of *très honorable*, despite its ultimate importance, the grade assigned is still interpreted as an appreciation for his contribution. Jean-Michel Courtault et al. point out in "On the Centenary of *Théorie de la spéculation*" that *honorable* was "the highest note which could be awarded for a thesis that was essentially outside mathematics and that had a number of arguments far from being rigorous."

For several years following the successful defense of his thesis, Bachelier further developed the theory of diffusion processes, and was published in prestigious journals. In 1909 he became a "free professor" at the Sorbonne. In 1914, he published a book, *Le Jeu, la Chance, et le Hasard* (Games, Chance, and Randomness), that sold over six thousand copies. With the support of the Council of the University of Paris, Bachelier was given a permanent professorship at the Sorbonne, but World War I intervened and Bachelier was drafted into the French army as a private. His army service ended on December 31, 1918.^{[4]} In 1919, he found a position as an assistant professor in Besançon, replacing a regular professor on leave.^{[4]} He married Augustine Jeanne Maillot in September 1920 but was soon widowed.^{[4]} When the professor returned in 1922, Bachelier replaced another professor at Dijon.^{[4]} He moved to Rennes in 1925, but was finally awarded a permanent professorship in 1927 at the University of Besançon, where he worked for 10 years until his retirement.^{[4]}

Besides the setback that the war had caused him, Bachelier was blackballed in 1926 when he attempted to receive a permanent position at Dijon. This was due to a "misinterpretation" of one of Bachelier's papers by Professor Paul Lévy, who—to Bachelier's understandable fury—knew nothing of Bachelier's work, nor of the candidate that Lévy recommended above him. Lévy later learned of his error, and reconciled himself with Bachelier.^{[5]}

Although Bachelier's work on random walks predated Einstein's celebrated study of Brownian motion by five years, the pioneering nature of his work was recognized only after several decades, first by Andrey Kolmogorov who pointed out his work to Paul Lévy, then by Leonard Jimmie Savage who translated Bachelier's thesis to English and brought the work of Bachelier to the attention of Paul Samuelson. Bachelier arguments used in his thesis also predate Eugene Fama's Efficient-market hypothesis, which is very closely related, as the idea of random walk is suited to predict the random future in a stock market where everyone has all the available information. His work in finance is recognized as one of the foundations for the Black–Scholes model.

- Bachelier 1900a,
*Théorie de la spéculation*

- Also published as a book, Bachelier 1900b
- Republished in a book of combined works, Bachelier 1995
- Translated into English, Cootner 1964, pp. 17–78
- Translated into English with additional commentary and background, Bachelier et al. 2006
- Translated into English, May 2011

- Bachelier 1901,
*Théorie mathématique du jeu*

- Republished in a book of combined works, Bachelier 1995

- Bachelier 1906,
*Théorie des probabilités continues* - Bachelier 1908a,
*Étude sur les probabilités des causes* - Bachelier 1908b,
*Le problème général des probabilités dans les épreuves répétées* - Bachelier 1910a,
*Les probabilités à plusieurs variables* - Bachelier 1910b,
*Mouvement d’un point ou d’un système matériel soumis à l’action de forces dépendant du hasard* - Bachelier 1912, (Book)
*Calcul des probabilités*^{[6]}

- Republished, Bachelier 1992

- Bachelier 1913a,
*Les probabilités cinématiques et dynamiques* - Bachelier 1913b,
*Les probabilités semi-uniformes* - Bachelier 1914, (Book)
*Le Jeu, la Chance et le Hasard*

- Republished, Bachelier 1993
- Translated into English, Harding 2017

- Bachelier 1915,
*La périodicité du hasard* - Bachelier 1920a,
*Sur la théorie des corrélations* - Bachelier 1920b,
*Sur les décimales du nombre* - Bachelier 1923,
*Le problème général de la statistique discontinue* - Bachelier 1925,
*Quelques curiosités paradoxales du calcul des probabilités* - Bachelier 1937, (Book)
*Les lois des grands nombres du Calcul des Probabilités*(Book) - Bachelier 1938, (Book)
*La spéculation et le Calcul des Probabilités* - Bachelier 1939, (Book)
*Les nouvelles méthodes du Calcul des Probabilités* - Bachelier 1941a,
*Probabilités des oscillations maxima*

- Erratum, Bachelier 1941b

- Black–Scholes equation
- Martingale
- Random walk
- Brownian Motion
- Louis Bachelier Prize
- Henri Poincaré
- Vinzenz Bronzin
- Jules Regnault

**^**Felix 1970, pp. 366–367**^**https://www.encyclopediaofmath.org/images/f/f1/LouisBACHELIER.pdf**^**Weatherall, James Owen (January 2, 2013).*The Physics of Wall Street: A Brief History of Predicting the Unpredictable*. Houghton Mifflin Harcourt. pp. 10–11. ISBN 978-0547317274.- ^
^{a}^{b}^{c}^{d}^{e}Jean-Michel Courtault; Yuri Kabanov; Bernard Bru; Pierre Crépel; Isabelle Lebon; Arnaud Le Marchand (2000). "Louis Bachelier on the Centenary of Théorie de la Spéculation".*Mathematical Finance*.**10**(3): 339–353. doi:10.1111/1467-9965.00098. **^**Mandelbrot, Benoit; Hudson, Richard L. (2014),*The Misbehavior of Markets: A Fractal View of Financial Turbulence*, Basic Books, pp. 48–49, ISBN 9780465004683.**^**Rietz H. L. (1914). "Review:*Calcul des Probabilités*by Louis Bachelier. Tome I" (PDF).*Bull. Amer. Math. Soc*.**20**(5): 268–273. doi:10.1090/s0002-9904-1914-02484-x.

- Philip Ball,
*Critical Mass*Random House, 2004 ISBN 0-09-945786-5, pp238–242. - Bachelier, L. (1900a), "Théorie de la spéculation" (PDF),
*Annales Scientifiques de l’École Normale Supérieure*,**3**(17), pp. 21–86

- Bachelier, L. (1900b),
*Théorie de la spéculation*, Gauthier-Villars

- Bachelier, L. (1901), "Théorie mathématique du jeu" (PDF),
*Annales Scientifiques de l’École Normale Supérieure*,**3**(18), pp. 143–210

- Bachelier, L. (1906), "Théorie des probabilités continues",
*Journal de Mathématiques Pures et Appliquées*,**6**(2), pp. 259–327

- Bachelier, L. (1908a), "Étude sur les probabilités des causes",
*Journal de Mathématiques Pures et Appliquées*,**6**(4), pp. 395–425

- Bachelier, L. (1908b), "Le problème général des probabilités dans les épreuves répétées",
*Comptes-rendus des Séances de l’Académie des Sciences*, Séance du 25 Mai 1908 (146), pp. 1085–1088

- Bachelier, L. (1910a), "Les probabilités à plusieurs variables" (PDF),
*Annales Scientifiques de l’École Normale Supérieure*,**3**(27), pp. 339–360

- Bachelier, L. (1910b), "Mouvement d'un point ou d'un système matériel soumis à l'action de forces dépendant du hasard",
*Comptes-rendus des Séances de l’Académie des Sciences*, Séance du 14 Novembre 1910, présentée par M. H. Poincaré (151), pp. 852–855

- Bachelier, L. (1912),
*Calcul des probabilités*,**1**, Gauthier-Villars

- Bachelier, L. (1913a), "Les probabilités cinématiques et dynamiques" (PDF),
*Annales Scientifiques de l’École Normale Supérieure*,**30**, pp. 77–119

- Bachelier, L. (1913b), "Les probabilités semi-uniformes",
*Comptes-rendus des Séances de l’Académie des Sciences*, Séance du 20 Janvier 1913, présentée par M. Appell (156), pp. 203–205

- Bachelier, L. (1914),
*Le Jeu, la Chance et le Hasard*, Bibliothèque de Philosophie scientifique, E. Flammarion

- Bachelier, L. (1915), "La périodicité du hasard",
*L’Enseignement Mathématique*,**17**, pp. 5–11, archived from the original on 2011-07-16

- Bachelier, L. (1920a), "Sur la théorie des corrélations",
*Bulletin de la Société Mathématique de France. Vie de la société. Comptes rendus des Séances*, Séance du 7 Juillet 1920 (48), pp. 42–44

- Bachelier, L. (1920b), "Sur les décimales du nombre ",
*Bulletin de la Société Mathématique de France. Vie de la société. Comptes rendus des Séances*, Séance du 7 Juillet 1920 (48), pp. 44–46

- Bachelier, L. (1923), "Le problème général de la statistique discontinue",
*Comptes-rendus des Séances de l’Académie des Sciences*, Séance du 11 Juin 1923, présentée par M. d’Ocagne (176), pp. 1693–1695

- Bachelier, L. (1925), "Quelques curiosités paradoxales du calcul des probabilités",
*Revue de Métaphysique et de Morale*,**32**, pp. 311–320

- Bachelier, L. (1937),
*Les lois des grands nombres du Calcul des Probabilités*, Gauthier-Villars

- Bachelier, L. (1938),
*La spéculation et le Calcul des Probabilités*, Gauthier-Villars

- Bachelier, L. (1939),
*Les nouvelles méthodes du Calcul des Probabilités*, Gauthier-Villars

- Bachelier, L. (1941a), "Probabilités des oscillations maxima",
*Comptes-rendus des Séances de l’Académie des Sciences*, Séance du 19 Mai 1941 (212), pp. 836–838

- Bachelier, L. (1941b), "Probabilités des oscillations maxima (Erratum)",
*Comptes-rendus des Séances de l’Académie des Sciences*(213), p. 220

- Bachelier, L. (1992),
*Reprint of Calcul des probabilités (1912)*,**1**, Editions Jacques Gabay, ISBN 2-87647-090-X

- Bachelier, L. (1993),
*Reprint of Le Jeu, la Chance et le Hasard (1914)*, Editions Jacques Gabay, ISBN 2-87647-147-7

- Bachelier, L. (1995),
*Combined volume prints of Théorie de la spéculation (1900b) and Théorie mathématique du jeu (1901)*, Editions Jacques Gabay, ISBN 2-87647-129-9

- Bachelier, L.; Samuelson, P. A.; Davis, M.; Etheridge, A. (2006),
*Louis Bachelier's Theory of Speculation: the Origins of Modern Finance*, Princeton NJ: Princeton University Press, ISBN 978-0-691-11752-2

- Cootner, P.H. (ed.) (1964),
*The Random Character of Stock Market Prices*, Cambridge, MA: MIT PressCS1 maint: Extra text: authors list (link)

- Bachelier, L.; May, D. (2011),
*Theory of Speculation*, Google Documents

- Courtault, J-M.; Kabanov, Y.; Bru, B.; Crépel, P.; Lebon, I.; Le Marchand, A. (2000), "On the Centenary of Théorie de la Spéculation" (PDF),
*Mathematical Finance*,**10**(3, July 2000), pp. 341–353, doi:10.1111/1467-9965.00098, archived from the original (PDF) on 2007-06-22

- Felix, L. (1970),
*Dictionary of Scientific Biography*,**1**, New York: Charles Scribner's Sons, ISBN 0-684-10114-9

- Taqqu, M.S. (2001),
*Bachlier and his Times: A Conversation with Bernard Bru*(PDF), Boston University, archived from the original (PDF) on 2007-06-27

- "Louis Bachelier, fondateur de la finance mathématique" Louis Bachelier webpage at the Université de Franche-Comté, Besançon / France. Text in French.
- Louis Bachelier at the Mathematics Genealogy Project
- O'Connor, John J.; Robertson, Edmund F., "Louis Bachelier",
*MacTutor History of Mathematics archive*, University of St Andrews. - Louis Bachelier par Laurent Carraro et Pierre Crepel
- Bachelier's theory of speculation is demonstrated by this 8 ft tall Probability Machine comparing stock market returns to the randomness of the beans dropping through the quincunx pattern on YouTube. also from Index Funds Advisors, this discussion of Bachelier's and other academic's contribution to financial science.

Amatino Manucci gave us the earliest extant accounting of double-entry bookkeeping.

Manucci kept the accounts for Giovanni Farolfi & Company, a merchant partnership based in Nîmes, France. Manucci was a partner for the Salon, France branch.The writing, entirely in Manucci’s hand, is neat, legible, and mostly well preserved. Financial records from 1299—1300 survive that he kept for the firm's branch in Salon, Provence. Although these records are incomplete, they show enough detail to be identified as double-entry bookkeeping. These details include the use of debits and credits and duality of entries. "No more is known of Amatino Manucci than this ledger that he kept." Manucci didn't invent the double entry system, that was a 100 year process (perhaps a 9,000 year process). If he didn't finish the process himself, it didn't occur long before, because it was clearly finished by the time he kept the books for his company.

Black–Scholes modelThe Black–Scholes or Black–Scholes–Merton model is a mathematical model for the dynamics of a financial market containing derivative investment instruments. From the partial differential equation in the model, known as the Black–Scholes equation, one can deduce the Black–Scholes formula, which gives a theoretical estimate of the price of European-style options and shows that the option has a unique price regardless of the risk of the security and its expected return (instead replacing the security's expected return with the risk-neutral rate). The formula led to a boom in options trading and provided mathematical legitimacy to the activities of the Chicago Board Options Exchange and other options markets around the world. It is widely used, although often with adjustments and corrections, by options market participants.Based on works previously developed by market researchers and practitioners, such as Louis Bachelier, Sheen Kassouf and Ed Thorp among others, Fischer Black and Myron Scholes demonstrated in the late 1960s that a dynamic revision of a portfolio removes the expected return of the security, thus inventing the risk neutral argument. In 1970, after they attempted to apply the formula to the markets and incurred financial losses due to lack of risk management in their trades, they decided to focus in their domain area, the academic environment. After three years of efforts, the formula named in honor of them for making it public, was finally published in 1973 in an article entitled "The Pricing of Options and Corporate Liabilities", in the Journal of Political Economy. Robert C. Merton was the first to publish a paper expanding the mathematical understanding of the options pricing model, and coined the term "Black–Scholes options pricing model". Merton and Scholes received the 1997 Nobel Memorial Prize in Economic Sciences for their work, the committee citing their discovery of the risk neutral dynamic revision as a breakthrough that separates the option from the risk of the underlying security. Although ineligible for the prize because of his death in 1995, Black was mentioned as a contributor by the Swedish Academy.The key idea behind the model is to hedge the option by buying and selling the underlying asset in just the right way and, as a consequence, to eliminate risk. This type of hedging is called "continuously revised delta hedging" and is the basis of more complicated hedging strategies such as those engaged in by investment banks and hedge funds.

The model's assumptions have been relaxed and generalized in many directions, leading to a plethora of models that are currently used in derivative pricing and risk management. It is the insights of the model, as exemplified in the Black–Scholes formula, that are frequently used by market participants, as distinguished from the actual prices. These insights include no-arbitrage bounds and risk-neutral pricing (thanks to continuous revision). Further, the Black–Scholes equation, a partial differential equation that governs the price of the option, enables pricing using numerical methods when an explicit formula is not possible.

The Black–Scholes formula has only one parameter that cannot be directly observed in the market: the average future volatility of the underlying asset, though it can be found from the price of other options. Since the option value (whether put or call) is increasing in this parameter, it can be inverted to produce a "volatility surface" that is then used to calibrate other models, e.g. for OTC derivatives.

Filip LundbergErnst Filip Oskar Lundberg (2 June 1876 – 31 December 1965) Swedish actuary, and mathematician. Lundberg is one of the founders of mathematical risk theory and worked as managing director of several insurance companies.

According to Harald Cramér, "Filip Lundberg's works on risk theory were all written at a time when no general theory of stochastic processes existed, and when collective reinsurance methods, in the present sense of the word, were entirely unknown to insurance companies. In both respects his ideas were far ahead of their time, and his works deserve to be generally recognized as pioneering works of fundamental importance."

Josef TeichmannJosef Teichmann (* 27 August 1972 in Lienz) is an Austrian mathematician and professor at ETH Zürich working on mathematical finance.

After studying mathematics at the University of Graz, he pursued his PhD at the University of Vienna. The title of his dissertation in 1999 under the supervision of Peter W. Michor was "The Theory of Infinite-Dimensional Lie Groups from the Point of View of Functional Analysis".

After working at the Vienna University of Technology, he obtained the Habilitation there in 2002. Since June 2009 he has been a Professor at the Department of Mathematics at ETH Zürich.

In 2005 he was awarded the Prize of the Austrian Mathematical Society and in 2006 the Start-Preis of the FWF.

In 2014 he was awarded the Louis Bachelier Prize by the French Academy of Sciences.

Jules RegnaultJules Augustin Frédéric Regnault (French: [ʁəɲo]; 1 February 1834, Béthencourt – 9 December 1894, Paris) was a French stock broker's assistant who first suggested a modern theory of stock price changes in Calcul des Chances et Philosophie de la Bourse (1863), using a random walk model. A key conclusion appears on Page 50: "l'écart des cours est en raison directe de la racine carrée des temps", in English: "the deviation of prices is directly proportional to the square root of time". He is also one of the first authors who tried to create a "stock exchange science" based on statistical and probabilistic analysis. His hypotheses were used by Louis Bachelier.

Leonard Jimmie SavageLeonard Jimmie Savage (born Leonard Ogashevitz; 20 November 1917 – 1 November 1971) was an American mathematician and statistician. Economist Milton Friedman said Savage was "one of the few people I have met whom I would unhesitatingly call a genius."He graduated from the University of Michigan and later worked at the Institute for Advanced Study in Princeton, New Jersey, the University of Chicago, the University of Michigan, Yale University, and the Statistical Research Group at Columbia University. Though his thesis advisor was Sumner Myers, he also credited Milton Friedman and W. Allen Wallis as statistical mentors.

His most noted work was the 1954 book The Foundations of Statistics, in which he put forward a theory of subjective and personal probability and statistics which forms one of the strands underlying Bayesian statistics and has applications to game theory.

During World War II, Savage served as chief "statistical" assistant to John von Neumann, the mathematician credited with describing the principles upon which electronic computers should be based. Later he was one the participants to the Macy conferences on cybernetics.One of Savage's indirect contributions was his discovery of the work of Louis Bachelier on stochastic models for asset prices and the mathematical theory of option pricing. Savage brought the work of Bachelier to the attention of Paul Samuelson. It was from Samuelson's subsequent writing that "random walk" (and subsequently Brownian motion) became fundamental to mathematical finance.

In 1951 he introduced the minimax regret criterion used in decision theory.

The Hewitt–Savage zero–one law is (in part) named after him, as is the Friedman–Savage utility function.

List of French scientistsThis is a list of notable French scientists.

List of Iranian mathematiciansThe following is a list of Iranian mathematicians including ethnic Iranian mathematicians.

List of Légion d'honneur recipients by name (B)The following is a list of some notable Légion d'honneur recipients by name. The Légion d'honneur is the highest order of France. A complete, chronological list of the members of the Legion of Honour nominated from the very first ceremony in 1803 to now does not exist. The number is estimated at one million including about 3,000 Grand Cross.

Louis Bachelier PrizeThe Louis Bachelier Prize is a biennial prize in applied mathematics jointly awarded by the London Mathematical Society, the Natixis Foundation for Quantitative Research and the Société de Mathématiques Appliquées et Industrielles (SMAI) in recognition for "exceptional contributions to mathematical modelling in finance, insurance, risk management and/or scientific computing applied to finance and insurance." The prize is named in honor of French mathematician Louis Bachelier, a pioneer in the field of probability and its use in financial modeling.

Mathematical financeMathematical finance, also known as quantitative finance, is a field of applied mathematics, concerned with mathematical modeling of financial markets. Generally, mathematical finance will derive and extend the mathematical or numerical models without necessarily establishing a link to financial theory, taking observed market prices as input. Mathematical consistency is required, not compatibility with economic theory. Thus, for example, while a financial economist might study the structural reasons why a company may have a certain share price, a financial mathematician may take the share price as a given, and attempt to use stochastic calculus to obtain the corresponding value of derivatives of the stock (see: Valuation of options; Financial modeling; Asset pricing). The fundamental theorem of arbitrage-free pricing is one of the key theorems in mathematical finance, while the Black–Scholes equation and formula are amongst the key results.Mathematical finance also overlaps heavily with the fields of computational finance and financial engineering. The latter focuses on applications and modeling, often by help of stochastic asset models (see: Quantitative analyst), while the former focuses, in addition to analysis, on building tools of implementation for the models. In general, there exist two separate branches of finance that require advanced quantitative techniques: derivatives pricing on the one hand, and risk- and portfolio management on the other.French mathematician Louis Bachelier is considered the author of the first scholarly work on mathematical finance, published in 1900. But mathematical finance emerged as a discipline in the 1970s, following the work of Fischer Black, Myron Scholes and Robert Merton on option pricing theory.

Today many universities offer degree and research programs in mathematical finance.

PSL Research UniversityUniversité PSL (officially Université de recherche Paris Sciences et Lettres in French and Paris Sciences & Lettres - PSL Research University or Paris Sciences & Lettres - PSL University in English) is a French decentralized collegiate university, currently organized as a ComUE (university community). PSL was formed in 2010 and is made up of 9 members. It has 10 associates and receives support from 3 national research entities. PSL is located in Paris, with its main sites in the Latin Quarter, at the Jourdan campus, at Porte Dauphine, in northern Paris, and at Carré Richelieu.

PSL awards Bachelor’s, Master’s, and PhD diplomas for its member schools & institutes. It offers an education based on research and interdisciplinary instruction, and its 20,000 students have access to a broad range of disciplines in science, engineering, humanities and social science, and the arts. Three of PSL University’s programs, from Bachelor's through PhD level, include CPES multidisciplinary undergraduate degree, ICFP-ENS, and SACRe doctoral program.

PSL has 181 laboratories and 101 ERC grants, and runs cross-cutting flagship programs such as the Scripta Interdisciplinary and Strategic Research Initiative (IRIS), the PSL Mathematics program, and the Q-Life Institut.

PSL students and researchers have access to 92 specialized and general libraries, archives, and photo libraries as well as online databases and journals. PSL has framework agreements with the University of Cambridge, UCL, EPFL, New York University, Columbia, Beijing University, Tsinghua University, and Hong Kong University of Science and Technology.

Random walk hypothesisThe random walk hypothesis is a financial theory stating that stock market prices evolve according to a random walk (so price changes are random) and thus cannot be predicted. It is consistent with the efficient-market hypothesis.

The concept can be traced to French broker Jules Regnault who published a book in 1863, and then to French mathematician Louis Bachelier whose Ph.D. dissertation titled "The Theory of Speculation" (1900) included some remarkable insights and commentary. The same ideas were later developed by MIT Sloan School of Management professor Paul Cootner in his 1964 book The Random Character of Stock Market Prices. The term was popularized by the 1973 book, A Random Walk Down Wall Street, by Burton Malkiel, a Professor of Economics at Princeton University, and was used earlier in Eugene Fama's 1965 article "Random Walks In Stock Market Prices", which was a less technical version of his Ph.D. thesis. The theory that stock prices move randomly was earlier proposed by Maurice Kendall in his 1953 paper, The Analysis of Economic Time Series, Part 1: Prices.

Revue de métaphysique et de moraleThe Revue de métaphysique et de morale is a French philosophy journal co-founded in 1893 by Léon Brunschvicg, Xavier Léon and Élie Halévy. The journal initially appeared six times a year, but since 1920 has been published quarterly. It was the leading French-language journal for philosophical debates at the 20th century, hosting articles by Victor Delbos, Bergson, etc., and still exists today.

Xavier Léon served as the first editor of the journal until his death in 1935, when he was succeeded by Dominique Parodi. On Parodi's death in 1955, the journal was headed by Jean Wahl.It published in 1906 Bertrand Russell's article on the Berry paradox, as well as articles by Louis Bachelier, the logicist Jean Nicod, the mathematician Henri Poincaré, Félix Ravaisson, Célestin Bouglé, Henri Delacroix (concerning William James), Louis Couturat, Sully Prudhomme, Henri Maldiney, Francine Bloch, Frédéric Rauh, Jean Cavaillès, Julien Benda, Georges Poyer, Maurice Merleau-Ponty, Georg Simmel, etc. More recently: Barbara Cassin, etc.

Société de Mathématiques Appliquées et IndustriellesThe Société de Mathématiques Appliquées et Industrielles (SMAI) is a French scientific society aiming at promoting applied mathematics, similarly to the Society for Industrial and Applied Mathematics (SIAM).

SMAI was founded in 1983 to contribute to the development of applied mathematics for research, commercial applications, publications, teaching, and industrial training. As of 2009, the society has nearly 1300 members, including both individuals and institutions.

SMAI is directed by an administration elected by the general assembly. Its chief activities are:

to organize conferences and workshops,

to publish the thrice-yearly bulletin Matapli, which contains overviews, book reviews, and information about theses and upcoming conferences,

to publish scholarly journals including Modélisation Mathématique et Analyse Numérique (M2AN), Contrôle Optimisation et Calcul des Variations (COCV), Probabilités et Statistiques (P&S), Recherche opérationnelle (RO), ESAIM: Proceedings and Surveys, and the cross-disciplinary journal MathematicS in Action (MathS in A.).

Stochastic calculusStochastic calculus is a branch of mathematics that operates on stochastic processes. It allows a consistent theory of integration to be defined for integrals of stochastic processes with respect to stochastic processes. It is used to model systems that behave randomly.

The best-known stochastic process to which stochastic calculus is applied is the Wiener process (named in honor of Norbert Wiener), which is used for modeling Brownian motion as described by Louis Bachelier in 1900 and by Albert Einstein in 1905 and other physical diffusion processes in space of particles subject to random forces. Since the 1970s, the Wiener process has been widely applied in financial mathematics and economics to model the evolution in time of stock prices and bond interest rates.

The main flavours of stochastic calculus are the Itô calculus and its variational relative the Malliavin calculus. For technical reasons the Itô integral is the most useful for general classes of processes, but the related Stratonovich integral is frequently useful in problem formulation (particularly in engineering disciplines). The Stratonovich integral can readily be expressed in terms of the Itô integral. The main benefit of the Stratonovich integral is that it obeys the usual chain rule and therefore does not require Itô's lemma. This enables problems to be expressed in a coordinate system invariant form, which is invaluable when developing stochastic calculus on manifolds other than Rn.

The dominated convergence theorem does not hold for the Stratonovich integral, consequently it is very difficult to prove results without re-expressing the integrals in Itô form.

Stochastic processIn probability theory and related fields, a stochastic or random process is a mathematical object usually defined as a collection of random variables. Historically, the random variables were associated with or indexed by a set of numbers, usually viewed as points in time, giving the interpretation of a stochastic process representing numerical values of some system randomly changing over time, such as the growth of a bacterial population, an electrical current fluctuating due to thermal noise, or the movement of a gas molecule. Stochastic processes are widely used as mathematical models of systems and phenomena that appear to vary in a random manner. They have applications in many disciplines including sciences such as biology, chemistry, ecology, neuroscience, and physics as well as technology and engineering fields such as image processing, signal processing, information theory, computer science, cryptography and telecommunications. Furthermore, seemingly random changes in financial markets have motivated the extensive use of stochastic processes in finance.Applications and the study of phenomena have in turn inspired the proposal of new stochastic processes. Examples of such stochastic processes include the Wiener process or Brownian motion process, used by Louis Bachelier to study price changes on the Paris Bourse, and the Poisson process, used by A. K. Erlang to study the number of phone calls occurring in a certain period of time. These two stochastic processes are considered the most important and central in the theory of stochastic processes, and were discovered repeatedly and independently, both before and after Bachelier and Erlang, in different settings and countries.The term random function is also used to refer to a stochastic or random process, because a stochastic process can also be interpreted as a random element in a function space. The terms stochastic process and random process are used interchangeably, often with no specific mathematical space for the set that indexes the random variables. But often these two terms are used when the random variables are indexed by the integers or an interval of the real line. If the random variables are indexed by the Cartesian plane or some higher-dimensional Euclidean space, then the collection of random variables is usually called a random field instead. The values of a stochastic process are not always numbers and can be vectors or other mathematical objects.Based on their mathematical properties, stochastic processes can be divided into various categories, which include random walks, martingales, Markov processes, Lévy processes, Gaussian processes, random fields, renewal processes, and branching processes. The study of stochastic processes uses mathematical knowledge and techniques from probability, calculus, linear algebra, set theory, and topology as well as branches of mathematical analysis such as real analysis, measure theory, Fourier analysis, and functional analysis. The theory of stochastic processes is considered to be an important contribution to mathematics and it continues to be an active topic of research for both theoretical reasons and applications.

University of BurgundyThe University of Burgundy (French: Université de Bourgogne, uB; also University of Dijon, Université de Dijon) is a university in Dijon, France.

The University of Burgundy is situated on a large campus called Campus Montmuzard, 15 minutes by bus from the city centre. The humanities and sciences are well represented on the main campus, along with law, medicine, and literature in separate buildings.

The IUT (Institute of technology) is also on the campus, providing specialist higher level diplomas in business, biology, communications and computer science.

The university counts 10 faculties, 4 engineering schools, 3 institutes of technology offering undergraduate courses, and 2 professional institutes providing post-graduate programmes.

With numerous student societies and good support services for international and disabled students, the campus is a welcoming place with numerous CROUS restaurants and canteens providing subsidised food and snacks.

University of Franche-ComtéThe University of Franche-Comté is a French university in the Academy of Besançon with five campuses: Besançon (Doubs), Belfort (named for Léon Delarbre), Montbéliard (Doubs), Vesoul (Haute-Saône), and Lons-le-Saunier (Jura).

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