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# Brownian motion

Applet adapted from the work of Michael Fowler (1998), Beams Professor at
Department of Physics ,University of Virginia.

It all started in the summer of 1827 when the biologist Robert Brown discovered that small particles from inside pollen grains, when suspended in water and observed in a microscope, moved in a highly irregular manner (see applet above for illustration). This phenomenon which we now call ``Brownian motion'' was described in Brown (1828,1829). An explanation of ``Brownian motion'' was not given until the end of the century; the highly irregular movement was caused by water molecules repeatedly hitting the small particles. The first more mathematical treatment of Brownian motion was given by Bachelier(1900), who was interested in modelling stock prices. A quantitative explanation of the phenomenon was given by Einstein (1905). Einstein also gave the density for the movement along the x-axis during a time-interval t, as well as a physical explanation of the scale parameter in the density function. In the following decades Brownian motion was put in a more strict mathematical framework through the works of Wiener (1923,1924) and Lévy (1939,1948). Stochastic integrals with respect to Brownian motion were first constructed by Ito (1944) although some work were done earlier and independently by Doeblin in 1940 (see Bru and Yor (2002)).

### Mathematical Brownian motion

Mathematically Brownian motion is defined as a process which has independent
and stationary Gaussian increments. So if we let $X$ be a Brownian motion
we have that X(t) has a Gaussian distribution with mean $\mu t$ and variance $\sigma t$.
The parameter $\mu$ is called drift and the parameter $\sigma$ is called diffusion constant or in mathematical finance context volatility. A standard Brownian motion has $\mu=0$ and $\sigma=1$.
Below you can see a Brownian motion in action.
It is possible to change the parameters μ and σ to see the impact on the trajectory.

Copyright (C) Magnus Wiktorsson 2019

Brownian motion is used as building block in models for a number different applications e.g. financial markets, turbulence, seismology, fatigue, neuronal activity and hydrology. Usually these models are formulated as stochastic differential equations.## References

It all started in the summer of 1827 when the biologist Robert Brown discovered that small particles from inside pollen grains, when suspended in water and observed in a microscope, moved in a highly irregular manner (see applet above for illustration). This phenomenon which we now call ``Brownian motion'' was described in Brown (1828,1829). An explanation of ``Brownian motion'' was not given until the end of the century; the highly irregular movement was caused by water molecules repeatedly hitting the small particles. The first more mathematical treatment of Brownian motion was given by Bachelier(1900), who was interested in modelling stock prices. A quantitative explanation of the phenomenon was given by Einstein (1905). Einstein also gave the density for the movement along the x-axis during a time-interval t, as well as a physical explanation of the scale parameter in the density function. In the following decades Brownian motion was put in a more strict mathematical framework through the works of Wiener (1923,1924) and Lévy (1939,1948). Stochastic integrals with respect to Brownian motion were first constructed by Ito (1944) although some work were done earlier and independently by Doeblin in 1940 (see Bru and Yor (2002)).

Copyright (C) Magnus Wiktorsson 2019

Brownian motion is used as building block in models for a number different applications e.g. financial markets, turbulence, seismology, fatigue, neuronal activity and hydrology. Usually these models are formulated as stochastic differential equations.

- Bachelier, L. (1900),
*Théorie de spéculation*, Annales Scientifiques de l'École Normale Supérieure**3**(17): 21-86. Article - Brown, R. (1828). A brief account of microscopical observations made in the months of June, July and Augusti 1827, on the particles contained in the pollen of plants; and on existence of active molecules in organic and inorganic bodies,
*Phil. Mag.***4**, 161-173. Article - Brown, R. (1829). Additional remarks on active molecules.
*Phil. Mag.***6**, 161-166. Article - Bernard Bru, Marc Yor (2002).Comments on the life and mathematical legacy of Wolfgang Doeblin.
*Finance and Stochastics***6**. 3-47. Article - Einstein, A. (1905). Über die von der molekularkinetischen Theorie der Wärme
geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen.
*Ann. Physik*,**17**, 549-560. Article - Itô, K. (1944). Stochastic integral.
*Proc. Imperial Acad. Tokyo*,**20**, 519-524. Article - Lévy, P. (1939). Sur certain processus stocastiques homogénes.
*Composito Math.***7**, 283-339.. Article - Lévy, P. (1948).
*Processus Stochastiques et Mouvement Brownien.*Gauthier-Villars, Paris. - Wiener, N. (1923). Differential space.
*J. Math. Phys.***2**, 131-174. - Wiener, N. (1924). Un problème de probabilités dénombrables.
*Bull. Soc. Math. France*,**52**, 569-578.

Questions: Magnus Wiktorsson

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