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BROWNIAN MOTION

Brownian Motion is a stochastic process W(t) which has the following four properties:

1] W(0) = 0,

2] almost surely, the trajectory of W(t) is continuous;

3] W(t) has independent increments: for any moments of time s < t < u, random variables W(t) - W(s) and W(u) - W(s) are independent;

4] W(t) has stationary increments: for any moments of time s < t and positive shift h, random variables W(s+h) - W(s) and W(t+h) - W(t) have the same distribution.

It follows from properties 3, 4 and the Central Limit Theorem that any finite-dimensional distributions of a Brownian motion are Gaussian (normal). For example, in the definition above, random variables W(t) - W(s), W(u) - W(s), W(s+h) - W(s) and W(t+h) - W(t) are jointly Gaussian.

Trajectories of Brownian motion are illustrated in this plot. Brownian motion is also known as "standard Wiener process".


BROWNIAN MOTION REFERENCES

Lawler, G. F. (1995). Introduction to Stochastic Processes. New York: Chapman and Hall/CRC.

Ross, S. M. (1995). Stochastic Processes (2nd ed). New York: Wiley.

Karlin, S., & Taylor, H. M. (1975). A First Course in Stochastic Processes (2nd ed). New York: Academic Press.

Karatzas, I., & Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus (2nd ed). New York: Springer.

Oksendal, B. K. (2002). Stochastic Differential Equations: An Introduction with Applications (5th ed). Springer-Verlag Berlin Heidelberg.

Durrett, R. (2004). Probability: Theory and Examples (3rd ed). Belmont, CA: Duxbury Press.

Protter, P. E. (2005). Stochastic Integration and Differential Equations (2nd ed). Springer-Verlag Berlin Heidelberg.

Gikhman, I. I., & Skorokhod, A. V. (2007). The Theory of Stochastic Processes III. Springer-Verlag Berlin Heidelberg.

Tsay, R. S. (2005). Analysis of Financial Time Series. Wiley-Interscience.


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