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Brownian Motion (also known as Standard Wiener Process) 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 visualized below.

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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