A random walk as the name implies have a tendency to

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A random walk, as the name implies, have a tendency to wander haphaz- ardly. However, if the variance of the white-noise process that is driving the random walk is small, then the values of the stochastic increments will also be small and the random walk will wander slowly. A first-order random walk over a surface is know as Brownian motion. For a physical example of Brownian motion, one can imagine small particles, such a pollen grains, floating on the surface of a viscous liquid. The viscosity might be expected to bring the particles to a halt quickly if they are in motion. However, if the particles are very light, then they will dart hither and thither on the surface of the liquid under the impact of its molecules, which are themselves in constant motion. The term Brownian motion has been adopted to describe a univariate processes. Wiener Processes A Wiener process is the consequence of allowing the in- tervals of a discrete-time random walk to tend to zero. The dates at which the process is defined become a continuum. The result is a process that is continuous almost everywhere but nowhere differentiable. The Wiener process has all of the characteristics of the random walk pro- cess that has been described above. When sampled at regular intervals, it has the same mathematical description as a discrete-time process. However, whereas the random walk is defined only on the set of integers, the Wiener process is defined for all points on a real line that represents continuous time. The generalisation can be achieved by replacing the integer h of equation (5) by an increment dt that can take infinitesimally small values. The equation can be rewritten accordingly as dw ( t ) = w ( t + dt ) w ( t ) = ζ ( t + dt ) dt (6) This equation describes a standard Wiener process. The process fulfils the following conditions: (a) w (0) = 0 , (b) E { w ( t ) } = 0 , for all t , (c) w ( t ) is normally distributed, (d) dw ( s ) , dw ( t ) for all t = s are independent stationary increments, (e) V { w ( t + h ) w ( t ) } = h for h > 0.
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