Explain the important properties of Brownian motion

Question

Explain the important properties of Brownian motion.

Answer 1

The significant properties of BM are as given here:

• Finiteness: the scaling of the variance along with the time step is crucial to Brownian motion remaining finite.
• Continuity: the ways are continuous, there are no discontinuities. Nonetheless, the path is fractal, and not any differentiable anywhere.
• Markov: the conditional distribution of Wt specified information up till τ< t depends only upon W_τ.
• Martingale: It given information up till τ< t the conditional expectation of W_t is W_τ.
• Quadratic variation: When we divide up the time 0 to t within a partition along with n + 1 partition points t_i = i.t/n after that

• Normality: Over finite time increments as t_i-1 to t_i, W_ti -W_ti-1 that is normally distributed along with mean zero and variance t_i - t_i-1.

You'll notice this 'W' in the form dW like the stochastic increment term in stochastic differential equations. This might also appear as dB or dX, various authors using various letters, and sometimes with a time subscript. Although these are all similar thing!

It's frequently easiest just to think of dW as being a random number drawn by a normal distribution along with the properties:

E[dW] = 0 and E[dW²] = dt.