# Difference between ito process, brownian motion and random walk

Can someone explain to a non-math person (myself) what is the difference between these three? If they are so different that a comparison does not even make sense, please point it out.

1.Ito process 2.Brownian motion 3.Random walk

The wikipedia articles are too in depth (yes!) for me at this point.

Assuming no math at all:

Using an Ito process we can describe the return of a stock with two components: an average level (the "drift") plus some uncertainty (the "volatility"). This uncertainty is represented by a Brownian Motion.

As written in Wikipedia,

A random walk is a mathematical formalization of a path that consists of a succession of random steps.

You can get the random steps by tossing a coin n times. If head, go up one step; if tail, go down one step. This is the "symmetric random walk". You can obtain a Brownian Motion from the symmetric random walk using a bit of mathematical machinery.

• Thanks. IS random walk similar to a martingale? May 29 '15 at 19:07
• For a random walk to be a martingale it requires p=q=0.5 i.e. a symmetric random walk, with equal chance of moving up or down in the next time step. May 29 '15 at 19:34

First, for Ito processes and Brownian motion. Ito process is a continuous-time trajectory with random evolution, so non-smooth and very kinky - also has a fractal look: no matter how much you'd zoom in, it will look similar. Ito process consists in fact of two parts: the drift part (deterministic evolution) and the diffusion part (where all the kinkiness and fractalness comes from). If Ito process does not posses the latter, it just looks like a continuous smooth trajectory. For example, $y = x^2$ is such an (purely drift) Ito process. In contrast, if Ito process has only diffusion component, you won't be able to spot it. That is, when the diffusion component is present, it's hard to say whether there is a drift component or not, because of the noise diffusion provides.

Brownian motion is a special case of an Ito process, and is the main building block for the diffusion component. In fact, any diffusion is just a time scaled Brownian motion. One important property of Brownian motion is that its increments are uncorrelated (in fact, they are independent) whereas in general Ito process there can be loads of cross-correlation happening.

Finally, formally random walk is a discrete-time process - hence not comparable with Ito processes which are continuous-time things. On the other hand, random walk must also have independent increments - that's why Brownian motion sometimes referred to as a (continuous-time) random walk.

A Brownian Motion is a continuous time series of random variables whose increments are i.i.d. normally distributed with 0 mean.

An Ito Process is a Brownian Motion with possibly nonzero mean.

A random walk is a discrete process whose increments are +/-1 with equal probability.

I would have added this as a comment to one of the answers but I don't have enough reputation for it. I recommend below lectures for this (they have been pretty useful to me at least):