# Why does the variance of a Brownian motion increase linearly with time?

Brownian motion is said to follow a path where each value is normally distributed with mean $\mu t$ and variance $\sigma^2 t$.

What is the basis for the relation that variance varies directly proportional to the 1st power of $t$?

• It follows directly from the definition of the process (specifically, the variance of the increments and the fact that they are independent). Are you asking for a formal proof of this fact? Or an intuitive explanation or illustration? – Chris Haug Aug 31 '18 at 12:09
• Hi: Hamilton's Time Series Analysis has a decent discussion of this question without the use of stochastic differential equation type math. What you want to look at depends heavily on your math background and also as chris haug said, on what you want. – mlofton Aug 31 '18 at 12:43
• @ChrisHaug How about both? I don't have any info on the proof or the intuition. – Dom Jo Aug 31 '18 at 13:09
• @mlofton I graduated in engineering. But i work in finance. I was trying to study Black Scholes equation from scratch. – Dom Jo Aug 31 '18 at 13:10
• Could you share with us the definition of Brownian Motion you are working with? For many of us, the question you ask is a key part of the definition. Evidently, then, you have a different definition in mind, so we need to know what it might be. – whuber Aug 31 '18 at 13:12