Brownian Motion

The Brownian Motion is the only continuous process with the following properties: 

 

  • It starts off in zero   --> W(0) = 0
  • All non-overlapping increments are independent    --> W(t+s) - W(s) is independent of W(t) 
  • All increments are stationary so they only depend on the length of the increment
  • All increments W(t+s) - W(s) are normally distributed with mean zero and variance s    --> dW≈ W(t+s) - W(s) ~ N(0,s) 

 

Translating the independent property into asset prices, this is saying that we cannot look at the historical path followed by the price and use this to predict how it will change. 

 

dW~ √dt N(0,1)      and       W~ √t N(0,1)

 

Brownian Motion has some very interesting properties: 

 

1. Continuity

A BM is continuous since its change in a small time interval dt is dWwhose mean is zero and variance dt is infinitesimal. 

 

2. Infinite jaggedness

In any time interval, the expected total up and down movements of Wt are infinite. 

 

3. Deterministic Sum Squared Returns 

 

It is certainly the most important and fascinating property of BM. The property that allows everything to work. 

 

When applied inside an integral, the square of the RV dWis deterministic and given by:      (dWt)= dt 

Note that this formula is true inside an integral and not in a more general algebraic sense. In the rest of the discussion, we take the approach that         (dWt)= dt only makes sense inside an integral, and so can be replaced by dt without further clarification. 

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