Brownian motion
Think of Brownian motion as a cloud of paths.
There are two kinds of properties: path properties, distributional properties
Path properties
- for “almost every”
- is a continuous function of for “almost every”
Distributional properties
- For
taking , we have
i.e. increments over non-overlapping intervals of time are independent.
Geometric Brownian motion
- is a martingale
Proof for being a martingale
Let be given
This argument works even if is negative.
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