The Markov Property

When we first motivated Brownian Motion, we insisted that the random BM increment dWt should be independent of the current state of the BM and its past history --> dWt is independent of {Ws: s ≤ t}. 

 

Markov Property: given the present, the future does not depend on the past. 

 

It is oftern helpful to introduce notation for the filtration Ft, which is the set of all events whose outcome is known at time t. 

The Markov property for Wt can then be written:   ℙ(Wu ∈ A ∣ Ft) = ℙ(Wu ∈ A ∣ Wt)

 

As the lognormal spot process is built from BM, it also has the Markov property. All knowledge that could be used to predict future spot prices is encoded in the current spot price. 

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