Saturday, April 18, 2015

stochastic calculus - Girsanov's Theorem - Change of Measure


I have trouble understanding Girsanov's theorem. The Radon Nikodym process Z is defined by:


Z(t)=exp(t0ϕ(u)dW(u)t0ϕ2(u)2du)


Now ˆP is a new probability measure. The trouble is I am not understanding how to go from old P to the new one. The old P is normally distributed with mean 0 and variance t. Now say I want to know the new probability for an infinitesimally small interval around 0.2. For that I need to know the value of Z at this interval (event you may say). And then I can multiply (integrate) the value of Z with old P, and get new ˆP.


Assume t is fixed.


I have no idea how to calculate the value of Z for this interval/event. Help would be appreciated.




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