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Given a stochastic process, how do we prove and generate the change-of-measure? I have been trying to prove the change-of-measure as under the Radon-Nikodym theorem and Girsanov Theorem, but struggling.

I read through the thread here : Radon-Nikodym derivative and risk natural measure as well.

Given a lognormal process : enter image description here

which is under the real-world measure.

We claim there is an equivalent probability measure (called the risk-neutral measure) where the return is the risk-free rate, with the Brownian Motion adjusted with a drift.

We set this equation to be : enter image description here

  1. So we have set the 'market-price of risk' $= (\mu - r)/\sigma$
  2. We set the risk-neutral Brownian motion $dW(Q) = (\mu - r)/\sigma * dt + dW$

My understanding of the Radon-Nikodym and Girsanov Theorem, is that we can now express any asset X (or S in this case) from one measure (real-world) into another measure (risk-neutral) by the Radon-Nikodym as per follows

  1. Given enter image description here

where EN is the expectation under risk-neutral world, X is asset X, EU is expectation under real-world' and dN/dU is the Radon-Nikodym derivative

  1. Is the term dN/dU merely the expectation of the stochastic exponential enter image description here
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  • $\begingroup$ What exactly is your question? $\endgroup$ – Sanjay Apr 1 at 7:49
  • $\begingroup$ question is, do I have the Radon-Nikodym derivative correct (2) according to the convention? $\endgroup$ – Kiann Apr 1 at 14:48

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