# Change of measure from physical to risk-neutral under Radon-Nikodym and Girsanov Theorem

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.

Given a lognormal process :

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 :

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

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
• What exactly is your question? – Sanjay Apr 1 '19 at 7:49
• question is, do I have the Radon-Nikodym derivative correct (2) according to the convention? – Kiann Apr 1 '19 at 14:48