You need to use the Cameron-Martin-Girsanov (CMG) Theorem from Chapter 15 Section 1. The CMG Theorem essentially tells us that the relationship between the standard Brownian motion (SBM) under the real-world "P" probabilities, Z(t), and the SBM under the risk-neutral "Q" probabilities, Z~(t), is: Z~(t) = Z(t) + gamma*t or equally: dZ~(t) = dZ(t) + gamma*dt where gamma is the market price of risk, which in the context of term structure models is: gamma = [m(t,T) - r(t)] / S(t,T) - see page 12 of Chapter 17. So, all you need to do is replace dZ(t) in the SDE for r(t) under P given in the question with: dZ(t) = dZ~(t) - gamma*dt to get the corresponding SDE for r(t) under Q. In this case: gamma = mu*r(t)/sigma and you obtain: dr(t) = sigma*dZ~(t) under Q.
