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In HJM model (framework), the drift of the forward is determined by its diffusion coefficient:

$$ \mu(t,s) = \sigma(t,s)\int_t^s \sigma(t,v)^Tdv $$

My understanding, is that the change of measure under Grisanov theorem for continuous-time semi-martingales only affect the finite variation part (i.e. drift for HJM). Thus, if we start with an SDE under a risk-neutral-measure $Q$

$$ df(t,s) = \mu^Q(t,s)dt + \sigma(t,s)dW_t^Q $$

and the change to the real-world measure $P$ changes this to

$$ df(t,s) = \mu^P(t,s)dt + \sigma(t,s)dW_t^P $$

does this then mean that $\mu^Q(t,s) = \mu^P(t,s)$ since they are both functions of $\sigma(t,s)$?

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Your statement at the beginning of the question is not correct. That's why you have the "contradiction" later. It should say: In HJM model (framework), the drift of the forward under the risk-neutral measure Q is determined by its diffusion coefficient: $$ \mu^Q(t,s) = \sigma(t,s)\int_t^s \sigma(t,v)^Tdv. $$ That formula is not a general formula to obtain the drift under any probability measure, only applies to $Q$. Note that later under the forward measure $Q^T$ the drift is zero and the volatility is terms are not zero.

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As you said, the change of measure only affects the finite variation part, which is the drift. It's not clear why this implies $\mu^Q(t,s)=\mu^P(t,s)$. These are the drifts under the two measures, so I don't think they have to be the same.

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  • $\begingroup$ Thank you for your reply. The drift under the risk-neutral measure is a function of the diffusion coefficient only, and if this also true for the drift under the real-world measure (i.e. that it is a function of the diffusion coefficient), then since the diffusion coefficient does not change with the change of measure, then this would suggest that the two drifts are the same. If this is not true, then either the real-world drift is not a function of the diffusion coefficient or it is a different function. $\endgroup$
    – Confounded
    Jun 1 '20 at 9:03
  • $\begingroup$ It's not clear how we can start with an SDE under the risk neutral measure and apply Girsanov's theorem to change to the actual measure. How we get to the drift under the risk-neutral measure is that we assume the evolution of the forward curve has some drift $\alpha(t,T)$ and volatility $\sigma(t,T)$, then we derive that under the risk neutral measure, the drift is what you had above. $\endgroup$ Jun 5 '20 at 1:09
  • $\begingroup$ Girsanov's theorem essentially says that if we start with some measure P with Brownian motion W, and define the density of another measure Q with respect to P to be of a specific form, then the Brownian motion under Q is related to W through a specific form dependent on the density (process). When we say we change from P to Q, it's more like we start with P and define a measure Q. Therefore, it's unclear how we can start with some measure Q and change measure to actual measure P, since by assumption, P isn't something that we define. $\endgroup$ Jun 5 '20 at 1:11
  • $\begingroup$ In $P$ the expected value of the rate at $T$ is not necessarily equal to the forward rate, so I don’t think you can derive the same drift formula. $\endgroup$
    – dm63
    Jul 1 '20 at 12:12

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