When working with diffusions, Girsanov′s Theorem uses the same likelihood ratios to relate Brownian motions under one probability to another. We will use the following version of it.^{1}

Let *B*_{1}(*t, ω*) be a Brownian motion under a probability measure *P*_{1}: *B*_{1}(*t, ω*) ~ *N*(0*, t*) under *P*_{1}. Define a new probability measure *P*_{2} related to *P*_{1} through their likelihood ratios as follows:
for a random process *γ* (*s, ω*). Then subject to some regularity conditions on *γ* (*s, ω*), the process *B*_{2}(*t, ω*) defined as
(or *dB*_{2}(*t, ω*) = *dB*_{1}(*t, ω*) + *γ* (*t, ω*)*dt* in differential format) is a Brownian motion under *P*_{2}: *B*_{2}(*t, ω*) ~ *N*(0*, t*) under *P*_{2}.

As for short-rate models, the original HJM formulation of their framework was in continuous time and in terms of *instantaneous* forward rates, related to discount factors as follows:

We assume the following general dynamics for the 1-factor (can easily be generalized to multifactor) ...

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