**APPENDIX C**

**Girsanov′s Theorem and Change of Numeraire**

**T**he change of numeraire formula in Chapter 11 relates the risk-neutral probability sets when switching numeraires via their likelihood ratios (Radon-Nikodym derivative):

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}**Girsanov′s Theorem**

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}.#
** CONTINUOUS-TIME, INSTANTANEOUS-FORWARDS HJM FRAMEWORK**

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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