Girsanov′s Theorem and Change of Numeraire
The 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
Let B1(t, ω) be a Brownian motion under a probability measure P1: B1(t, ω) ~ N(0, t) under P1. Define a new probability measure P2 related to P1 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 B2(t, ω) defined as
(or dB2(t, ω) = dB1(t, ω) + γ (t, ω)dt in differential format) is a Brownian motion under P2: B2(t, ω) ~ N(0, t) under P2.
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) ...