I am studying the calibration of the 2 factors Hull White model on Brigo and Mercurio's book. They point out that, using cap volatilities, the value of $\rho$ is almost minus one and this means that the HW 2 factors model tends to degenerate into a one-factor model. So far, so good.
Now, the authors claim that this one-factor model is non-Markov. Could you explain to me why? I know (more or less) what a Markov process is, but, still, I don't fully understand their statement.
They also go further, stating the following:
"the degenerate process for the short rate is still non-Markovian (if $a \ne b$), which explains what really makes the G2++ model outperform its one-factor version"
There are many things I don't get about this sentence.
First, why are they using the word "still"? Does it mean that even the non-degenerate Hull White 2 factors is non-Markovian? How can I say quickly whether an interest rate model is Markovian or not?
Second, why on earth (at least in financial modelling) should a non-Markovian process outperfom a Markovian one? Their statement looks very strong...
Since we are talking about Markov/non-Markov interest rates, I would also like to ask a question concerning the HJM framework. One of the main problems with this approach is that the dynamics of the short rate is often messy and non-Markovian. Again, by simply looking at the dynamics, how can we say that? The drift of the short rate involves some (deterministic and stochastic) integrals and I read somewhere that if the variable $t$ appears only as one of the extremes of the integral, then the short rate is Markovian; if, on the other hand, $t$ appears also inside the integral, the Markovianity is lost.
Is it true? If so, could you give me an intuitive and/or rigorous explanation?
Thank you very, very much for any help you can provide!!!
P.S. Here is the model :)
$d x_t=-a x_t\ dt+\sigma dW^1_t $
$d y_t=-b y_t\ dt+\eta d W^2_t $
$d W^1_t d W^2_t = \rho dt$
where $\phi_t$ is the deterministic shift chosen so as to fit any initial forward curve