Hi all,I am implementing a BGM model with 3-dimsional Brownian motion via PCA of historical Correlation matrix.The SDE of some Libor Rate (under appropriate measure) is the following :$dL_t(T)/L_t(T)=\ksi(t,T).c(T-t)^*dW_t$ where -L_t(T) the Libor rate value at time t with fixing date T -c(T-t) is the vector used for modeling instantaneous correlation beetwen the three coordonates of W_t-\ksi(t,T) is the intantaneous volatility of the Libor Rate L(t,T) ( which has some parametrization of its own)As I am using (and want to use) vectors c(T-t) issued from the PCA of an historical correlation matrix, I only have the value of the vector c(T-t) for fixed values of T-t=integer * Period of Libor Rates.This implies that I need to "interpolate" in some suitable sense the vector c(T-t) for T-t within two periods. I would like to know what kind of interpolation is used in practice when using this kind of specification.Of course I can interpolate c(T-t) between C( (n-1).Period) and C( n.Period) and renormalize it with its euclidian norm (so I still have a one dimensional brownian motion with the scalar product (c(T-t)^*dW_t)). But I think the appropriate way would be to get the angle between C( (n-1).Period) and C( n.Period) and to interpolate this angle linearly.It would be easier to implement the first method but maybe more hazardous. Intuition (geometrically) tells me that the path followed by both interpolation method is the same but is run at different speed, and I wonder how this could affect the quality of interpolation which I would like in sprit to be as close as possible of a linear interpolation.So basically I would need either another method, or a way of measuring the difference between those two interpolation without implementing the "angle" one. Any idea is welcome

Hi!According to my experience (I've implemeted the LMM in my Master thesis) I used Rebonato (1999) approach which is very easy to implement. It basically gives you vectors (with arbitrarly chosen number of factors, for example 3 as in your case) you can use to distribute the instantaneous volatility of each forward rate to each Brownian motion.For further details see BRigo and Mercurio or directly Rebonato 1999 paper "On the simultaneous calibration of multifactor lognormal interest rate models to Black volatilities and to the correlation matrix"Regards PS Please write in lateX formulas, they will be much more understandable!