For a given tenor structure \(T_0,T_1,...,T_n\) evenly spaced with \(\tau = T_{i+1} - T_i, \forall i=1,...,n\), and a number of factors \(F\), we evolve the LIBOR rates for all \(n\) maturities with the following stochastic equation:
\[L_i(T_{j+1})=L_i(T_j)exp[\tau_{j+1}(\mu_i(T_j)-\frac{1}{2}\sigma_i(T_j)^2)+\sigma_i(T_j)\sqrt{\tau_{i+1}}dW_i]\]
Where \(\sigma_i(t)\) are calibrated volatilities, \(dW_i\) is a Brownian motion scaled by the pseudo-sqrt of the correlations matrix and \(\mu_i(t)\) is the drift defined in terms of the volatilities and correlations between tenors:
\[dW_i=\sum_{k=1}^{F}\eta_{i,k}W_k\]
\[\mu_i(t)=-\sigma_i(t)\sum_{m=i+1}^{n}\frac{\tau_mL_m(t)\sigma_m(t)\tilde{\rho}_{i,m}}{1+\tau_iL_m(t)}\]