I am working on a SABR-LMM model with the following system of SDEs under a numeraire $N$:
$$ \begin{align} &\mathrm{d} F_i(t) = \sigma_i (t) (F_i(t) + s)^{\beta} \Big( \mu^f_i (t) \mathrm{d}t + \mathrm{d} W^{N}_i(t) \Big) \ , \notag \\ &\mathrm{d} \sigma_i(t) = \nu_i \sigma_i(t) \Big( \mu^\sigma_i (t) \mathrm{d}t + \mathrm{d} Z^{N}_i(t) \Big) \ , \notag \\ &E \big\{\mathrm{d} W_i(t) \mathrm{d} Z_j(t)\big\} = \rho_{ij} \mathrm{d} t \ , \label{Eq. SABR-LMM} \\ &E \big\{\mathrm{d} W_i(t) \mathrm{d} W_j(t) \big\} = \gamma_{ij} \mathrm{d}t \ , \notag \\ &E \big\{\mathrm{d} Z_i(t) \mathrm{d} Z_j(t) \big\} = \phi_{ij} \mathrm{d}t \ , \notag \end{align} $$
where $F_i (t)$ is the forward LIBOR rate fixing at $T_{i-1}$. Assuming that I already have a way to calibrate the model and that I have chosen a numeraire, what is the best way to perform a large-step Monte-Carlo simulation (timestep > 6M)? Until now I have performed a simple Euler-discretization with predictor-corrector approach for approximating the drifts. This seems to work relatively well for 6M steps when I price swaptions, however I fail to find a good and fast simulation scheme for steps larger than 6 months.
