LIBOR Market Model: notation and specification

Reference: Bjork Ch.27

Two ways of specifying a LMM:
1) dL_i = L_i sigma_i dW^i
where
L_i = the i-th forward rate
sigma_i = a VECTOR of deterministic volatility
dW^i = k-dimensional Wiener process

Here dW^i is under the i-th forward measure. However, all forward rates are driven by the SAME Wiener process, albeit martingale is attained only when the rate is expressed under its own forward measure.

2) dL_i = L_i sigma_i dW_i
where
L_i = the i-th forward rate
sigma_i = a SCALAR of deterministic volatility
dW_i = 1-dimensional Wiener process


Note:
In 2), a correlation structure dW_i dW_j = rho_{i,j} is also required. Here, each forward rate is driven by its own Wiener process dW_i.