I would like to establish the equations of forward libors under risk neutral measure. Here is how I do it, and what I get :
Under the $P_{T_j} $ measure, forward Libor $L_j$ is martingale. Thus:
$$ dL_j = L_j \times \sigma_j(t) \times dW^{T_j} $$
The change of numéraire implies that:
$$ \frac{dQ^{T_j}}{dQ^*} = \frac{P(t,T)\times\exp{\left(-\int_0^T{r(s)ds}\right)}}{P(0,T)} $$
Under $Q^*$, $P(t,T)$ discounted is martingale, which means that:
$$ \frac{dP(t,T)}{P(t,T)} = r(t) dt + \eta(t) dW^* $$
Solving this gives:
$$ P(t,T) = P(0,T) \times \exp\left(\int_0^T\left(r(s)-\frac{1}{2} \eta(s)^2\right)ds + \int_0^T{\eta(s)}dW^{T_j}\right) $$
Thus:
$$ \frac{dQ^{T_j}}{dQ^*} = \exp\left(-\int_0^T{\frac{1}{2}} \eta(s)^2ds + \int_0^T{\eta(s)}dW^{T_j}\right) $$
Girsanov shows that:
$$ dW^{T_j} = dW^* - \eta(t) dt $$
Under $Q^*$:
$$ \frac{dL_j}{L_j} = \alpha(t) dt + \sigma_j(t) dW^{*} $$
Writing $P(t,T) $ as $\exp(-\int_t^T{f(t,s)ds}) = \exp(Y_t)$:
$$ \frac{dP(t,T)}{P(t,T)} = dY_t +\frac{1}{2}<Y_t>dt $$
with $ dY_t = f(t,t) dt -\int_t^T{\alpha(t) dt ds } - \int_t^T{\sigma(t)dt dW_s}$. Identifying, I conclude that:
$$ \eta(t) = \sigma(t) $$
Finally:
$$ dL_j(t) = -L_j(t) \sigma_j(t) \int_t^{T_j}{\sigma_j(s)ds} dt + L_j\sigma_j(t) dW^*.$$
Is it correct ?