HJM framework: Girsanov transformation between forward measure and risk neutral measure

Consider a HJM framework $d f(t, T) = \sigma (t, T) d W_t^T$ which is a SDE of instantaneous forward rates on $T$ -forward measure, and let $P (t, T) = \exp (-\int_t^T f (t, u) d u)$ $B (t) = \exp (\int_0^t f(u, u) d u)$ be a $T$ -discount bond price and a continuously compounded money market account. By definition, \begin{align} P(t, T) &= \exp (-\int_t^T f(0, u) d u - \int_t^T \int_0^t \sigma (s, u) d W_s^T d u) \ &= \exp (-\int_t^T f(0, u) d u - \int_0^t \int_t^T \sigma (s, u) d u d W_