In page 364 of the article Diffusion processes with continuous coefficients I (Stroock Varadhan - 1969), one finds in lemma 3.5:
The question is: how do we prove that
$$ Y^s_\theta(t) = \exp \big\{ \langle \theta, \eta(t)\rangle - \frac{1}{2} \int_s^t \langle \theta, \sigma^*(u) a(u) \sigma(u)\rangle \, du \big\} ?$$
**Attempt **
In the proof of lemma 3.5 one reads
So we come back to theorem 3.2
Now we try to see how to obtain the martingale property of $$ Y^s_\theta(t) = \exp \big\{ \langle \theta, \eta(t)\rangle - \frac{1}{2} \int_s^t \langle \theta, \sigma^*(u) a(u) \sigma(u)\rangle \, du \big\} $$
as a consequence of Theorem 3.2
First we note that
$$\langle \theta, \eta(t) \rangle = \sum_{j} \theta_j \eta_j(t) $$
but since $\eta(t) = \int_s^t\sigma(u)d \xi(u)$
$$ \eta_j(t) = \sum_k \int_s^t \sigma_{j,k}(u) d\xi_k(u)$$ so
$$ \langle \theta, \eta(t) \rangle = \sum_j \sum_k \theta_j\int_s^t \sigma_{j,k}(u) d\xi_k(u) \\ = \sum_k \int_s^t \sum_j \sigma*_{k,j}(u)\theta_j d\xi_k(u)\\ = \sum_k \int_s^t (\sigma*(u)\theta)_k d\xi_k(u)\\ $$
So applying theorem 3.2 we get $$ Y^s_\theta(t) = \exp \big\{ \langle \theta, \eta(t)\rangle - \frac{1}{2} \int_s^t \langle \sigma^*(u)\theta, a(u) \sigma^*(u)\theta\rangle \, du \big\}\\ = \exp \big\{ \langle \theta, \eta(t)\rangle - \frac{1}{2} \int_s^t \langle \theta, \sigma(u) a(u) \sigma^*(u)\theta\rangle \, du \big\} $$
So this result is a bit different than what is stated in theorem 3.2.
Is this a typo?