In the page 199 of the paper A Martingale approach to Infinite Systems of Interacting Processes one reads:
I suspect that (1.5) is incorrect
To simplify notation, let's write $X_\theta^s(t) = \exp(F(\lambda))$ where $$F(\lambda) = \int_s^{s\vee t} \lambda\theta(u) d\tilde\alpha(u) - \sum_k \int_s^{t\vee s} c_k(u) (e^{-2\lambda\theta_k(u)\alpha_k(u)} - 1 + 2 \lambda\theta_k(u) \alpha_k(u))\, du $$
Now taking the derivative yields: $$\frac{d}{d\lambda} X_\theta^s(t) = \exp(F(\lambda))F'(\lambda)\\ \frac{d^2}{(d\lambda)^2} X_\theta^s(t) = \exp(F(\lambda))(F'(\lambda))^2 + \exp(F(\lambda))F''(\lambda) \\ $$
Every object here is also a martingale. We note that $F(0) = 0$ so the exponential terms vanish. Now we need to compute $F'$ and $F''$ $$ F'(\lambda) = \int_s^{s\vee t} \theta(u) d\tilde\alpha(u) \\- \sum_k \int_s^{t\vee s} c_k(u) \big((-2\theta_k(u)\alpha_k(u))e^{-2\lambda\theta_k(u)\alpha_k(u)} + 2 \theta_k(u) \alpha_k(u)\big)\, du$$ $$ F''(\lambda) = - \sum_k \int_s^{t\vee s} c_k(u) ((4\theta^2_k(u)\alpha^2_k(u))e^{-2\lambda\theta_k(u)\alpha_k(u)} )\, du$$
So $$F'(0) = \int_s^{s\vee t} \theta(u) d\tilde\alpha(u) $$ and
$$ F''(0) = - 4 \sum_k \int_s^{t\vee s} c_k(u) \theta^2_k(u)\alpha^2_k(u) \, du$$
So $|\theta(u)|_{c(u)}^2 $ should be
$$|\theta(u)|_{c(u)}^2 = \sum_k c_k(u) \alpha^2_k(u)\theta^2_k(u) $$
Therefore (1.5) is incorrect,
and for a better notation we should write $$|\theta(u)|_{c(u),\alpha(u)}^2 = \sum_k c_k(u) \alpha^2_k(u)\theta^2_k(u)$$ Or am I missing something?