Self-financing condition and funding, collateral and discounting

Question

I'm reading "Illustrating a problem in the self-financing condition in two 2010-2011 papers on funding, collateral and discounting" paper.

1. Is it just me or authors have a typo in their main warning (***): $\theta$'s seem to have the superscript $A$ on one side, but not the other?

$$d\left(\sum_{i=1}^n \theta_i^A P_i^A \right) \not= \sum_{i=1}^n \theta_i dP_i^A $$

2. The claim $Y$ has a value process, $V^Y$, but also a gain process (!?) $G^Y$. What is it (I understand gain processes for assets, price and cumulative dividend, does the concept extend to claims too?) and is it needed to complete the paper's arguments, as equality $G^Y=G^\Pi$ doesn't seem to be used anywhere?

Thank you for taking your time to look into my questions.