Dividend and Gain process in paper by Brigo, Buescu, Pallavicini and Liu

Question

In the paper Illustrating a problem in the self-financing condition in two 2010-2011 papers on funding, collateral and discounting (2012) (link) from Brigo, Buescu, Pallavicini and Liu, the above authors give a detailed description on how to work with the gain process, and how it differs from the price process.

Namely, given an asset $A$, we identify two processes with it, $P^{A}$ (price process) and $D^{A}$ (dividend process). The Gain process is then defined by $G^{A}=P^{A} + D^{A}$. Further ahead, they apply this definition to particular assets (I modified slightly their notation): $A_{1}$ a repo contract (with risky underlying asset having price $S_{t}$), $A_{2}$ a collateral position (with price being $C$) and $A_{3}$ a funding position, whose price is denoted by $\alpha$.

Now, the authors identify $P^{A_{1}} = 0$ (I guess here the repo rate chosen is such that the price of the contract is zero), $P^{A_{2}}=C$ and $P^{A_{3}}=\alpha$.

Then they claim: and the gain processes $dG^{A_{1}} = dS + (r_{D}-r_{R})Sdt$, $dG^{A_{2}}=r_{C}Cdt$, $dG^{A_{3}}=r_{F}\alpha dt$, where $r_{*}$ is some rate (not relevant for the question).

The question is, how do they get this? I thought, for example for $A_{3}$, that $dG^{A_{3}} = dP^{A_{3}}+dD^{A_{3}} = d\alpha + r_{F}\alpha dt$.

I hope my question is clear.

Answer 1

By definition of the authors, repos are defined as pure dividend processes where the price performance of $S$ is continuously exchanged against a funding cost $r_RS$, whereas collateral and the funding account are pure price processes which continuously earn and compound interest: \begin{align} &A_1:\begin{cases} dP^{A_1}=0, \quad P^{A_1}_0=0 \\ dD^{A_1}=dS+(r_D-r_R)Sdt,\quad D^{A_1}_0=0 \\ dG^{A_1}=dP^{A_1}+dD^{A_1}=dS+(r_D-r_R)Sdt,\quad G^{A_1}_0=P^{A_1}_0+D^{A_1}_0=0 \end{cases}\tag{1} \\[10pt] &A_2:\begin{cases} dP^{A_2}=r_CCdt, \quad P^{A_2}_0=C_0 \\ dD^{A_2}=0,\quad D^{A_2}_0=0 \\ dG^{A_2}=r_CCdt,\quad G^{A_2}_0=P_0^{A_2} \end{cases}\tag{2} \\[10pt] &A_3:\begin{cases} dP^{A_3}=r_F\alpha dt, \quad P^{A_3}_0=\alpha_0 \\ dD^{A_3}=0,\quad D^{A_3}_0=0 \\ dG^{A_3}=r_F\alpha dt,\quad G^{A_3}_0=P_0^{A_3} \end{cases}\tag{3} \end{align}