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

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.

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}