What is the no ponzi condition for this representative consumer problem? Tags: Initial condition, no-ponzi game

This question is following Farhi et.al. 2014‘s consumer budget constraint modified to add a government bond and removing money. A simplified version is given below which is pertinent to the question.

$$ sum_{jin J_{t}} Q_{t}^{j} B_{t+1}^{j} + B_{gt+1} + P_{t}C_{t} = (1+i_{t})B_{Gt} + sum_{jin J_{t-1}} (Q_{t}^{j}+ D_{t}^{j}) B_{t+1}^{j} + Y_{t}$$

where $B_{Gt}$ is a one period government bond which in only traded in the home country and pays a certain nominal rate of interest $i_{t}$ and ${B_{t}^{j}}_{jin J_{t}}$ is an internationally traded bond where $jin J_{t}$ is a subset of assets available at time $t$ to each consumer. The pricing kernel $Q_{t}^{j}$ is such that we obtain complete markets.

The problem:

1. What is the no-ponzi condition for the two assets?
   My solution is to treat $Y_{t} + (1+i_{t})B_{Gt} – P_{t}C_{t} – B_{gt+1} = tilde{Y_{t}}$ and obtain a no ponzi for the international assets as $$ lim_{srightarrow infty}frac{sum_{jin J_{t}} Q_{t}^{j} B_{t+1}^{j}}{P_{qt} prod_{s=-infty}^{0} (1+tilde{i}_{t-s})} = 0,$$
   where $P_{qt}$ is just an index of pricing kernel $Q_{t}^{j}$ so traded and $tilde{i}_{t}$ is an index of nominal return for all the assets that were traded.
   But this solution does not have any restrictions on the consumer’s side for the government bonds. It just says that consumer cannot accumulate internationally traded bonds more than what he can pay off in his lifetime. (Consumer could be she).

2. What would be the initial condition for the internationally traded securities if I assume that my country could start with some positive debt level?
   My solution: I have used backward induction on the budget constraint to obtain an initial condition for the internationally traded security on the basis of which I get the following:
   $$ B_{-1} = – sum_{s = -infty}^{0} frac{tilde{Y}_{t-s}}{P_{qt-s} ( 1+tilde{i}_{t-s})}$$

Any help would be appreciated. I have referred a few macro textbooks, but I am not clear on this issue myself. The cleanest I could get to understanding was using Acemoglou’s growth textbook.