What is the distribution of the time-to-ruin for a gambler's ruin problem that allows “pauper bets”?

In another question on this site I have derived the distribution for the time-to-ruin in the gambler's ruin problem where the wealth of the gambler follows a discrete-time random walk. In this standard problem, we assume that the gambler starts with an amount of wealth that is a positive integer multiple of the betting amount, such that there is no "residual wealth" lower than the betting amount when the gambler gets down near to zero wealth.

In the present question I would like to examine a variant of this standard problem. Here we suppose that (until ruin) the gambler has some positive wealth $$w>0$$ and bets in unit increments as normal. However, if the wealth of the gambler gets into the range $$0 (i.e., less than the standard betting amount but more than nothing) then the gambler is allowed a "pauper bet" where the gambler bets his remaining wealth (even though it is less than one unit). The pauper bet is treated the same as the standard bets, in the sense that a loss loses the bet amount and a win returns the bet amount as profit. Thus, when the gambler makes a "pauper bet" his wealth either goes to zero or to $$2w$$ (with equal probability). Note that the gambler is only allowed to make a pauper bet whenever his wealth is less than one unit.

The stochastic process for this variant of the gambler's ruin can be described formally as follows. Let $$w_0 > 0$$ be the gambler's starting wealth. Unlike in the standard problem, this value can be any positive real value; not just an integer. Let $$\{ Q_t | t \in \mathbb{N} \}$$ denote the sign values for the bet outcomes. The time-series for the gambler's wealth is described recursively by:

$$w_{t+1} = w_t + Q_i \cdot \min(1, w_t) \quad \quad \quad \mathbb{P}(Q_i = -1) = \mathbb{P}(Q_i = 1) = \frac{1}{2}.$$

The time-to-ruin is defined as:

$$T \equiv \min \{ t \in \mathbb{N} | w_t = 0 \}.$$

I would like to derive the distribution of the time-to-ruin for this variant of the gambler's ruin problem. The remainder amounts in the "pauper bets" complicate the situation substantially, so I am looking for techniques to deal with this variation in the derivation of the distribution. Can anyone show how to derive the distribution of the time-to-ruin in this case?

As a smaller ---but still interesting--- question, are the time-to-ruin probabilities even continuous with respect to the starting wealth $$w_0$$? Intuitively it seems certain that they would be continuous at non-integer values, but it might be possible that there would be a "jump" at the integer values. I think they would be continuous, but a proof of this would be nice.