I'm already very familiar with the expected value of a round of Martingale betting, where a player increases their bet by some percent in order to cover their losses plus make a profit equal to the original target profit (provided they eventually win before going bankrupt).
If we assume we have full control over both the odds and the bet increase on loss, we can determine how much we need to increase our bet by to satisfy these conditions using:
$$ \textit{Increase Bet By} = \frac{1}{Bet Odds - 1} $$
With bet odds of $2$, we increase our bet by $1$ (100%), bet odds of $3$ require an increase of $0.5$ (50%), and so on...
Calculating expected value of one round of the strategy is fairly straightforward as we only need to know our $\textit{Potential Profit}$, $\textit{Max Successive Losses}$ (the total number of losses in a row that our balance can handle), $\textit{Potential Loss}$ (the total amount lost if we were to lose $\textit{Max Successive Losses}$ in a row), and the probability of losing $\textit{Max Successive Losses}$ times in a row (our $\textit{Bankruptcy Probability}$).
The potential profit is always
$$ \textit{Potential Profit} = \textit{Initial Bet} \times \textit{Bet Odds} - \textit{Initial Bet} $$
The potential loss is always
$$ \textit{Potential Loss} = \sum_{i=1}^{\textit{Max Successive Losses}} \textit{Initial Bet} \times (1 + \textit{Increase Bet By})^{i-1} $$ $$ = \textit{Initial Bet} \times \frac{(1 + \textit{Increase Bet By})^{\textit{Max Successive Losses}} - 1}{\textit{Increase Bet By}} $$
The odds of losing $\textit{Max Successive Losses}$ times in a row, assuming a 5% house edge, is
$$ \textit{Bankruptcy Probability} = (1 - \frac{0.95}{\textit{Bet Odds}})^{\textit{Max Successive Losses}} $$ $$ \textit{Win Probability} = 1 - \textit{Bankruptcy Probability} $$
Making the total expected value of a round
$$ \textit{Expected Value} = \textit{Win Probability} \times \textit{Potential Profit} - \textit{Bankruptcy Probability} \times \textit{Potential Loss} $$
I would like to extend this calculation to cover any percent increase on loss (i.e., if bet odds are $2$ and each loss we increase our bet by $200$% instead of $100$%).
My thought is that this would only affect our $\textit{Potential Profit}$, as it is no longer fixed (meaning the $\textit{Potential Loss}$, $\textit{Bankruptcy Probability}$, and $\textit{Win Probability}$ formulas go unchanged).
I know that our potential profit increases depending on the number of successive losses we face before our win (which is always $\leq \textit{Max Successive Losses}$), and that each $i$-th potential profit is affected by the probability of losing $i$ times in a row, but don't have the necessary math / statistics background to translate my assumptions into a correct, definitive formula.
I'm sure there is some common statistics formula that is used for this kind of thing, but I don't know the right terminology needed to search online for what I'm looking for.
Any help extending the potential profit formula to cover any percent increase on loss would be greatly appreciated!
Also, I say "Martingale-like" because, from what I understand, the Martingale betting system should always leave you with your original intended profit after winning. Is there a name for the type of betting system I've described? Or is it still considered Martingale even if the resulting profit may be greater than the initial intended profit?