Classic ruin theory assumes that the income is constant and that only the losses are random with an underlying distribution.
Suppose we want to determine the risk of ruin for a game (for example poker), where also the winnings are random.
Let $\psi(u)$ denote the risk of ruin, starting from initial surplus $u$. We assume that the winnings/losses in each game follow a normal distribution with mean $\mu>0$. The game is played until the player is broke or his surplus goes to infinity.
Classical ruin theory doesn't seem to apply because of the non-constant income. I am grateful for any advice on how to approach this problem.
According to risk of ruin wikipedia page,
The formula for risk of ruin for such setting can be approximated by
$$\left( \frac{2}{1+\frac{\mu}{r}}-1 \right)^{u/r}=\left(\frac{1-\frac{\mu}{r}}{1+\frac{\mu}{r}} \right)^{u/r}$$
where $$r=\sqrt{\mu^2+\sigma^2}$$
It is described that the approximation formula is obtained by using binomial distribution and law of large numbers.
I have written the formula in the form of proposed by Perry Kaufman
$$\left( \frac{1-\text{edge}}{1+\text{edge}}\right)^{\text{capital units}}$$