Asymmetric Gambler's Ruin Absorption Probability

Scenario In quantitative trading and quantitative research, candidates are frequently evaluated on probability modeling, specifically regarding Gambler's Ruin / Difference Equations.

Your Task
A gambler starts with an initial capital of \i$and bets$$1$at each step. In each independent game, the gambler wins$$1$with probability$p$and loses$$1$with probability$q = 1-p$, where$p \ne 1/2$. The game terminates when the gambler reaches$$N$(win) or$$0$(ruin). (1) Derive the exact probability$P_i$of reaching$$N$before ruin. (2) Derive the expected duration$\mathbb{E}[T_i]$ of the game.

Constraints

• All variables are defined on a standard probability space $(\Omega, \mathcal{F}, \mathbb{P})$.
• Analytical expressions must be in closed form, and empirical simulations should converge within standard statistical tolerances.