Martingale Analysis of Fair Coin Doubling Strategy

Scenario In quantitative trading and quantitative research, candidates are frequently evaluated on probability modeling, specifically regarding Martingales / Optional Stopping Theorem.

Your Task
A gambler plays a fair coin game ($\mathbb{P}(\text{Win}) = 1/2, \mathbb{P}(\text{Loss}) = 1/2$). The gambler starts betting \1$, and doubles their bet after every loss ($$1, $2, $4, \dots, $2^{k-1}$) until the first win, at which point they stop. (1) Show that if the gambler has unlimited wealth, their net profit at stopping time$\tau$is always exactly$+$1$. (2) Show that the gambler's accumulated wealth process$M_n$is a Martingale, but explain why the Optional Stopping Theorem ($\mathbb{E}[M_\tau] = M_0 = 0$) does NOT hold, identifying which condition of Doob's theorem is violated.

Constraints

• All trials are mutually independent and identically distributed.
• Probabilities lie strictly within $(0, 1)$ unless degenerate.