Consider a symmetric random walk: X 0 = u and X t = u+ 1 +...+ t

Consider a symmetric random walk: X₀ = u and X_t = u+ξ₁+...+ξ_t, for t = 1,2, ..., where an integer u > 0 and ξ’s are independent and assume the values ±1 with equal probabilities. Let τ be the moment of ruin, that is, τ = min{t : X_t = 0}. Let Y_t = X_t if t ≤ τ, and Y_t = 0 if t ≥ τ. We may think about a gambler who starts with the initial capital u, and quits at the moment of ruin. (We know that P(τ < ∞) = 1; Show that Y_t is a martingale. Using (5.1), estimate the probability that before ruin, the gambler will reach a level a ≥ u.