(The PDF of the FPT)

(i) Prove that

Conclude that the first passage time to a given point is finite with probability 1 but its mean is infinite. This is the continuous time version of the Gambler’s Ruin “Paradox”: gambling with even odds against an infinitely rich adversary leads to sure ruin in a finite number of games, but on the average, the gambler can play forever (see [72, Ch. XIV.3]).

(ii) Use the above result to conclude that the one-dimensional MBM is recurrent in the sense that Pr{w(t, ω) = x for some t > T} = 1 for every x and every T. This means that the MBM returns to every point infinitely many times for arbitrary large times.

(iii) Consider two independent Brownian motions, w1(t) and w2(t) that start at x1 and x2 on the positive axis and denote by τ1 and τ2 their first passage times to the origin, respectively. Define τ = τ1 ∧ τ2, the first passage time of the first Brownian motion to reach the origin. Find the PDF and mean value of τ

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