Let X(t) = μt + σB(t) be a Brownian motion with drift, where μ > 0. Suppose that a signal is triggered whenever M(t) = sups≤t X(s) reaches the levels 0, 1, 2,.... So the nth signal is triggered at...

Let X(t) = μt + σB(t) be a Brownian motion with drift, where μ > 0. Suppose that a signal is triggered whenever M(t) = sups≤t X(s) reaches the levels 0, 1, 2,.... So the nth signal is triggered at time τn = inf{t : M(t) = n}, n ≥ 0. Show that these times of the signals form a renewal process and find the mean, variance and Laplace transform of the times between signals. In addition, obtain this information under the variation in which a signal is triggered whenever M(t) reaches the levels 0 = L_0
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Ln − Ln−1 are independent exponential random variables with rate λ