Exercise 14.7 For a standard Brownian motion {z(t)}, let

The joint distribution of (z(T),M(T)) is given, from the result in Exercise 12.9, as

Consider now X(t) = z(t)+μt, and let Mμ(T) = max0≤t≤T X(t). Let Y (T) = exp{μz(T) − μ2T/2}. Applying Girsanov’s theorem (Proposition 14.2), show that

In general, for a Brownian motion with drift μ and diffusion coefficient σ, the distribution function is given by

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M(T) = max(t). 0