Exercises to Chapter 3

Exercise 3.1: X = ? + AZ for the vector ? and the vector of independent standard Normal random variables Z. For t = ( t ₁, , t _n)

where ? is the characteristic function of the vector Z. By independence Hence Finally .

Exercise 3.2: E X = ? ^? ₀ xdF(x) = ? ^? ₀ ? ^x ₀ dtdF (x) = ? ^? ₀ ? ^? _t dF (x)dt = ? ^? ₀ (1 ? F(t))dt.

Exercise 3.3: If f(t) is non-increasing then . Now apply the previous result.

Exercise 3.9: For x ? 0, by using the distribution of M(t) = max _{s? t} B(s) P( B(t) > x)= P( B(t) > x)+P( B(t) < ? x) = 2P( B(t) > x)= P( M(t) > x).

Exercise 3.10: By Theorem 3.18, . The integral converges at infinity for any r. At zero it converges only for .

Exercise 3.11: f _M (y) = ? _?? ^? f _B,M (x,y) dx = by (3.16).

Exercise 3.12: min _s?t B(s) = -max _s?t ? B(s). Let W(t) = ?B(t), then it is also a Brownian motion and we have P(B(t) ? x, min _s?t B(s)