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Question 6 In this question assume that {B(t), t = 0} is the standard Brownian motion (BM). (a) Find P{B(1) + B(2)

> 2}. (b) For 0 < s < t, write a formula for the conditional p.d.f. of B(t) given B(s) = x. (c) For 0 < s < t, write a formula for the conditional p.d.f. of B(s) given B(t) = x. d) Using conditions defining the BM,give a detailed proof of the Brownian scaling property: {X(t), t ≥ 0} defined by X(t) = √ cB(t/c) is also a BM, for c > 0 fixed. (e) Which of the following defines a BM: −2B(t/4), √ tB(1), B(2t) − B(t), B(t + 1) − B(1)? (f) For the geometric BM {X(t) = e B(t) , t ≥ 0}, calculate E[X(t)] and V ar[X(t)].

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