(12 marks) The process {Xt}te[0,1] is called non-random and simple if there exists a partition II = = {tn}N-, of [0, T],

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12 Marks The Process Xt Te 0 1 Is Called Non Random And Simple If There Exists A Partition Ii Tn N Of 0 T 1 (136.64 KiB) Viewed 32 times

(12 marks) The process {Xt}te[0,1] is called non-random and simple if there exists a partition II = = {tn}N-, of [0, T], where 0 = to st2 <... <tn =T, such that, for every n, 0 <n<N, Xtn is non-random and te [0, tı), Xti te (t1, t2), Xt = XtN-1 teſt N-1,tn). Xto For teſtk, tk+1), define the random sum k-1 It = Xtn (Wtnt1 – Wtn) + Xtx (W+ – Wtx). n=0 a. (2 marks) Show that whenever 0 < s <t <T, the increment It – Ig is independent of Fs. b. (4 marks) Show that whenever 0 < s < t < T, the increment It I, is a normally distributed random variable. Find its mean and variance. c. (6 marks) For each of the following process, mathematically justify whether or not it is a martingale: ({Itte[]; 1}; (b.1) {1}10 10,7), (b.2) {f}}c(0.71 (1,3) {1} – L. (X.)'du}.com 12 ?