Problem 2. We consider an N-step random walk with the standard notation seen in class. In partic- ular, we consider the random walk (Sn)A-0 and the standard filtration (Fn)-0. Out of the following processes, determine which ones are (i) adapted; (ii) previsible. • X (1) • • n := max Sj. 0

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Problem 2.
We consider an N-step random walk with the standard notation seen in class. In partic-
ular, we consider the random walk (Sn)A-0 and the standard filtration (Fn)-0. Out of
the following processes, determine which ones are (i) adapted; (ii) previsible.
• X (1)
•
•
n
:= max Sj.
0<j<n
X(2) = SN - Sn
X(3) = en - n.
X(4)
n
-
1
if Sn-1 Sn-2 = 1,
==
0
N
• X (5)
n
otherwise.
:= E[SN | Fn].
One of them is a martingale, find which.
Transcribed Image Text:Problem 2. We consider an N-step random walk with the standard notation seen in class. In partic- ular, we consider the random walk (Sn)A-0 and the standard filtration (Fn)-0. Out of the following processes, determine which ones are (i) adapted; (ii) previsible. • X (1) • • n := max Sj. 0<j<n X(2) = SN - Sn X(3) = en - n. X(4) n - 1 if Sn-1 Sn-2 = 1, == 0 N • X (5) n otherwise. := E[SN | Fn]. One of them is a martingale, find which.
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