Sheet 4 - Fakultät für Mathematik

Universität Wien WS 2016/17
Fakultät für Mathematik
Prof. Walter Schachermayer
Ludovic Tangpi
Mathematical Finance 1
Exercise sheet 4
Throughout this exercise sheet, (Ω, F, P ) is supposed to be a finite probability space.
1. Solve Exercise 3 of Sheet 3.
2. Let X be a random variable in L(Ω, F, P ) and {Ai , i ∈ {1, . . . , n}} a partition of
Ω with P (Ai ) > 0. Let G denote the σ-algebra generated by {Ai , i ∈ {1, . . . , n}}.
(i) Show that the conditional expectation E[X | G] of X with respect to G is
given by
n
X
E[X1Ai ]
E[X|G] =
1Ai .
P (Ai )
i=1
(ii) Let Ω := {ω1 , . . . , ω6 }, F = P(Ω), the power set. Put P (ωi ) = 1/8 for i =
1, . . . , 4 and P (ωi ) = 1/4 for i = 5/6 and G := {∅, Ω, {ω1 , ω2 }, {ω3 , . . . , ω6 }}
and X(ωi ) = i. Compute E[X | G].
3. Let X, Y be a random variables in L(Ω, F, P ) and H ⊂ G ⊂ F. Prove the
following properties of the conditional expectation:
E[E[X|G]] = E[X],
E[aX + b|G] = aE[X|G] + b, a, b ∈ R
X ≤ Y ⇒ E[X|G] ≤ E[Y |G],
E[E[X|G]|H] = E[X|H].

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