Stochastic Processes
Ariel Yadin
Exercise Sheet - Conditional Expectation
Exercise 1. Let X be an integrable random variable on (Ω, F, P). Let G ⊂ F be a sub-σ-algebra.
Then,
• If X ∈ G then E[X|G] = X. [ The average value of X given X is X itself. ]
• If G = {∅, Ω} then E[X|G] = E[X]. [ Given no information, the average value of X is
E[X]. ]
• If X = c for c a constant, then X is measurable with respect to the trivial σ-algebra
{∅, Ω} ⊂ G, so E[c|G] = c.
• If X is independent of G then E[X|G] = E[X]. [ Given no information about X, the
average value of X is E[X]. ]
• E[E[X|G]] = E[X].
Exercise 2. If Y = Y 0 a.s. then E[X|Y ] = E[X|Y 0 ]. [ Changing by measure 0 does not change
the conditioning. ]
Hint: Consider Z := E[X|σ(Y ) ∩ σ(Y 0 )]. What can be said about an event A ∈ σ(Y )4σ(Y 0 )?
Use this to show that Z = E[X|Y ] and Z = E[X|Y 0 ].
Exercise 3. E[aX + Y |G] = a E[X|G] + E[Y |G] a.s.
Exercise 4. If X ≤ Y then E[X|G] ≤ E[Y |G].
Exercise 5. Let G ∈ G. Show that for any event A with P[A] > 0,
P[G|A] =
E[P[A|G]1G ]
.
P[A]
Exercise 6 (Monotone Convergence). If (Xn )n is a monotone non-decreasing sequence of nonnegative integrable random variables, such that Xn % X for some integrable X, then E[Xn |G] %
E[X|G] a.s.
Hint: Let Yn = X − Xn . Consider the sequence E[Yn |G]. Is this a monotone sequence? What
can one say about Z = lim inf n E[Yn |G]? Use Fatou’s Lemma to show that Z = 0 a.s.
1
2
Exercise 7. Suppose that (Ω, F, P) is a probability space with Ω =
U
k∈I
Ak where Ak ∈ F for
all k ∈ I, with I some countable (possibly finite) index set. Show that
]
σ((Ak )k∈I ) =
Ak : J ⊂ I .
k∈J
Hint: Show that any set in the right-hand side must be in σ((Ak )k∈I ). Show that the righthand side is a σ-algebra.