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Conditional Expectation as an L² Projection
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Probability
Hilbert Spaces
Conditional Expectation
1
Article
2
Lab
3
Notebook
4
Quiz
Quick Quiz
1.
The characterisation of
E
[
X
∣
G
]
\mathbb{E}[X\mid\mathcal{G}]
E
[
X
∣
G
]
as the best predictor of
X
X
X
via an orthogonal projection requires which space structure?
L
1
(
F
)
L^1(\mathcal{F})
L
1
(
F
)
, under the mean-absolute-error norm
Any complete metric space of random variables
C
0
(
Ω
)
C^0(\Omega)
C
0
(
Ω
)
, the continuous functions on
Ω
\Omega
Ω
L
2
(
F
)
L^2(\mathcal{F})
L
2
(
F
)
, a Hilbert space with
⟨
X
,
Y
⟩
=
E
[
X
Y
]
\langle X,Y\rangle=\mathbb{E}[XY]
⟨
X
,
Y
⟩
=
E
[
X
Y
]
2.
Which property uniquely characterises
X
^
=
E
[
X
∣
G
]
\hat{X}=\mathbb{E}[X\mid\mathcal{G}]
X
^
=
E
[
X
∣
G
]
among all
G
\mathcal{G}
G
-measurable square-integrable variables?
X
^
\hat{X}
X
^
has the same variance as
X
X
X
X
^
\hat{X}
X
^
converges to
X
X
X
almost surely as information accumulates
The residual
X
−
X
^
X-\hat{X}
X
−
X
^
is orthogonal to every
Z
∈
L
2
(
G
)
Z\in L^2(\mathcal{G})
Z
∈
L
2
(
G
)
X
^
\hat{X}
X
^
is the maximum-likelihood estimator of
X
X
X
given
G
\mathcal{G}
G
3.
E
[
X
∣
G
]
=
arg
min
Y
∈
L
2
(
G
)
(
⋅
)
\mathbb{E}[X\mid\mathcal{G}]=\arg\min_{Y\in L^2(\mathcal{G})}(\,\cdot\,)
E
[
X
∣
G
]
=
ar
g
min
Y
∈
L
2
(
G
)
(
⋅
)
. Which objective is minimised?
P
(
X
≠
Y
)
\mathbb{P}(X\neq Y)
P
(
X
=
Y
)
E
[
∣
X
−
Y
∣
]
\mathbb{E}\,[\,|X-Y|\,]
E
[
∣
X
−
Y
∣
]
E
[
(
X
−
Y
)
2
]
\mathbb{E}\,[(X-Y)^2]
E
[(
X
−
Y
)
2
]
V
a
r
(
X
−
Y
)
\mathrm{Var}(X-Y)
Var
(
X
−
Y
)
4.
To verify
X
^
=
E
[
X
∣
G
]
\hat{X}=\mathbb{E}[X\mid\mathcal{G}]
X
^
=
E
[
X
∣
G
]
it suffices to check
E
[
(
X
−
X
^
)
1
A
]
=
0
\mathbb{E}[(X-\hat{X})\mathbf{1}_A]=0
E
[(
X
−
X
^
)
1
A
]
=
0
for every
A
∈
G
A\in\mathcal{G}
A
∈
G
, rather than for all
Z
∈
L
2
(
G
)
Z\in L^2(\mathcal{G})
Z
∈
L
2
(
G
)
.
True
False
5.
In Longstaff-Schwartz least-squares Monte Carlo, the continuation value
E
Q
[
C
k
+
1
∣
S
t
k
]
\mathbb{E}^{\mathbb{Q}}[C_{k+1}\mid S_{t_k}]
E
Q
[
C
k
+
1
∣
S
t
k
]
is estimated by:
Solving the Black-Scholes PDE separately along each simulated path
Regressing
C
k
+
1
C_{k+1}
C
k
+
1
on basis functions of
S
t
k
S_{t_k}
S
t
k
— an
L
2
L^2
L
2
projection
Taking the maximum discounted payoff across all simulated paths (perfect foresight)
Averaging
C
k
+
1
C_{k+1}
C
k
+
1
across every path, ignoring the current state
S
t
k
S_{t_k}
S
t
k
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