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Stochastic Calculus & SDEs
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01 — Article
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Quiz
Quiz: Brownian Motion and Quadratic Variation
Module 1 of 4 · Medium
Quick Quiz
1.
Which of the following is NOT one of the defining properties of standard Brownian motion?
W
0
=
0
W_0 = 0
W
0
=
0
almost surely
Almost surely continuous sample paths
Almost surely differentiable sample paths
Independent increments with
W
t
−
W
s
∼
N
(
0
,
t
−
s
)
W_t - W_s \sim \mathcal{N}(0, t-s)
W
t
−
W
s
∼
N
(
0
,
t
−
s
)
2.
For
Q
n
=
∑
i
=
1
n
(
W
t
i
−
W
t
i
−
1
)
2
Q_n=\sum_{i=1}^n (W_{t_i}-W_{t_{i-1}})^2
Q
n
=
∑
i
=
1
n
(
W
t
i
−
W
t
i
−
1
)
2
on an equal-mesh partition of
[
0
,
t
]
[0,t]
[
0
,
t
]
with
h
=
t
/
n
h=t/n
h
=
t
/
n
, what is
V
a
r
(
Q
n
)
\mathrm{Var}(Q_n)
Var
(
Q
n
)
?
t
2
/
n
t^2/n
t
2
/
n
2
t
2
/
n
2
2t^2/n^2
2
t
2
/
n
2
2
t
2
/
n
2t^2/n
2
t
2
/
n
2
t
2
2t^2
2
t
2
3.
Why can
∫
0
T
f
t
d
W
t
\int_0^T f_t\,dW_t
∫
0
T
f
t
d
W
t
not be defined pathwise as a Lebesgue-Stieltjes integral for Brownian motion?
Because
W
W
W
has infinite total variation
Because
W
W
W
is not a measurable process
Because
W
W
W
has infinite quadratic variation
Because
W
W
W
is not a martingale
4.
For a continuously differentiable function
f
:
[
0
,
T
]
→
R
f:[0,T]\to\mathbb{R}
f
:
[
0
,
T
]
→
R
, the quadratic variation
[
f
]
T
=
0
[f]_T = 0
[
f
]
T
=
0
.
True
False
5.
Lévy's characterisation says a continuous local martingale
M
M
M
with
M
0
=
0
M_0=0
M
0
=
0
is a standard Brownian motion if and only if:
M
t
∼
N
(
0
,
t
)
M_t\sim\mathcal{N}(0,t)
M
t
∼
N
(
0
,
t
)
for each fixed
t
t
t
M
M
M
has independent increments
[
M
]
t
=
t
[M]_t=t
[
M
]
t
=
t
for all
t
≥
0
t\geq 0
t
≥
0
E
[
M
t
2
]
=
t
\mathbb{E}[M_t^2]=t
E
[
M
t
2
]
=
t
for all
t
≥
0
t\geq 0
t
≥
0
6.
Which of these processes is a martingale with respect to the Brownian filtration?
W
t
2
−
t
W_t^2 - t
W
t
2
−
t
W
t
2
+
t
W_t^2 + t
W
t
2
+
t
W
t
2
W_t^2
W
t
2
e
W
t
e^{W_t}
e
W
t
Submit
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