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Itô's Lemma: Derivation and Applications
Hard
·
22 min read
Stochastic Calculus
Itô's Lemma
Stochastic Differential Equations
1
Article
2
Quiz
Quick Quiz
1.
In the Itô multiplication table, what is
(
d
W
t
)
2
(dW_t)^2
(
d
W
t
)
2
?
d
W
t
dW_t
d
W
t
(
d
t
)
2
(dt)^2
(
d
t
)
2
d
t
dt
d
t
0
2.
With
d
S
t
=
μ
S
t
d
t
+
σ
S
t
d
W
t
dS_t=\mu S_t\,dt+\sigma S_t\,dW_t
d
S
t
=
μ
S
t
d
t
+
σ
S
t
d
W
t
, applying Itô's lemma to
f
(
S
)
=
ln
S
f(S)=\ln S
f
(
S
)
=
ln
S
gives
d
(
ln
S
t
)
=
d(\ln S_t)=
d
(
ln
S
t
)
=
1
S
t
d
S
t
\tfrac{1}{S_t}\,dS_t
S
t
1
d
S
t
μ
d
t
+
σ
d
W
t
\mu\,dt+\sigma\,dW_t
μ
d
t
+
σ
d
W
t
(
μ
−
σ
2
2
)
d
t
+
σ
d
W
t
\left(\mu-\tfrac{\sigma^2}{2}\right)dt+\sigma\,dW_t
(
μ
−
2
σ
2
)
d
t
+
σ
d
W
t
(
μ
+
σ
2
2
)
d
t
+
σ
d
W
t
\left(\mu+\tfrac{\sigma^2}{2}\right)dt+\sigma\,dW_t
(
μ
+
2
σ
2
)
d
t
+
σ
d
W
t
3.
The Itô isometry states that for a square-integrable adapted process
σ
\sigma
σ
:
E
[
∫
0
T
σ
s
d
W
s
]
=
∫
0
T
E
[
σ
s
]
d
s
\mathbb{E}\!\left[\int_0^T \sigma_s\,dW_s\right]=\int_0^T \mathbb{E}[\sigma_s]\,ds
E
[
∫
0
T
σ
s
d
W
s
]
=
∫
0
T
E
[
σ
s
]
d
s
V
a
r
(
∫
0
T
σ
s
d
W
s
)
=
(
∫
0
T
σ
s
d
s
)
2
\mathrm{Var}\!\left(\int_0^T \sigma_s\,dW_s\right)=\left(\int_0^T \sigma_s\,ds\right)^2
Var
(
∫
0
T
σ
s
d
W
s
)
=
(
∫
0
T
σ
s
d
s
)
2
∫
0
T
σ
s
2
d
s
=
T
E
[
σ
0
2
]
\int_0^T \sigma_s^2\,ds=T\,\mathbb{E}[\sigma_0^2]
∫
0
T
σ
s
2
d
s
=
T
E
[
σ
0
2
]
E
[
(
∫
0
T
σ
s
d
W
s
)
2
]
=
E
[
∫
0
T
σ
s
2
d
s
]
\mathbb{E}\!\left[\left(\int_0^T \sigma_s\,dW_s\right)^2\right]=\mathbb{E}\!\left[\int_0^T \sigma_s^2\,ds\right]
E
[
(
∫
0
T
σ
s
d
W
s
)
2
]
=
E
[
∫
0
T
σ
s
2
d
s
]
4.
The Stratonovich integral
∫
0
T
f
(
W
t
)
∘
d
W
t
\int_0^T f(W_t)\circ dW_t
∫
0
T
f
(
W
t
)
∘
d
W
t
obeys the classical (Newton-Leibniz) chain rule, with no Itô correction term.
True
False
5.
Solving
d
S
t
=
μ
S
t
d
t
+
σ
S
t
d
W
t
dS_t=\mu S_t\,dt+\sigma S_t\,dW_t
d
S
t
=
μ
S
t
d
t
+
σ
S
t
d
W
t
with
S
0
S_0
S
0
given, the solution is
S
T
=
S_T=
S
T
=
S
0
e
μ
T
+
σ
W
T
S_0\,e^{\mu T+\sigma W_T}
S
0
e
μ
T
+
σ
W
T
S
0
+
μ
T
+
σ
W
T
S_0+\mu T+\sigma W_T
S
0
+
μ
T
+
σ
W
T
S
0
e
(
μ
−
σ
2
/
2
)
T
+
σ
W
T
S_0\,e^{(\mu-\sigma^2/2)T+\sigma W_T}
S
0
e
(
μ
−
σ
2
/2
)
T
+
σ
W
T
S
0
e
(
μ
+
σ
2
/
2
)
T
+
σ
W
T
S_0\,e^{(\mu+\sigma^2/2)T+\sigma W_T}
S
0
e
(
μ
+
σ
2
/2
)
T
+
σ
W
T
6.
For which regularity class of
f
f
f
does the classical statement of Itô's lemma hold?
f
f
f
merely measurable
f
f
f
Lipschitz in
x
x
x
f
f
f
continuous in
(
t
,
x
)
(t,x)
(
t
,
x
)
f
∈
C
1
,
2
f\in C^{1,2}
f
∈
C
1
,
2
— once in
t
t
t
, twice in
x
x
x
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