THE CHINESE UNIVERSITY OF HONG KONG
Department of Mathematics
MATH4210 Financial Mathematics 2020-2021 T1
Assignment 3
Due date: 28 November 2020 11:59 p.m.
Please submit this assignment on blackboard. If you have any questions re-
garding this assignment, please email TA Yang Fan ([email protected]).
1. Apply Itoˆ formula to represent the following process (Xt)t≥0 as an Itoˆ
process, i.e., Xt = X0 +
∫ t
0
bs ds+
∫ t
0
σs dBs for some process (bs, σs)s≥0.
(a) Xt = Bt.
(b) Xt = B
2
t .
(c) Xt = B
3
t .
(d) Xt = exp(−σ2t2 + σBt).
(e) Xt = sin(Bt).
2. Let f : [0, T ] → R be a bounded continuous (deterministic) function
and fn(t) =
n−1∑
i=0
αi1(ti,ti+1](t) for some deterministic constants αi and
discrete time interval 0 = t0 ≤ t1 ≤ t2 ≤ ... ≤ tn = T . Let
In :=
∫ T
0
fn(t) dBt and I :=
∫ T
0
f(t) dBt.
(a) Prove that In =
n−1∑
i=0
αi(Bti+1 −Bti) and
In ∼ N
(
0,
n−1∑
i=0
α2i (ti+1 − ti)
)
= N
(
0,
∫ T
0
fn(t)
2 dt
)
.
(b) Assume that fn
L2([0,T ])−−−−−→ f .
Prove that In
L2(Ω)−−−→ I and I ∼ N
(
0,
∫ T
0
f(t)2 dt
)
.
(c) Compute the law of the following random variables (defined by
stochastic integration):∫ T
0
t dBt,
∫ T
0
et dBt,
∫ T
0
cos(t) dBt.

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