Sheet 1

Humboldt-Universität zu Berlin
Institut für Mathematik
Prof. Dr. Ulrich Horst
Continuous Time Finance
Exercise Sheet 1
Summer Term 2016
Exercise 1. The quadratic variation hXit up to time t ≥ 0 of an Itô processes
Z
Xt = X0 +
t
νs dWs
0
is defined by
n
X
hXit := lim
|Π|→0
t
Z
us ds +
0
(Xti+1 − Xti )2
in L2 (P),
i=0
where 0 = t0 < t1 < · · · tn = t and |Π| = max{|ti − ti−1 | : i = 1, . . . , n}. Show that
Z
hXit =
t
νs2 ds.
0
Exercise 2. Let g ∈ C 1,2 ([0, ∞) × R) with bounded derivatives and
Z
Xt = X0 +
t
Z
u(s, ω) ds +
0
t
v(s, ω) dWs (ω)
0
Rt
be an 1-dimensional Itô processes with ν ∈ V and 0 |u(s, ω)| ds < +∞ for all t ≥ 0 almost
surely. Proof Itô’s formula for the process Yt = g(t, Xt ):
dYt =
∂
∂
∂2
g(t, Xt ) dt +
g(t, Xt ) dXt + 2 g(t, Xt ) (dXt )2 ,
∂t
∂xi
∂x
where (dXt )2 is computed according to dWt dWt = dt and dWt dt = dt dWt = dt dt = 0.
Exercise 3. Let (Xt )t≥0 be an Itô process with quadratic variation hXit , t ≥ 0. Show that
1
1
S(X)t = exp
hXit sin(Xt ),
C(X)t = exp
hXit cos(Xt ),
2
2
define in the framework of Itô calculus analogs of the classical trigonometric functions. That is,
dS(X) = C(X) dX,
dC(X) = −S(X) dX.

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