Stochastic Processes II Summer term 2008 (Stochastische Analysis)
Prof. Dr. Uwe K¨ uchler Dr. Irina Penner
Exercises, 23rdth April
2.1 (2 points) Let (B
ti, t ≥ 0), i = 1, 2 be two independent Standard Brow- nian motions on some probability space (Ω, F , P ). Prove that
B
t:= 1
√ 2 (B
1t+ B
t2), t ≥ 0 is also a Standard Brownian motion.
2.2 (2+2 points) Let a be a continuous function of finite variation and let x be a continuous function with continuous quadratic variation hxi on [0, T ].
a) Show that x + a has continuous quadratic variation and hx + ai = hxi.
b) Show that for f ∈ C
1[0, T ] and t ∈ [0, T ] there exists the Itˆ o-Integral Z
t0
f(x
s+ a
s)dx
sand for y := x + a it holds
Z
t0
f (x
s+ a
s)dx
s= Z
t0
f(y
s)dy
s− Z
t0
f(y
s)da
s.
2.3 (2+2 points) Let (B
t, t ≥ 0) be a Standard Brownian motion on some probability space (Ω, F , P ).
a) Compute
Z
t0
B
nsdB
s,
where n ∈ N .
b) Let
W
t:= σB
t+ µt, t ≥ 0 for some σ 6= 0, µ ∈ R . Show that
X
t:= exp
σB
t+ µt − σ
22 t
, t ≥ 0 is a solution of a stochastic differential equation
dX
t= X
tdW
t, X
0= 1 in the Itˆ o-sense.
2.4 (1+2+2 points) Let (B
t)
t≥0be a Standard Brownian motion on some probability space (Ω, F , P ).
a) Let h(s) (0 ≤ s ≤ 1) be a continuous function of finite variation and with h(1) = 0. Using Itˆ o’s product formula show that for P -almost each trajectory B (ω) the equality
Z
10
h(s)dB
s(ω) = − Z
10
B
s(ω)dh(s), (1) holds, and conclude that
E Z
10
h(s)dB
s= 0.
b) Wiener and Paley have used (1) for definition of stochastic integral on the left-hand side (the right-hand side is well defined in classical measure theory). Moreover, they have extended the definition of the integral for arbitrary deterministic integrals h ∈ L
2[0, 1] using the isometry
E
"
Z
10
h(s)dB
s 2#
= Z
10
h(s)
2ds. (2) Prove (2) and sketch the construction of the “Wiener-integral” via isometry.
c) Show that the random variable M
t:= R
t0
h(s)dB
sis normally dis- tributed with expectation 0 and variance R
t0
