Mathematisch-
Naturwissenschaftliche Fakultät
Fachbereich Mathematik
Prof. Dr. Andreas Prohl Dr. Ananta Kumar Majee
Stochastische Differentialgleichungen
Summer Semester 2017 Tübingen, 04.05.2017
Homework 3
Problem 1.Show that for anyf, g∈M2, there holds
E h
I(f)I(g) i
=E hZ ∞
0
f(s)g(s)ds i
.
Hint:Write the left-hand side in terms ofE h
I(f) +I(g)
2i andE
h
I(f)−I(g)
2i
, the right-hand side in terms ofE
hR∞ 0
f(s) +g(s)
2dsi andE
hR∞ 0
f(s)−g(s)
2dsi .
Problem 2.Show thatIT :Mstep2 →L2(Ω)is a linear map, i.e., for anyf, g∈Mstep2 and anyα, β ∈R
IT(αf+βg) =αIT(f) +βIT(g).
Generalize this property to Itô’s integralIT :MT2 →L2(Ω).
Problem 3.Show that each stochastic step processf ∈Mstep2 belongs toMT2 for allT >0, and that
IT(f) = Z T
0
f(s)dWs
is a martingale.
Problem 4.Show thatW2=
Wt2;t≥0 belongs toMT2 for eachT >0and verify the equality
Z T
0
Ws2dWs= 1 3WT3 −
Z T
0
Wsds,
where the integral on the right-hand side is a Riemann integral.
Hint:Use a partition of[0, T]to define an approximately stochastic process ofW2. The identity
a2(b−a) = 1
3(b3−a3)−a(b−a)2−1
3(b−a)3 can be applied to transform the sums approximating the stochastic integral.
Date of submission: 10.05.2017.
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