Mathematisch-
Naturwissenschaftliche Fakultät
Fachbereich Mathematik
Prof. Dr. Andreas Prohl Dr. Ananta Kumar Majee
Stochastische Differentialgleichungen
Summer Semester 2017 Tübingen, 09.05.2017
Homework 4
Problem 1. Let(Ω,F,F,P) be a filtered probability space, whereF =
Ft;t ≥ 0 is such thatFt = σ
Ws; 0≤s≤t is generated by the Wiener processW. a) Show thatX=
|Wt|2−t;t≥0 is anF-martingale.
b) Show thatX=
Xt;t≥0 withXt= exp(Wt−2t)for allt≥0is anF-martingale.
Problem 2. Show that the exponential martingale X from Problem1,b) is an Itô process and verify that it satisfies the equation
dXt=XtdWt.
Hint:Don’t forget to show thatX∈MT2 for allT >0before applying Itô’s formula.
Problem 3.Letα >0andσ∈Rbe fixed. DefineY ={Yt;t≥0}via
Yt=σexp(−αt) Z t
0
exp(αs)dWs.
Show thatY satisfies
dYt=−αYtdt+σdWt (t≥0).
The processY is known as Ornstein-Uhlenbeck process.
Problem 4. Let(Ω,F,F,P) be the filtered probability space from Problem1, andX =
Xt; 0 ≤ t ≤ T ⊂ MT2. Show that there exists a continuous modification of Y =
Yt; 0 ≤ t ≤ T where Yt = Rt
0XsdWs=It(X).
Hint: Consider an approximating sequence
Xj;j ∈ N ⊂ Mstep2 . Use Doob’s inequality and Itô isometry to study convergence of
I(Xj);j ∈N , whereI(Xj) =
It(Xj); 0≤ t≤ T . Then involve the Tschebycheff inequality and the Borel-Cantelli lemma.
Date of submission: 17.05.2017.
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