Mathematisch-
Naturwissenschaftliche Fakultät
Fachbereich Mathematik
Prof. Dr. Andreas Prohl Dr. Ananta Kumar Majee
Stochastische Differentialgleichungen
Summer Semester 2017 Tübingen, 17.05.2017
Homework 5
Problem 1.Let θn
∞
n=1 ⊂L1(0, T)be a sequence of functions such that(n∈N) θn+1(t)≤f(t) +β
Z t
0
θn(s)ds ∀t∈[0, T]. (1)
Show that
θn+1(t)≤f(t) +β Z t
0
f(ξ) exp β[t−ξ]
dξ+βn Z t
0
(t−ξ)n−1
(n−1)! θ1(ξ)dξ.
Remark:In the lecture, we used this result forf(t)≡αandθ1(t)≡Cin (1), which yields the estimate θn+1(t)≤αexp(βt) +C(βt)n
n! .
Problem 2.Letτ1, τ2 be twoF-stopping times. Then show that i)
τ1< t ∈ Ft, and also
τ1 =t ∈ Ftfor allt≥0.
ii) min
τ1, τ2 and max
τ1, τ2 areF-stopping times.
Problem 3.We considerXto be the strong solution of the SDE considered in the lecture. LetE ⊂Rn be nonempty, open or closed. Then the hitting time
τ := inf
t≥0|Xt∈ E is anF-stopping time.
Hint:Consider the case whereE is open independent from the case whereEis closed.
Problem 4.Letf ∈MT2, andτ be anF-stopping time such that0≤τ ≤T. We define Z τ
0
fsdWs:=
Z T
0
1{s≤τ}fsdWs.
Show that i) E
hRτ
0 fsdWs
i
= 0.
ii) E hRτ
0 fsdWs
2i
=E hRτ
0 fs2ds i
.
Date of submission: 24.05.2017.
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