L¨ohr/Winter Winter term 2015/16
Exercises to the lecture Probability Theory II
Exercise sheet 12
Brownian motion II
Let (Bt)t≥0 be a standard Brownian motion.
Exercise 12.1 (Quadratic variation).
Lett >0,tn,k:=tk2−n, and ∆n,k:=Btn,k −Btn,k−1 (n∈N,k= 1, . . . ,2n). Prove that
2n
X
k=1
∆2n,k → t a.s. and inL2.
Hint: Show first L2-convergence. Then use Borel-Cantelli.
Remark: It is possible to show that the classical quadratic variation of a Brownian path (the supremum over all finite partitions of [0, t]) is a.s. infinite.
The Optional-Stopping Theorem is also valid for continuous martingales in continuous time:
If (Xt)t≥0 is a martingale with continuous paths, and τ a stopping time, then (Xτ∧t)t≥0 is a martingale. This may be used in the following.
Exercise 12.2 (Brownian motion and optional sampling).
(a) Let a <0,b >0 and τ := inf t≥0
Bt∈ {a, b} . Calculate P {Bτ =b} . (b) Let τ as in (a). Calculate E(τ).
(c) Let τb = inf{t≥0|Bt=b}. Show that
E e−sτb
= e−b√2s ∀s≥0.
Hint: Use that similar to Exercise 11.1, (exBt−12x2t)t≥0 is a martingale.
Exercise 12.3.
Show: limt→0 1
tBt= 0 a.s.
Hint: Use the reflection principle and the Markov property.
Please turn
Donsker’s Theorem: Let C [0,1]
be the space of continuous functions from [0,1] to R, equipped with the sup-norm. LetY1, Y2, . . . be independent, identically distributed random variables with E(X1) = 0, Var(X1) =σ2, 0< σ2<∞. Fort∈[0,1], let
Stn:=
P⌊nt⌋ i=1 Yi
√σ2n . Then
(Stn)t∈[0,1] n=→∞⇒ (Bt)t∈[0,1], where the convergence in law is on the spaceC [0,1]
. Exercise 12.4 (Donsker for the Brownian bridge).
LetU1, U2, . . .be independent and uniformly distributed on [−1,1], and for n∈N
Zt(n) := (1− nt)
⌊t⌋
X
k=1
Uk−nt n
X
k=⌊t⌋+1
Uk+ t− ⌊t⌋ U⌊t⌋+1.
Show that
q3 nZnt(n)
t∈[0,1]
n=→∞⇒ (Xt)t∈[0,1],
where (Xt)t≥0 is a Brownian bridge and the convergence is convergence in law on the space C [0,1]
.
Hint: Use Donsker and a suitable map F:C [0,1]
→ C [0,1]
.
Das Blatt wird am 03.02. in der ¨Ubung besprochen, braucht aber nicht abgegeben zu werden und wird nicht korrigiert.
Probability Seminar:
26.01.: Mihail Zervos (London School of Economics) .
02.02.: Georgiy Shevchenko (Taras Shevchenko University Kiev)
Fine integral representations through small deviations of quadratic variation
Tue, 16:15 – 17:15in WSC-S-U-3.03