Günter Last
Institut für Stochastik
Karlsruher Institut für Technologie
Unbiased shifts of Brownian motion
and balancing random measures
Günter Last (Karlsruhe)
joint work with
Peter Mörters (Bath) and Hermann Thorisson (Reykjavik)
presented at the
10th German Probability and Statistics Days 2012
Stochastik-Tage Mainz
08.03.2012
1. Definition of an unbiased shift
Setting
B = (Bt )t∈R is a two-sided standard Brownian motion starting in
0 (B0 = 0) defined on its canonical probability space (Ω, A, P0 ).
Definition
An unbiased shift (of B) is a random time T (negative values
are allowed) such that:
B (T ) := (BT +t − BT )t∈R is a Brownian motion,
B (T ) is independent of BT .
Günter Last
Unbiased shifts of Brownian motion
Example
If T ≥ 0 is a stopping time, then (BT +t − BT )t≥0 is a one-sided
Brownian motion independent of BT . However, the example
T := inf{t ≥ 0 : Bt = a}
shows that (BT +t − BT )t∈R need not be a two-sided Brownian
motion.
Example
Consider a deterministic T ≡ t0 . Then B (T ) = (Bt0 +t − Bt0 )t∈R is
(T )
a two-sided Brownian. However, since B−t0 = −Bt0 , this
two-sided motion is not independent of BT = Bt0 .
Günter Last
Unbiased shifts of Brownian motion
Remark
An unbiased shift with BT = 0 is characterized by
d
(BT +t )t∈R = B.
According to Mandelbrot (The Fractal Geometry of Nature)
...the process of Brownian zeros is stationary in a weakened
”
form.“ He is using the (non-rigorous) concept of conditional
stationarity.
However, the stopping time
T := inf{t ≥ 1 : Bt = 0}
has the property BT = 0. But clearly B (T ) is not a Brownian
motion. Something is missing!
Günter Last
Unbiased shifts of Brownian motion
Definition
Let `0 be the local time (random measure) at zero. Its
right-continuous (generalised) inverse is defined as
(
sup{t ≥ 0 : `0 [0, t] = r },
r ≥ 0,
Tr :=
0
sup{t < 0 : ` [t, 0] = −r },
r < 0.
Theorem
Let r ∈ R. Then Tr is an unbiased shift.
Idea of the proof: The intervals [Tn , Tn+1 ], n ∈ Z, split B into
iid-cycles.
Günter Last
Unbiased shifts of Brownian motion
2. Local time
Definition
The local time measure `x at x ∈ R can be defined by
Z
1
x
` (C) := lim
1{s ∈ C, x ≤ Bs ≤ x + h}ds.
h→0 h
Hence
Z
ZZ
f (Bs , s)ds =
f (x, s)`x (ds)dx
P0 -a.s.
for all measurable f : R2 → [0, ∞).
Günter Last
Unbiased shifts of Brownian motion
Definition
For t ∈ R the shift θt : Ω → Ω is given by
(θt ω)s := ωt+s ,
s ∈ R.
For x ∈ R let
Px := P0 (B + x ∈ ·),
x ∈ R,
where B is the identity on Ω.
Remark
It is a possible to choose a perfect version of local times, that is
a (measurable) kernel satisfying for all x ∈ R and Px -a.e. that `x
is diffuse and
`x (θt ω, C − t) = `x (ω, C),
C ∈ B, t ∈ R,
`x (B, ·) = `0 (B − x, ·).
Günter Last
Unbiased shifts of Brownian motion
3. Palm measures and allocation rules
Definition
Let
Z
P :=
Px dx
be the distribution of a Brownian motion with a uniformly
”
distributed“ starting value.
Remark
A consequence of the stationary increments of B is the
invariance property
P = P ◦ θs ,
Günter Last
s ∈ R.
Unbiased shifts of Brownian motion
Definition
A random measure ξ on R is a kernel from Ω to R such that
ξ(ω, B) < ∞ for P-a.e. ω and all compact B ⊂ R. It is called
invariant if P-a.e.:
ξ(θt ω, C − t) = ξ(ω, C),
C ∈ B, t ∈ R.
In this case its Palm measure Qξ is defined by
Z
Qξ (A) := E 1[0,1] (s)1A (θs B)ξ(ds), A ∈ A.
If the intensity λξ := EP ξ[0, 1] = Qξ (Ω) is positive and finite,
then the Palm probability measure of ξ is defined by
Q0ξ := λ−1
ξ Qξ .
Günter Last
Unbiased shifts of Brownian motion
Theorem (Geman and and Horowitz ’73)
The Palm (probability) measure of the local time `x is Px .
Definition
Let ν be a probability measure on R. Define
Z
Z
ν
Pν := Px ν(dx),
` := `x ν(dx).
Corollary
Pν is the Palm probability measure of `ν .
Remark
In the language of stochastic analysis `ν is a continuous
additive functional with Revuz measure ν.
Günter Last
Unbiased shifts of Brownian motion
Definition
An allocation rule is a measurable mapping τ : Ω × R → R that
is equivariant in the sense that
τ (θt ω, s − t) = τ (ω, s) − t,
s ∈ R, t ∈ R, P-a.e. ω ∈ Ω.
Theorem (L. and Thorisson ’09)
Let ξ and η be two invariant random measures with positive and
finite intensities. Let τ be an allocation rule and define
T := τ (·, 0). Then
Qξ (θT B ∈ ·) = Qη
iff τ is balancing ξ and η, that is
Z
1{τ (s) ∈ ·}ξ(ds) = η
Günter Last
P-a.e.
Unbiased shifts of Brownian motion
4. Unbiased shifts and balancing allocation rules
Definition (Skorokhod embedding problem)
Let ν be a probability measure on R. A random time T embeds
ν if BT has distribution ν.
Theorem
Let T be a random time and ν be a probability measure on R.
Then T is an unbiased shift embedding ν if and only if the
allocation rule τ defined by τT (s) := T ◦ θs + s is balancing `0
and `ν .
Günter Last
Unbiased shifts of Brownian motion
Example
Let r > 0. Then
τ (s) := inf{t > s : `0 ([s, t]) = r },
s ∈ R.
Then τ is an allocation rule balancing `0 with itself. Hence
Tr = τ (·, 0) is an unbiased shift (embedding δ0 ).
Günter Last
Unbiased shifts of Brownian motion
5. Existence of unbiased shifts
Theorem
Let ν be a probability measure on R with ν{0} = 0. Then the
stopping time
T := inf{t > 0 : `0 [0, t] = `ν [0, t]}
embeds ν and is an unbiased shift.
Remark
The above stopping time above was introduced in Bertoin and
Le Jan (1992) as a solution of the Skorokhod embedding
problem.
Günter Last
Unbiased shifts of Brownian motion
Theorem
Let ν be a probability measure on R. Then there exists a
nonnegative stopping time that is an unbiased shift embedding
ν.
Theorem
Assume that ν{0} < 1. Then any unbiased shift T embedding ν
satisfies
P0 {T = 0} = 0.
If, however, ν = δ0 , then for any p ∈ [0, 1] there is an unbiased
shift T ≥ 0 embedding ν and such that P0 {T = 0} = p.
Günter Last
Unbiased shifts of Brownian motion
Theorem
Let ξ and η be jointly stationary orthogonal diffuse random
measures on R with finite and equal intensities. Then the
mapping τ : Ω × R → R, defined by
τ (s) := inf{t > s : ξ[s, t] = η[s, t]},
s ∈ R,
is an allocation rule balancing ξ and η.
Remark
The previous theorem holds in a more general stationary
setting. The assumption of equal intensities has to be replaced
by
E ξ[0, 1]I = E η[0, 1]I P-a.e.,
where I is the invariant σ-field. In the Brownian setting, P is
trivial on I. (If A ∈ I then either P(A) = 0 or P(Ac ) = 0.)
Günter Last
Unbiased shifts of Brownian motion
6. Moment properties of unbiased shifts
Theorem
If T is an unbiased shift embedding a probability measure
ν 6= δ0 , then
p
E0 |T | = ∞.
Idea of the proof:
Take an x > 0 such that ν[x, ∞) = P(BT > x) > 0.
On the event {BT > x}, T can be bounded from below by
the minimum of two independent hitting times for −x,
independent of BT .
Use the moment properties of hitting times.
Günter Last
Unbiased shifts of Brownian motion
Theorem
Suppose ν is a distribution with ν{0} = 0, and finite absolute
mean. If the stopping time T ≥ 0 is an unbiased shift
embedding ν, then
E0 T 1/4 = ∞.
Theorem
Let ν be as above and let T be the Bertoin/Le Jan stopping
time. Then, for all β ∈ [0, 1/4),
E0 T β < ∞.
Günter Last
Unbiased shifts of Brownian motion
Theorem
Let T ≥ 0 be an unbiased shift embedding δ0 . If T is non-trivial,
that is P0 {T = 0} < 1, then
E0 T = ∞.
Example
There is an unbiased shift T 6= 0 embedding δ0 , such that
Eeλ|T | < ∞ for some λ > 0.
Günter Last
Unbiased shifts of Brownian motion
7. Minimality
Definition
A random time T ≥ 0 is called minimal unbiased shift if it is an
unbiased shift and if for any other unbiased shift S embedding
P(BT ∈ ·) and such such that P0 (0 ≤ S ≤ T ) = 1 we have that
P0 (S = T ) = 1.
Theorem
The Bertoin-Le Jan stopping time T is a minimal unbiased shift.
Günter Last
Unbiased shifts of Brownian motion
8. References
J. Bertoin and Y. Le Jan (1992). Ann. Probab. 20,
538–548.
A.E. Holroyd, R. Pemantle, Y. Peres, and O. Schramm
(2009). Poisson Matching. Annales de l’institut Henri
Poincaré (B) 45, 266–287.
G. Last, P. Mörters and H. Thorisson (2011). Unbiased
shifts of Brownian motion. arXiv:1112.5373v1
G. Last and H. Thorisson (2009). Invariant transports of
stationary random measures and mass-stationarity. Ann.
Probab. 37, 790–813.
B. Mandelbrot (1982). The Fractal Geometry of Nature.
Freeman and Co., San Francisco.
Günter Last
Unbiased shifts of Brownian motion