Proces Wienera

.
Proces Wienera
Karol Dziedziul
Gdansk
Karol Dziedziul
Proces Wienera
.
A schedule of the lecture
Hitting times
Reflection principle
Hitting time of a two sided barrier for
BM
Karol Dziedziul
Proces Wienera
.
Wiener process
Defintion
Let Wt be a stochastic process which satisfies the following conditions
(a) It starts at zero W (0) = 0
(b) It has independent increments: for any partition
0 ≤ t0 < t1 < · · · < tm ≤ 1 and any m the random variables
Wt1 − Wt0 , . . . , Wtm − Wtm−1 are independent.
(c) For every 0 ≤ s < t ≤ 1, Wt − Ws has a Gaussian distribution with
mean zero and variance σ 2 = t − s.
♣ Use Program 20 to create the estimation of a sample path of the
√
Wiener process. In algorithm we use a random walk with a δ = 1/ n
step. We start from 0, Sn (0) = 0. At the discrete moment t = kδ we are
in position Sn (t) and we decide where to go in a moment
t + δ = (k + 1)δ. Hence, we use the rule
ξ
Sn (t + δ) = Sn−1 (t) + √ = Sn−1 (t) + δξ
n
where the next step ξ ∈ N(0, 1) is independent of the past by the time t .
Next, we join two points with an interval.
Karol Dziedziul
Proces Wienera
.
Random walk
Namely, we create a piecewise linear process given by
Sn (t) =
ξ1 + · · · ξ[nt]
ξ[nt]+1
√
+ (nt − [nt]) √ ,
n
n
where ξj , j = 1, 2, ... are independent with the standard normal
distribution N(0, 1). Note that
Sn : Ω → C ([0, ∞)),
where C ([0, T ]) denotes the space of continuous function defined on
interval [0, 1].
To formulate the next construction of a Wiener process Wt , t ∈ [0, 1] we
need to define a Haar system.
Karol Dziedziul
Proces Wienera
.
Haar system
A wavelet system on R starts with one

 1 for
−1 for
h1 (t) =

0
function. Let
0 ≤ t < 2−1
2−1 ≤ t ≤ 1
t ∈ R \ [0, 1]
We define a wavelet (Haar) system on R: for all k, j ∈ Z
hk,j (x) = 2k/2 h1 (2k x − j),
x ∈ R.
This system is an orthonormal basis in L2 (R). This concept will be
explained later. Note that
j j +1
supphk,j := {x ∈ R : hk,j (x) 6= 0} = k , k .
2
2
Note that supphk,j ⊂ [0, 1] if and only if 0 ≤ j < 2k and k ≥ 0.
Karol Dziedziul
Proces Wienera
.
We define a Haar system on interval [0, 1] by choosing the functions hk,j ,
where 0 ≤ j < 2k and k ≥ 0 and restricting these functions to the
interval [0, 1]. We have to add also h0 , a constant function
h0 (t) = 1,
t ∈ [0, 1].
The equivalent definition for Haar system on [0, 1] is the following: for
any natural number n ≥ 1 there is a unique representation of n by
n = 2k + j,
0 ≤ j < 2k ,
and k, j ∈ N. For example
13 = 23 + 5,
20 = 24 + 4.
A Haar function with an index n ≥ 1 is defined for t ∈ [0, 1] by

k
2j
2j+1

 2 2 for 2k+1 ≤ t < 2k+1
k
hn (t) = hk,j (t)|[0,1] =
−2 2 for 2j+1
≤ t ≤ 2j+2
2k+1
2k+1


0
otherwise
and h0 is defined above.
Karol Dziedziul
Proces Wienera
.
We define an inner product for square integrable functions
Z 1
< f , g >=
f (x)g (x)dx.
0
By a square integrable function f we understand a function such that
Z 1
|f (x)|2 dx < ∞.
0
The space of such functions is denoted by L2 [0, 1]. Note that for all
n 6= m
< hn , hm >= 0
and for all n
< hn , hn >= 1.
The Haar system is an orthonormal basis in L2 [0, 1], i.e. for any
f ∈ L2 [0, 1] we have following representation called a reconstruction
formula
∞
X
f =
< f , hn > hn .
n=0
The convergence is in the norm of L2 [0, 1].
Karol Dziedziul
Proces Wienera
.
2
Orthonormal basis in L [0, 1]2
There is a similar Haar system Hn is in L2 [0, 1]2 . When transmitting data
(a picture f ) by internet, we send N coefficient < f , Hn > and then we
reconstruct f by fN
f ≈ fN =
N
X
< f , Hn > Hn .
n=0
If we want to enlarge a picture (in google earth, for instance), it is
sufficient to transmit only the next coefficients:
f ≈ fN+M =
N+M
X
< f , Hn > Hn = fN +
n=0
N+M
X
n=N+1
Karol Dziedziul
Proces Wienera
< f , Hn > Hn .
.
Ciesielski’s construction of Wiener process
A sample path of the Wiener process is piecewiese linear. Ciesielski’s
construction of Wiener process allows us to overcome the difficulty of
estimation.
Let {ξj } be a sequence of independent variables with the standard normal
distribution. Then
Z t
∞
∞
X
X
W (t) =
ξn
hn (s)ds =
ξn ϕn (t)
n=0
0
n=0
is the Wiener process. The series converges a.s. in C [0, 1]. The functions
Z t
hn (s)ds
ϕn (t) =
0
are known as Schauder’s functions. They are continuous and piecewise
linear. Moreover, if n = 2k + j, 0 ≤ j < 2k , then for all 0 ≤ m ≤ 2k ,
m 6= 2j + 1
m 2j + 1
= 1.
ϕn k+1 = 0, ϕn
2
2k+1
Karol Dziedziul
Proces Wienera
.
Let us assume that we have a process
ψK (t) :=
K
2X
−1
ξn ϕn (t).
n=0
To construct a piecewise linear trajectory of the process ψK (t) it is
sufficient to know the value of the trajectory in points
t=
i
,
2K
0 ≤ i ≤ 2K .
Note that
ψK +1 (t) =
+1
2KX
−1
ξn ϕn (t) = ψK (t) +
+1
2KX
−1
ξn ϕn (t).
n=2K
n=0
To draw a piecewise linear trajectory of the process ψK +1 we ONLY have
to know the value of a trajectory in points
t=
2i + 1
,
2K +1
Karol Dziedziul
0 ≤ i < 2K .
Proces Wienera
.
Why? The reason is simple
ψK (t) = ψK +1 (t),
for
t=
i
,
2K
0 ≤ i ≤ 2K .
From the definition of ϕn we obtain that for
t=
2i + 1
,
2K +1
0 ≤ i < 2K ,
we have
ψK +1
2i + 1
2K +1
=
ψK
i
2K
+ ψK
2
i+1
2K
+
1
√
2
K +1
ξ,
where a random variable ξ has the normal standard distribution N(0, 1)
and ξ is independent of previous generated random variables. This is a
key formula in Program 21.
♣ Use Program 21 to create a sample path of Wiener process.
Karol Dziedziul
Proces Wienera
.
Multivariate Gaussian distribution
We consider two dimensional vector
T
(X1 , X2 ) =
X1
X2
Let Σ denote the covariance matrix of (X1 , X2 )
a11 a12
Σ=
a21 a22 ,
where ai,j = Cov (Xi , Xj ) = EXi Xj − EXi EXj . Recall that
det Σ = a11 a22 − a12 a21 .
When Σ is nonsingular (det Σ 6= 0), then the density of Gaussian
distribution is defined for (x, y ) ∈ R 2 by
f (x, y ) =
T
1
1
√
e − 2 (x−µ1 ,y −µ2 ))Q(x−µ1 ,y −µ2 ) ,
2π detΣ
where µj = EXj .
Karol Dziedziul
Proces Wienera
.
The matrix Q = Σ−1 is invertible to matrix Σ, i..e.
1 0
QΣ = ΣQ =
.
0 1.
From algebra we know that there exits a nonnegative symmetric square
root of Σ, i.e. a matrix A such that
AAT = Σ
If (Z1 , Z2 ) are independent and have the standard normal distribution,
then a random vector A(Z1 , Z2 )T + µ has the same distribution as the
vector (X1 , X2 )T . This is clue formula how to generate a sample.
♣ Use Program 22 to generate a sample of (X1 , X2 )T , where µ = (1, 2)
1 −1
A=
.
0 2
Observe a histogram and a smooth estimator of the density.
Karol Dziedziul
Proces Wienera
.
Karol Dziedziul
Proces Wienera
.
Brownian motion
Let (Zt1 , Zt2 ) be two independent copies of the Wiener process. A sample
path of
t → (Zt1 , Zt2 ) ∈ R 2
is a visualization of a Brownian motion.
♣ Use Program 27 to generate sample paths of two-dimensional
Brownian motion.
Karol Dziedziul
Proces Wienera
.
Two-dimensional correlated Wiener process
We create two independent copies of the Wiener process Zt1 and Zt2 .
Next, we define a process
Wt = ρ1 Zt1 + ρ2 Zt2 ,
where ρ1 , ρ2 > 0 and
ρ21 + ρ22 = 1.
It turns out that Wt is the Wiener process. For each i = 1, 2 processes
(Zti , Wt ) define two dimensional correlated Wiener process. Therefore,
(Zti , Wt ) for i = 1, 2 are two dimensional Gaussian vectors and such that
Cov (Zti , Wt ) = ρi t, i = 1, 2.
♣ Use Program 23 to generate sample paths of these three Wiener
processes. Change the parameter ρi .
Karol Dziedziul
Proces Wienera
.
Laws of the iterated logarithm
Theorem (Laws of the iterated logarithm)
Let Wt be the Wiener process. Then
Wt
lim sup p
=1
t→∞
2t log log t
♣ Use Program 24 to check the thesis of Theorem
Theorem (Maximum process)
Let Wt be the Wiener process. Let
Mt = sup Ws .
0≤s≤t
Then for all a ≥ 0
P(Mt > a) = 2P(Wt > a) = P(|Wt | > a).
Karol Dziedziul
Proces Wienera
.
Karol Dziedziul
Proces Wienera
.
♣ Use Program 25 to check that random variables M1 and |W1 | have the
same distribution.
♣ Use Program 3 to create the arcsine law. This distribution is an
example of the beta distribution with parameter a = b = 0.5.
Theorem ( Levy Theorem)
Let Wt be the Wiener process, t ∈ [0, 1]. Then the optional times
τ1 = inf{t ∈ [0, 1] : Wt = M1 },
τ2 = sup{t ∈ [0, 1] : Wt = 0}
have the arcsine distribution.
♣ Use Program 26 to check the statement of the Theorem for the
optional time τ2 .
Karol Dziedziul
Proces Wienera
.
Hitting times
Let Wt be a Brownian motion. For Wt we denote by Ta the first hitting
time of a level (a barrier) a > 0 by Wt , i.e.
Ta = inf{t > 0 : Wt = a}.
Program 39. Ta is a random variable such that for every scenario ω ∈ Ω,
Ta (ω) = t,
where Wt (ω) = a is the first moment of reaching the level a.
Question: what is a distribution of RV Ta for given a? Let us create an
empirical distribution of Ta for a = 0.5. We observed that there is a
chance that a trajectory does not reach the level a. If we observe
trajectories in time interval [0, 10] and if a trajectory does not reach the
level a = 0.5 then in this case we will put t = 10.5. Program 40.
Karol Dziedziul
Proces Wienera
.
Reflection principle
To calculate the distribution of Ta we need to use a reflection principle.
Let us define the maximum process of the Brownian motion by
Mt = sup Wt .
s≤t
Note that for y ≥ 0
{Mt ≥ y } = {Ty ≤ t}.
Obviously if Ty (ω) ≤ t it means that the Brownian motion reached the
level y to the time t, hence the maximum Mt ≥ y .
Theorem
For y ≥ 0, x ≤ y we have
P(Wt ≤ x, Mt ≥ y ) = P(Wt ≥ 2y − x).
Consequently if we take in the above formula x = y then for y ≥ 0
P(Wt ≤ y , Mt ≥ y ) = P(Wt ≥ y ).
Karol Dziedziul
Proces Wienera
.
The distribution function of hitting time
On the other hand for y ≥ 0
P(Wt > y , Mt ≥ y ) = P(Wt > y ) = P(Wt ≥ t).
The sets {Wt > y , Mt ≥ y } and {Wt ≤ y , Mt ≥ y } are disjoint. Thus
for y ≥ 0
P(Mt ≥ y ) = P(Wt > y , Mt ≥ y ) + P(Wt ≤ y , Mt ≥ y ) = 2P(Wt > y ).
√
But Wt ∈ N(0, t). Hence for y ≥ 0
√
√
√
P(Wt > y ) = P(Wt / t > y / t) = 1 − Φ(y / t),
where Φ is a distribution function of the standard normal distribution.
Hence for y ≥ 0
√
Fy (t) = P(Ty ≤ t) = P(Mt ≥ y ) = 2(1 − Φ(y / t))
Karol Dziedziul
Proces Wienera
.
The density function of hitting time
The density function for t ≥ 0
√
f (t) = Fy0 (t) = φ(y / t)yt −3/2 ,
where φ is the density function of the standard normal distribution. Recall
2
1
φ(x) = √ e −x /2 .
2π
We can compare the empirical distribution with the theoretical one.
Program 41 y = 0.5. For given y > 0 the expected time of reaching the
barrier y is infinity i.e.
Z ∞
Z ∞
t −y 2 /(2t) y
1
√ dt = ∞,
ETy =
e
dt
∼
3/2
2π
t
t
0
1
There is not a singularity at t = 0, since
√
e −1/t
lim+ √ = lim e −u u = 0.
u→∞
t→0
t
Karol Dziedziul
Proces Wienera
.
Hitting time of a two sided barrier for BM
Let a < 0 < b and Ta , Tb the two hitting times. Recall
Ta = inf{t > 0 : Wt = a}.
Use Program 42 and 43 to see the following phenomena
Theorem
Let T ∗ = min{Ta , Tb } then
P(Ta < Tb ) =
b
.
a−b
Moreover
ET ∗ = −ab.
Compare two cases |a| > |b| and |a| < |b|. The density function is a
rather complicated.
Karol Dziedziul
Proces Wienera