The Integrated Brownian
Motion for the study of the
atomic clock error
Gianna Panfilo
Istituto Elettrotecnico Nazionale “G. Ferraris”
Politecnico of Turin
Patrizia Tavella
Istituto Elettrotecnico Nazionale “G. Ferraris”
Turin
Stochastic Methods in Mathematical Finance
15 September 2005
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In the past
This work started in 2001 with my graduate thesis developed in
collaboration between the University “La Sapienza” (Bruno Bassan) and
IEN “Galileo Ferraris” (Patrizia Tavella), one of the Italian metrological
institutes.
G.Panfilo, B.Bassan, P.Tavella. “The integrated Brownian motion for the study of
the atomic clock error”. VI Proceedings of the “Società Italiana di Matematica
Applicata e Industriale” (SIMAI). Chia Laguna 27-31 May 2002
Now
I have continued this work in my Doctoral study in “Metrology” at Turin
Polytechnic and IEN “Galileo Ferraris” also in collaboration with BIPM
(Bureau International des Poids et Measures)
“The mathematical modelling of the atomic clock error with application
to time scales and satellite systems”
Stochastic Methods in Mathematical Finance
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The aim:
We are interested in the evaluation of the probability
that the clock error exceeds an allowed limit a certain time after
synchronization.
clock error
n
T(-m,n)
t
-m
The atomic clock error can be modelled by stochastic processes
T(-m,n) the first passage time of a stochastic process across two
fixed constant boundaries
Survival probability
Stochastic Methods in Mathematical Finance
15 September 2005
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Summary
 The stochastic model of the atomic clock error obtained by the
solution of the stochastic differential equations.
 Survival probability.
 Link between the stochastic differential equations (SDE) and
partial differential equations (PDE): infinitesimal generator.
the
 Numerical solution:
Monte Carlo method for SDE
Finite Differences Method for PDE
Finite Elements Method for PDE.
 Application: Model of the atomic clock error and Integrated Brownian
motion.
 Application to rubidium clock used in spatial and industrial
applications.
Stochastic Methods in Mathematical Finance
15 September 2005
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The atomic clock model
The atomic clock model can be expressed by the solution of the
following stochastic differential equation:
dX 1 t   X 2 t dt   1 dW1 t 

dX 2 t   dt   2 dW2 t 
The exact
solution is:
 X 1 0  x0

 X 2 0  y0
with initial conditions
t

t2
 X 1 t   x0  y0t     1 W1 t    2 0 W2 s ds
2

 X t   y   t   W t 
0
2
2
 2
The stochastic processes involved in this model are:
Brownian Motion (BM)
Integrated Brownian Motion (IBM)
Observation: The IBM is given by the same system without the term
1W1 which represents the contribution of the BM.
Stochastic Methods in Mathematical Finance
15 September 2005
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…and iterative form
The solution can be expressed in an iterative form useful for exact
simulation
t k 1

2
 X 1 t k 1   X 1 t k   X 2 t k      1 W1 t k 1   W1 t k    2 t W2 s ds
k
2

 X t   X t       W t   W t 
2 k
2
2 k 1
2 k
 2 k 1
Innovation
where   tk 1  tk
clock error
X1 t 
t3
   t 
3
2
1
100
2
2
50
10
-50
20
30
40
t
-100
-150
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The infinitesimal generator
The infinitesimal generator A of a homogeneous Markov process Xt , for
t0  t  T , is defined by:
1
Ag ( x)  lim Tt g ( x)  g ( x)
t 0  t
Ag(x) is interpreted as the mean infinitesimal rate
of change of g(Xt) in case Xt=x
where:
•Tt is an operator defined as:
Tt g ( x)   g ( y ) f (t , x, dy )  E x g ( X t )
•g is a bounded function
•Xt is a realization of a homogeneous stochastic Markov process
• f t , x, B  P X t  s  B| X s  x is the transition probability density
function
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Link between the stochastic differential equations
and the partial differential equations for diffusions
Stochastic differential equation:
dX t  b( X t )dt   ( X t )dWt
Infinitesimal generator Lt:

1 m
Lt    T
2 i , j 1

i, j
m
2

  bi ( x)
xi x j i 1
xi
Partial differential equation for the
transition probability f:
(Kolmogorov’s backward equation)
f
 Lt f  0
t
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The survival probability
Other functionals verify the same partial differential equation but with
different boundary conditions.
Example: the survival probability p(x,t):
pt , x   PT m,n   t | X 0  x 
p

L
p

on D  0, T 
 t
t

 p(0, x)  1D
 p(t , x)  0 on D  0, T 


1 D \ D
where: •1D is the indicator function
1D  
0 D
•[0,T]- time domain
•D- spatial domain
• D - boundary of the domain D
Stochastic Methods in Mathematical Finance
15 September 2005
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PDE for the clock survival probability
For the complete model (IBM+BM):
p
p
2 1 p
2 1 p
 y  1
2  2
t
x
2 x
2 y 2
2
2
t [0, T ]
x [ m, n]
y R
=0
Integrated Brownian motion
Brownian Motion
It is not always possible to derive the analytical solution!!!
Monte Carlo Method
applied to SDE.
Numerical Methods applied to PDE:
a) Finite Differences Method
b) Finite Elements Method
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Example: The Integrated Brownian Motion
The Integrated Brownian motion is defined by the following Stochastic
Differential Equation:
t

t2
 X 1 t   x  yt     2 0 W2 s ds
dX 1 (t )  X 2 (t )dt
2


 X t   y  t   W t 
dX 2 (t )  dt   2 dW2 (t )
2 2
 2
To have the survival probability we have to solve:
2
 p
p
1

p
2

y


t  [0, T ] ( x, y )  [  m, n]  R  D
2

2
x
2 y
 t
 p (t , n, y )  0 y  0

 p (t , m, y )  0 y  0
 p (t , x,)  0
Numerical Methods:

 p (t , x, )  0
A) Monte Carlo

 p (0, x, y )  1D
B) Finite Differences
It doesn’t exist the analytical solution
C) Finite Elements
Stochastic Methods in Mathematical Finance
15 September 2005
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=0
SDE
PDE
The survival probability for IBM
It’s not possible to solve analytically the PDE for the survival probability
of the IBM process. Appling the Monte Carlo method to SDE and
difference finites method to PDE we obtain:
p
p
1
Monte Carlo
Finite Differences ht=0.05
Finite Differences ht=0.01
1
0.9
0.8
0.95
0.7
0.6
hx = 0.04
hy = 0.5
ht = 0.05
ht = 0.01
0.9
0.5
0
0.5
1
t
0.4
0.3
0.2
σ 1
N =105 trajectories
τ = 0.01 discretization step
0.1
0
-m=n = 1
0
2
4
6
8
10
12
14
t
The two numerical methods agree to a large extent.
Difficulties arises in managing very small discretization steps.
Stochastic Methods in Mathematical Finance
15 September 2005
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IBM:Application to atomic clocks
Atomic Clock: Rubidium
IBM
Considering different values for the
boundaries m and for the survival
probabilities:
Experimental data
p
m [ns] \ p
10
30
50
100
300
500
1
0.9
0.8
n=-m= 350 ns
0.7
0.6
90%
0.5
0.9
1.3
2.1
4.4
6.1
95%
0.4
0.8
1.2
1.9
3.9
5.5
99%
0.3
0.7
1
1.6
3.3
4.6
0.5
0.4
For example
0.3
0.2
±10 ns
0.1
0
0
5
10
15
20
25
30
35
0.4 days (0.95)
40
t [days]
Stochastic Methods in Mathematical Finance
15 September 2005
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Complete Model (IBM+BM): Survival Probability
By the numerical methods we obtain the survival probability of the
complete model:
p
1
0.8
0.85
0.7
0.6
0.5
0.7
0
0.15
0.35
N =105 trajectories
τ = 0.01 discretization step
0.5
t
0.4
0.3
σ1  σ 2  1
0.2
0.1
0
0
0.5
1
1.5
hx = 0.2
hy = 0.5
ht = 0.01
Finite Differences
Finite Elements
Monte Carlo
1
p
0.9
2
2.5
3
t
-m=n = 1
h = 0.01
For the finite elements hx = 0.02
y
method
ht = 0.003
The Monte Carlo method and the finite elements method agree for
any discretization step. For the difference finites method the
difficulties arises in managing very small discretization steps.
Stochastic Methods in Mathematical Finance
15 September 2005
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Complete Model (IBM+BM):Application to atomic clocks
Atomic Clock: Rubidium
IBM
Complete Model (IBM+BM)
Experimental data
p
1
0.9
m = 8 ns
0.8
0.7
0.6
Considering different values for the
boundaries m and for the survival
probabilities:
m [ns] \ p
10
30
50
100
300
500
90%
0.24
0.8
1.2
2
4.4
6.1
95%
0.2
0.7
1
1.8
3.9
5.5
99%
0.1
0.6
0.9
1.5
3.2
4.5
0.5
For example
0.4
0.3
±10 ns
0.2
0.2 days (0.95)
0.1
0
0
0.5
1
1.5
2
2.5
3
3.5
4
t [days]
Stochastic Methods in Mathematical Finance
15 September 2005
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Applications
In GNSS (GPS, Galileo) the localization accuracy
depends on error of the clock carried by the
satellite. When the error exceeds a maximum
available level, the on board clock must be resynchronized.
Our model estimates that we are confident with
probability 0.95 that the atomic clock error is
inside the boundaries of 10 ns for 0.2 days (about 5
hours) in case of Rubidium clocks.
Calibration interval : In industrial measurement process the
measuring instrument must be periodically calibrated. Our model
estimates how often the calibration is required.
Stochastic Methods in Mathematical Finance
15 September 2005
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Perspectives
It’s necessary to use other stochastic process to describe the
behaviour of different atomic clock error.
We have considered the Ornstein-Uhlembeck process to model the
filtered white noise.
x
20
15
30 realizations of the
Brownian Motion (red) and
Ornstein-Uhlembeck (blue)
10
5
0
-5
-10
-15
-20
0
5
10
15
20
25
30
35
40
45
50
t
Other stochastic processes used to metrological application can be
1. The Integrated Ornstein-Uhlembeck
2. The Fractional Brownian Motion
Stochastic Methods in Mathematical Finance
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Conclusions
• Stochastic differential equations helps in modelling the atomic
clock errors
• Using the atomic clock model
clock behavior prediction
• By the SDE or related PDE the survival probability of a
stochastic process is obtained.
•The use of the model of the atomic clock error and the survival
probability are very important in many applications like the
space and industrial applications.
The authors thank Laura Sacerdote and Cristina Zucca from
University of Turin for helpful suggestions, support and
collaboration.
Stochastic Methods in Mathematical Finance
15 September 2005
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