the integrated brownian motion for the study of the atomic clock error
DESCRIPTION
The Integrated Brownian Motion for the study of the atomic clock error. Gianna Panfilo Istituto Elettrotecnico Nazionale “G. Ferraris” Politecnico of T urin Patrizia Tavella Istituto Elettrotecnico Nazionale “G. Ferraris” Turin. In the past. - PowerPoint PPT PresentationTRANSCRIPT
Stochastic Methods in Mathematical FinanceStochastic Methods in Mathematical Finance 15 September 200515 September 2005
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The Integrated Brownian The Integrated Brownian Motion for the study of Motion for the study of the atomic clock errorthe atomic clock error
Gianna PanfiloGianna PanfiloIstituto Elettrotecnico Nazionale “G. Ferraris”
Politecnico of Turin
Patrizia TavellaPatrizia TavellaIstituto Elettrotecnico Nazionale “G. Ferraris”Turin
Stochastic Methods in Mathematical FinanceStochastic Methods in Mathematical Finance 15 September 200515 September 2005
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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 Now
In the pastIn the past
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 FinanceStochastic Methods in Mathematical Finance 15 September 200515 September 2005
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The aim:The aim: We are interested in the evaluation of the probability that the clock error exceeds an allowed limit a certain time after synchronization.
Survival probabilitySurvival probability
T(-m,n) the first passage time of a stochastic process across two fixed constant boundaries
T(-m,n) the first passage time of a stochastic process across two fixed constant boundaries
n
-m
clock error
t
T(-m,n)
The atomic clock error can be modelled by stochastic The atomic clock error can be modelled by stochastic processesprocesses
The atomic clock error can be modelled by stochastic The atomic clock error can be modelled by stochastic processesprocesses
Stochastic Methods in Mathematical FinanceStochastic Methods in Mathematical Finance 15 September 200515 September 2005
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The stochastic model of the atomic clock error obtained by the solution of the stochastic differential equations.
SummarySummary
Numerical solution: Monte Carlo methodMonte Carlo method for SDEFinite Differences MethodFinite Differences Method for PDEFinite Elements MethodFinite 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.
Link between the stochastic differential equations (SDE) and the partial differential equations (PDE): infinitesimal generator.
Survival probability.
Stochastic Methods in Mathematical FinanceStochastic Methods in Mathematical Finance 15 September 200515 September 2005
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The atomic clock modelThe atomic clock model
tdWdttdX
tdWdttXtdX
222
1121
02
01
0
0
yX
xXwith initial conditions
tWtytX
dssWtWt
tyxtXt
2202
0 2211
2
001 2
The exact solution is:
The atomic clock model can be expressed by the solution of the following stochastic differential equation:
Observation: The IBM is given by the same system without the term 1W1 which represents the contribution of the BM.
Brownian Motion (BM)Integrated Brownian Motion (IBM)
The stochastic processes involved in this model are:
The stochastic processes involved in this model are:
Stochastic Methods in Mathematical FinanceStochastic Methods in Mathematical Finance 15 September 200515 September 2005
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Innovation
The solution can be expressed in an iterative form useful for exact simulation
where kk tt 1
……and iterative formand iterative form
10 20 30 40
-150
-100
-50
50
100
t
tX1
clock error
3
322
21
tt
kkkk
t
tkkkkk
tWtWtXtX
dssWtWtWtXtXtXk
k
2122212
221111
2
2111
1
2
Stochastic Methods in Mathematical FinanceStochastic Methods in Mathematical Finance 15 September 200515 September 2005
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)()(1
lim)(0
xgxgTt
xAg tt
The infinitesimal generatorThe infinitesimal generatorThe infinitesimal generator A of a homogeneous Markov process Xt , for , is defined by:
where:
Ag(x) is interpreted as the mean infinitesimal rate of change of g(Xt) in case Xt=x
Ag(x) is interpreted as the mean infinitesimal rate of change of g(Xt) in case Xt=x
)(),,()()( tx
t XgEdyxtfygxgT •Tt is an operator defined as:
•g is a bounded function•Xt is a realization of a homogeneous stochastic Markov process• is the transition probability density function
f t x B P X B X xt s s, , |
Ttt 0
Stochastic Methods in Mathematical FinanceStochastic Methods in Mathematical Finance 15 September 200515 September 2005
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Link between the stochastic differential equations Link between the stochastic differential equations and the partial differential equations for diffusionsand the partial differential equations for diffusions
Stochastic differential equation: tttt dWXdtXbdX )()(
0
fLt
ft
Partial differential equation for the transition probability f:
(Kolmogorov’s backward equation)
m
ji
m
i ii
jiji
Tt x
xbxx
L1, 1
2
, )(2
1
Infinitesimal generator Lt:
Stochastic Methods in Mathematical FinanceStochastic Methods in Mathematical Finance 15 September 200515 September 2005
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The survival probabilityThe survival probabilityOther functionals verify the same partial differential equation but with different boundary conditions.
Example: the survival probability p(x,t):
xXtTPxtp nm 0, |,
TDonxtp
xp
TDont
ppL
D
t
,00),(
1),0(
,0
•1D is the indicator function
•[0,T]- time domain•D- spatial domain• - boundary of the domain D
D
DDD 0
\11where:
D
Stochastic Methods in Mathematical FinanceStochastic Methods in Mathematical Finance 15 September 200515 September 2005
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PDE for the clock survival probabilityPDE for the clock survival probability
For the complete model (IBM+BM):
pt
ypx
px
py
12
2
2 22
2
2
12
12
t T
x m n
y
[ , ]
[ , ]
0
R
Integrated Brownian motion Brownian
MotionIt is not always possible to derive the analytical solution!!!It is not always possible to derive the analytical solution!!!
Numerical Methods applied to PDE:
a) Finite Differences Methodb) Finite Elements Method
Numerical Methods applied to PDE:
a) Finite Differences Methodb) Finite Elements Method
Monte Carlo Method applied to SDE.
Monte Carlo Method applied to SDE.
=0
Stochastic Methods in Mathematical FinanceStochastic Methods in Mathematical Finance 15 September 200515 September 2005
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Example: The Integrated Brownian MotionExample: The Integrated Brownian Motion
Dyxp
xtp
xtp
yymtp
yyntp
DRnmyxTty
p
x
py
t
p
1),,0(
0),,(
0),,(
00),,(
00),,(
],[),(],0[2
12
222
)()(
)()(
222
21
tdWdttdX
dttXtdX
tWtytX
dssWt
ytxtXt
222
0 22
2
1 2
The Integrated Brownian motion is defined by the following Stochastic Differential Equation:
Numerical Methods:
A) Monte Carlo
B) Finite Differences
SDE
C) Finite ElementsPDE
To have the survival probability we have to solve:
It doesn’t exist the analytical solution
=0
Stochastic Methods in Mathematical FinanceStochastic Methods in Mathematical Finance 15 September 200515 September 2005
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The two numerical methods agree to a large extent.Difficulties arises in managing very small discretization
steps.
The two numerical methods agree to a large extent.Difficulties arises in managing very small discretization
steps.
The survival probability for IBMThe survival probability for IBMIt’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:
0 2 4 6 8 10 12 140
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1p
t0 2 4 6 8 10 12 14
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1p
t
p
t0 0.5 1
0.9
0.95
1
hx = 0.04
hy = 0.5
ht = 0.05
ht = 0.01
Monte Carlo
Finite Differences ht=0.05
Finite Differences ht=0.01
N =105 trajectoriesτ = 0.01 discretization step
1σ-m=n = 1
Stochastic Methods in Mathematical FinanceStochastic Methods in Mathematical Finance 15 September 200515 September 2005
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0 5 10 15 20 25 30 35 400
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
t [days]
p
n=-m= 350 ns
IBM:Application to atomic clocksIBM:Application to atomic clocks
For example
±10 ns 0.4 days (0.95)
Considering different values for the boundaries m and for the survival probabilities:
m [ns] \ p 90% 95% 99%10 0.5 0.4 0.330 0.9 0.8 0.750 1.3 1.2 1
100 2.1 1.9 1.6300 4.4 3.9 3.3500 6.1 5.5 4.6
Atomic Clock: Rubidium IBM
Experimental data
Stochastic Methods in Mathematical FinanceStochastic Methods in Mathematical Finance 15 September 200515 September 2005
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By the numerical methods we obtain the survival probability of the complete model:
Complete Model (IBM+BM): Survival ProbabilityComplete Model (IBM+BM): Survival Probability
0 0.5 1 1.5 2 2.5 30
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
t
p
p
t0 0.15 0.35 0.5
0.7
0.85
1
Finite Differences
Finite Elements
Monte Carlo
The Monte Carlo method and the finite elements method agree for any discretization step. For the
difference finites method thedifficulties arises in managing very small discretization
steps.
The Monte Carlo method and the finite elements method agree for any discretization step. For the
difference finites method thedifficulties arises in managing very small discretization
steps.
N =105 trajectoriesτ = 0.01 discretization step
hx = 0.01
hy = 0.02
ht = 0.003
For the finite elements method
121 σσ -m=n = 1
hx = 0.2
hy = 0.5
ht = 0.01
Stochastic Methods in Mathematical FinanceStochastic Methods in Mathematical Finance 15 September 200515 September 2005
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0 0.5 1 1.5 2 2.5 3 3.5 40
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
t [days]
p
m = 8 ns
Complete Model (IBM+BM):Application to atomic clocksComplete Model (IBM+BM):Application to atomic clocks
For example
±10 ns 0.2 days (0.95)
Considering different values for the boundaries m and for the survival probabilities:
Atomic Clock: Rubidium
IBMComplete Model (IBM+BM)Experimental data m [ns] \ p 90% 95% 99%
10 0.24 0.2 0.130 0.8 0.7 0.650 1.2 1 0.9
100 2 1.8 1.5300 4.4 3.9 3.2500 6.1 5.5 4.5
Stochastic Methods in Mathematical FinanceStochastic Methods in Mathematical Finance 15 September 200515 September 2005
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ApplicatioApplicationsns
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 re-synchronized.
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 intervalCalibration 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 FinanceStochastic Methods in Mathematical Finance 15 September 200515 September 2005
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PerspectivesPerspectives
It’s necessary to use other stochastic process to describe the behaviour of different atomic clock error.
Other stochastic processes used to metrological application can be
1. The Integrated Ornstein-Uhlembeck2. The Fractional Brownian Motion
We have considered the Ornstein-Uhlembeck process to model the filtered white noise.
0 5 10 15 20 25 30 35 40 45 50-20
-15
-10
-5
0
5
10
15
20
t
x
0 5 10 15 20 25 30 35 40 45 50-20
-15
-10
-5
0
5
10
15
20
t
x
30 realizations of the Brownian Motion (red) and Ornstein-Uhlembeck (blue)
Stochastic Methods in Mathematical FinanceStochastic Methods in Mathematical Finance 15 September 200515 September 2005
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ConclusionsConclusions
• By the SDE or related PDE the survival probability of a stochastic process is obtained.
• Using the atomic clock model
clock behavior prediction
• Stochastic differential equations helps in modelling the atomic clock errors
The authors thank Laura Sacerdote and Cristina Zucca from University of Turin for helpful suggestions, support and collaboration.
The authors thank Laura Sacerdote and Cristina Zucca from University of Turin for helpful suggestions, support and collaboration.
•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.