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Stochastic Methods in Mathematical Stochastic Methods in Mathematical Finance Finance 15 September 2005 15 September 2005 1 The Integrated Brownian The Integrated Brownian Motion for the study of the Motion for the study of the atomic clock error atomic clock error Gianna Panfilo Gianna Panfilo Istituto Elettrotecnico Nazionale “G. Ferraris” Politecnico of Turin Patrizia Tavella Patrizia Tavella Istituto Elettrotecnico Nazionale “G. Ferraris” Turin

Stochastic Methods in Mathematical Finance 15 September 2005 1 The Integrated Brownian Motion for the study of the atomic clock error Gianna Panfilo Istituto

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Page 1: Stochastic Methods in Mathematical Finance 15 September 2005 1 The Integrated Brownian Motion for the study of the atomic clock error Gianna Panfilo Istituto

Stochastic Methods in Mathematical FinanceStochastic Methods in Mathematical Finance 15 September 200515 September 2005

1

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

Page 2: Stochastic Methods in Mathematical Finance 15 September 2005 1 The Integrated Brownian Motion for the study of the atomic clock error Gianna Panfilo Istituto

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”

Page 3: Stochastic Methods in Mathematical Finance 15 September 2005 1 The Integrated Brownian Motion for the study of the atomic clock error Gianna Panfilo Istituto

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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

Page 4: Stochastic Methods in Mathematical Finance 15 September 2005 1 The Integrated Brownian Motion for the study of the atomic clock error Gianna Panfilo Istituto

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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.

Page 5: Stochastic Methods in Mathematical Finance 15 September 2005 1 The Integrated Brownian Motion for the study of the atomic clock error Gianna Panfilo Istituto

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:

Page 6: Stochastic Methods in Mathematical Finance 15 September 2005 1 The Integrated Brownian Motion for the study of the atomic clock error Gianna Panfilo Istituto

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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

Page 7: Stochastic Methods in Mathematical Finance 15 September 2005 1 The Integrated Brownian Motion for the study of the atomic clock error Gianna Panfilo Istituto

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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

Page 8: Stochastic Methods in Mathematical Finance 15 September 2005 1 The Integrated Brownian Motion for the study of the atomic clock error Gianna Panfilo Istituto

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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:

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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

Page 10: Stochastic Methods in Mathematical Finance 15 September 2005 1 The Integrated Brownian Motion for the study of the atomic clock error Gianna Panfilo Istituto

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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

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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

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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

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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

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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

Page 15: Stochastic Methods in Mathematical Finance 15 September 2005 1 The Integrated Brownian Motion for the study of the atomic clock error Gianna Panfilo Istituto

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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

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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.

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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)

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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.