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 Elettrotecnico Nazionale “G. Ferraris” Politecnico of Turin Patrizia Tavella Istituto Elettrotecnico Nazionale “G. Ferraris” Turin

2. Stochastic Methods in Mathematical Finance 15 September 2005 2 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 In 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”

3. Stochastic Methods in Mathematical Finance 15 September 2005 3 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 probability 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 processes

4. Stochastic Methods in Mathematical Finance 15 September 2005 4 The stochastic model of the atomic clock error obtained by the solution of the stochastic differential equations. Summary Numerical solution: Monte Carlo method Monte Carlo method for SDE Finite Differences Method Finite Differences Method for PDE Finite Elements Method 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. Link between the stochastic differential equations (SDE) and the partial differential equations (PDE): infinitesimal generator. Survival probability.

5. Stochastic Methods in Mathematical Finance 15 September 2005 5 The atomic clock model with initial conditions 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  1 W 1 which represents the contribution of the BM. Brownian Motion (BM) Integrated Brownian Motion (IBM) The stochastic processes involved in this model are:

6. Stochastic Methods in Mathematical Finance 15 September 2005 6 Innovation The solution can be expressed in an iterative form useful for exact simulation where …and iterative form 10203040 -150 -100 -50 50 100 clock error

7. Stochastic Methods in Mathematical Finance 15 September 2005 7 The infinitesimal generator The infinitesimal generator A of a homogeneous Markov process X t, for, is defined by: where: Ag(x) is interpreted as the mean infinitesimal rate of change of g(X t ) in case X t =x T t is an operator defined as: g is a bounded function X t is a realization of a homogeneous stochastic Markov process is the transition probability density function

8. Stochastic Methods in Mathematical Finance 15 September 2005 8 Link between the stochastic differential equations and the partial differential equations for diffusions Stochastic differential equation: Partial differential equation for the transition probability f: (Kolmogorov’s backward equation) Infinitesimal generator L t :

9. Stochastic Methods in Mathematical Finance 15 September 2005 9 The survival probability Other functionals verify the same partial differential equation but with different boundary conditions. Example: the survival probability p(x,t): 1 D is the indicator function [0,T]- time domain D- spatial domain - boundary of the domain D where :

10. Stochastic Methods in Mathematical Finance 15 September 2005 10 PDE for the clock survival probability For the complete model (IBM+BM): Integrated Brownian motion Brownian Motion It is not always possible to derive the analytical solution!!! Numerical Methods applied to PDE: a)Finite Differences Method b)Finite Elements Method Numerical Methods applied to PDE: a)Finite Differences Method b)Finite Elements Method Monte Carlo Method applied to SDE.  =0

11. Stochastic Methods in Mathematical Finance 15 September 2005 11 Example: The Integrated Brownian Motion The Integrated Brownian motion is defined by the following Stochastic Differential Equation: Numerical Methods: A) Monte Carlo B) Finite Differences SDE C) Finite Elements PDE To have the survival probability we have to solve: It doesn’t exist the analytical solution  =0

12. Stochastic Methods in Mathematical Finance 15 September 2005 12 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 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 t h x = 0.04 h y = 0.5 h t = 0.05 h t = 0.01 Monte Carlo Finite Differences h t =0.05 Finite Differences h t =0.01 N =10 5 trajectories τ = 0.01 discretization step -m=n = 1

13. Stochastic Methods in Mathematical Finance 15 September 2005 13 t [days]p n=-m= 350 ns IBM: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: Atomic Clock: Rubidium IBM Experimental data

14. Stochastic Methods in Mathematical Finance 15 September 2005 14 By the numerical methods we obtain the survival probability of the complete model: Complete Model (IBM+BM): Survival Probability t p p t 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 the difficulties 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 the difficulties arises in managing very small discretization steps. N =10 5 trajectories τ = 0.01 discretization step h x = 0.01 h y = 0.02 h t = 0.003 For the finite elements method -m=n = 1 h x = 0.2 h y = 0.5 h t = 0.01

15. Stochastic Methods in Mathematical Finance 15 September 2005 15 t [days] p m = 8 ns Complete 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 IBM Complete Model (IBM+BM) Experimental data

16. Stochastic Methods in Mathematical Finance 15 September 2005 16 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 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 interval Calibration interval : In industrial measurement process the measuring instrument must be periodically calibrated. Our model estimates how often the calibration is required.

17. Stochastic Methods in Mathematical Finance 15 September 2005 17 Perspectives 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-Uhlembeck 2.The Fractional Brownian Motion We have considered the Ornstein-Uhlembeck process to model the filtered white noise. 30 realizations of the Brownian Motion (red) and Ornstein-Uhlembeck (blue)

18. Stochastic Methods in Mathematical Finance 15 September 2005 18 Conclusions 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 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.