Stochastic Controledited Part 2 doc

When the Kolmogorov’s forward partial differ-ential equationsFokker-Planck equation is interpreted as an advection diffusion equation, the associated set of stochastic differential equat

, w )

, G(

w is a standard normal variable The dimension of the phase space of the

stochastic problem of concern here is four, since the state vector

4 3

four and the number of entries in the variance matrix of the state is sixteen Since the state

vector is a real-valued vector stochastic process, the condition P ij Pjiholds The total

number of distinct entries in the variance matrix w’d be ten The initial conditions are

chosen as

and variable

state

for the

, 0 ) 0 ( P rad/TU, 1

1 ) 0 ( , AU/TU 01

0 ) 0 ( rad, 1 ) 0 ( ,

v

The initial conditions considered here are in canonical system of units Astronomers adopt a

normalized system of units, i.e ‘canonical units’, for the simplification purposes In

canonical units, the physical quantities are expressed in terms of Time Unit (TU) and

Astronomical Unit (AU) The diffusion parameters 2

3(TU) 0121

 are chosen for numerical simulations Here we consider a set of

deterministic initial conditions, which implies that the initial variance matrix w’d be zero

Note that random initial conditions lead to the non-zero initial variance matrix The system

is deterministic at t  t0and becomes stochastic at t  t0 because of the stochastic

perturbation This makes the contribution to the variance evolution coming from the ‘system

non-linearity coupled with ‘initial variance terms’ will be zero at t  t1. The contribution

to the variance evolution att  t1 comes from the perturbation term ( GGT)( xt, t ) only

For t  t1, the contribution to the variance evolution comes from the system non-linearity

as well as the perturbation term This assumption allows to study the effect of random

perturbations explicitly on the dynamical system The values of diffusion parameters are

selected so that the contribution to the force coming from the random part is smaller than

the force coming from the deterministic part It has been chosen for simulational

The Itô calculus for a noisy dynamical system 33

, w

) ,

w is a standard normal variable The dimension of the phase space of the

stochastic problem of concern here is four, since the state vector

4 3

four and the number of entries in the variance matrix of the state is sixteen Since the state

vector is a real-valued vector stochastic process, the condition P ij Pjiholds The total

number of distinct entries in the variance matrix w’d be ten The initial conditions are

chosen as

and variable

state

for the

, 0

) 0

( P

rad/TU, 1

1

) 0

( ,

AU/TU 01

0

) 0

( rad,

1 )

0 (

v

The initial conditions considered here are in canonical system of units Astronomers adopt a

normalized system of units, i.e ‘canonical units’, for the simplification purposes In

canonical units, the physical quantities are expressed in terms of Time Unit (TU) and

Astronomical Unit (AU) The diffusion parameters 2

3(TU)

0121

4)

 are chosen for numerical simulations Here we consider a set of

deterministic initial conditions, which implies that the initial variance matrix w’d be zero

Note that random initial conditions lead to the non-zero initial variance matrix The system

is deterministic at t  t0and becomes stochastic at t  t0 because of the stochastic

perturbation This makes the contribution to the variance evolution coming from the ‘system

non-linearity coupled with ‘initial variance terms’ will be zero at t  t1. The contribution

to the variance evolution att  t1 comes from the perturbation term ( GGT)( xt, t ) only

For t  t1, the contribution to the variance evolution comes from the system non-linearity

as well as the perturbation term This assumption allows to study the effect of random

perturbations explicitly on the dynamical system The values of diffusion parameters are

selected so that the contribution to the force coming from the random part is smaller than

the force coming from the deterministic part It has been chosen for simulational

) , (

~

~ 2

t q

t x f x

The Itô calculus for a noisy dynamical system 35

) , (

~

~ 2

t q

t x f x

) ,

( 2

t pq

t x f

t pq

t x f

involves the variance

termPpq. The evolution of the variance termPij encompasses the contributions from the

preceding variances, partials of the system non-linearity, the diffusion coefficient

t x GG

q pq



Significantly, the variance terms are also accounted for in the mean trajectory This explains

the second-order approximation leads to the perturbed mean trajectory This section

discusses very briefly about the numerical testing for the mean and variance evolutions

derived in the previous section A greater detail is given in the Author’s Royal Society

contribution This chapter is intended to demonstrate the usefulness of the Itô theory for

stochastic problems in dynamical systems by taking up an appealing case in satellite

mechanics

4 Conclusion

In this chapter, the Author has derived the conditional moment evolutions for the motion of

an orbiting satellite in dust environment, i.e a noisy dynamical system The noisy

dynamical system was modeled in the form of multi-dimensional stochastic differential

equation Subsequently, the Itô calculus for ‘the Brownian motion process as well as the

dynamical system driven by the Brownian motion’ was utilized to study the stochastic

problem of concern here Furthermore, the Itô theory was utilized to analyse the resulting

stochastic differential equation qualitatively The Markovian stochastic differential system

can be analysed using the Kolmogorov-Fokker-Planck Equation (KFPE) as well The

KFPE-based analysis involves the definition of conditional expectation, the adjoint property of the

Fokker-Planck operator as well as integration by part formula On the other hand, the Itô

differential rule involves relatively fewer steps, i.e Taylor series expansion, the Brownian

motion differential rule It is believed that the approach of this chapter will be useful for

analysing stochastic problems arising from physics, mathematical finance, mathematical

control theory, and technology

Appendix 1

The qualitative analysis of the non-linear autonomous system can be accomplished by

taking the Lie derivative of the scalar function, where : U  R , U is the phase space

of the non-linear autonomous system and  ( xt)  R The function  is said to be the first

integral if the Lie derivative Lv vanishes (Arnold 1995) The problem of analysing the

non-linear stochastic differential system qualitatively becomes quite difficult, since it

involves multi-dimensional diffusion equation formalism The Itô differential rule (Liptser

and Shirayayev 1977, Sage and Melsa 1971) allows us to obtain the stochastic evolution of

the function Equation (6) of this chapter can be re-written as

2

) , ( ) ( 2

1 ) , ( ) ( )

(

i

t t

ii T i

t i

t t

x

x t

x GG t

x f x

x dt

t j

T

x x

x t

1 ,



B t x G x

x

t i r

1 ) ( )

(

i

t t

ii T i

i

t t

x

x E t x GG x

x

x

E dt

j i

t j

T

x x

x E t x

) , , , ( )

, , , ( ) , , , ( ) , , , ( ) , , , (

2 2

r t

ii T j

r r

r

r r

r r

x

v r E t x GG

v r E v v

v r E v

r E r r

v r E dt

v r dE

( (

)) ( (

) , , , (

2 2 2

2 2 2

' 2 '

v r v r V r v r V r dt

v r dE

r

r

r r

r r

Thus the derivative of the energy function for the stochastic system of concern here will not vanish leading to the non-conservative nature of the energy function

Appendix 2

The Fokker-Planck equation has received attention in literature and found applications for

developing the prediction algorithm for the Itô stochastic differential system Detailed

The Itô calculus for a noisy dynamical system 37

)

,

( 2

t pq

t x

t pq

t x

involves the variance

termPpq. The evolution of the variance termPij encompasses the contributions from the

preceding variances, partials of the system non-linearity, the diffusion coefficient

t x

Significantly, the variance terms are also accounted for in the mean trajectory This explains

the second-order approximation leads to the perturbed mean trajectory This section

discusses very briefly about the numerical testing for the mean and variance evolutions

derived in the previous section A greater detail is given in the Author’s Royal Society

contribution This chapter is intended to demonstrate the usefulness of the Itô theory for

stochastic problems in dynamical systems by taking up an appealing case in satellite

mechanics

4 Conclusion

In this chapter, the Author has derived the conditional moment evolutions for the motion of

an orbiting satellite in dust environment, i.e a noisy dynamical system The noisy

dynamical system was modeled in the form of multi-dimensional stochastic differential

equation Subsequently, the Itô calculus for ‘the Brownian motion process as well as the

dynamical system driven by the Brownian motion’ was utilized to study the stochastic

problem of concern here Furthermore, the Itô theory was utilized to analyse the resulting

stochastic differential equation qualitatively The Markovian stochastic differential system

can be analysed using the Kolmogorov-Fokker-Planck Equation (KFPE) as well The

KFPE-based analysis involves the definition of conditional expectation, the adjoint property of the

Fokker-Planck operator as well as integration by part formula On the other hand, the Itô

differential rule involves relatively fewer steps, i.e Taylor series expansion, the Brownian

motion differential rule It is believed that the approach of this chapter will be useful for

analysing stochastic problems arising from physics, mathematical finance, mathematical

control theory, and technology

Appendix 1

The qualitative analysis of the non-linear autonomous system can be accomplished by

taking the Lie derivative of the scalar function, where : U  R , U is the phase space

of the non-linear autonomous system and  ( xt)  R The function  is said to be the first

integral if the Lie derivative Lv vanishes (Arnold 1995) The problem of analysing the

non-linear stochastic differential system qualitatively becomes quite difficult, since it

involves multi-dimensional diffusion equation formalism The Itô differential rule (Liptser

and Shirayayev 1977, Sage and Melsa 1971) allows us to obtain the stochastic evolution of

the function Equation (6) of this chapter can be re-written as

2

) , ( ) ( 2

1 ) , ( ) ( )

(

i

t t

ii T i

t i

t t

x

x t

x GG t

x f x

x dt

t j

T

x x

x t

1 ,



B t x G x

x

t i r

1 ) ( )

(

i

t t

ii T i

i

t t

x

x E t x GG x

x

x

E dt

j i

t j

T

x x

x E t x

) , , , ( )

, , , ( ) , , , ( ) , , , ( ) , , , (

2 2

r t

ii T j

r r

r

r r

r r

x

v r E t x GG

v r E v v

v r E v

r E r r

v r E dt

v r dE

( (

)) ( (

) , , , (

2 2 2

2 2 2

' 2 '

v r v r V r v r V r dt

v r dE

r

r

r r

r r

Thus the derivative of the energy function for the stochastic system of concern here will not vanish leading to the non-conservative nature of the energy function

Appendix 2

The Fokker-Planck equation has received attention in literature and found applications for

developing the prediction algorithm for the Itô stochastic differential system Detailed

discussions on the Fokker-Planck equation, its approximate solutions and applications in

sciences can be found in Risken (1984), Stratonovich (1963) The Fokker-Planck equation is

also known as the Kolmogorov forward equation The Fokker-Planck equation is a special

case of the stochastic equation (kinetic equation) as well The stochastic equation is about the

evolution of the conditional probability for given initial states for non-Markov processes

The stochastic equation is an infinite series Here, we explain how the Fokker-Planck

equation becomes a special case of the stochastic equation The conditional probability

density

).

( ) (

)

, , , ( ) , , , ( ) , ,

)

( ) ( ) , , ,

( x1 x2 xn p x1x2 p x2 x3 p xn 1xn p xn

Thus,

), ( ) , ( )

, ( ) , ( )

, , ,

( x1 x2 xn qt1,t2 x1 x2 qt2,t3 x2 x3 qt 1,t xn 1 xn q xn

where qt i1,t i( xi1, xi)is the transition probability density, 1  i  n and t i1 ti.The

transition probability density is the inverse Fourier transform of the conditional

characteristic function, i.e

2

1 ) ,

x x iu x x iu i

i t

 (11) For deriving the stochastic equation, we consider the conditional probability

densityp ( x1 x2),where

).

( ) ( ) , ( x1 x2 p x1x2 p x2

n

n n

u x u

where  ( u x , )can be regarded as the generating function of the sequence{ n( x )} As a

n

n x

du dx x p x x n

iu e

x iu

2 2

) (

!

) ( ( 2

1 )

n

n x

x iu n

!

1

2 2 2

1

dx x p x x x x x n

n n

Consider the random variables xt1and xt2,where t 1 t2 The time instants t1andt2can

be taken as t1  t   , t2  t For the short hand notation, introducing the notion of the stochastic process, takingx1  x, x2  x, equation (15) can be recast as

dx x p x x x x x

n x

n

) (

!

1 )

(0

!

10

The Itô calculus for a noisy dynamical system 39

discussions on the Fokker-Planck equation, its approximate solutions and applications in

sciences can be found in Risken (1984), Stratonovich (1963) The Fokker-Planck equation is

also known as the Kolmogorov forward equation The Fokker-Planck equation is a special

case of the stochastic equation (kinetic equation) as well The stochastic equation is about the

evolution of the conditional probability for given initial states for non-Markov processes

The stochastic equation is an infinite series Here, we explain how the Fokker-Planck

equation becomes a special case of the stochastic equation The conditional probability

density

).

( )

( )

, , ,

( )

, , ,

( )

( )

( )

( )

, , ,

( x1 x2 xn p x1x2 p x2 x3 p xn 1xn p xn

Thus,

), (

) ,

( )

, (

) ,

( )

, , ,

( x1 x2 xn qt1,t2 x1 x2 qt2,t3 x2 x3 qt 1,t xn 1 xn q xn

where qt i1,t i( xi1, xi)is the transition probability density, 1  i  n and t i1 ti.The

transition probability density is the inverse Fourier transform of the conditional

characteristic function, i.e

2

1 )

x x

iu x

x iu

i i

t

 (11) For deriving the stochastic equation, we consider the conditional probability

densityp ( x1x2),where

).

( )

( )

, ( x1 x2 p x1 x2 p x2

n

n n

u x u

where  ( u x , )can be regarded as the generating function of the sequence{ n( x )} As a

n

n x

du dx x p x x n

iu e

x iu

2 2

) (

!

) ( ( 2

1 )

n

n x

x iu n

!

1

2 2 2

1

dx x p x x x x x n

n n

Consider the random variables xt1and xt2,where t 1 t2 The time instants t1andt2can

be taken as t1  t   , t2  t For the short hand notation, introducing the notion of the stochastic process, takingx1  x, x2  x, equation (15) can be recast as

dx x p x x x x x

n x

n

) (

!

1 )

(0

!

10

where ( x  x )n  kn( x )



 and the time interval condition   0 leads to

) ( ) ( ) (

!

1 )

( ) (

1

x n

x p x p

(1

x p x k x n x

The above equation describes the evolution of conditional probability density for given

initial states for the non-Markovian process The Fokker-Plank equation is a stochastic

equation withki( x )  0 , 2  i Suppose the scalar stochastic differential equation of the

form

, ) , ( ) ,

), , ( )

and the higher-order coefficients of the stochastic equation will vanish as a consequence of

the Itô differential rule Thus, the Fokker-Planck equation

).

( ) ,

( 2

1 ) ( ) , ( )

x

t x g x

p t x f x x

I express my gratefulness to Professor Harish Parthasarathy, a Scholar and Author, for

introducing me to the subject and explaining cryptic mathematics of stochastic calculus

5 References

Arnold, V I (1995) Ordinary Differential Equations, The MIT Press, Cambridge and

Massachusetts

Dacunha-Castelle, D & Florens-Zmirou, D (1986) Estimations of the coefficients of a

diffusion from discrete observations, Stochastics, 19, 263-284

Jazwinski, A H (1970) Stochastic Processes and Filtering Theory, Academic Press, New York

and London

Karatzas, I & Shreve, S E (1991) Brownian Motion and Stochastic Calculus (graduate text in

mathematics), Springer, New York

Kloeden, P E & Platen, E (1991) The Numerical Solutions of Stochastic Differential Equations

(applications of mathematics), Springer, New York, 23

Landau, I D & Lifshitz, E M (1976) Mechanics (Course of Theoretical Physics, Vol 1),

Butterworth-Heinemann, Oxford, UK

Liptser, R S & Shiryayev, A N (1977) Statistics of Random Processes 1, Springer, Berlin Protter, Philip E (2005) Stochastic Integration and Differential Equations, Springer, Berlin,

Heidelberg, New York

Pugachev, V S & Sinitsyn, I N ( 1977) Stochastic Differential Systems (analysis and filtering),

John-Wiley and Sons, Chichester and New York

Revuz, D & Yor, M (1991) Continuous Martingales and Brownian Motion, Springer-Verlag,

Berlin, Heidelberg

Risken, H (1984) The Fokker-Planck Equation: Methods of Solution and Applications,

Springer-Verlag, Berlin

Sage, A P & Melsa, M L (1971) Estimation Theory with Applications to Communications and

Control, Mc-Graw Hill, New York

Stratonovich, R L (1963) Topics in the Theory of Random Noise (Vol 1and 2), Gordan and

Breach, New York

Shambhu N Sharma & Parthasarathy, H (2007) Dynamics of a stochastically perturbed

two-body problem Pro R Soc A, The Royal Society: London, 463, pp.979-1003,

(doi: 10.1080/rspa.2006.1801)

Shambhu N Sharma (2009) A Kushner approach for small random perturbations of a

stochastic Duffing-van der Pol system, Automatica (a Journal of IFAC, International

Federation of Automatic Control), 45, pp 1097-1099

Strook, D W & Varadhan, S R S (1979) Multidimensional Diffusion Processes (classics in

mathematics), Springer, Berlin, Heidelberg, New York

Campen, N G van (2007) Stochastic Processes in Physics and Chemistry, Elsevier, Amsterdam,

Boston, London

Wax, N (ed.) (1954) Selected Papers on Noise and Stochastic Processes, Dover Publications, Inc,

New York

The Itô calculus for a noisy dynamical system 41

where ( x  x )n  kn( x )



 and the time interval condition   0 leads to

) (

) (

) (

!

1 )

( )

(

1

x n

x p

x p

(1

x p

x k

x n

The above equation describes the evolution of conditional probability density for given

initial states for the non-Markovian process The Fokker-Plank equation is a stochastic

equation withki( x )  0 , 2  i Suppose the scalar stochastic differential equation of the

form

, )

, (

) ,

( )

(

), ,

( )

and the higher-order coefficients of the stochastic equation will vanish as a consequence of

the Itô differential rule Thus, the Fokker-Planck equation

).

( )

,

( 2

1 )

( )

, (

)

x

t x

g x

p t

x f

x x

I express my gratefulness to Professor Harish Parthasarathy, a Scholar and Author, for

introducing me to the subject and explaining cryptic mathematics of stochastic calculus

5 References

Arnold, V I (1995) Ordinary Differential Equations, The MIT Press, Cambridge and

Massachusetts

Dacunha-Castelle, D & Florens-Zmirou, D (1986) Estimations of the coefficients of a

diffusion from discrete observations, Stochastics, 19, 263-284

Jazwinski, A H (1970) Stochastic Processes and Filtering Theory, Academic Press, New York

and London

Karatzas, I & Shreve, S E (1991) Brownian Motion and Stochastic Calculus (graduate text in

mathematics), Springer, New York

Kloeden, P E & Platen, E (1991) The Numerical Solutions of Stochastic Differential Equations

(applications of mathematics), Springer, New York, 23

Landau, I D & Lifshitz, E M (1976) Mechanics (Course of Theoretical Physics, Vol 1),

Butterworth-Heinemann, Oxford, UK

Liptser, R S & Shiryayev, A N (1977) Statistics of Random Processes 1, Springer, Berlin Protter, Philip E (2005) Stochastic Integration and Differential Equations, Springer, Berlin,

Heidelberg, New York

Pugachev, V S & Sinitsyn, I N ( 1977) Stochastic Differential Systems (analysis and filtering),

John-Wiley and Sons, Chichester and New York

Revuz, D & Yor, M (1991) Continuous Martingales and Brownian Motion, Springer-Verlag,

Berlin, Heidelberg

Risken, H (1984) The Fokker-Planck Equation: Methods of Solution and Applications,

Springer-Verlag, Berlin

Sage, A P & Melsa, M L (1971) Estimation Theory with Applications to Communications and

Control, Mc-Graw Hill, New York

Stratonovich, R L (1963) Topics in the Theory of Random Noise (Vol 1and 2), Gordan and

Breach, New York

Shambhu N Sharma & Parthasarathy, H (2007) Dynamics of a stochastically perturbed

two-body problem Pro R Soc A, The Royal Society: London, 463, pp.979-1003,

(doi: 10.1080/rspa.2006.1801)

Shambhu N Sharma (2009) A Kushner approach for small random perturbations of a

stochastic Duffing-van der Pol system, Automatica (a Journal of IFAC, International

Federation of Automatic Control), 45, pp 1097-1099

Strook, D W & Varadhan, S R S (1979) Multidimensional Diffusion Processes (classics in

mathematics), Springer, Berlin, Heidelberg, New York

Campen, N G van (2007) Stochastic Processes in Physics and Chemistry, Elsevier, Amsterdam,

Boston, London

Wax, N (ed.) (1954) Selected Papers on Noise and Stochastic Processes, Dover Publications, Inc,

New York

Application of coloured noise as a driving force in the stochastic differential equations 43

Application of coloured noise as a driving force in the stochastic differential equations

W.M.Charles

0

Application of coloured noise as a driving force

in the stochastic differential equations

W.M.Charles

University of Dar-es-salaam, College of Natural and Applied sciences,

Department of Mathematics, P.O.Box 35062 Dar-es-salaam, Tanzania

Abstract

In this chapter we explore the application of coloured noise as a driving force to a set of

stochastic differential equations(SDEs) These stochastic differential equations are sometimes

called Random flight models as in A W Heemink (1990) They are used for prediction of

the dispersion of pollutants in atmosphere or in shallow waters e.g Lake, Rivers etc Usually

the advection and diffusion of pollutants in shallow waters use the well known partial

differ-ential equations called Advection diffusion equations(ADEs)R.W.Barber et al (2005) These

are consistent with the stochastic differential equations which are driven by Wiener processes

as in P.E Kloeden et al (2003) The stochastic differential equations which are driven by

Wiener processes are called particle models When the Kolmogorov’s forward partial

differ-ential equations(Fokker-Planck equation) is interpreted as an advection diffusion equation,

the associated set of stochastic differential equations called particle model are derived and are

exactly consistent with the advection-diffusion equation as in A W Heemink (1990); W M

Charles et al (2009) Still, neither the advection-diffusion equation nor the related traditional

particle model accurately takes into account the short term spreading behaviour of particles

This is due to the fact that the driving forces are Wiener processes and these have independent

increments as in A W Heemink (1990); H.B Fischer et al (1979) To improve the behaviour of

the model shortly after the deployment of contaminants, a particle model forced by a coloured

noise process is developed in this chapter The use of coloured noise as a driving force unlike

Brownian motion, enables to us to take into account the short-term correlated turbulent fluid

flow velocity of the particles Furthermore, it is shown that for long-term simulations of the

dispersion of particles, both the particle due to Brownian motion and the particle model due

to coloured noise are consistent with the advection-diffusion equation

Keywords: Brownian motion, stochastic differential equations, traditional particle models,

coloured noise force, advection-diffusion equation, Fokker-Planck equation

1 Introduction

Monte carlo simulation is gaining popularity in areas such as oceanographic, atmospheric as

well as electricity spot pricing applications White noise is often used as an important

pro-cess in many of these applications which involve some error prediction as in A W Heemink

3

(1990); H.B Fischer et al (1979); J R Hunter et al (1993); J.W Stijnen et al (2003) In these

types of applications usually the deterministic models in the form of partial differential

equa-tions are available and employed The solution is in most cases obtained by discretising the

partial differential equations as in G.S Stelling (1983) Processes such as transport of

pol-lutants and sediments can be described by employing partial differential equations(PDEs)

These well known PDEs are called advection diffusion equations In particular when applied

in shallow water e.g River, Lakes and Oceans, such effects of turbulence might be

consid-ered However when this happens, it results into a set of partial differential equations These

complicated set of PDEs are difficult to solve and in most cases not easy to get a closed

so-lution In this chapter we explore the application coloured noise a a driving force to a set of

stochastic differential equations(SDEs) These stochastic differential equations are sometimes

called Random flight models They are used for prediction of the dispersion of pollutants

in atmosphere or in shallow waters e.g Lake, Rivers J R Hunter et al (1993); R.W.Barber et

al (2005) Usually the advection and diffusion of pollutants in shallow waters use the well

known partial differential equations called Advection diffusion equations(ADEs) These are

consistent with the stochastic differential equations which are driven by Wiener processes

as in C.W Gardiner (2004); P.E Kloeden et al (2003) The stochastic differential equations

which are driven by Wiener processes are called particle models When the Kolmogorov’s

forward partial differential equations(Fokker-Planck equation) is interpreted as an advection

diffusion equation, the associated with this set of stochastic differential equations called

par-ticle model are derived and are exactly consistent with the advection-diffusion equation as

in W M Charles et al (2009) Still, neither the advection-diffusion equation nor the related

traditional particle model accurately takes into account the short term spreading behaviour of

particles This is due to the fact that the driving forces are Wiener processes and these have

independent increment To improve the behaviour of the model shortly after the deployment

of contaminants, a particle model forced by a coloured noise process is developed in this

ar-ticle The use of coloured noise as a driving force unlike Brownian motion, enables to us to

take into account the short-term correlated turbulent fluid flow velocity of the particles

Fur-thermore, it is shown that for long-term simulations of the dispersion of particles, both the

particle due to Brownian motion and the particle model due to coloured noise are consistent

with the advection-diffusion equation

To improve the behaviour of the model shortly after the deployment of contaminants, a

ran-dom flight model forced by a coloured noise process are often used The scheme in Figure 1,

shows that for long term simulation both models, advection diffusion equation and the

ran-dom flight models have no difference, such situation better to use the well known ADE The

use of coloured noise as a driving force unlike Brownian motion, enables to us to take into

account only the short-term correlated turbulent fluid flow velocity of the particles as in A W

Heemink (1990); W M Charles et al (2009) An exponentially coloured noise process can also

be used to mimic well the behaviour of electricity spot prices in the electricity market

Further-more, when the stochastic numerical models are driven by the white noise, in most cases their

order of accuracy is reduced Such models consider that particles move according to a simple

random walk and consequently have independent increment as in A.H Jazwinski (1970); D.J

Thomson (1987) The reduction of the order of convergence happens because white noise is

nowhere differentiable However, one can develop a stochastic numerical scheme and avoid

the reduction of the order of convergence if the coloured noise is employed as a driving force

as in A W Heemink (1990); J.W Stijnen et al (2003); R.W.Barber et al (2005); P.S Addison et

al (1997)

Advection−DiffusionEquation (ADE) Consistentwith (Traditional Particle Model)Random Walk Model

Dispersion in Coastal WatersModelling of Long Term Scale

(Random Flight model)

Fig 1 A schematic diagram showing that for t >> T Lboth the ADEs and Random flightmodels are consistent

The application of coloured noise as a driving force to improve the model prediction of thedispersion of pollutants soon after deployment is discussed in this chapter For it is well-known that the advection-diffusion equation describes the dispersion of particles in turbulentfluid flow accurately if the diffusing cloud of contaminants has been in the flow longer than

a certain Lagrangian time scale and has spread to cover a distance that is larger in size thanthe largest scale of the turbulent fluid flow as in H.B Fischer et al (1979) The Lagrangiantime scale(T L)is a measure of how long it takes before a particle loses memory of its initialturbulent velocity therefore, both the particle model which is driven by Brownian force andthe advection-diffusion model are unable to accurately describe the short time scale correlatedbehaviour which is available in real turbulent flows at sub-Lagrangian time Thus, a randomflight model have been developed for any length of the coloured noise This way, the parti-cle model takes correctly into account the diffusion processes over short time scales when theeddy(turbulent) diffusion is less than the molecular diffusion The inclusion of several param-eters in the coloured noise process allows for a better match between the auto-covariance ofthe model and the underlying physical processes

2 Coloured noise processes

In this part coloured noise forces are introduced and represent the stochastic velocities of theparticles, induced by turbulent fluid flow It is assumed that this turbulence is isotropic andthat the coloured noise processes are stationary and completely described by their zero meanand Lagrangian auto covariance functionH.M Taylor et al (1998); W M Charles et al (2009)

2.1 The scalar exponential coloured noise process

The exponentially coloured noise are represented by a linear stochastic differential equation.The exponential coloured noise represent the velocity velocity of the particle;

du1(t) = − T1L u1(t)dt+α1dW(t) (1)

u1(t) = u0e −t TL +α1

Application of coloured noise as a driving force in the stochastic differential equations 45

(1990); H.B Fischer et al (1979); J R Hunter et al (1993); J.W Stijnen et al (2003) In these

types of applications usually the deterministic models in the form of partial differential

equa-tions are available and employed The solution is in most cases obtained by discretising the

partial differential equations as in G.S Stelling (1983) Processes such as transport of

pol-lutants and sediments can be described by employing partial differential equations(PDEs)

These well known PDEs are called advection diffusion equations In particular when applied

in shallow water e.g River, Lakes and Oceans, such effects of turbulence might be

consid-ered However when this happens, it results into a set of partial differential equations These

complicated set of PDEs are difficult to solve and in most cases not easy to get a closed

so-lution In this chapter we explore the application coloured noise a a driving force to a set of

stochastic differential equations(SDEs) These stochastic differential equations are sometimes

called Random flight models They are used for prediction of the dispersion of pollutants

in atmosphere or in shallow waters e.g Lake, Rivers J R Hunter et al (1993); R.W.Barber et

al (2005) Usually the advection and diffusion of pollutants in shallow waters use the well

known partial differential equations called Advection diffusion equations(ADEs) These are

consistent with the stochastic differential equations which are driven by Wiener processes

as in C.W Gardiner (2004); P.E Kloeden et al (2003) The stochastic differential equations

which are driven by Wiener processes are called particle models When the Kolmogorov’s

forward partial differential equations(Fokker-Planck equation) is interpreted as an advection

diffusion equation, the associated with this set of stochastic differential equations called

par-ticle model are derived and are exactly consistent with the advection-diffusion equation as

in W M Charles et al (2009) Still, neither the advection-diffusion equation nor the related

traditional particle model accurately takes into account the short term spreading behaviour of

particles This is due to the fact that the driving forces are Wiener processes and these have

independent increment To improve the behaviour of the model shortly after the deployment

of contaminants, a particle model forced by a coloured noise process is developed in this

ar-ticle The use of coloured noise as a driving force unlike Brownian motion, enables to us to

take into account the short-term correlated turbulent fluid flow velocity of the particles

Fur-thermore, it is shown that for long-term simulations of the dispersion of particles, both the

particle due to Brownian motion and the particle model due to coloured noise are consistent

with the advection-diffusion equation

To improve the behaviour of the model shortly after the deployment of contaminants, a

ran-dom flight model forced by a coloured noise process are often used The scheme in Figure 1,

shows that for long term simulation both models, advection diffusion equation and the

ran-dom flight models have no difference, such situation better to use the well known ADE The

use of coloured noise as a driving force unlike Brownian motion, enables to us to take into

account only the short-term correlated turbulent fluid flow velocity of the particles as in A W

Heemink (1990); W M Charles et al (2009) An exponentially coloured noise process can also

be used to mimic well the behaviour of electricity spot prices in the electricity market

Further-more, when the stochastic numerical models are driven by the white noise, in most cases their

order of accuracy is reduced Such models consider that particles move according to a simple

random walk and consequently have independent increment as in A.H Jazwinski (1970); D.J

Thomson (1987) The reduction of the order of convergence happens because white noise is

nowhere differentiable However, one can develop a stochastic numerical scheme and avoid

the reduction of the order of convergence if the coloured noise is employed as a driving force

as in A W Heemink (1990); J.W Stijnen et al (2003); R.W.Barber et al (2005); P.S Addison et

al (1997)

Advection−DiffusionEquation (ADE) Consistentwith (Traditional Particle Model)Random Walk Model

Dispersion in Coastal WatersModelling of Long Term Scale

(Random Flight model)

Fig 1 A schematic diagram showing that for t >> T Lboth the ADEs and Random flightmodels are consistent

The application of coloured noise as a driving force to improve the model prediction of thedispersion of pollutants soon after deployment is discussed in this chapter For it is well-known that the advection-diffusion equation describes the dispersion of particles in turbulentfluid flow accurately if the diffusing cloud of contaminants has been in the flow longer than

a certain Lagrangian time scale and has spread to cover a distance that is larger in size thanthe largest scale of the turbulent fluid flow as in H.B Fischer et al (1979) The Lagrangiantime scale(T L)is a measure of how long it takes before a particle loses memory of its initialturbulent velocity therefore, both the particle model which is driven by Brownian force andthe advection-diffusion model are unable to accurately describe the short time scale correlatedbehaviour which is available in real turbulent flows at sub-Lagrangian time Thus, a randomflight model have been developed for any length of the coloured noise This way, the parti-cle model takes correctly into account the diffusion processes over short time scales when theeddy(turbulent) diffusion is less than the molecular diffusion The inclusion of several param-eters in the coloured noise process allows for a better match between the auto-covariance ofthe model and the underlying physical processes

2 Coloured noise processes

In this part coloured noise forces are introduced and represent the stochastic velocities of theparticles, induced by turbulent fluid flow It is assumed that this turbulence is isotropic andthat the coloured noise processes are stationary and completely described by their zero meanand Lagrangian auto covariance functionH.M Taylor et al (1998); W M Charles et al (2009)

2.1 The scalar exponential coloured noise process

The exponentially coloured noise are represented by a linear stochastic differential equation.The exponential coloured noise represent the velocity velocity of the particle;

du1(t) = − T1L u1(t)dt+α1dW(t) (1)

u1(t) = u0e TL −t+α1

where u1is the particle’s velocity, α1 > 0 is constant, and T Lis a Lagrangian time scale For

t > s it can be shown as in A.H Jazwinski (1970), that the scalar exponential coloured noise

process in Eqn (2) has mean, variance and Lagrangian auto-covariance of respectively,

2

1T L

where α1 > 0 is constant, and T L is a Lagrangian time scale For t > s it can be shown A.H.

Jazwinski (1970), that the scalar exponential coloured noise process in eqn.(2)has mean,

vari-ance and Lagrangian auto-covarivari-ance of respectively,

2T L

2.2 The general vector coloured noise force

The general vector form of a linear stochastic differential equation for coloured noise processes

as in A.H Jazwinski (1970); H.M Taylor et al (1998) is given by

du(t) = Fu(t)dt+G(t)dW(t), dv(t) =Fv(t)dt+G(t)dW(t) (5)

Where u(t)and v(t)are vectors of length n, F and G are n × n respectively n × m matrix

functions in time and{ W(t); t ≥0 is an m-vector Brownian process with E[dW(t)dW(t)T] =

Q(t)dt In this chapter, a special case of the Ornstein-Uhlenbeck process C.W Gardiner (2004);

H.M Taylor et al (1998) is extended and repeatedly integrate it to obtain the coloured noise

forcing along the x and y-directions:

As you keep increasing the length of the coloured noise, an auto-covariance of the velocity

processes is modelled more realistically to encompasses the characteristics of an isotropic

ho-mogeneous turbulent fluid flow

Figure 2 in an example of Wiener path and that of a coloured noise process The sample path

of the coloured noise are smoother that that of Wiener process

The vector Langevin equation (6) generates a stationary, zero-mean, correlated Gaussian

pro-cess denoted by(u n(t), v n(t)) The Lagrangian time scale TLindicates the time over which the

process remains significantly correlated in time The linear system in eqn.(6), is the same in

−4

−2 0 2 4

6 Coloured noise of different lengths

t W(t)

dt= 0.0025

Fig 2 Sample paths of coloured noise (a) and sample path of Wiener process (b)

the Itô and the Stratonovich sense because the diffusion function is not a function of state butonly of time In order to get more accurate results the stochastic differential equation driven

by the coloured is integrated by the Heun scheme (see e.g., G.N Milstein (1995); J.W Stijnen

et al (2003); P.E Kloeden et al (2003))

The main purpose of this chapter is the application of coloured noise forcing in the dispersion

of a cloud of contaminants so as to improve the short term behaviour of the model while ing the long term behaviour unchanged Being the central part of the model, it is important

leav-to study the properties of coloured noise processes in more detail Coloured noise is a sian process and it is well known that these processes can be completely described by theirmean and covariance functions see L Arnold (1974) From eqn.(2) and from Figure 3(a), it iseasily seen that the mean approaches zero throughout and therefore requires little attention

Gaus-The covariance, however, depends not only on time but also on the initial values of u n(0)and

v n(0) This immediately gives rise to the question of how to actually choose or determine

these values Let’s consider the covariance matrix of the stationary process u in the stochastic

differential equations of the form (5) It is known (see e.g.,A.H Jazwinski (1970)) that ance function can now be described by

covari-dP

The equation (7) can be equated zero so as to find the steady state covariance matrix ¯P which

will then be used to generate instances of coloured noise processes Sampling of instances of

u vector by using a steady state matrix, ensures that the process is sampled at its stationary

phase thus removing any artefacts due to a certain choice of start values that would otherwise

be used The auto-covariance is depicted in Figure 3(c) Note that the behaviour of a physicalprocess in this case depends on the parameters in the Lagrangian auto-covariance Of courseshort term diffusion behaviour is controlled by the auto-covariance function This provides

room for the choice of parameters e.g.,α1, α2· · · The mean, variance and the auto-covariance

are not stationary for a finite time t but as t →∞,they approach the limiting stationary bution values as shown in Figure 3(a)–(c)

distri-Application of coloured noise as a driving force in the stochastic differential equations 47

where u1is the particle’s velocity, α1 > 0 is constant, and T Lis a Lagrangian time scale For

t > s it can be shown as in A.H Jazwinski (1970), that the scalar exponential coloured noise

process in Eqn (2) has mean, variance and Lagrangian auto-covariance of respectively,

Cov[u1(t), u1(s)] = α

2

1T L

where α1> 0 is constant, and T L is a Lagrangian time scale For t > s it can be shown A.H.

Jazwinski (1970), that the scalar exponential coloured noise process in eqn.(2)has mean,

vari-ance and Lagrangian auto-covarivari-ance of respectively,

Cov[u1(t), u1(s)] = α

2T L

2.2 The general vector coloured noise force

The general vector form of a linear stochastic differential equation for coloured noise processes

as in A.H Jazwinski (1970); H.M Taylor et al (1998) is given by

du(t) = Fu(t)dt+G(t)dW(t), dv(t) =Fv(t)dt+G(t)dW(t) (5)

Where u(t)and v(t)are vectors of length n, F and G are n × n respectively n × m matrix

functions in time and{ W(t); t ≥0 is an m-vector Brownian process with E[dW(t)dW(t)T] =

Q(t)dt In this chapter, a special case of the Ornstein-Uhlenbeck process C.W Gardiner (2004);

H.M Taylor et al (1998) is extended and repeatedly integrate it to obtain the coloured noise

forcing along the x and y-directions:

As you keep increasing the length of the coloured noise, an auto-covariance of the velocity

processes is modelled more realistically to encompasses the characteristics of an isotropic

ho-mogeneous turbulent fluid flow

Figure 2 in an example of Wiener path and that of a coloured noise process The sample path

of the coloured noise are smoother that that of Wiener process

The vector Langevin equation (6) generates a stationary, zero-mean, correlated Gaussian

pro-cess denoted by(u n(t), v n(t)) The Lagrangian time scale TLindicates the time over which the

process remains significantly correlated in time The linear system in eqn.(6), is the same in

−4

−2 0 2 4

6 Coloured noise of different lengths

t W(t)

dt= 0.0025

Fig 2 Sample paths of coloured noise (a) and sample path of Wiener process (b)

the Itô and the Stratonovich sense because the diffusion function is not a function of state butonly of time In order to get more accurate results the stochastic differential equation driven

by the coloured is integrated by the Heun scheme (see e.g., G.N Milstein (1995); J.W Stijnen

et al (2003); P.E Kloeden et al (2003))

The main purpose of this chapter is the application of coloured noise forcing in the dispersion

of a cloud of contaminants so as to improve the short term behaviour of the model while ing the long term behaviour unchanged Being the central part of the model, it is important

leav-to study the properties of coloured noise processes in more detail Coloured noise is a sian process and it is well known that these processes can be completely described by theirmean and covariance functions see L Arnold (1974) From eqn.(2) and from Figure 3(a), it iseasily seen that the mean approaches zero throughout and therefore requires little attention

Gaus-The covariance, however, depends not only on time but also on the initial values of u n(0)and

v n(0) This immediately gives rise to the question of how to actually choose or determine

these values Let’s consider the covariance matrix of the stationary process u in the stochastic

differential equations of the form (5) It is known (see e.g.,A.H Jazwinski (1970)) that ance function can now be described by

covari-dP

The equation (7) can be equated zero so as to find the steady state covariance matrix ¯P which

will then be used to generate instances of coloured noise processes Sampling of instances of

u vector by using a steady state matrix, ensures that the process is sampled at its stationary

phase thus removing any artefacts due to a certain choice of start values that would otherwise

be used The auto-covariance is depicted in Figure 3(c) Note that the behaviour of a physicalprocess in this case depends on the parameters in the Lagrangian auto-covariance Of courseshort term diffusion behaviour is controlled by the auto-covariance function This provides

room for the choice of parameters e.g.,α1, α2· · · The mean, variance and the auto-covariance

are not stationary for a finite time t but as t →∞,they approach the limiting stationary bution values as shown in Figure 3(a)–(c)

0 0.2 0.4 0.6 0.8 1 1.2 1.4

time

Variance of coloured processes

variance of u1(t) variance of u2(t) variance of u3(t) variance of u4(t)

Fig 3 (a)Shows that the mean goes to zero, while (b)-(c) shows that the variance and

auto-covariance of coloured noise processes started from non-stationary to stationary state

2.3 The particle model forced by coloured noise

The prediction of the dispersion of pollutants in shallow waters are modeled by the random

flight which is driven by coloured as in A W Heemink (1990) In this work, an extension

to the work by A W Heemink (1990) has been done by generalising the cloured noise to any

length that is, to(u n(t), v n(t)) The coloured noise processes stand for the velocity of the

par-ticle at time t in respectively the x and y directions This way the Lagrangian auto-covariance

processes can be modelled more realistically by taking into account the characteristics of the

turbulent fluid flow for t  T L By using the following set of equations the random flight

model remains consistent with the advection-diffusion equation for t >> T Lwhile modelling

realistically the short term correlation of the turbulent fluid flows In this application, unlike

in W M Charles et al (2005), Longer length of the coloured noise have been chosen, that is

n = 6 and more experiments are carried out in the whirl pool ideal domain for simulations

of the advection and diffusion of pollutants in shallow waters Thus the following coloured

noise are used

(u6(0), v6(0))with zero mean and variance that agrees with covariance matrix ¯P at a steady

state For instance in this chapter, the following covariance matrix was obtained when theparameters shown in Table 1 were used in the simulation;

3 The spreading behaviour of a cloud of contaminants

The characteristics of a spreading cloud of contaminants due to Brownian motion andcoloured noise processes are discussed in the following sections

3.1 Long term spreading behaviour of clouds of particles due Brownian motion force

Consider, the following 1 dimensional stochastic differential equation in the Itô sense

dX(t) Itˆo= f(t, X t)dt+g(t, X t)dW(t) (10)

where f(t, X t)is the drift coefficient function and where g(t, X t)is the diffusion coefficient

function If it assumed that there is no drift term in eqn.(10) that is, f(X(t), t) =0, gives

con-Application of coloured noise as a driving force in the stochastic differential equations 49

0 0.2 0.4 0.6 0.8 1 1.2 1.4

time

Variance of coloured processes

variance of u1(t) variance of u2(t) variance of u3(t) variance of u4(t)

Fig 3 (a)Shows that the mean goes to zero, while (b)-(c) shows that the variance and

auto-covariance of coloured noise processes started from non-stationary to stationary state

2.3 The particle model forced by coloured noise

The prediction of the dispersion of pollutants in shallow waters are modeled by the random

flight which is driven by coloured as in A W Heemink (1990) In this work, an extension

to the work by A W Heemink (1990) has been done by generalising the cloured noise to any

length that is, to(u n(t), v n(t)) The coloured noise processes stand for the velocity of the

par-ticle at time t in respectively the x and y directions This way the Lagrangian auto-covariance

processes can be modelled more realistically by taking into account the characteristics of the

turbulent fluid flow for t  T L By using the following set of equations the random flight

model remains consistent with the advection-diffusion equation for t >> T Lwhile modelling

realistically the short term correlation of the turbulent fluid flows In this application, unlike

in W M Charles et al (2005), Longer length of the coloured noise have been chosen, that is

n = 6 and more experiments are carried out in the whirl pool ideal domain for simulations

of the advection and diffusion of pollutants in shallow waters Thus the following coloured

noise are used

(u6(0), v6(0))with zero mean and variance that agrees with covariance matrix ¯P at a steady

state For instance in this chapter, the following covariance matrix was obtained when theparameters shown in Table 1 were used in the simulation;

3 The spreading behaviour of a cloud of contaminants

The characteristics of a spreading cloud of contaminants due to Brownian motion andcoloured noise processes are discussed in the following sections

3.1 Long term spreading behaviour of clouds of particles due Brownian motion force

Consider, the following 1 dimensional stochastic differential equation in the Itô sense

dX(t) Itˆo= f(t, X t)dt+g(t, X t)dW(t) (10)

where f(t, X t)is the drift coefficient function and where g(t, X t)is the diffusion coefficient

function If it assumed that there is no drift term in eqn.(10) that is, f(X(t), t) =0, gives

con-Theorem 1. Let g(x)be continuous function and { W(t), t ≥0 be the standard Brownian motion

process H.M Taylor et al (1998) For each t > 0, there exits a random variable

H.M Taylor et al (1998) for example

3.2 Long term spreading behaviour of clouds of contaminants subject to coloured noise

forcing

As discussed in earlier, where for example, the first an exponential coloured u1(t)from eqn (2)

is used as forcing coloured noise, if it is assumed that there is no background flow, the position

of a particle at time t is given by

process u1(t)behaves much like the one driven by Brownian motion with variance parameter

σ2α21T L2 Hence, the dispersion coefficient is related to variance parameters σ2α21T L2 = 2D.

Clarification are done by considering eqn.(14), where the second part is u1(t)itself;

X(t) =σT L[α1W(t)− u1(t)], where u1(t) =α1

0 e − TL1(t−k) dW(k)

Let us now rescale the position process in order to better observe the changes over large time

spans By doing so, for N >0, yields,

N remains a standard Brownian motion process For sufficiently large N it

becomes clear that eqn.(15) behaves like Brownian motion as in H.M Taylor et al (1998); W

M Charles et al (2009):

X N(t) ≈ σα1T L W˜(t)

3.3 The analysis of short term spreading behaviour of a cloud of contaminants

The analysis of the coloured noise processes usually starts with a scalar coloured noise, it can

be shown using eqn.(4) that

Cov[u t+τ u t] = E[u t+τ u t] =E[v t+τ v t] = 1

2α2T L e −|τ| TL (16)From equation(16), It follows that,

The integration of equation(17)can easily be yielded by separately considering the regions

τ < s and τ > s, and it can be shown that

Since the short time analysis, eqn (18) are of interest in this section and is considered only for

very small values of t in a sense that for t  T Lthe variance of a cloud of particles shortlyafter deployment is then given by the following equation: