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AM216StochasticDifferentialEquationsLecture06

CopyrightbyHongyunWang,UCSC

Listoftopicsinthislecture

• LongroadtoconcludingdW/dtisawhitenoise:Energyspectrumdensity(ESD),powerspectrumdensity(PSD);

Stationarystochasticprocess,auto-correlationfunctionR(t);Wiener-Khinchintheorem(PSDisFouriertransformofR(t));

dW/dthasauniformPSD(definitionofwhitenoise).

• ConstrainedWienerprocess,BayesTheorem

RecapTheshortstoryofwhitenoise:

1)Z(t)≡ dW

dtisnotaregularfunction.

2) E Z(t)Z(s)( ) = δ(t − s) 3) exp(−i2πξt)E Z(t)Z(0)( )dt∫ =1 4) Z(t)isawhitenoise(wewillclarifywhatthismeans).

Planforthelongstory

Wewillfirstaddressthedefinitionofwhitenoiseinitem4).Wediscussageneralstationarystochasticprocessandinthatcontextdefinewhitenoiseinstepslistedbelow

• Energyspectrumdensity(ESD)

• Powerspectrumdensity(PSD)

• AgeneralstationarystochasticprocessanditsPSD

• RelationbetweenPSDandauto-correlationfunction

• DefinitionofwhitenoisebasedonPSD

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Beforeweproceedwiththeplan,wefinishthepropertiesofFouriertransform.

PropertiesofFouriertransform

1)F ρ

N(0,σ2 )(t)⎡⎣⎤⎦ = F

12πσ2

exp −t2

2σ2⎛

⎝⎜⎞

⎠⎟⎡

⎣⎢⎢

⎤

⎦⎥⎥= exp −2π2σ2ξ2( )

2) F[δ(x)]=1

3) F[1]=δ(ξ)4) Parseval’stheorem

y(t)2dt∫ = ŷ(ξ)

2dξ∫

Proof:

ŷ(ξ)2dξ∫ = ŷ(ξ) ŷ(ξ)dξ∫

= exp(−i2πξt)y(t)dt∫ exp(i2πξ s)y(s)ds∫

⎛

⎝⎜

⎞

⎠⎟ dξ∫

= exp −i2πξ(t − s)( ) y(t)y(s)dtds∫∫

⎛

⎝⎜

⎞

⎠⎟ dξ∫

Changetheorderofintegration

= y(t)y(s) exp −i2πξ(t − s)( )dξ∫

F[1]=δ(t−s)! "#### $####

⎛

⎝⎜

⎞

⎠⎟dtds∫∫

= y(t)y(s)δ(t − s)dtds∫∫

= y(s)y(s)ds∫ = y(s)

2ds∫

Thelongstoryofwhitenoise

Energyspectrumdensity(ESD)

Inphysicsproblems,

Energy∝ y(t)2dt∫ = ŷ(ξ)

2dξ∫

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Note: Here“energy”referstotheenergydissipated.

Examples:y(t)=electriccurrent

Energy = R ⋅ y(t)2dt∫ , R=electricalresistance

y(t)=velocity

Energy = b ⋅ y(t)2dt∫ , b=viscousdragcoefficient

Formathematicalconvenience,wescaletheenergytodefine

Energy = y(t)2dt∫ = ŷ(ξ)

2dξ∫

Weliketoknowhowtheenergyisdistributedinthefrequencydimension.

Definitionofenergyspectrumdensity(ESD)

ESD≡ ŷ(ξ)2 = exp(−i2πξt)y(t)dt∫

2

Caution: ŷ(ξ)2isanunnormalizeddensity.

ŷ(ξ)2dξ∫ = y(t)

2dt∫ =Energy ≠1

Otherexamplesofunnormalizeddensity:

Populationdensity: Xnumberofpersonspersquaremile

Pollutiondensity: XamountofchemicalsperunitvolumeofairorwaterCardensity: Xnumberofcarsperthousandpersons

Xnumberofcarspersquaremile

Caution: (differentdefinitionsofenergyspectrumdensity)

Inelectricalengineering(EE),energyspectrumdensityisdefinedas

ESD≡Φ(ω)= 1

2πexp(−iωt)y(t)dt∫

2

Φ(ω)and ŷ(ξ)2arerelatedby

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Φ(ω)= 12π ŷ(ξ)

2 , ξ = ω2π

PowerspectrumdensityEnergyspectrumdensityismeaningfulonlywhen

Energy = y(t)2dt∫ = finite

Example:

y(t)=electriccurrent=y0, constantintime

Energydissipated = R ⋅ y02dt∫ =∞

Whenthetotalenergyisnotfinite,welookattheenergyperunittime.

Power = Energytime = finite

Definitionofpowerspectrumdensity(PSD)

PSD≡ lim

T→∞

exp(−i2πξt)y(t)dt−T

T

∫2

2T

Expressionofpowerspectrumdensity(PSD)

WewritePSDintoamoreworkableexpression.

PSD= lim

T→∞

exp(−i2πξt)y(t)dt−T

T

∫ exp(i2πξ s)y(s)ds−TT

∫2T

= limT→∞

exp(−i2πξ(t − s))y(t)y(s)dt−T

T

∫ ds−TT

∫2T

changeofvariableτ=t–s

= limT→∞

exp(−i2πξτ)y(τ+ s)y(s)dτ−T−s

T−s

∫ ds−TT

∫2T

Drawtheintegrationregionins-τplane.Foreachs,therangeforτis[–T–s,T–s].

Foreachτ,therangeforsis[a(τ),b(τ)]

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where

a(τ)= −T − τ, τ∈[−2T , 0]

−T , τ∈[0, 2T ]⎧⎨⎪

⎩⎪

b(τ)= T , τ∈[−2T , 0]

T − τ, τ∈[0, 2T ]⎧⎨⎪

⎩⎪

Changetheorderofintegration

PSD= lim

T→∞

exp(−i2πξτ) y(τ+ s)y(s)dsa(τ)

b(τ)∫ dτ−2T

2T∫

2T (PSD01)

Sofar,weworkedwithdeterministicprocessy(t).Nextweintroducestochasticprocessandstationarystochasticprocess.

StationarystochasticprocessDefinitionofstochasticprocess

Astochasticprocessisafunctionoftimethatvarieswiththerandomoutcomeofanexperiment.

y(t)

Shortnotation! = y(t ,ω)

Fullnotation"#$ %$ ω=randomoutcomeofanexperiment

DefinitionofstationarystochasticprocessLety(t)beastochasticprocess.Wesayy(t)isstationaryifforanysetoftimeinstances(t1,t2,…,tk),thejointdistributionof(y(t+t1),y(t+t2),…,y(t+tk))isindependentoft.

Example:

• W(t)isastochasticprocesses.

• Z(t)= dW(t)

dtisastochasticprocess.

Note: Forfinitedt,dW/dtiswelldefined,withnocomplicationatall.

Example:

• W(t)isnotstationary.E W(t)2( ) = t varieswitht.

• Z(t)= dW(t)

dtisstationary

Note: Thejointdistributionisinvariantunderashift.

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PropertiesofstationarystochasticprocessForastationarystochasticprocess,wehave

• E(y(t))=E(y(0))• E y(s + τ)y(s)( ) = E y(τ)y(0)( )

Caution:

Thesearenecessaryconditionsforastationarystochasticprocess.Theyarenotsufficientconditions.

Auto-correlationfunctionDefinitionofauto-correlationfunction

Forastationarystochasticprocess,

R(τ)≡ E y(s + τ)y(s)( ) = E y(τ)y(0)( ) iscalledtheauto-correlationfunction.

Note: R(τ)isafunctionofτonly,independentofs.Caution: becarefulwiththeterm“auto-correlation”

Auto-correlationcoefficientisdefinedas

ρ(τ)≡

E y(τ)−E( y(0))⎡⎣ ⎤⎦ y(0)−E( y(0))⎡⎣ ⎤⎦( )var( y(0))

Auto-correlationfunctionisdefinedas

R(τ)≡ E y(τ)y(0)( )

Wiener-Khinchintheorem(relationbetweenPSDandauto-correlationfunction)

Forastationarystochasticprocess,thepowerspectrumdensity(PSD)is

PSD≡ s(ξ)NewnotationforPSD

!"# = limT→∞E exp(−i2πξt)y(t)dt

−T

T

∫2⎛

⎝⎜⎞⎠⎟

2T

Weuse(PSD01)obtainedabovetorewrites(ξ)as

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s(ξ)= lim

T→∞

E exp(−i2πξτ) y(τ+ s)y(s)dsa(τ)

b(τ)∫ dτ−2T

2T∫⎛⎝

⎞⎠

2T

Changetheorderofintegrationandtakingaverage

= limT→∞

exp(−i2πξτ) E y(τ+ s)y(s)( )dsa(τ)b(τ)∫ dτ−2T2T∫2T

= limT→∞

exp(−i2πξτ)R(τ)(b(τ)−a(τ))dτ−2T

2T∫

2T

Theterm(b(τ)–a(τ))hastheexpression:

b(τ)−a(τ)= 2T + τ , τ∈[−2T , 0]2T − τ , τ∈[0, 2T ]

⎧⎨⎪

⎩⎪=2T − τ

Substitutingitintotheexpressionofs(ξ)yields

s(ξ)= lim

T→∞exp(−i2πξτ)R(τ) 1− τ2T

⎛⎝⎜

⎞⎠⎟dτ

−2T

2T∫

TakingthelimitasT→∞,wearriveat

s(ξ)= exp(−i2πξτ)R(τ)dτ−∞+∞

∫ WejustderivedtheWiener-Khinchintheorem.

Wiener-Khinchintheorem:Forastationarystochasticprocessy(t),thepowerspectrumdensity,s(ξ),andtheauto-correlationfunction,R(t),arerelatedby

s(ξ)= exp(−i2πξt)R(t)dt−∞+∞

∫ Inotherwords,thePSDisFouriertransformoftheauto-correlationfunction.

DefinitionofwhitenoiseLety(t)beastationarystochasticprocess.Wesayy(t)isawhitenoiseif

s(ξ)= const Inotherwords,thepowerofawhitenoiseisuniformlydistributedinthefrequencydimension.

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TheWiener-Khinchintheoremestablishedabovetellsusthatawhitenoisehastwoequivalentdefiningcharacters.

s(ξ)= const ⇐⇒ R(t)= E y(t)y(0)( )∝δ(t)

WorkingoutitemsintheshortstoryWere-writetheshortstoryintermsoftheauto-correlationfunvtionR(τ)andpowerspectrumdensitys(ξ).

1)Z(t)≡ dW

dtisnotaregularfunction.

2) R(τ)= E Z(s + τ)Z(s)( ) = δ(τ) 3) s(ξ)= exp(−i2πξt)R(t)dt∫ =1 4) Z(t)isawhitenoise.

• ToshowZ(t)isawhitenoise(item4),weonlyneedtoshows(ξ)=const(item3).• Toshows(ξ)=1(item3),weonlyneedtoshowR(t)=δ(t)(item2)

Thus,theremainingtaskistoshowitem2,whichwedonow.

DerivationofR(t)=δ(t) (a“formal”derivation”)

WefirstcalculateE(W(t)W(s))fort≥s.

E W(t)W(s)( ) = E (W(t)−W(s)+W(s))W(s)( )

= E (W(t)−W(s))W(s)( )+E W(s)2( ) =0+ s = s

SinceE(W(t)W(s))=E(W(s)W(t)),weobtain

E W(t)W(s)( ) =min(t ,s) Next,inthecalculationofE(Z(t)Z(s)),we“formally”exchangetheorderoftakingderivativesandtakingaverage.

E Z(t)Z(s)( ) = E ∂∂s

∂∂t

W(t)W(s)( )⎛⎝⎜

⎞⎠⎟

= ∂∂s

∂∂tE W(t)W(s)( ) = ∂∂s

∂∂tmin(t ,s)

Asafunctionoft,wehave

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min(t ,s)= t , t < s

s , t > s⎧⎨⎪

⎩⎪

Differentiatingwithrespecttot,andthenwritingitasafunctionofs,weget

∂∂tmin(t ,s)= 1, t < s

0, t > s⎧⎨⎪

⎩⎪ (asafunctionoft)

=

0, s < t1, s > t

⎧⎨⎪

⎩⎪ (asafunctionofs)

Differentiatingwithrespecttos,wearriveat

∂∂s

∂∂tmin(t ,s)= δ(s −t)

Therefore,weconclude

E Z(t)Z(s)( ) = δ(t − s) ==> R(τ)= E Z(s + τ)Z(s)( ) = δ(τ)

ExpressionsofR(t)ands(ξ)forfinitedt

(Justificationofthe“formal”derivationsabove)

Toemphasizethefinitetimestep,letΔt≡dt.Wehave

Z(t)=W(t +Δt)−W(t)

Δt awelldefinedstationarystochasticprocess

E Z(t)Z(s)( ) = E W(t +Δt)−W(t)Δt ⋅

W(s +Δt)−W(s)Δt

⎛⎝⎜

⎞⎠⎟

=0 , |t − s |> ΔtΔt−|t − s |(Δt)2 , |t − s |≤ Δt

⎧

⎨⎪

⎩⎪

(derivationnotincluded)

R(τ)= E Z(s + τ)Z(s)( ) =0 , |τ|> ΔtΔt−|τ|(Δt)2 , |τ|≤ Δt

⎧

⎨⎪

⎩⎪

TakingtheFouriertransformofR(t),weobtain

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s(ξ)= exp(−i2πξt)R(t)dt∫ = exp(−i2πξt)

Δt−|t |(Δt)2 dt−Δt

Δt

∫

=2cosh(i2πξΔt)−1(i2πξΔt)2

(derivationnotincluded)

Now,afterweobtains(ξ)forfiniteΔt,wetakethelimitasΔt→0.

Atanyfixedξ,asΔt→0,wehave

limΔt→0

s(ξ)= limΔt→0

2cosh(i2πξΔt)−1(i2πξΔt)2 =1

Observation:

• Mathematically,workingwithfinitedtuntiltakingthelimitattheendisarigorousapproachinwhicheverystepisproperlyjustified.Butderivingexpressionswithfinitedtisalsomathematicallymuchmoreelaborated.

• “Formal”derivationsaremuchsimpler.Buttheyarenotrigorous.

Aclassofcolorednoise:InthesubsequentdiscussionofOrnstein-Uhlenbeckprocess(OU),wewillseethatitsauto-correlationhastheform:

R(t)= E y(t)y(0)( )∝exp(−β|t |) Thecorrespondingpowerspectrumdensityis

s(ξ)= exp(−i2πξt)R(t)dt∫ ∝ exp(−i2πξt)exp(−β|t |)dt∫= 2ββ2 +4π2ξ2

Endofdiscussionofwhitenoise

ConstrainedWienerprocess

ForanunconstrainedWienerprocess,wehave

W(0)=0 and W(t1)~N(0,t1)

WhathappensifitisconstrainedbyW(t1+t2)=y?

W(t1)isstillrandom.

Weliketoknowtheconditionaldistribution(W(t1)|W(t1+t2)=y).

Forthatdiscussion,weneedtointroduceBayestheorem.

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BayesTheorem

ConsidertwoeventsAandB.WewritePr(AandB)intwoways.Pr(AandB)=Pr(A|B)Pr(B)

Pr(AandB)=Pr(B|A)Pr(A)Equatingthetwo,weget

Pr(A|B)Pr(B)=Pr(B|A)Pr(A)

ExpressPr(A|B)intermsofPr(B|A),wearriveatBayesTheoremforevents:

Pr(A B)=

Pr(B A)Pr(A)Pr(B)

ThisisBayestheoremforevents.

ToderiveBayestheoremfordensities,weconsider

A=“x≤X<x+ΔxB=“y≤Y<y+Δy

Wewriteprobabilitiesintermsofdensities

Pr(A B) ≈ ρ(X = x Y = y)Δx

Pr(B A) ≈ ρ(Y = y X = x)Δy

Pr(A) ≈ ρ(X = x)Δx

Pr(B) ≈ ρ(Y = y)Δy SubstitutingthesetermsintoBayestheorem,weobtainBayestheoremfordensities.

Bayestheoremfordensities

ρ(X = x Y = y) =

ρ(Y = y X = x)⋅ρ(X = x)ρ(Y = y)

Ausefultrick:

Indensityρ(X=x|Y=y),xistheindependentvariableandyisaparameter.ρ(Y=y)ontheRHSisafunctionofyonly,independentofx.ItsimplyservesasanormalizingfactortomaketheRHSintegrateto1.Thus,wedon’tneedtoexplicitlykeeptrackofρ(Y=y).WecanwriteBayestheoremconvenientlyas

ρ(X = x Y = y) ∝ ρ(Y = y X = x)⋅ρ(X = x)

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wheretheRHSneedsapropernormalizingfactortomakeitintegrateto1.

Thistrickisespeciallyconvenientfornormaldistributions.Oncewefind

ρ(X = x) ∝ exp −(x −µ)

2

2σ2⎛

⎝⎜⎞

⎠⎟,

wecanconcludeX~N(μ,σ2).

Conditionaldistribution(W(t1)|W(t1+t2)=y)

ThetargetisaprobabilityconditionedonWatalatertime.WeuseBayestheoremtowriteitintermsofaprobabilityconditionedonWatanearliertime.

LetX=W(t1)andY=W(t1+t2).Wehave

W(t1)~N(0,t1)

==>ρ W(t1)= x( )~N(0,t1)∝exp −x

2

2t1⎛

⎝⎜⎞

⎠⎟

W(t1 +t2) = W(t1)+ W(t1 +t2)−W(t1)( )

~N(0,t2)! "### $###

==> W(t1 +t2)W(t1)= x( ) ~ x +N(0,t2)

==>ρ W(t1 +t2)= y W(t1)= x( ) ∝ exp −( y − x)

2

2t2⎛

⎝⎜⎞

⎠⎟

TheBayestheoremgivesus

ρ W(t1)= x W(t1 +t2)= y( ) ∝ ρ W(t1 +t2)= y W(t1)= x( )⋅ρ W(t1)= x( )

∝exp −( y − x)

2

2t2⎛

⎝⎜⎞

⎠⎟exp −x

2

2t1⎛

⎝⎜⎞

⎠⎟

(Wedon’tneedtokeeptrackoffactorsthatareindependentofx!)

∝exp − 12t1

+ 12t2⎛

⎝⎜⎞

⎠⎟x2 −2 y2t2

x⎛

⎝⎜

⎞

⎠⎟

⎛

⎝⎜

⎞

⎠⎟

(Completingthesquare)

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∝exp− x −

t1 yt1 +t2

⎛

⎝⎜⎞

⎠⎟

2

2 t1t2t1 +t2

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

~ N t1 yt1 +t2

, t1t2t1 +t2

⎛

⎝⎜⎞

⎠⎟

Weconclude

ρ W(t1)= x W(t1 +t2)= y( ) ~ N t1 yt1 +t2 ,

t1t2t1 +t2

⎛

⎝⎜⎞

⎠⎟

Forthegeneralcase,wehave

ρ W(a+t1)= x W(a)= ya andW(a+t1 +t2)= yb( ) ~ N t1 yb +t2 yat1 +t2 ,

t1t2t1 +t2

⎛

⎝⎜⎞

⎠⎟

Aspecialcase: t1=t2=h

ρ W(a+h)= x W(a)= ya andW(a+2h)= yb( ) ~ N ya + yb2 ,

h2

⎛

⎝⎜⎞

⎠⎟