First hitting time of Double Integral Processes to curved boundaries

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boundaries

Jonathan Touboul, Olivier Faugeras

To cite this version:

Jonathan Touboul, Olivier Faugeras. First hitting time of Double Integral Processes to curved boundaries. [Research Report] RR-6264, INRIA. 2008, pp.37. �inria-00166335v3�

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a p p o r t

d e r e c h e r c h e

9-6399ISRNINRIA/RR--6264--FR+ENG

Thème BIO

First hitting time of Double Integral Processes to curved boundaries

Jonathan Touboul and Olivier Faugeras

N° 6264

January 28, 2008

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Jonathan Toubouland OlivierFaugeras

ThèmeBIOSystèmesbiologiques

ProjetOdyssée

∗

Rapportdereherhe n°6264January28, 200837pages

Abstrat: TheproblemofndingthersthittingtimeofaDoubleIntegralProess(DIP)

suh asthe Integrated Wiener Proes (IWP) hasbeen aentral and diult endeavor in

stohastialulusandhasappliationsinmanyeldsofphysis(rstexittimeofapartile

in a noisy fore eld) orin biology and neurosiene (spike time of an integrate-and-re

neuron with exponentially deaying synapti urrent). The only results available so far

were an approximation of the stationnarymean rossing time and the distribution of the

rst hitting time of the IWP to a onstant boundary. In this paper, we generalize those

resultsandndananalytialformulaforthersthittingtimeoftheIWPtopieewiseubi

boundaries.Weusethisformulatoapproximatethelawofthersthittingtimeofageneral

DIPtoasmoothurvedboundary,andweprovideanestimationoftheonvergeneofthis

method. This approximationformula isthe rstanalytialdesriptionof thehitting time

of aDIP to aurvedboundary, and allowsus to inferproperties of this random variable

andprovidesawayforomputingauratelyitslaw. Theaurayoftheapproximationis

omputedinthegeneralasefortheIWPandthealulationofrossingprobabilityanbe

arriedoutthroughaMonte-Carlomethod.

Key-words: Firsthitting time, rstpassagetime, urved boundary, integrated Wiener

proess, Double integral proess, seond-order stohasti dierential equation, numerial

omputation.

∗

OdysséeisajointprojetbetweenENPC-ENSUlm-INRIA

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Résumé : Le problème de trouver le premier temps d'atteinte d'un proessus double-

integral (DIP) tel que le mouvement brownien intégré (IWP) est un probleme entral et

diile dansledomainedualul stohastiqueet adesappliationsdansbeauoupdedo-

mainesphysiques(premiertempsde sortied'unepartiuledans unhampdeforebruité)

ouenbiologieetenneurosienes(tempsdespiked'unneuroneintegre-et-tireavedesou-

rantssynaptiàdéroissaneexponentielle). Lesseulsrésultatsdisponiblesjusqu'iiétaient

ladistributiondupremiertempsd'atteintedel'IWPauneonstanteetuneapproximation

delamoyennedeettevariablealéatoire. Dansetartile,nousgénéralisonsesrésultatset

trouvonsuneformuleanalytiquepourladistributiondupremiertempsd'atteintedel'IWP

aune ourbeubique parmoreaux. Nous utilisons ette formule pour approximer laloi

deprobabilitédupremiertempsd'atteinted'unDIPgénéralàuneourbelisse,etdonnons

une estimation de la onvergene de ette méthode. Cette approximationest la première

desriptionanalytiquedupremiertempsd'atteinted'unDIPàuneourbe,etnouspermet

detrouverdespropriétésdeettevariable aléatoireetune méthodeandelaalulere-

aement. Lapréisionde etteméthode est estimée dansle asgénéralpourl'IWP et le

alulnumériquedeetteloipeut-etreeetuéenutilisantunalgorithmedeMonte-Carlo.

Mots-lés : premiertempsd'atteinte,premiertempsdepassage,frontiereourbe,primi-

tivedumouvementbrownien, proessusdouble-integral,equation dierentielle stohatique

duseond order,alulnumérique

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Introdution

Firstpassagetimeproblemsforone-dimensionaldiusionproessesthroughatime-dependent

boundaryhave reeived alot of attention over thelast three deades. Unfortunately, the

evaluationof the rstpassagetime probabilitydistribution funtion (pdf) througha on-

stantortimedependentboundaryisin generalanarduoustaskwhihhasstillnotreeived

asatisfatory solution. Analytial results are sare and fragmentary, even if losed form

solutions exist for some verypartiular ases. Sine no analytial method seem to solve

theproblem,oneisled eitherto thestudyoftheasymptotibehaviorofthisfuntion and

of itsmoments(see e.g. [23, 24℄), orto the useof somewhatad-ho numerialproedures

yielding approximate evaluations of therst passage time distributions. Suh proedures

anbe lassiedas follows: (i) those that are based on probabilisti approahes(see e.g.

[4,6,7,21,29,30℄),and(ii)purelynumerialmethods,suhasthewidelyusedMonte-Carlo

method whih applies withoutany restrition, but whose results are generally too oarse

(fornumerialmethods, seee.g. [1,9,11,15℄).

Intwoandhigherdimensions,theproblemisevenmoreomplexandresultsanhardly

befound. For thesimplest Double IntegralProess (DIP), the Integrated WienerProess

(IWP)denedin(1.10),MKean[20℄Goldman[12℄,Lahal[16,17,18℄foundtheprobability

distributionofthersthittingtimetoaonstantboundaryusingstohastialulusmeth-

ods. Lefebvreused theKolmogorov(Fokker-Plank)equation tondin somespeial ases

losed-formsolutions[19℄. Generalizationsoftheseformulasto otherboundariesandother

kindsofproessesaresimplynotavailable. Inthepresentpaper,weproposealosed-form

solutionforthersthitting timeof theIWPto apieewiseubi funtion,andapply this

formulato nd anapproximationofthe rsthitting time of aDIPto anysmoothurved

boundary. Wealsoprovideanestimationoftherateofonvergeneofthisapproximation.

Inthe rstsetion,weintrodueamotivationof thisstudy, dene theDoubleIntegral

Proessand provethemain properties whih willbeusefulforus in therest ofthepaper.

Intheseondsetion,westudythersthittingtimesoftheIWPandprovidealosed-form

formulaforthersthittingtimeof thisproessto apieewiseubi funtion. Inthethird

setion,weintroduetheapproximationmethod ofthersthittingtimeoftheIWPtoany

smooth urvedboundary, and ndthe rate ofonvergene of this method. Finally in the

lastsetionweprovideanapproximationformulaforthersthittingtimeofageneralDIP

toaurvedboundary. ThefthsetiondesribesbrieyanumerialMonte-Carloalgorithm

whih anbeusedto omputetheprobabilityrepartitionfuntion eiently.

1 The Double Integral Proess

InthissetionweintroduetheDoubleIntegralProess(DIP)andprovesomeusefulprop-

erties. Butbeforethemathematialstudyoftheproblem,wemotivatethistheoretialwork

by a spei problem arising in neurosiene: the distribution of the spike times for an

integrate-and-reneuronwithexponentiallydeayingsynaptiurrents.

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1.1 Motivation

Thedenition oftheDIP andthe studyof itsrsthittingtimes ofurvedboundaries has

been motivated by numerous physial and biologial problems. For instane a problem

arising in neurosiene is to haraterize the probability distribution of the spike (ation

potentials) timesin preseneofsynaptinoise(see[10℄ foranintrodutionof theneuronal

modelizationofspikingneuronsand[33℄forreviewoftheproblemofspiketimedistribution).

A lassial neuron model is the leaky integrate-and-re model, where the membrane

potentialV(t)ôfâ^neuralêllîntegratesêxternalînputsând^the^noiseât^the^synapses,ând

emitsaspikewhenthemembranepotentialreahesadeterministithresholdfuntion θ(t)

(whih isonstantin general). Hene inthis model, themembranepotentialissolutionof

theequation:

τmdV(t) = −(V(t)−V^rest) +Ie(t)

dt+dIs(t) ^(1.1)

Inthis equationτm ^is^theharateristitimeofintegration ofthemembranepotential,

V^rest^is^the^rest^potential^of^the^neuron,Ie^representsdeterministiexternalinputsandIs^the

noisysynaptiinputs(seeforinstane[8,10,33℄). Thesimplestmodelofsynaptinoiseisa

standardBrownianmotion,ifweneglettheintegrationtimeofthesynapse. Nevertheless,

realpost-synaptiurrentshaveaveryshortrisetimeandalargerdeaytime.

If wetakeinto aount thedeay time of the synapseτs^, ^then ^the ^synapti^urrent^is

solutionofthestohastidierentialequation:

τsdIs(t) =−Is(t)dt+σdWt

Weanintegratethissystemofstohastidierentialequationsasfollows. Theequation

governingthemembranepotentialyields

V(t) =V^rest(1−e⁻

t τm) +_τ¹

m

Rt 0e

s−t

τmIe(s)ds+_τ¹

m

Rt 0e

s−t

τmIs(s)ds,

andthesynaptiurrentequationanbeintegratedas

Is(t) =Is(0)e⁻

t τs + σ

τs

Z t 0

e

s−t τs dWs,

where Is(0) ^is ^a ^given ^random ^variable. ^We ^dene _α¹ = _τ¹

m − τ¹s

. Replaing in the rst

equationIs(t)^byîts^valueⁱⁿ^the^seondêquation^weôbtain V(t) =Vrest(1−e⁻

t

τm) +_τ¹_mRt 0e

s−t

τmIe(s)ds+

Is(0) 1−^ττ^ms

(e⁻^τ^t^s −e⁻

t

τ_m) + σ τmτs

e⁻

t τ_m

Z t 0

e^α^s Z s

0

e

s^′ τ_sdWs^′

ds

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Thetime ofthespikeemissionisthe rsthittingtimeof V(t)^to ^the^thresholdθ(t)^, ^so

thersthittingtimeofthestohastiproess,whihisapartiularaseofwhatwewillall

inthesequelDoubleIntegralProess(DIP)

Xt= Z t

0

e^α^s Z s

0

e

s^′ τsdWs^′

ds ^(1.2)

tothedeterministiurvedboundary

a(t) =θ(t)−

Vrest

1−e⁻

t τm

+_τ¹_mRt 0e

s−t

τmIe(s)ds+₁₋^I^s⁽⁰⁾τm τs

e⁻

t τs −e⁻

t τm

.

1.2 Denition and main properties of DIPs

In this setion, we dene a lass of stohasti proesses inluding the proess (1.2), and

provesomeusefulpropertiesoftheseproesses.

Denition 1.1 (DIP). Letf ∈ L²(^R) ^and g∈L¹(^R)^. ^LetMt ^be^the^martingale ^dened

byMt:=Rt

0f(s)dWs^.

Thedoubleintegral proess(DIP) assoiatedtothefuntions f ândgîs^dened ^forâllt

by:

Xt= Z t

0

g(s)Msds= Z t

0

g(s) Z s

0

f(u)dWu

ds ^(1.3)

Proposition 1.1. Thetwo-dimensionalproess(Xt, Mt)^is^a^Gaussian ^Markov^proess.

Proof. Firstof all, note that if Ft^X ^(resp. Ft^M⁾ ^denes ^the^anonial ^ltration ^assoiated

to theproessX ^(resp. M⁾ ^then ^it ^is^lear ^that ∀t≥0^, Ft^X ⊂ Ft^M^. ^Hene ^the ^ltration

assoiatedtothepair(Xt, Mt)t≥0^is^simply(Ft^M)t≥0^,^whih^we^denoteⁱⁿ^the^sequel(F^t)t≥0

ItisalsolearthatM ^is^amartingale,andsatisestheMarkovproperty. Lets≤t^. ^We

have:

Xt= Z t

0

g(u)Mudu

= Z s

0

g(u)Mudu+ Z t

s

g(u)Mudu

Xt=Xs+ Z t

s

g(u)(Mu−Ms)du+Ms

Z t s

g(u)du ^(1.4)

ConditionallytoMs^,^the^proessRt

sg(u)(Mu−Ms)du ^isindependentofFs^M ^so^the^law

ofXt^knowing(Xs, Ms)îsindependentofthesigma-algebra(F^t)^,ând^soîsM^,^soêventually

thepair(X, M)^is^Markov.

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The pair is learly a Gaussian proess sine its two omponents are. Indeed, M ^is

Gaussian asthe limit of theRiemann sumsof Brownianinrements, whih are Gaussian,

andX îsâlso^the^limitôf^Riemann^sumsôfâ^Gaussian^proess,^namelyM^,^with^the^weights

givenbyg^.

Remark 1. In theproof ofproposition 1.1, we proved alsothat onditionally toMs^, ^the

inrements(Xt−Xs, Mt−Ms)^areindependentoftheσ^-eld F^s^.

Proposition 1.2. Foreah value of t ≥0^, ^the ^random^variable Yt := (Xt, Mt) ^is^a ^two-

dimensionalGaussianvariableofparameters:





E[Yt] = (0,0)

E

Y_t^T ·Yt

=

ρX(0, t) C_(X,M)(0, t) C_(X,M)(0, t) ρM(0, t)

(1.5)

wherethefuntionsρX(s, t)^,C(X,M)(s, t)^andρM(s, t)^are^dened ^by:







ρM(s, t) =Rt

sf(u)²du ρX(s, t) = 2Rt

sg(u) Ru

s g(v)ρM(s, v)dv du C_(X,M)(s, t) =Rt

sg(u)ρM(s, u)du

(1.6)

ThetransitionmeasureoftheMarkovproess(Yt)t^has^a^Gaussian^density^w.r.t. ^Lebesgue's

measure:

N

x_s+m_sRt sg(u)du ms

,C(s, t)˜

(1.7)

wheretheorrelationmatrixC(s, t)˜ ^reads:

C(s, t) =˜

ρX(s, t) C_(X,M)(s, t) C_(X,M)(s, t) ρM(s, t)du

(1.8)

Proof. Thealulationsareessentiallystraightforward. Toompute thetransitiondensity

funtion,weusetheequation(1.4)andwrite:

Xt

Mt

=

Xs+MsRt sg(u)du Ms

+

Rt

sg(u)(Mu−Ms)du Mt−Ms

(1.9)

Therstterminthesumintherighthandsideof(1.9)isF^smeasurable. GivenXs=xs

andMs=ms^,îtîsêqual^to

xs+msRt sg(u)du ms

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TheseondtermisindependentofF^s ^and^is^Gaussian.

Eventually, the proess Yt ^knowing Ys = (xs, ms) ^has ^the ^same ^law ^as ^the ^Gaussian

proess:

N

xs+ms

Rt sg(u)du ms

,C(s, t)˜

Denition1.2 (IWP). The IntegratedWiener Proess isaspeialaseofthe DIPwhere

thefuntionsf ândgâreîdentially êqual^to1 ^: Xt=

Z t 0

Wsds Ms=Ws ^(1.10)

Fromproposition1.2,weknowthatitstransitionmeasurereads:

P

h

Xt+s∈du, Wt+s∈dvX_s=x, Ws=yi_def

= pt(u v;x, y)du dv=

√3 πt²exph

− 6

t³(u−x−ty)²+ 6

t²(u−x−ty)(v−y)−2

t(v−y)²i

du dv ^(1.11)

Lemma1.3. Let(Xt)t≥0 ^beâ^DIP^dened ^by^(1.3). Âssume ^that f(s)6= 0 ^forâlls≥0^.

ThestudyofthehittingtimesoftheDIPXîsêquivalent^to^the^studyôf^the^simpler^proess:

X˜t= Z t

0

˜

g(s)Wsds,

whereg˜^is^denedⁱⁿ ^the^proof.

Proof. Let(Mt)t^be^the^martingale^dened^by:

Mt= Z t

0

f(s)dWs

Dubins-Shwarz'theorem 1

ensuresusthatthere existsaBrownianmotion(Wt)t^suh ^that

almostsurely

Mt=W_hM_i_t

Wenote

Φ(t) =hMi^t= Z t

0

f²(s)ds

1

EventhoughhMi∞6=∞^beauseôfôur^hypothesisônf^,^see^[14℄.

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0 1 2 3 4 5 6 7 8 9 10

−10

−5 0 5 10 15

Xt W

t a(t)

Figure1: AsamplepathoftheproessUt= (Xt, Wt)^where X îs â^standardÎWPândW

astandardBrownianmotion,andaboundaryurvea(t)^. ^TheÎWPXtîs^reset^to0 ^whenît

rossestheboundary.

Φîs ôntinuousând ^sine^weâssumed ^that f(s)6= 0 ^forâll s≥0^, ^strily înreasing, ^so ît

isabijetion. Its derivativeΦ^′(t)êxistsând îs^nonzero^forâllt≥0^. ^Weûse^the^hangeôf

variableu= Φ(s)^. ^We^have:

Xt= Z t

0

g(s)Msds

=L

Z t 0

g(s)WΦ(s)ds

=

Z Φ⁻¹(t) 0

g(Φ⁻¹(u)) Φ^′ Φ⁻¹(u)Wudu

HenethehittingtimeofageneralDIPanbededuedfromthehittingtimeoftheproess

X˜t=X_Φ(t) ^whih ^is^of^typeRt

0g(s)W˜ sds^,^where˜g(t) =_Φ^g(Φ_′_(Φ⁻₋¹1^(t))(t))

2 Hitting time of the integrated Wiener proess

Weonsider thespeial ase(Wt)t≥0^, â ^standard^Brownian^motion. ^Weâre înterested ⁱⁿ

thersthittingtimeto aurvedboundarya(t)^of^the^stohasti^proess:

Xt= Z t

0

Wsds ^(2.1)

This problem hasbeenwidely studied andhas reeived nosatisfatory solutionso far.

OneofthemaindiultiesomesfromthefatthattheproessisnonMarkov,sowehave

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torefertotheunderlyingWienerproess. ClassialapproahesbasedonVolterraequations

orDurbin's method,workfortheBrownianmotion,but failin providingasolutiontothis

problem(seeforinstane[33℄forareview). Toahievetheprogramofharaterizingthose

hittingtimes,werstreallexistingresultsonthersthittingtimestoonstantboundaries,

generalize them to ubi and pieewise ubi boundaries, to end with the approximation

formulaforgeneralboundaries.

2.1 First hitting time to a onstant boundary

Lahal in [16℄ studies this problem in the ase where the boundary is a onstant. More

preisely, in thissetion we studythe proessU_t= (Xt+x+ty, W_t+y) ^where X_t ^is^the

standardIWP.Wedenoteby

τa = inf

t >0 ; Xt+x+ty=a

therstpassagetimeat aôf ^the^rstômponent ôf^thebidimensionalMarkovproessUt^.

TheworkofLahal[16℄followstheworkofMKean[20℄,wherethejointlawoftheproess

(τa, Wτa)îsômputedⁱⁿ^theâsex=a^. ^The^resultîs:

P

h

τa ∈dt; |Wτa| ∈dzU₀= (a, y)i_def

= P(a, y)(τa ∈dt;|Wτa| ∈dz)

= 3z π√

2t²e^−(2/t)(y²^−|y|z+z²⁾

Z 4|y|z/t 0

e^−3θ/2 dθ

√πθ

!

1[0,+∞)(z)dzdt ^(2.2)

Wedenotethisdensitybym^a(t, y, z)^.

Later,Goldmanin [12℄omputedthedistribution oftherandomvariableτa ⁱⁿ ^the^ase

wherex < aândy≤0 ândôbtained^the^formula:

P

h

τa ∈dtU0= (x, y)i

=dthr 3 8πt³

3(a−x) t −y

e^{−3(a−x−ty)}²^/(2t³⁾ +

Z +∞

0

zdz Z t

0

Z ∞ 0

P

h

τ0∈ds; |Wτ0| ∈dµU₀= (0, z)i

qt−s(x, y;a, z)i

(2.3)

whereqt(x, y;u, v) =pt(x, y;u, v)−pt(x, y;u,−v)^.

Lastly,Lahalin[16℄extendedalltheseresultsandgavethejointdistributionofthepair

(τa, W_τ_a)ⁱⁿâllâses. ^The^quiteômplex^formula^reads:

P(x,y)[τa ∈dt; Wτa∈dz] =|z|h

pt(x, y;a, z)− Z t

0

Z +∞

0

m⁰(s,−|z|, µ)pt−s(x, y;a,−εµ)dµ dsi

1A(z)dzdt ^(2.4)