Extreme values and the level-crossing problem. An application to the Feller process

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by  Jaume Masoliver

Extreme values and the level-crossing problem.
An application to the Feller process
Jaume Masoliver∗
Departament de F´ısica Fonamental, Universitat de Barcelona,
Diagonal, 647, E-08028 Barcelona, Spain
(Dated: January 21, 2014)
Wereviewthequestionoftheextremevaluesattained byarandomprocess. Werelateittolevel
crossings either to one boundary (first-passage problems) and two boundaries (escape problems).
Theextremesstudiedarethemaximum,theminimum,themaximumabsolutevalueandtherange
or span. We specialize in diffusion processes and present detailed results for the Wiener and Feller
processes.
4
PACSnumbers: 89.65.Gh,02.50.Ey,05.40.Jc,05.45.Tp
1
0
2
I. INTRODUCTION the reasons why, to my knowledge, few exact analytical

n approacheshaveappearedexceptfor the Wiener process
a and,toalessextend,fortheOrnstein-Uhlenbeckprocess
J Level-crossing problems –including first-passage and
[4,18,19]. Despitethe intrinsicdifficulty thereare,how-
0 escape problems– have a long and standing tradition in
ever, recent works investigating this kind of problems in
2 physics,engineeringandnaturalsciences,withgreatthe-
subdiffusions and other anomalousdiffusion processes as
oretical interest in, for instance, bistability and phase
well (see [21] and references therein).
] transitions and countless practical applications ranging
h
from meteorology, seismology, reliable theory, structural
c
e andelectricalengineeringandfinance,justtonameafew Inarecentpaper[22]wehavestudiedthefirst-passage
m [1–15]. problem for the Feller process and presented a complete
solution of it, including first-passage and exit probabil-
- Thelevelcrossingproblemiscloselyrelatedtothethe-
t ities and mean first-passage and mean exit times. One
a ory of extremes, the latter initiated in the late nineteen
t twenties by the works of Frechet, Fisher and Tippet and of our goals here is to apply those results to obtain the
s
extreme values attained by the Feller process. Another
. subsequently developed by Gnedenko and Gumbel later
t objectiveistoreviewthelinkbetweenlevelcrossingsand
a in the forties and early fifties [16]. It applied to series of
m extremesby presentinga complete accountofthe results
independent random variables and the central result is
involved (some of them in a new and simpler form) be-
- theFrechet-Tippettheoremwhichstatesthatundersuit-
d causethe connectionamongbothproblemsis notwidely
able conditions the asymptotic distribution of extremes
n known in the current physics literature.
are restricted to be of three types (Gumbel, Frechet and
o
c Weibull) [4, 16, 17]. As remarked in Refs. [4] and [6],
[ when extreme events are rare (which is often the case) In level-crossing problems the issue of primary inter-

they can be approximately treated as independent vari- est is to ascertainthe statistical informationonthe time
1
ables for which the Fisher-Tippet theorem holds. This takenbyarandomprocesstoreach,orreturnto,agiven
v
9 approximation,however,reduces the question to a prob- boundary for the first time. If the boundary consists of
3 lemofstatisticsandtimeseriesanalysisandneglectsthe only one point –which we usually call critical value or
9 underlying dynamics and the correlations induced by it. threshold–onedealswitha first-passageorhitting prob-
4 Theextreme-valueproblembasicallyincludesthemax- lem. If the boundary consists of two points we have an
.
1 imum and minimum values attained by a given random escapeorexitproblemoutoftheintervalspannedbythe
0 process during a certain time interval. It also encom- boundarypoints. Aswewillseemaximumandminimum
4 passestherangeorspandefinedasthedifferencebetween aretheextremesrelatedtothehittingproblemwhilethe
1 the maximum and the minimum. In physics this prob- maximumabsolute value and the spanarerelatedto the
:
v lem has been traditionally related to level crossings and exit problem.
Xi first-passagetimes andithasbeenbasicallyrestrictedto
diffusionprocesses[4,18,19](seealso[20]forsimilarde- Thepaperisorganizedasfollows. InSec. IIwereview
r
a velopmentsaimedalsotodiffusionprocessesbutoriented the relationshipbetween first-passageandextreme-value
to the pure mathematician). problems. In Sec. III we review the link between the
This is acomplicatedbusiness because obtaining first- escape problem and, both, the maximum absolute value
passageprobabilitiesisessentiallydifficult. Thisisoneof and the span. In Secs. IV and V we explicitly obtain
these results for the Wiener and Feller processes respec-
tively. A short summary of main results is presented in
the last section. Some mathematical proofs and more
∗Electronicaddress: [email protected] technical details are in appendices.
2
II. FIRST PASSAGE AND EXTREMES x isimpossibleandhaszeroprobability. Inotherwords,
}
Φ (ξ,tx)=0,ifξ <x. Wesummarizebothcasesinto
max
|
The hitting problem of a random process X(t) is the single expression:
solved if we know the first-passage probability, W (tx),
of reaching for the first time threshold xc when thce p|ro- Φmax(ξ,t|x)=Sξ(t|x)Θ(ξ−x), (5)
cess starts at x=X(t ) at some initial time t (in what
0 0 whereΘ(x)istheHeavisidestepfunction. Bytakingthe
followswedealwithtime-homogeneousprocessessothat
derivative with respect to ξ and recalling that S (tx)=
x
t = 0). In terms of the hitting probability the survival |
0 0 (survival is impossible starting at the boundary) we
probability –i.e., the probability S (tx) that at time t,
c get the following expression for the probability density
|
orduringanyprevioustime, the processhasnotreached
function (PDF) ϕ (ξ,tx) of the maximum
max
x – is simply given by |
c
∂S (tx)
ξ
Sc(t|x)=1−Wc(t|x). (1) ϕmax(ξ,t|x)= ∂ξ| Θ(ξ−x). (6)
For one-dimensional diffusion processes charaterized
Letusdenoteby M(t) x the meanmaximumvalue,
by drift f(x) and diffusion coefficient D(x), the hitting
probability satisfies the Fokker-Planck equation (FPE) D (cid:12) E
[1, 22] (cid:12)∞
M(t) x = ξϕ (ξ,tx)dξ. (7)
max
|
∂ W (tx)=f(x)∂ W (tx)+ 1D(x)∂2 W (tx), (2) D (cid:12) E Z−∞
t c | x c | 2 xx c | We have (cid:12)
with initial and boundary conditions given by ∞ ∂Sξ(tx)
M(t) x = ξ | dξ. (8)
∂ξ
Wc(0x)=0, Wc(txc)=1. (3) D (cid:12) E Zx
| | Atfirstsightthisex(cid:12)pressioncanbesimplifiedbyaninte-
Equation (1) shows that the survival probability obeys gration by parts. This is, however, not possible because
the same FPE but with initial and boundary conditions S 1 as ξ leading to a divergent result. The
ξ
reversed. situ→ation can b→e a∞mended using W instead of S . Sub-
ξ ξ
We will now relate the first-passage problem with stituting Eq. (1) into Eq. (8) followed by an integration
the extreme values (the maximum and the minimum) by parts then yields
reached by the process during a given interval of time.
∞
There are other extremes, such as the range or span,
M(t) x =x+ W (tx)dξ, (9)
which will be discussed in the next section. ξ |
D (cid:12) E Zx
where we have ass(cid:12)umed that W decreases faster than
ξ
A. The maximum 1/ξ (i.e., ξWξ 0 as ξ ). Attending that Wξ is
→ → ∞
always positive this equation shows, the otherwise obvi-
ous result, that the mean maximum is greater than the
We denote by M(t) the maximum value reached by
initial value.
X(t) over the time span (0,t). Formally,
Followingananalogousreasoningwecaneasilyseethat
M(t)=max X(τ);0 τ t . the moments of the maximum, defined by
{ ≤ ≤ }
∞
Note that M(t) is a random quantity whose value de- Mn(t) x = ξnϕ (ξ,tx)dξ, (10)
max
pends on the particular trajectory of X(t) and its distri- |
bution function is defined by D (cid:12) E Z−∞
are given by (cid:12)
Φ (ξ,tx)=Prob M(t)<ξ X(0)=x . (4)
max | { | } ∞
Mn(t) x =xn+n ξn−1W (tx)dξ, (11)
ξ
Inordertorelatethis functionwiththe hitting probabil- |
ity we distinguish two cases: ξ > x and ξ < x. Suppose D (cid:12) E Zx
(n = 1,2,3,.(cid:12)..). In writing this equation we have as-
first that the value of the maximum ξ is greater than
sumed that ξnW 0 as ξ which is the condition
the initial value, ξ > x, in this case the process X(t) ξ
→ → ∞
imposed on W for moments to exist.
has notcrossedthresholdξ attime t andthe probability ξ
of the event M(t) < ξ X(0) = x equals the survival
{ | }
probability S (tx). That is
ξ
| B. The minimum
Φ (ξ,tx)=S (tx), (ξ >x).
max ξ
| |
We denote by
If on the other hand the value of the maximum is lower
thantheinitialpoint,ξ <x,theevent M(t)<ξ X(0)= m(t)=min X(τ);0 τ t
{ | { ≤ ≤ }
3
the minimum value attained by X(t) during the time problemissolvedwhenoneknowstheescapeprobability
interval (0,t), and let W (tx), which is defined as the probability of leaving
a,b
|
(a,b) at time t (or before) for the first time and starting
Φmin(ξ,t|x)=Prob{m(t)<ξ|X(0)=x} at x ∈ (a,b). Closely related to the Wa,b is the survival
probability,
be its distribution function. Note that if ξ <x the event
m(t)<ξ X(0)=x impliesthattheprocesshascrossed S (tx)=1 W (tx), (18)
a,b a,b
{ | } | − |
thresholdξattimetorbefore. Hencethethedistribution
givingtheprobabilitythat,startinginside(a,b),thepro-
function agrees with the hitting probability to level ξ,
cess has not exited this interval at time t or before.
i.e. Φ (ξ,tx) = W (tx). On the other hand, when
min ξ
| | For one dimensional diffusion processes, the escape
ξ > x the event m(t) < ξ X(0) = x is certain and
{ | } probability satisfies the FPE [1, 22]
Φ (ξ,tx)=1. Summing up
min
|
1
Φmin(ξ,t|x)=Θ(ξ−x)+Wξ(t|x)Θ(x−ξ). (12) ∂tWa,b(t|x)=f(x)∂xWa,b(t|x)+ 2D(x)∂x2xWa,b(t|x),
(19)
Let us denote by ϕ (ξ,tx) the PDF of the minimum
min | with initial and boundary conditions given by
m(t). Taking the derivative with respect to ξ of Φ
min
and noting that Wξ(t|x)δ(x−ξ) = δ(x−ξ) (recall that Wa,b(0|x)=0, Wa,b(t|a)=Wa,b(t|a)=1. (20)
W (tξ)=1) we get
ξ
| NotethatS (tx)alsoobeysEq. (19)butwithinitial
a,b
|
∂W (tx) and boundary conditions reversed; that is,
ξ
ϕ (ξ,tx)= | Θ(x ξ). (13)
min
| ∂ξ −
S (0x)=1, S (ta)=S (ta)=0.
a,b a,b a,b
| | |
The mean minimum value, defined as
Extreme values related to the escape probability are
∞ essentially two: the maximum absolute value and the
m(t) x = ξϕmin(ξ,tx)dξ, (14) span. Let us next address them.
|
D (cid:12) E Z−∞
is then given by (cid:12)
A. The maximum absolute value
x ∂W (tx)
ξ
m(t) x = ξ | dξ. (15)
∂ξ We now consider the maximum absolute value at-
D (cid:12) E Z−∞ tained by X(t) during the time span (0,t). Denote by
An integration by (cid:12)parts yields G (ξ,tx) its distribution function,
max
|
x G (ξ,tx)=Prob max X(τ) <ξ X(0)=x , (21)
m(t) x =x lim [ξWξ(tx)] Wξ(tx)dξ. max |
D (cid:12) E −ξ→−∞ | −Z−∞ | where 0 ≤ τ ≤ t an(cid:8)d ξ >(cid:12)(cid:12) 0. (cid:12)(cid:12)Certa(cid:12)(cid:12)inly ξ can(cid:9)not be
Because(cid:12)W (tx)=0(i.e., hittinganinfinitethreshold negative and hence
−∞
|
is impossible) then, if we also assume that W decreases
ξ G (ξ,tx)=0, (ξ <0).
max
faster than 1/ξ , we have ξW 0 as ξ and |
ξ
| | → →−∞
In order to connect this distribution function with the
x
escape problem we must distinguish two cases according
m(t) x =x W (tx)dξ, (16)
− ξ | to which the initial point is inside or outside the inter-
D (cid:12) E Z−∞ val ( ξ,ξ) spanned by the level ξ > 0 of the absolute
which shows that(cid:12)the mean minimum value is indeed maxi−mum. For the first case where ξ <x<ξ, we have
lower than the initial value. −
Analogously to the maximum value, the moments of max X(τ) <ξ;0 τ t X(0)=x
≤ ≤
the minimum are given by
n (cid:12) (cid:12) (cid:12) o
= ξ(cid:12) <X(cid:12)(τ)<ξ;0 τ (cid:12) t X(0)=x ,
x − ≤ ≤
mn(t) x =xn n ξn−1Wξ(tx)dξ, (17) n (cid:12) o
D (cid:12) E − Z−∞ | hmaesanniontglethftatthdeuirnintegrtvhael t(imξe,ξs)p.anH(e0n,cte(cid:12)),tthheepdrioscterisbsuXti(otn)
as long as Wξ(cid:12) decreases faster than ξ −n as ξ . function (21) coincides wi−th the survival probability
| | →−∞
G (ξ,tx)=S (tx), (x <ξ).
max −ξ,ξ
| | | |
III. EXTREMES AND THE ESCAPE PROBLEM
Note that whenthe initial value is outside the interval
( ξ,ξ),theevent maxX(τ) <ξ X(0)=x (0 τ t)
The escape, or exit, problem addresses the question − { | | | } ≤ ≤
is impossible and
ofwhether ornota givenprocessX(t)starting inside an
interval(a,b)hasdepartedfromitforthefirsttime. The G (ξ,tx)=0, (x >ξ).
max
| | |
4
Therefore, B. The range or span
Gmax(ξ,tx)=S−ξ,ξ(tx)Θ(ξ x), (22) The range or span (also termed as “the oscillation”)
| | −| |
of a random process X(t) over the time interval (0,t) is
(ξ > 0). The PDF of the absolute maximum is defined defined as the difference between the maximum and the
by minimum:
∂ R(t)=M(t) m(t). (27)
g (ξ,tx)= G (ξ,tx). −
max max
| ∂ξ |
This random quantity is either characterized by the dis-
tribution function,
Substituting for Eq. (22) and noting that
F (r,tx)=Prob R(t)<rX(0)=x ,
R
S (tx)δ(ξ x)=S (tx)δ(ξ x)=0, | { | }
−ξ,ξ −|x|,|x|
| −| | | −| |
or by the PDF
we get
∂
f (r,tx)= F (r,tx). (28)
R R
∂S (tx) | ∂r |
−ξ,ξ
g (ξ,tx)= | Θ(ξ x), (23)
max
| ∂ξ −| |
Wecanrelatethespandistributiontotheescapeprob-
lem out of a variable interval. This connection is a bit
(ξ > 0). In terms of the escape probability W this
−ξ,ξ convoluted and we show in Appendix A that
PDF can be written as
x ∂2S (tx)
v,r+v
∂W (tx) f (r,tx)= | dv, (29)
g (ξ,tx)= −ξ,ξ | Θ(ξ x). (24) R | ∂r2
max | − ∂ξ −| | Zx−r
(r > 0), where S (tx) is the survival probability in
v,r+v
Let us next evaluate the mean value of the absolute the (variable) interval (v|,r+v).
maximum defined by Having the expression for the span PDF we next ad-
dress the issue of the mean span:
∞
max X(t) x = ξgmax(ξ,tx)dξ. ∞
| | |
D (cid:12) E Z0 R(t) x = rfR(r,t|x)dr. (30)
From Eq. (24) we ha(cid:12)ve D (cid:12) E Z0
(cid:12)
Unfortunately the introduction of Eq. (29) into this def-
∞ ∂W (tx) inition leads to indeterminate boundary terms as the
−ξ,ξ
max X(t) x = ξ | dξ. readercan easily check. In the Appendix B we present a
| | − ∂ξ
D (cid:12) E Z|x| way of avoiding these inconsistencies and the final result
(cid:12) reads
Integration by parts yields
∞ ∂S (tx)
ξ
∞ R(t) x = ξ | dξ, (31)
∂ξ
max|X(t)| x =|x|+ W−ξ,ξ(t|x)dξ, (25) D (cid:12) E Z−∞
D (cid:12) E Z|x| where Sξ(tx) if th(cid:12)e survival probability up to thresh-
(cid:12) |
old ξ. Let us incidentally note the curious fact that the
where wehavetakeninto accountthat W (tx)=1
−|x|,|x|
| complete probability distribution of the span is deter-
and made the reasonable assumption that the escape
mined by the escape problem out of the variable inter-
probability W decreases faster than 1/ξ, that is,
−ξ,ξ
val (v,v +r) where x r < v < x. However, the first
ξW−ξ,ξ 0 as ξ . −
→ →∞ moment of this distribution depends only on the first-
Again, the moments of the maximum absolute value
passage problem of a varying threshold <ξ < .
can be written as −∞ ∞
In terms of the the hitting probability W (tx) the ex-
ξ
|
∞ pression above for the mean span is greatly simplified.
max|X(t)| n x =|x|n+n ξn−1W−ξ,ξ(t|x)dξ, Indeed, substituting Sξ =1−Wξ into Eq. (31), followed
D(cid:0) (cid:1) (cid:12) E Z|x| (26) by an integration by parts, yield
(cid:12)
(n = 1,2,3,...). These moments exist as long as W−ξ,ξ ∞ ∂Wξ(tx)
decreases faster than ξ −n as ξ . R(t) x = ξ | dξ
lutWeevafilunealilsymreemanairnkglt|ehs|ast, foobrtat|hini|si→nvga∞ltuheeims ninoitmaumranadbosmo- D (cid:12)(cid:12) E −Z−∞ ξ∂=ξ+∞ ∞
= ξW (tx) + W (tx)dξ.
ξ ξ
variable: it is always zero. − | |
(cid:12)ξ=−∞ Z−∞
(cid:12)
(cid:12)
(cid:12)
5
However, W 0 as ξ (i.e., crossing becomes (2)-(3) with f(x) = 0 and D(x) = D. The time Laplace
ξ
→ → ±∞
impossibleasthresholdgrows). If,inaddition,weassume transform
that this decay is faster than 1/ξ, i.e., ξW 0 (ξ
ξ ∞
→ →
), we have Wˆ (sx)= e−stW (tx)dt
±∞ c | c |
Z0
∞
R(t) x = W (tx)dξ. (32) leads to the following boundary-value problem
ξ
|
It is worth nDoticin(cid:12)(cid:12)gEthatZo−n∞e can arriveat this expres- d2Wˆc =(2/D)sWˆ , Wˆ (sx )=1/s. (34)
dx2 c c | c
sion in a more direct way. In effect, recalling the defini-
tionofthe rangeasthedifferencebetweenthe maximum Thesolutiontothisproblemthatisfiniteforbothx>x
c
and the minimum, we have and x<x is straightforwardand reads
c
R(t) x = M(t) x m(t) x , (33) 1 2s
− Wˆ (sx)= exp x x .
c c
D (cid:12) E D (cid:12) E D (cid:12) E | s (−rD | − |)
and substituting(cid:12)for Eqs. (9)(cid:12)and (16) we(cid:12)get
Laplace inversion yields [23]
∞ x
R(t) x = W (tx)dξ+ W (tx)dξ,
D (cid:12) E Zx ξ | Z−∞ ξ | Wc(t|x)=Erfc |x√−2Dxtc| , (35)
(cid:12) (cid:20) (cid:21)
which is Eq. (32).
Thereisnosimpleexpressions,besideEq. (32),forthe where Erfc(z) is the complementary error function. The
span higher moments as it is for the other extremes. In PDF of the maximum value is then given by Eq. (6) or,
the present case moments have to be evaluated through equivalently, by
their definition and the use of Eq. (29)
∂W (tx)
ξ
ϕ (ξ,tx)= | Θ(ξ x),
∞ max | − ∂ξ −
Rn(t) x = rnf (r,tx)dr
R
|
D (cid:12) E Z0∞ x ∂2S (tx) whichresultsinthefollowingtruncatedGaussiandensity
(cid:12) = rndr v,r+v | dv.
Z0 Zx−r ∂r2 ϕ (ξ,tx)= 2 1/2e−(ξ−x)2/2DtΘ(ξ x). (36)
max
| πDt −
This is quite unfortunate because the evaluation of span (cid:18) (cid:19)
moments becomes a complicated business even numeri- Themeanmaximumis thengivenby(cfEqs. (7)or(9))
cally. The reason for not having a more convenient ex-
pression lies in the fact that maxima and minima are 2Dt 1/2
generallycorrelatedquantitiesandthesecorrelationsap- M(t) x =x+ , (37)
π
pear in all moments greater than the first one. D (cid:12) E (cid:18) (cid:19)
Likewise, the PDF(cid:12) of the minimum value is given by
(cf Eq. (13))
IV. THE WIENER PROCESS
ϕ (ξ,tx)= 2 1/2e−(x−ξ)2/2DtΘ(x ξ), (38)
We now illustrate the expressions obtained above by min
| πDt −
reviewingoneofthe simplest,albeitveryrelevant,cases: (cid:18) (cid:19)
the Wiener process or free Brownian motion, a diffusion and the mean minimum reads
process with zero drift andconstantdiffusion coefficient.
1/2
Although some results related to first-passage and ex- 2Dt
m(t) x =x . (39)
tremesfortheBrownianmotioncanbetracedasfarback − π
as to Bechelier, Levy and Feller [18], many results are D (cid:12) E (cid:18) (cid:19)
found scattered in the mathematics and physics litera- Noticethatbothext(cid:12)remevaluesgrowliket1/2 ast ,
→∞
ture[18,19]. Itis,therefore,usefultohaveasummaryof the otherwise typical behavior of the Wiener process.
the main results about the extreme values of the Wiener These results can be generalized to include any mo-
process. ment of the maximum and the minimum. By combining
Eqs. (10) and (36) we easily see that
A. The maximum and the minimum Mn(t) x =
D1 n (cid:12) nE k+1
The first-passage probability Wc(tx) to some thresh- (cid:12) Γ (2Dt)k/2xn−k (40)
old x will be determined by the so|lution of the FPE √π k 2
c k=0(cid:18) (cid:19) (cid:18) (cid:19)
X
6
(n = 1,2,3,...). Following an analogous reasoning we the asymptotic time behavior of that average. It turns
show that the moments of of the minimum are out to be more efficient to proceed from the Laplace
transform of the average. We thus define
mn(t) x
=D 1 (cid:12)(cid:12)nE( 1)k n Γ k+1 (2Dt)k/2xn−k(41) µˆ(s|x)=L max|X(t)| x
√π − k 2 nD (cid:12) Eo
k=0 (cid:18) (cid:19) (cid:18) (cid:19) as the (time) Laplace transform of t(cid:12)he mean absolute
X
maximum. Transforming Eq. (25) yields
(n = 1,2,3,...). With increasing n these expressions
become rather clumsy. We can get, however,simpler ex- 1 ∞
pressions if instead of the maximum or the minimum we µˆ(sx)= x + Wˆ−ξ,ξ(sx)dξ.
| s| | |
consider their “distance” from the initial position. This Z|x|
is defined by M(t) x in the case ofthe maximumor by
− PluggingEq. (43)we see thatthe resultingintegralscan
x m(t) for the minimum. We have
− be done in close form and write
n n
M(t) x (t) x = x m(t) (t) x 1 √2D
− − µˆ(sx)= x + cosh x 2s/D
D(cid:0) (cid:1)1 (cid:12)nE+1D(cid:0) (cid:1) (cid:12) E | s| | s3/2
= √πΓ (cid:12) 2 (2Dt)n/2. (cid:12) (42) π arcta(cid:16)nepx√2s/D(cid:17) (45)
(cid:18) (cid:19) × 2 −
h i
Both distances are equal showing the otherwise obvious
Wenowusethisexactexpressionfortheasymptoticanal-
symmetry of the process.
ysis of the mean because, as Tauberian theorems prove
[24], the long time behavior of the mean is determined
by the small s behavior of its Laplace transform. It is a
B. The maximum absolute value
matter of simple algebra to show that as s 0 we have
→
As shown in the previous section in order to charac- 1 π√2D 1
terize both the maximum absolute value and the span, µˆ(sx)= x + +O ,
| s| | 4 s3/2 s1/2
weneedtoknowtheescapeprobability,W (tx), outof (cid:18) (cid:19)
a,b
|
an interval (a,b). For the maximum absolute value the
whichafterLaplaceinversionyields the asymptotic form
interval is symmetric while for the span is asymmetric.
of the mean absolute maximum
TheLaplacetransformoftheexitprobabilityobeysthe
same equation than that of the first-passage probability, πDt 1/2 1
Eq. (34), but with two boundary points: max X(t) x x + +O , (46)
| | ≃| | 2 t1/2
1 D (cid:12) E (cid:18) (cid:19) (cid:18) (cid:19)
Wˆa,b(sa)=Wˆa,b(sb)= . showing again(cid:12)the t1/2 growth.
| | s
The solution to this problem is
C. The span
cosh 2s/D[x (a+b)/2]
Wˆ (tx)= − . (43)
a,b
| scopsh 2s/D[(a b)/2] Let us finally describe the span of the Wiener process.
−
Asbeforewebetter workwithLaplacetransforms. Thus
TheLaplacetransformcanpbeeasilyinverted[23]. Inthe
from Eq. (29) we write
caseofasymmetricinterval( ξ,ξ)theinversetransform
−
is somewhat simpler yielding [18, 23] ∂2 x
fˆ (r,sx)= Wˆ (sx)dv,
W (tx)=1 2 ∞ (−1)n e−D(n+1/2)2π2t/ξ2 R | −∂r2 Zx−r v,r+v |
−ξ,ξ
| − π nX=0 n+1/2 (r > 0), where the escape probability Wˆv,r+v(s|x) is
cos (n+1/2)πx/ξ . (44) given by Eq. (43) (note that the second derivative can
× be pulled out of the integral because the lower limit is
The PDF for the maxim(cid:2)um absolute (cid:3)value, linear in r).
g (ξ,tx), is readily obtained by introducing Eq. For the Wiener process the escape probability is given
max
|
(44) into Eq. (24) (we will not write this expression). by Eq. (43) and the integral above can be done in close
Likewise the mean absolute maximum can be obtained form yielding
from this form of the escape probability after substitut-
ing it into Eq. (25). The resulting expression is given fˆ (r,sx)= (2D)1/2 ∂2 1 tanh s 1/2r .
by complicated infinite sums of exponential functions of R | − ∂r2 s3/2 2D
little practical use, since from it is hard to figure out (cid:20) (cid:16) (cid:17) (cid:21)(47)
7
TheLaplacetransformofthemeanspanisthengivenby if θ >1 such a probability is zero which renders the ori-
gin unaccessible (see [22] for a simple proof and more
∞
R(t) x = rfˆ (r,sx)dr = details).
R
L | The linear drift f(x) = (x θ) drives the process
nD (cid:12) E∞o ∂Z20 1 s 1/2 towards level θ, a determin−istic−pull which is increased
(2D)1/2(cid:12) r tanh r dr. near the origin where the noise term is very small. In
− ∂r2 s3/2 2D
Z0 (cid:20) (cid:16) (cid:17) (cid:21) effect, the state-dependent diffusion coefficient D(x) =
Integration by parts yields 2x for large values of x enhances the the effect of noise
while as x goes to zero this effect vanishes. Therefore,
(2D)1/2 when the process reaches the origin the drift drags it
R(t) x = ,
L s3/2 towards θ and since θ is positive the process remains
nD (cid:12) Eo always positive. The very fact that X(t) never attains
and after inversion we ge(cid:12)t the exact result negativevaluesmakestheprocessasuitablecandidatefor
modeling a number of phenomena in natural and social
1/2
2Dt sciences [22].
R(t) x =2 , (48)
π WenowstudytheextremevaluesattainedbytheFeller
D (cid:12) E (cid:18) (cid:19) process. We will basically obtain expressions for the
whichis,ofcourse,th(cid:12)edifferencebetweenthemeanmax-
maximumandminimumvaluesbecause,duethepositive
imum (37) and the mean minimum (39) (see Eq. (33)).
characterof the process,extremes such as the maximum
An interesting fact to note is that the long-time ratio
absolute value coincide with the maximum.
betweenthemeanabsolutemaximum(46)andthemean
For X(t)described by Eq. (49) the first-passageprob-
span is fixed and given by
ability to some threshold ξ is the solution of the Fokker-
Planck equation (cf. Eqs. (2)-(3))
max X(t) x
π
lim | | = , ∂ W (tx)= (x θ)∂ W (tx)+x∂2 W (tx), (50)
t→∞D R(t) x (cid:12) E 4 t ξ | − − x ξ | xx ξ |
(cid:12)
with initial and boundary conditions given by
D (cid:12) E
which means that at long tim(cid:12)es the mean maximum ab-
W (0x)=0, W (tξ)=1. (51)
solute value is always smaller than the mean span. ξ | ξ |
Wehaverecentlyprovedthatthesolutiontothisprob-
lemforthetimeLaplacetransformofW isgivenby[22]
ξ
V. EXTREMES OF THE FELLER PROCESS
F(s,θ,x)
, ξ x,
The Feller processis anotherexample ofdiffusion pro- Wˆ (sx)= sF(s,θ,ξ) ≥ (52)
cess having linear drift and linear diffusion coefficient ξ |  U(s,θ,x)
vanishingattheorigin[25]. Theprocesshasbeenapplied sU(s,θ,ξ), ξ ≤x,
not only to the modeling of socio-economic systems (the
where F and U are confluent hypergeometric (Kummer)
CIR-Heston model [26]) but also in theoretical biology
functions of first and second kind respectively [33].
suchaspopulationdynamicsandneuronfiringprocesses
[27,28]. Ithasbeenrecentlyappliedtoreproducecholera
epidemics as a susceptible-infected-recoveredmodel [29].
A. The maximum
It is also a significant model for single neuron dynamics
where functionals of the first-passage time are employed
to characterize the parameters of the model [30, 31]. The distribution function of the maximum is related
The process is governed by a stochastic differential to the survival probability Sξ(tx) by Eq. (5) which we
|
equation which in non-dimensional units (see [22]) can write in terms of the hitting probability, Wξ(tx), as
|
be written as
Φ (ξ,tx)=[1 W (tx)]Θ(ξ x).
max ξ
| − | −
dX(t)= [X(t) θ]dt+ 2X(t)dW(t), (49)
− − IntermsofWξ the meanmaximumisgivenby Eq. (16):
p
where W(t) is the Wiener process and θ >0 is a dimen- ∞
sionless parameter –called saturation or normal level– M(t) x =x+ W (tx)dξ.
ξ
|
representing the value to which X(t) is attracted. This D (cid:12) E Zx
parameterhasakeyroleinthebehavioroftheprocessfor Looking at Eq. ((cid:12)52) we see that for the Feller process
it is related to the important question of the possibility the time Laplace transform of the distribution function
of reachingthe origin(which, for instance, in population and that of the mean are respectively given by
dynamics would imply extinction [32]). Indeed, if θ 1
≤
theprobabilityofreachingtheoriginisgreaterthanzero 1 F(s,θ,x)
Φˆ (ξ,sx)= 1 Θ(ξ x). (53)
andx=0 is anaccessibleboundary. Onthe otherhand, max | s − F(s,θ,ξ) −
(cid:20) (cid:21)
8
and process T (x) = and the approximation given by Eq.
ξ
∞
(59)ismeaningless. FortheFellerprocessthistimeexists
1 ∞ dξ
Mˆ(sx)= x+F(s,θ,x) , (54) and, as we have proved in [22], reads
| s F(s,θ,ξ)
(cid:20) Zx (cid:21)
ξ
(1/θ) F(1,1+θ,z)dz, ξ >x,
where x
T (x)= (60)
ξ  xU(R1,1+θ,z)dz, ξ <x.
Mˆ(sx)= M(t) x  ξ
| L
nD (cid:12) Eo If the meanRfirst-passage time exists, the distribution
is the time Laplace transform of the(cid:12)mean maximum. function of the maximum and its mean are, as t ,
→ ∞
The PDF of the maximum is readily obtained by tak- approximately given by
ing the derivative with respect to ξ of the distribution
function (53). We have Φmax(ξ,tx) e−t/Tξ(x)Θ(ξ x), (61)
| ≃ −
F(s,θ,x)F′(s,θ,ξ) and
ϕˆ (ξ,sx)= Θ(ξ x), (55)
max | sF2(s,θ,ξ) − ∞
M(t) x x+ 1 e−t/Tξ(x) dξ, (62)
≃ −
where [33] D (cid:12) E Zx h i
where here (cid:12)
d s
F′(s,θ,ξ)= F(s,θ,ξ)= F(s+1,θ+1,ξ). (56) 1 ξ
dξ θ
T (x)= F(1,1+θ,z)dz
ξ
θ
Unfortunately the analyticalinversionofthese expres- Zx
sions to get their values in real time seems to be beyond since the maximum is always greater than the initial
reach, even though numerical inversion is always possi- point(ξ >x). Notethat 1 e−t/Tξ(x) 0 asξ be-
− → →∞
ble. Wewillfind, nonetheless,someapproximationsthat causethemeanfirst-passagetimetoaninfinitethreshold
may be appropriate in practical cases. is infinite and the integral in Eq. (62) converges [34].
Letusfirstshowthat,likeBrownianmotion,themean Equation (62) is a compact expression that may be
maximum value of the Feller process diverges as t suitable for the numerical evaluation of the mean maxi-
. Onemighthavethoughtthatsince–unlikeBrownia→n mum for large values of time. As far as I can see it is,
∞motion– the Feller process possesses a force drifting the however, of little use for further analytical approxima-
process towards the value θ, the mean maximum would tions.
tend to a finite value (not far from θ) as time increases. Let us thus obtain another asymptotic expansion of
Let us show that this is not the case. Indeed, recalling the maximum value which is valid for large values of the
the following property of the Laplace transform [24]: initial position x. Our starting point is the time Laplace
transformof the mean maximum given by Eq. (54). As-
lim f(t)= lim sfˆ(s) . (57) sume now that x we can then use the following
→ ∞
t→∞ s→0 approximation[33]
h i
andthe value of the Kummer function F(s=0,θ,z)=1 Γ(θ)
[33], we see that the limit s 0 in (54) leads to F(s,θ,x)= exxs−θ 1+O x−1 , (63)
→ Γ(s)
∞ (cid:2) (cid:0) (cid:1)(cid:3)
lim sMˆ(sx) =x+ dξ = . andsinceξ >xthenξ isalsolargeandwehaveananalo-
s→0h | i Zx ∞ gous expressionfor F(s,θ,ξ). Substituting both approx-
imations into Eq. (54) we get as x
Whence →∞
1 ∞
Mˆ(sx) x+exxs−θ e−ξξθ−sdξ .
M(t) x , (t ) (58)
| ≃ s
→∞ →∞ (cid:20) Zx (cid:21)
D (cid:12) E
and the mean maxim(cid:12) um diverges as time increases. But the integral can written in terms of the incomplete
We next refine this asymptotic behavior. As is well Gamma function Γ(1 + θ s,x) and within the same
−
known [1–3] the long-time expressions of first-passage approximationwe have [33]
probabilities are related to the mean first-passage time
∞
by (see also [22] for a simple derivation) e−ξξθ−sdξ =Γ(1+θ s,x) e−xxθ−s 1+O x−1 .
− ≃
Zx
W (tx) 1 e−t/Tξ(x), (t ), (59) (cid:2) (cid:0) (cid:1)(cid:3)
ξ | ≃ − →∞ Substituting into the previous equation yields Mˆ(sx)
| ≃
whereT (x)isthemeanfirst-passagetimetothresholdξ (x+1)/s+O(x−1) which after Laplace inversionresults
ξ
starting from x. Obviously this asymptotic expression is in the simple asymptotic approximation:
valid as long as the mean firs-passage time exists which
is not alwaysthe case. Thus, for instance, in the Wiener M(t) x x+1+O x−1 . (64)
≃
D (cid:12) E (cid:0) (cid:1)
(cid:12)
9
Despite its appeal, this approximations merely means and from Eq. (57) we conclude that
thatthemeanmaximumvaluegrowsatthesamepaceas
it does the starting value, as can be otherwise seen from m(t) x 0, (t ). (71)
→ →∞
Eq. (9).
D (cid:12) E
The mean minimum(cid:12)thus convergesto the originas time
increases.
B. The minimum Wenextrefinethis crudeestimateforlarge,butfinite,
values of time. When t and after using the asymp-
→∞
We recall from Sec. II that in terms of the hitting totic form of the hitting probability given in Eq. (59),
probability the distribution function of the minimum is we get
(see Eq. (12))
Φ (ξ,tx) 1 e−t/Tξ(x)Θ(x ξ), (72)
min
| ≃ − −
Φ (ξ,tx)=Θ(ξ x)+W (tx)Θ(x ξ).
min ξ
| − | − (t ), whereT (x) is the MFPTto thresholdξ which
ξ
→∞
The mean minimum is given in Eq. (16) where, due to when ξ <x is given by (cf. Eq. (60))
the positive character of the Feller process, we replace
x
in the lower limit of integration by 0: T (x)= U(1,1+θ,z)dz, (ξ <x).
−∞ ξ
x Zξ
m(t) x =x W (tx)dξ. (65)
ξ Substituting Eq. (59)intoEq. (65)wefindthefollow-
− |
D (cid:12) E Z0 ing long-time approximation of the mean minimum
Taking into acco(cid:12)unt Eq. (52), the time Laplace trans-
x
form of these quantities reads m(t) x e−t/Tξ(x)dξ, (t ). (73)
≃ →∞
1 U(s,θ,x) D (cid:12) E Z0
Φˆmin(ξ,sx)= Θ(ξ x)+ Θ(x ξ) , Likewise(cid:12)thelong-timebehaviorofthemaximumvalue
| s − U(s,θ,ξ) −
(cid:20) (cid:21) discussed above, these asymptotic expressions related to
(66)
the minimum value are more appropriate for numerical
and
evaluationratherthanforobtainingfurtherpracticalan-
1 x dξ alytical approximations.
mˆ(sx)= x U(s,θ,x) , (67)
| s − U(s,θ,ξ) We will find, nonetheless, approximationsof the mean
(cid:20) Z0 (cid:21)
minimum when the initial value x is small and close to
where mˆ(sx) is the time Laplace transformof the mean the origin. Our starting point is the expression of the
|
minimum and the U’s are Kummer functions of second Laplace transform of the mean minimum given in Eq.
kind [33]. (67). We next assume that x is smallthen fromEq, (70)
Taking the ξ-derivative of Eq. (66) we get the PDF of and the fact that F(a,b,x)=1+O(x) [33] we write
the minimum
Γ(1 θ)
U(s,θ,x)U′(s,θ,ξ) U(s,θ,x) = − 1+O(x)
ϕˆ (ξ,sx)= Θ(x ξ), (68) Γ(s+1 θ)
min | − sU2(s,θ,ξ) − −
Γ(θ 1) (cid:2) (cid:3)
+ − x1−θ 1+O(x) . (74)
where [33] Γ(s)
(cid:2) (cid:3)
d Note that the leading termin this expansiondepends on
U′(s,θ,ξ)= U(s,θ,ξ)= sU(s+1,θ+1,ξ). (69)
whether θ > 1 or θ < 1. We, therefore, distinguish the
dξ −
cases:
Starting form Eq. (67) and using the property given (i)θ >1(recallthatinthiscasetheoriginisunattain-
in Eq. (57) we canobtain the limiting value of the mean ablebythedynamicalevolutionoftheprocess[22]). Now
minimum when t . We begin with the relationship Eq. (74) yields the approximation
→ ∞
between Kummer functions U and F [33]:
Γ(θ 1)
U(s,θ,x) − x1−θ 1+O(x) . (75)
Γ(1 θ) ≃ Γ(s)
U(s,θ,x) = − F(s,θ,x) (70)
Γ(1+s θ) (cid:2) (cid:3)
− Since the integral in Eq. (67) runs from ξ = 0 to ξ = x
Γ(θ 1)
+ − x1−θF(1+s θ,2 θ,x). when x is small ξ is also small. We can thus use approx-
Γ(s) − − imation (75) for U(s,θ,ξ) inside the integral and write
Recallingthat ass 0 F(s,θ,x) 1andΓ(s) we x dξ Γ(s) x
see that U(s,θ,x) →1. Hence → →∞ ξθ−1dξ
→ Z0 U(s,θ,ξ) ≃ Γ(θ−1)Z0
x Γ(s) xθ
lim[smˆ(sx)]=x dξ =0, = .
s→0 | −Z0 Γ(θ−1) θ
10
PluggingthisapproximationalongwithEq. (75)intoEq. transform of the mean minimum
(67) we get mˆ(sx) x(1 1/θ)/s which after Laplace
inversion yields | ≃ − 1 x2−θ
mˆ(sx) = x (80)
| s" − Γ(1 θ)
1 −
m(t) x 1 x, (x 0). (76)
≃ − θ → ∞ z s
(ii) θ <D1 (th(cid:12)(cid:12)eEorigi(cid:18)n is att(cid:19)ainable [22]). In this case × Z0 e−xzz−θ(cid:18)1+z(cid:19) dz#+O(x3−2θ).
Eq. (74) provides the following consistent expansion
In the Appendix C we invert this equation and obtain
the power law
Γ(1 θ) Γ(θ 1)
U(s,θ,x)= − + − x1−θ +O(x). (77)
Γ(1+s θ) Γ(s)
− m(t) x A(t)x2−θ, (x 0), (81)
≃ →
SubstitutingthisintotheintegralinEq. (67),expanding D (cid:12) E
the denominatorto the lowestorderin ξ (recallthat ξ < where (cid:12)
x is small when x is small) and integrating we obtain
1 e−t 1−θ
x dξ = Γ(1+s−θ)x+O(x2−θ). (78) A(t)= Γ(2−θ)(cid:18)1−e−t(cid:19) . (82)
U(s,θ,ξ) Γ(1 θ)
Z0 −
We finally note that θ < 1 implies 2 θ > 1 and the
In order to proceed further it is more convenient to mean minimum (81) decays sharper tha−n the linear law
use an integral representation for the Kummer function (76), the latter applicable when θ > 1. This is a some-
U(s,θ,x) (which multiplies the integral in Eq. (67)) in- what intuitive and interesting behavior meaning that as
stead of using the expansion (77). Thus, taking into ac- the process starts near the origin the average minimum
count the transformation formula [33] tends faster to x = 0 if the boundary is accessible than
otherwise.
U(s,θ,x)=x1−θU(s+1 θ,2 θ,x),
− − C. The span
and using the integral representation [33]
As shown in Sec. III the PDF of the range or span is
1 ∞ given by Eq. (29) which in terms of the escape proba-
U(a,b,x)= e−xzza−1(1+z)b−a−1dz,
bility and taking the Laplace transform with respect to
Γ(a)
Z0 time reads
we get
x ∂2Wˆ (sx)
U(s,θ,x)= x1−θ ∞e−xzz−θ z sdz. fˆR(r,s|x)=−Zx−r v∂,rr+2v | dv. (83)
Γ(1+s θ) 1+z
− Z0 (cid:18) (cid:19)
(79) We have proved elsewhere [22] that in the Feller process
Substituting Eqs. (78) and (79) into Eq. (67) results the Laplace transform of the escape probability is given
in the following approximate expression for the Laplace by
U(s,θ,v+r) U(s,θ,v) F(s,θ,x) F(s,θ,v+r) F(s,θ,v) U(s,θ,x)
Wˆ (sx)= − − − . (84)
v,v+r
| s F(s,θ,v)U(s,θ,v+r) F(s,θ,v+r)U(s,θ,v)
(cid:2) (cid:3) −(cid:2) (cid:3)
(cid:2) (cid:3)
Unfortunately the introductionofEq. (84)intoEq. (83) form of Eq. (32) we get
does not lead to an expression amenable to further ana-
lytical simplifications, being only suitable for numerical
work.
1 x dξ
Rˆ(sx)= U(s,θ,x)
| s" Z0 U(s,θ,ξ)
The mean span is simpler because we only need to
∞ dξ
knowthehittingprobabilityinsteadoftheescapeproba- +F(s,θ,x) , (85)
bility. ThussubstitutingEq. (52)intotheLaplacetrans- Zx F(s,θ,ξ)#