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 respectively. A short summary of main results is presented in the last section. Some mathematical proofs and more ∗Electronicaddress: jaume.masoliver@ub.edu technical details are in appendices. 2 II. FIRST PASSAGE AND EXTREMES x isimpossibleandhaszeroprobability. Inotherwords, } Φ (ξ,tx)=0,ifξ 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 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 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 distribution function, Substituting for Eq. (22) and noting that F (r,tx)=Prob R(t) 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 address 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 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 substituting 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 origin 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). 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 approximations 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 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 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 analytical 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)#

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

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