text: 2 of collisions. The rest of this paper is organized as follows. In Sec. II, we study the DPS contributions to the two particle production in the dilute-dense collisions and derive a formula for the ‘elliptic ﬂow’ parameter v . The result is relevant to pp and pA experiments at RHIC and the LHC. In Sec. III, we 2 numerically evaluate v in a model which incorporates the saturation eﬀect in the target. We point 2 out some generic features of the DPS contributions which can be compared to the experimental observations. We summarize our paper in Sec. IV. II. DOUBLE PARTON SCATTERING CONTRIBUTIONS IN THE DILUTE-DENSE COLLISIONS In order to describe the near-side two particle correlations in pp and pA collisions, we introduce the impact parameter dependence in the DPS framework. Similarly to the derivation of DPS in Refs. [35, 36], we write down the generic expression for the diﬀerential cross section of two parton production as (cid:12) (cid:12) (cid:12) (cid:12) (cid:90) dσ dy d2k ⊥dy d2k ⊥ 1 1 2 2 DPS · · (cid:48) (cid:48) = d2x⊥d2y⊥d2b ⊥d2b ⊥eik1⊥ x⊥eik2⊥ y⊥F (x , x ; z⊥)F (x , x ;(cid:126)b ⊥,(cid:126)b ⊥; (cid:126)x⊥, (cid:126)y⊥) , (1) 1 2 A p p B A A 1 2 | − | where z⊥ = (cid:126)b ⊥ (cid:126)b ⊥ , and (cid:126)b ⊥ and (cid:126)b ⊥ denote the two hard scattering positions with respect 1 2 1 2 to the center of the target. The ‘dipole sizes’ x⊥ and y⊥ are Fourier-conjugate variables to the partons’ outgoing transverse momentum k ⊥ and k ⊥, respectively. The longitudinal momentum (cid:48) (cid:48) 1 2 fractions x , x , x , and x are determined by the ﬁnal state kinematics. The physics picture p p A A is that two partons from the incoming proton encounter multiple scattering oﬀ the target, and fragment into two ﬁnal state particles. The multiple scattering is described in the CGC framework or in the color-dipole model. For a large n(cid:16)ucleus, (cid:17)we can(cid:16) assume(cid:17)a factorized form ≈ F S (cid:126)b ⊥, (cid:126)x⊥ S (cid:48) (cid:126)b ⊥, (cid:126)y⊥ , (2) B xA 1 xA 2 where S is the dipole S-matrix which may be in the fundamental or adjoint representation depending on the partonic channels involved in the DPS. The terms neglected in (2) are of order 1/N2. It has been argued [34] that these color-suppressed, but ‘connected’ contributions can give c rise to nonvanishing v in pp and pA collisions. Moreover, if the the target is small, as in pp 2 collisions, factorization (2) is violated even in the large-N limit due to the small-x evolution in the c target. (In the case of a dipole target, this can be shown analytically [43, 44].) Such factorization breaking eﬀects have been considered as another source of v in small systems [20–22]. 2 Here we show that, even if the factorization (2) holds strictly, there exist non-trivial angular correlations between the two outgoing particles due to the angular correlation between (cid:126)b1⊥ and (cid:126)x⊥ in the S-matrix. It should be mentioned that the idea that the correlation between impact parameter and dipole orientation generates anisotropy in the ﬁnal state has been previously studied in the context of single [16, 17] (see also, [46]) and double [32, 33] parton scattering. Thus, the approach here is essentially the same as in [32, 33]. Yet, our formulation is considerably more concise and clearly establishes the connection to the elliptic gluon Wigner distribution which is a fundamental object in the tomographic study of the nucleon/nucleus. For this purpose, let us write (1) as ##LTLine##

text: of

text: collisions.

text: The

text: rest

text: of

text: this

text: paper

text: is

text: organized

text: as

text: follows.

text: In

text: Sec.

text: II,

text: we

text: study

text: the

text: DPS

text: contributions

text: to

text: the

text: two

text: particle

text: production

text: in

text: the

text: dilute-dense

text: collisions

text: and

text: derive

text: a

text: formula

text: for

text: the

text: ‘elliptic

text: ﬂow’

text: parameter

text: v

text: The

text: result

text: is

text: relevant

text: to

text: pp

text: and

text: pA

text: experiments

text: at

text: RHIC

text: and

text: the

text: LHC.

text: In

text: Sec.

text: III,

text: we

text: numerically

text: evaluate

text: v

text: in

text: a

text: model

text: which

text: incorporates

text: the

text: saturation

text: eﬀect

text: in

text: the

text: target.

text: We

text: point

text: out

text: some

text: generic

text: features

text: of

text: the

text: DPS

text: contributions

text: which

text: can

text: be

text: compared

text: to

text: the

text: experimental

text: observations.

text: We

text: summarize

text: our

text: paper

text: in

text: Sec.

text: IV.

text: II.

label: section

text: DOUBLE

text: PARTON

text: SCATTERING

text: CONTRIBUTIONS

text: IN

text: THE

text: DILUTE-DENSE

text: COLLISIONS

text: In

text: order

text: to

text: describe

text: the

text: near-side

text: two

text: particle

text: correlations

text: in

text: pp

text: and

text: pA

text: collisions,

text: we

text: introduce

text: the

text: impact

text: parameter

text: dependence

text: in

text: the

text: DPS

text: framework.

text: Similarly

text: to

text: the

text: derivation

text: of

text: DPS

text: in

text: Refs.

text: we

text: write

text: down

text: the

text: generic

text: expression

text: for

text: the

text: diﬀerential

text: cross

text: section

text: of

text: two

text: parton

text: production

text: as

text: dσ

text: dy

text: ⊥dy

text: DPS

text: y⊥F

text: (x

text: x

text: z⊥)F

text: (x

text: x

text: A

text: p

text: p

text: B

text: A

text: A

text: where

text: z⊥

text: and

text: and

text: denote

text: the

text: two

text: hard

text: scattering

text: positions

text: with

text: respect

text: to

text: the

text: center

text: of

text: the

text: target.

text: The

text: ‘dipole

text: sizes’

text: x⊥

text: and

text: y⊥

text: are

text: Fourier-conjugate

text: variables

text: to

text: the

text: partons’

text: outgoing

text: transverse

text: momentum

text: k

text: and

text: k

text: respectively.

text: The

text: longitudinal

text: momentum

text: fractions

text: x

text: x

text: x

text: and

text: x

text: are

text: determined

text: by

text: the

text: ﬁnal

text: state

text: kinematics.

text: The

text: physics

text: picture

text: p

text: p

text: A

text: A

text: is

text: that

text: two

text: partons

text: from

text: the

text: incoming

text: proton

text: encounter

text: multiple

text: scattering

text: oﬀ

text: the

text: target,

text: and

text: fragment

text: into

text: two

text: ﬁnal

text: state

text: particles.

text: The

text: multiple

text: scattering

text: is

text: described

text: in

text: the

text: CGC

text: framework

text: or

text: in

text: the

text: color-dipole

text: model.

text: For

text: a

text: large

text: factorized

text: form

text: F

text: S

text: S

text: B

text: xA

text: xA

text: where

text: S

text: is

text: the

text: dipole

text: S-matrix

text: which

text: may

text: be

text: in

text: the

text: fundamental

text: or

text: adjoint

text: representation

text: depending

text: on

text: the

text: partonic

text: channels

text: involved

text: in

text: the

text: DPS.

text: The

text: terms

text: neglected

text: in

text: are

text: of

text: order

text: It

text: has

text: been

text: argued

text: that

text: these

text: color-suppressed,

text: but

text: ‘connected’

text: contributions

text: can

text: give

text: c

text: rise

text: to

text: nonvanishing

text: v

text: in

text: pp

text: and

text: pA

text: collisions.

text: Moreover,

text: if

text: the

text: the

text: target

text: is

text: small,

text: as

text: in

text: pp

text: collisions,

text: factorization

text: is

text: violated

text: even

text: in

text: the

text: large-N

text: limit

text: due

text: to

text: the

text: small-x

text: evolution

text: in

text: the

text: c

text: target.

text: (In

text: the

text: case

text: of

text: a

text: dipole

text: target,

text: this

text: can

text: be

text: shown

text: analytically

text: Such

text: factorization

text: breaking

text: eﬀects

text: have

text: been

text: considered

text: as

text: another

text: source

text: of

text: v

text: in

text: small

text: systems

text: Here

text: we

text: show

text: that,

text: even

text: if

text: the

text: factorization

text: holds

text: strictly,

text: there

text: exist

text: non-trivial

text: angular

text: correlations

text: between

text: the

text: two

text: outgoing

text: particles

text: due

text: to

text: the

text: angular

text: correlation

text: between

text: and

text: in

text: the

text: S-matrix.

text: It

text: should

text: be

text: mentioned

text: that

text: the

text: idea

text: that

text: the

text: correlation

text: between

text: impact

text: parameter

text: and

text: dipole

text: orientation

text: generates

text: anisotropy

text: in

text: the

text: ﬁnal

text: state

text: has

text: been

text: previously

text: studied

text: in

text: the

text: context

text: of

text: single

text: (see

text: also,

text: and

text: double

text: parton

text: scattering.

text: Thus,

text: the

text: approach

text: here

text: is

text: essentially

text: the

text: same

text: as

text: in

text: Yet,

text: our

text: formulation

text: is

text: considerably

text: more

text: concise

text: and

text: clearly

text: establishes

text: the

text: connection

text: to

text: the

text: elliptic

text: gluon

text: Wigner

text: distribution

text: which

text: is

text: a

text: fundamental

text: object

text: in

text: the

text: tomographic

text: study

text: of

text: the

text: nucleon/nucleus.

text: For

text: this

text: purpose,

text: let

text: us

text: write

text: as

text: ##LTLine##


text: (cid:16) (cid:17) ON VIX FUTURES IN THE ROUGH BERGOMI MODEL 13 N Proof. Since, by assumption ∆1/2VIX =∆ exp (cid:16)µ+σ (0,1) ,(cid:17)the price of a V(cid:18)IX futur(cid:19)e directly reads T 2 E |F − E − µ σ2 (VIX ) = ∆ 1/2 ∆1/2VIX = ∆ 1/2 exp + , T 0 T 2 8 (cid:3) and the second case follows analogously. As opposed to the simulation schemes, the log-normal approximation does depend on the qualitative proper- · ties of the initial forward variance curve ξ ( ). Hence, diﬀerent curves should be analysed to check the robustness 0 of the method. We now exploit the approximation in Assumption 3.11 to obtain a closed-form Black-Scholes type formulae for European options on the VIX. ≤ ≤ E |F Lemma 3.14. For 0 t T, let V (t) := (VIX ) denote the price at time t of a VIX future maturing T T t (cid:115) at T. Then, under Assumption 3.11, a European Call(cid:90)option on a VIX(cid:18)future(cid:19)maturing at T is worth CV E − |F − T+∆ −σ2 − (cid:101) T := [(VT ((cid:82)T) K)+ 0] = ∆ 1/2 (cid:101)ξ0(t)dt exp 8 (cid:101)Φ(d1) KΦ(d2), T − − − where K := [log(K2∆) log T+∆ ξ (t)dt+σ2/2]/σ, d := K+ 1σ and d := K, with σ2 as in Proposition 3.8 T 0 1 2 2 (cid:90) (cid:104) (cid:16) (cid:17) (cid:105) (cid:18) (cid:19) (cid:90) (cid:16) (cid:17) Proof. Under Assumption 3.11, the lemma follows directly from the following trivial computations: ∞ ∞ E − (cid:101) − µ σz − √1 µ σ2 (cid:101) − σ − (V (T) K) = ∆ 1/2 exp + K φ(z)dz = exp + φ z dz KΦ(d ). T + K 2 2 + ∆ 2 8 K 2 2 (cid:3) We perform a numerical analysis of the approximation by pricing VIX Futures using the parameters in (3.5). We consider 106 simulations and the three qualitative scenarios introduced in (3.4) for the forward variance − curve. Figures 5 suggest that the approximation is accurate being the diﬀerence of order 10 3 or less. The oscillating nature of the diﬀerence is also a good sign since it is probably caused by the Monte Carlo error and · does not show any monotonicity. Furthermore, the method shows to be robust for diﬀerent type of curves ξ ( ). 0 Moreover, the log-normal approximation seems to converge to the true mean, avoiding the oscillations of the Truncated Cholesky method. Therefore, we may conclude that the log-normal approximation produces a good output, with desired smoothness properties. On the other hand, we also analyse European VIX Call options repeating the previous parameters and considering the ﬂat forward variance curve from Scenario 1. Figure 6 shows that the log-normal approximation is accurate for at-the-money options. Nevertheless, the diﬀerence increases with maturity, and hence one must be careful when using this approximation to price options with long maturities. However, in practice, on Equity markets, liquid maturities are only up to two years. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: ON

text: VIX

text: FUTURES

text: IN

text: THE

text: ROUGH

text: BERGOMI

text: MODEL

text: N

text: Proof.

text: Since,

text: by

text: assumption

text: exp

text: price

text: of

text: a

text: directly

text: reads

text: T

text: E

text: |F

text: E

text: µ

text: σ2

text: (VIX

text: exp

text: T

text: T

text: and

text: the

text: second

text: case

text: follows

text: analogously.

text: As

text: opposed

text: to

text: the

text: simulation

text: schemes,

text: the

text: log-normal

text: approximation

text: does

text: depend

text: on

text: the

text: qualitative

text: proper-

text: ties

text: of

text: the

text: initial

text: forward

text: variance

text: curve

text: ξ

text: Hence,

text: diﬀerent

text: curves

text: should

text: be

text: analysed

text: to

text: check

text: the

text: robustness

text: of

text: the

text: method.

text: We

text: now

text: exploit

text: the

text: approximation

text: in

text: Assumption

text: to

text: obtain

text: a

text: closed-form

text: Black-Scholes

text: type

text: formulae

text: for

text: European

text: options

text: on

text: the

text: VIX.

text: E

text: |F

text: Lemma

text: For

text: t

text: T,

text: let

text: V

text: (t)

text: (VIX

text: denote

text: the

text: price

text: at

text: time

text: t

text: of

text: a

text: VIX

text: future

text: maturing

text: T

text: T

text: t

text: at

text: T.

text: Then,

text: under

text: Assumption

text: a

text: European

text: on

text: a

text: at

text: T

text: is

text: worth

text: CV

text: E

text: |F

text: T+∆

text: −σ2

text: T

text: [(VT

text: K)+

text: exp

text: T

text: where

text: K

text: log

text: T+∆

text: ξ

text: d

text: K+

text: 1σ

text: and

text: d

text: K,

text: with

text: σ2

text: as

text: in

text: Proposition

text: T

text: Proof.

text: Under

text: Assumption

text: the

text: lemma

text: follows

text: directly

text: from

text: the

text: following

text: trivial

text: computations:

text: E

text: µ

text: σz

text: √1

text: µ

text: σ2

text: σ

text: (V

text: (T)

text: K)

text: exp

text: K

text: φ(z)dz

text: exp

text: φ

text: z

text: dz

text: KΦ(d

text: T

text: K

text: K

text: We

text: perform

text: a

text: numerical

text: analysis

text: of

text: the

text: approximation

text: by

text: pricing

text: VIX

text: Futures

text: using

text: the

text: parameters

text: in

text: We

text: consider

text: simulations

text: and

text: the

text: three

text: qualitative

text: scenarios

text: introduced

text: in

text: for

text: the

text: forward

text: variance

text: curve.

text: Figures

text: suggest

text: that

text: the

text: approximation

text: is

text: accurate

text: being

text: the

text: diﬀerence

text: of

text: order

text: or

text: less.

text: The

text: oscillating

text: nature

text: of

text: the

text: diﬀerence

text: is

text: also

text: a

text: good

text: sign

text: since

text: it

text: is

text: probably

text: caused

text: by

text: the

text: Monte

text: Carlo

text: error

text: and

text: does

text: not

text: show

text: any

text: monotonicity.

text: Furthermore,

text: the

text: method

text: shows

text: to

text: be

text: robust

text: for

text: diﬀerent

text: type

text: of

text: curves

text: ξ

text: Moreover,

text: the

text: log-normal

text: approximation

text: seems

text: to

text: converge

text: to

text: the

text: true

text: mean,

text: avoiding

text: the

text: oscillations

text: of

text: the

text: Truncated

text: Cholesky

text: method.

text: Therefore,

text: we

text: may

text: conclude

text: that

text: the

text: log-normal

text: approximation

text: produces

text: a

text: good

text: output,

text: with

text: desired

text: smoothness

text: properties.

text: On

text: the

text: other

text: hand,

text: we

text: also

text: analyse

text: European

text: VIX

text: Call

text: options

text: repeating

text: the

text: previous

text: parameters

text: and

text: considering

text: the

text: ﬂat

text: forward

text: variance

text: curve

text: from

text: Scenario

text: Figure

text: shows

text: that

text: the

text: log-normal

text: approximation

text: is

text: accurate

text: for

text: at-the-money

text: options.

text: Nevertheless,

text: the

text: diﬀerence

text: increases

text: with

text: maturity,

text: and

text: hence

text: one

text: must

text: be

text: careful

text: when

text: using

text: this

text: approximation

text: to

text: price

text: options

text: with

text: long

text: maturities.

text: However,

text: in

text: practice,

text: on

text: Equity

text: markets,

text: liquid

text: maturities

text: are

text: only

text: up

text: to

text: two

text: years.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: ON VIX FUTURES IN THE ROUGH BERGOMI MODEL 19 VIX futures term structure Absolute difference 0.210 0.0018 Log-normal approx. Absolute difference 0.205 0.0016 Observed VIX futures term structure 0.200 0.0014 0.195 0.0012 0.190 0.0010 0.185 0.0008 0.180 0.0006 0.175 0.0004 0.170 0.0002 0.1650.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.00000.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Maturity Maturity Figure 10. VIX Futures calibration on 4/12/2015. Optimal parameters: (H, ν) = (0.09237, 1.004). VIX futures term structure Absolute difference 0.0012 0.215 0.0010 0.210 0.0008 0.0006 0.205 0.0004 Log-normal approx. 0.0002 0.200 Observed VIX futures term structure Absolute difference 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.00000.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Maturity Maturity Figure 11. VIX Futures calibration on 22/2/2016. Optimal parameters: (H, ν) = (0.10093, 1.00282). VIX futures term structure Absolute difference 0.210 0.0016 Log-normal approx. Absolute difference 0.208 Observed VIX futures term structure 0.0014 0.206 0.0012 0.204 0.0010 0.202 0.0008 0.200 0.0006 0.198 0.0004 0.196 0.0002 0.194 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.00000.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Maturity Maturity Figure 12. VIX Futures calibration on 4/1/2016. Optimal parameters: (H, ν) = (0.0509, 1.2937). only misspeciﬁed for short maturities (as is the case of the eSSVI), this aﬀects the whole term structure of the VIX process (for details we refer the reader to Section ). Therefore, an improved ξ estimation would not only 0 increase the accuracy of the model for short maturities, but also for the whole term structure. Remark 3.19. Our calibration involves two diﬀerent data sets, SPX options and VIX Futures: the Vanilla quotes are extracted from the CBOE delayed option quotes page2 and the VIX Futures from the CBOE VIX futures historical data page3. We perform an aggregation of the diﬀerent Future quotes, since they are quoted 2CBOE delayed option quotes: http://www.cboe.com/delayedquote/quotetable.aspx 3CBOE VIX futures historical data: http://cfe.cboe.com/data/historicaldata.aspx ecirp serutuf XIV ecirp serutuf XIV ecirp serutuf XIV ecnereffid etulosbA ecnereffid etulosbA ecnereffid etulosbA ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: ON

text: VIX

text: FUTURES

text: IN

text: THE

text: ROUGH

text: BERGOMI

text: MODEL

text: VIX

font: default

text: futures

text: term

text: structure

text: Absolute

text: difference

text: Log-normal

text: approx.

text: Absolute

text: difference

text: Observed

text: VIX

text: futures

text: term

text: structure

text: Maturity

text: Maturity

text: Figure

text: VIX

text: Futures

text: calibration

text: on

text: Optimal

text: parameters:

text: (H,

text: ν)

text: VIX

text: futures

text: term

text: structure

text: Absolute

text: difference

text: Log-normal

text: approx.

text: Observed

text: VIX

text: futures

text: term

text: structure

text: Absolute

text: difference

text: Maturity

text: Maturity

text: Figure

text: VIX

text: Futures

text: calibration

text: on

text: Optimal

text: parameters:

text: (H,

text: ν)

text: VIX

text: futures

text: term

text: structure

text: Absolute

text: difference

text: Log-normal

text: approx.

text: Absolute

text: difference

text: Observed

text: VIX

text: futures

text: term

text: structure

text: Maturity

text: Maturity

text: Figure

text: VIX

text: Futures

text: calibration

text: on

text: Optimal

text: parameters:

text: (H,

text: ν)

text: only

text: misspeciﬁed

text: for

text: short

text: maturities

text: (as

text: is

text: the

text: case

text: of

text: the

text: eSSVI),

text: this

text: aﬀects

text: the

text: whole

text: term

text: structure

text: of

text: the

text: VIX

text: process

text: (for

text: details

text: we

text: refer

text: the

text: reader

text: to

text: Section

text: Therefore,

text: an

text: improved

text: ξ

text: estimation

text: would

text: not

text: only

text: increase

text: the

text: accuracy

text: of

text: the

text: model

text: for

text: short

text: maturities,

text: but

text: also

text: for

text: the

text: whole

text: term

text: structure.

text: Remark

text: Our

text: calibration

text: involves

text: two

text: diﬀerent

text: data

text: sets,

text: SPX

text: options

text: and

text: VIX

text: Futures:

text: the

text: Vanilla

text: quotes

text: are

text: extracted

text: from

text: the

text: CBOE

text: delayed

text: option

text: quotes

text: and

text: the

text: VIX

text: Futures

text: from

text: the

text: CBOE

text: VIX

text: futures

text: historical

text: data

text: We

text: perform

text: an

text: aggregation

text: of

text: the

text: diﬀerent

text: Future

text: quotes,

text: since

text: they

text: are

text: quoted

text: delayed

text: option

text: quotes:

text: VIX

text: futures

text: historical

text: data:

text: ecirp

text: serutuf

text: XIV

text: ecirp

text: serutuf

text: XIV

text: ecirp

text: serutuf

text: XIV

text: ecnereffid

text: etulosbA

text: ecnereffid

text: etulosbA

text: ecnereffid

text: etulosbA

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: ON VIX FUTURES IN THE ROUGH BERGOMI MODEL 21 Algorithm 4.2 (Calibration algorithm for SPX options via VIX Futures). (i) Calibrate H and ξ using the VIX Futures; 0 {V } { } (ii) compute M paths of the Volterra process, (u) M and extract the Brownian motions Z(u) M driving u=1 u=1 { ⊥ } each process. Also, compute independent Brownian motions Z (u) M ; (cid:16) (cid:17) u=1 (iii) evaluate the Call prices in each calibration step: (cid:112) E V Vt(u) = ξ0(t) 2νC(cid:113)H t(u)(cid:16) , (cid:17) u = 1, . . . , M, − ⊥ W(u) = ρZ(u) + 1 ρ2Z (u), u = 1, . . . , M, − S(u) = S(u) + S(u) v(u) W(u) W(u) , u = 1, . . . , M; t+∆ t t t t+∆ t { } { } (iv) compute the Call price for ea(cid:88)ch available maturity T , . . . , T and set of strikes K(1), . . . , K(L) : 1 N 1 M − C = (S(u) K(j)) , for i = 1, . . . , N and j = 1, . . . , L; Ti,j M Ti + u=1 L (v) minimise over (ν, ρ) the objective function C(ν, ρ) in (4.1). Remark 4.3. Item (v) in the algorithm above may change the optimal values for ν, which was initially calibrated in (i) from the VIX Futures. Backtesting however shows that the calibration in (i) is not really aﬀected by this. We calibrate the model on December 4, 2015, ﬁxing H = 0.09237 obtained previously through VIX, and plot the ﬁt in Figure 13. The model is not fully consistent for short maturities, which may follow from the inability of ξ to fully capture the smiles for these maturities, but the ﬁt greatly improves with maturity. Interestingly, we 0 observe a 20% diﬀerence between the the parameter ν obtained through VIX calibration and the one obtained through SPX. This suggests that the volatility of volatility in the SPX market is 20% higher when compared to VIX, revealing potential data inconsistencies (arbitrage?). Nevertheless, we emphasise the importance of an accurate ξ curve to improve the ﬁt to SPX and to provide an eﬃcient joint calibration. 0 Conclusion Following the path set by Bayer, Friz and Gatheral [3], we developed here a relatively fast algorithm to calibrate VIX Futures and the VIX smile, consistently with the SPX smile, in the rough Bergomi model. The clear strength of this model is that only a few parameters are needed, making the (re)calibration robust and stable. From a trader’s point of view, we highlight some potential market discrepancy between the VIX and the SPX, and leave a reﬁned analysis thereof for future research. Appendix A. The hybrid scheme We brieﬂy recall the hybrid scheme developed in [4]. Following the notation in Deﬁnition 2.1, we consider a B ∈ N (truncated) Brownian semistationary process (α, W), and introduce the truncation parameter κ . On an ##LTLine## ##LTLine## ##LTLine##

text: ON

text: VIX

text: FUTURES

text: IN

text: THE

text: ROUGH

text: BERGOMI

text: MODEL

text: Algorithm

text: (Calibration

text: algorithm

text: for

text: SPX

text: options

text: via

text: VIX

text: Futures).

text: (i)

label: list

text: Calibrate

text: H

text: and

text: ξ

text: using

text: the

text: VIX

text: Futures;

text: {V

text: (ii)

text: compute

text: M

text: paths

text: of

text: the

text: Volterra

text: process,

text: (u)

text: M

text: and

text: extract

text: the

text: Brownian

text: motions

text: Z(u)

text: M

text: driving

text: each

text: process.

text: Also,

text: compute

text: independent

text: Brownian

text: motions

text: Z

text: (u)

text: M

text: (iii)

text: evaluate

text: the

text: Call

text: prices

text: in

text: each

text: calibration

text: step:

text: E

text: V

text: Vt(u)

text: u

text: M,

text: W(u)

text: ρZ(u)

text: (u),

text: u

text: M,

text: S(u)

text: S(u)

text: S(u)

text: v(u)

text: W(u)

text: W(u)

text: u

text: M;

text: t+∆

text: t

text: t

text: t

text: t+∆

text: t

text: (iv)

text: compute

text: the

text: Call

text: price

text: for

text: available

text: maturity

text: T

text: T

text: and

text: set

text: of

text: strikes

text: K(L)

text: N

text: M

text: C

text: (S(u)

text: K(j))

text: for

text: i

text: N

text: and

text: j

text: L;

text: Ti,j

text: M

text: Ti

text: L

text: (v)

text: minimise

text: over

text: (ν,

text: ρ)

text: the

text: objective

text: function

text: C(ν,

text: ρ)

text: in

text: Remark

text: Item

text: (v)

text: in

text: the

text: algorithm

text: above

text: may

text: change

text: the

text: optimal

text: values

text: for

text: ν,

text: which

text: was

text: initially

text: calibrated

text: in

text: (i)

text: from

text: the

text: VIX

text: Futures.

text: Backtesting

text: however

text: shows

text: that

text: the

text: calibration

text: in

text: (i)

text: is

text: not

text: really

text: aﬀected

text: by

text: this.

text: We

text: calibrate

text: the

text: model

text: on

text: December

text: ﬁxing

text: H

text: obtained

text: previously

text: through

text: VIX,

text: and

text: plot

text: the

text: ﬁt

text: in

text: Figure

text: The

text: model

text: is

text: not

text: fully

text: consistent

text: for

text: short

text: maturities,

text: which

text: may

text: follow

text: from

text: the

text: inability

text: of

text: ξ

text: to

text: fully

text: capture

text: the

text: smiles

text: for

text: these

text: maturities,

text: but

text: the

text: ﬁt

text: greatly

text: improves

text: with

text: maturity.

text: Interestingly,

text: we

text: observe

text: a

text: diﬀerence

text: between

text: the

text: the

text: parameter

text: ν

text: obtained

text: through

text: VIX

text: calibration

text: and

text: the

text: one

text: obtained

text: through

text: SPX.

text: This

text: suggests

text: that

text: the

text: volatility

text: of

text: volatility

text: in

text: the

text: SPX

text: market

text: is

text: higher

text: when

text: compared

text: to

text: VIX,

text: revealing

text: potential

text: data

text: inconsistencies

text: (arbitrage?).

text: Nevertheless,

text: we

text: emphasise

text: the

text: importance

text: of

text: an

text: accurate

text: ξ

text: curve

text: to

text: improve

text: the

text: ﬁt

text: to

text: SPX

text: and

text: to

text: provide

text: an

text: eﬃcient

text: joint

text: calibration.

text: Conclusion

text: Following

text: the

text: path

text: set

text: by

text: Bayer,

text: Friz

text: and

text: Gatheral

text: we

text: developed

text: here

text: a

text: relatively

text: fast

text: algorithm

text: to

text: calibrate

text: VIX

text: Futures

text: and

text: the

text: VIX

text: smile,

text: consistently

text: with

text: the

text: SPX

text: smile,

text: in

text: the

text: rough

text: Bergomi

text: model.

text: The

text: clear

text: strength

text: of

text: this

text: model

text: is

text: that

text: only

text: a

text: few

text: parameters

text: are

text: needed,

text: making

text: the

text: (re)calibration

text: robust

text: and

text: stable.

text: From

text: a

text: trader’s

text: point

text: of

text: view,

text: we

text: highlight

text: some

text: potential

text: market

text: discrepancy

text: between

text: the

text: VIX

text: and

text: the

text: SPX,

text: and

text: leave

text: a

text: reﬁned

text: analysis

text: thereof

text: for

text: future

text: research.

text: Appendix

text: A.

text: The

text: hybrid

text: scheme

text: We

text: brieﬂy

text: recall

text: the

text: hybrid

text: scheme

text: developed

text: in

text: Following

text: the

text: notation

text: in

text: Deﬁnition

text: we

text: consider

text: a

text: B

text: N

text: (truncated)

text: Brownian

text: semistationary

text: process

text: (α,

text: W),

text: and

text: introduce

text: the

text: truncation

text: parameter

text: κ

text: On

text: an

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: TROISIE`ME GROUPE DE COHOMOLOGIE NON RAMIFIE´E DES HYPERSURFACES DE FANO par J.-L. Colliot-Th´el`ene R´esum´e. — Sur un corps alg´ebriquement clos et sur un corps ﬁni, on ´etablit de nouveaux r´esultats d’annulation pour la cohomologie non ramiﬁ´ee de degr´e 3 des hypersurfaces de Fano. Abstract. — We establish the vanishing of degree three unramiﬁed coho- mology for several new types of Fano hypersurfaces when the ground ﬁeld is either ﬁnite or algebraically closed of arbitrary characteristic. Soit X une vari´et´e projective, lisse, g´eom´etriquement connexe sur un corps (cid:54) ≥ ∈ Z k et (cid:96) = car(k) un nombre premier. Pour tout couple d’entiers i 0 et j , Q Z le groupe de cohomologie non ramiﬁ´ee Hi (X, / (j)) est par d´eﬁnition nr (cid:96) (cid:96) le groupe des sections globales du faisceau pour la topologie de Zariski sur X ⊂ associ´e au pr´efaisceau qui `a un ouvert U X associe le groupe de cohomologie Q Z ´etale Hi (U, / (j)) de U `a valeurs dans le groupe des racines (cid:96)-primaires de ´et (cid:96) (cid:96) l’unit´e tordues j fois. Les propri´et´es g´en´erales de ces groupes sont d´ecrites dans Q Z le rapport [5]. Le groupe H2 (U, / (1)) est la composante (cid:96)-primaire du ´et (cid:96) (cid:96) Q Z groupe de Brauer de X. Les groupes Hi (X, / (j)) sont des invariants k- nr (cid:96) (cid:96) birationnels des vari´et´es projectives et lisses. On a une application naturelle du Q Z Q Z groupe de cohomologie galoisienne Hi(k, / (j)) = Hi (k, / (j)) dans Q Z (cid:96) (cid:96) ´et (cid:96) (cid:96) le groupe Hi (X, / (j)), application qui est un isomorphisme si X est nr (cid:96) (cid:96) k-birationnelle `a un espace projectif Pm. kQ Z On s’int´eresse ici au groupe H3 (X, / (2)). Ce groupe joue un rˆole im- nr (cid:96) (cid:96) portant dans l’´etude [9, 12, 7, 6] de l’application “cycle” sur le groupe de Chow des cycles de codimension 2 ⊗ Z → Z cyc : CH2(X) H4 (X, (2)). X (cid:96) ´et (cid:96) ⊂ Pour une hypersurface cubique lisse X PnC sur le corps des complexes, Q Z n = 4 et n = 5, on sait que l’on a H3 (X, / (2)) = 0 pour tout (cid:96). C’est une nr (cid:96) (cid:96) ##LTLine## ##LTLine##

text: TROISIE`ME

text: GROUPE

text: DE

text: COHOMOLOGIE

text: NON

text: RAMIFIE´E

text: DES

text: HYPERSURFACES

text: DE

text: FANO

text: par

text: J.-L.

label: author

text: Colliot-Th´el`ene

text: R´esum´e.

text: Sur

label: abstract

text: un

text: corps

text: alg´ebriquement

text: clos

text: et

text: sur

text: un

text: corps

text: ﬁni,

text: on

text: ´etablit

text: de

text: nouveaux

text: r´esultats

text: d’annulation

text: pour

text: la

text: cohomologie

text: non

text: ramiﬁ´ee

text: de

text: degr´e

text: des

text: hypersurfaces

text: de

text: Fano.

text: Abstract.

text: We

text: establish

text: the

text: vanishing

text: of

text: degree

text: three

text: unramiﬁed

text: coho-

text: mology

text: for

text: several

text: new

text: types

text: of

text: Fano

text: hypersurfaces

text: when

text: the

text: ground

text: ﬁeld

text: is

text: either

text: ﬁnite

text: or

text: algebraically

text: closed

text: of

text: arbitrary

text: characteristic.

text: Soit

text: X

text: une

text: vari´et´e

text: projective,

text: lisse,

text: g´eom´etriquement

text: connexe

text: sur

text: un

text: corps

text: Z

text: k

text: et

text: car(k)

text: un

text: nombre

text: premier.

text: Pour

text: tout

text: couple

text: d’entiers

text: i

text: et

text: j

text: Q

text: Z

text: le

text: groupe

text: de

text: cohomologie

text: non

text: ramiﬁ´ee

text: Hi

text: (X,

text: (j))

text: est

text: par

text: d´eﬁnition

text: nr

text: le

text: groupe

text: des

text: sections

text: globales

text: du

text: faisceau

text: pour

text: la

text: topologie

text: de

text: Zariski

text: sur

text: X

text: associ´e

text: au

text: pr´efaisceau

text: qui

text: `a

text: un

text: ouvert

text: U

text: X

text: associe

text: le

text: groupe

text: de

text: cohomologie

text: Q

text: Z

text: ´etale

text: Hi

text: (U,

text: (j))

text: de

text: U

text: `a

text: valeurs

text: dans

text: le

text: groupe

text: des

text: racines

text: de

text: ´et

text: l’unit´e

text: tordues

text: j

text: fois.

text: Les

text: propri´et´es

text: g´en´erales

text: de

text: ces

text: groupes

text: sont

text: d´ecrites

text: dans

text: Q

text: Z

text: le

text: rapport

text: Le

text: groupe

text: (U,

text: est

text: la

text: composante

text: du

text: ´et

text: Q

text: Z

text: groupe

text: de

text: Brauer

text: de

text: X.

text: Les

text: groupes

text: Hi

text: (X,

text: (j))

text: sont

text: des

text: invariants

text: k-

text: nr

text: birationnels

text: des

text: vari´et´es

text: projectives

text: et

text: lisses.

text: On

text: a

text: une

text: application

text: naturelle

text: du

text: Q

text: Z

text: Q

text: Z

text: groupe

text: de

text: cohomologie

text: galoisienne

text: Hi(k,

text: (j))

text: Hi

text: (k,

text: (j))

text: dans

text: Q

text: Z

text: ´et

text: le

text: groupe

text: Hi

text: (X,

text: (j)),

text: application

text: qui

text: est

text: un

text: isomorphisme

text: si

text: X

text: est

text: nr

text: k-birationnelle

text: `a

text: un

text: espace

text: projectif

text: Pm.

text: kQ

text: Z

text: On

text: s’int´eresse

text: ici

text: au

text: groupe

text: (X,

text: Ce

text: groupe

text: joue

text: un

text: rˆole

text: im-

text: nr

text: portant

text: dans

text: l’´etude

text: de

text: l’application

text: “cycle”

text: sur

text: le

text: groupe

text: de

text: Chow

text: des

text: cycles

text: de

text: codimension

text: Z

text: Z

text: cyc

text: (X,

text: X

text: ´et

text: Pour

text: une

text: hypersurface

text: cubique

text: lisse

text: X

text: PnC

text: sur

text: le

text: corps

text: des

text: complexes,

text: Q

text: Z

text: n

text: et

text: n

text: on

text: sait

text: que

text: l’on

text: a

text: (X,

text: pour

text: tout

text: C’est

text: une

text: nr

text: ##LTLine##

text: ##LTLine##


text: 4 J.-L. COLLIOT-THE´LE`NE ≤ (ii) Pour n = 4 et d 4, l’application cycle ⊗ Z → Z cyc : CH2(X) H4(X, (2)) X (cid:96) (cid:96) est surjective. (cid:54) ≤ Q Z (iii) Pour n = 5 et d n, on a H3 (X, / (2)) = 0. nr (cid:96) (cid:96) D´emonstration. — E´tablissons (i). Pour n = 3, la classe de tout k-point de Z Z (cid:39) Z X engendre le -module H4(X, (2)) . L’´enonc´e (i) est donc clair pour (cid:96) (cid:96) (cid:96) n = 3. ≥ \ Supposons n 4. Soit U = Pn X. Pour tout entier m > 0, on a la suite k exacte de cohomologie ´etale `a supports propres [15, III.1.30] : Z → Z → Z → Z H4(U, /(cid:96)m(2)) H4(Pn, /(cid:96)m(2)) H4(X, /(cid:96)m(2)) H5(U, /(cid:96)m(2)). c c Z − Z − Les groupes ﬁnis Hi(U, /(cid:96)m(2)) et H2n i(U, /(cid:96)m(2n 2)) sont duaux (dua- c lit´e de Poincar´e [15, VI.11.2]). ≥ − − Pour n 6, on a 2n 4 > 2n 5 > n. Le th´eor`eme de Lefschetz aﬃne [15, − Z − − Z − VI.7.2] donne H2n 4(U, /(cid:96)m(2n 2)) = 0 et H2n 5(U, /(cid:96)m(2n 2)) = 0. Z → Z La ﬂ`eche de restriction H4(Pn, /(cid:96)m(2)) H4(X, /(cid:96)m(2)) est donc un isomorphisme de groupes ﬁnis pour tout m. La ﬂ`eche de restriction Z Z → Z = H4(Pn, (2)) H4(X, (2)) (cid:96) (cid:96) (cid:96) est donc un isomorphisme. Ceci implique que l’application cycle ⊗ Z → Z cyc : CH2(X) H4(X, (2)) X (cid:96) (cid:96) ≥ est surjective. Ceci ´etablit (i) pour n 6. Pour n > 4, la consid´eration de la suite exacte Z → Z → Z H3(Pn, /(cid:96)m(2)) H3(X, /(cid:96)m(2)) H4(U, /(cid:96)m(2)), c la dualit´e de Poincar´e et le th´eor`eme de Lefschetz aﬃne donnent alors Z Z H3(X, /(cid:96)m(2)) = 0 pour tout m et donc H3(X, (2)) = 0. Ceci sera utilis´e (cid:96) dans la d´emonstration du th´eor`eme 3.1 ci-apr`es. E´tablissons l’´enonc´e (ii). Soit donc n = 4. L’argument qui suit corrige celui donn´e dans [6, Thm. 5.6 (vi)]. Pour tout degr´e d, et tout entier m > 0, la ﬂ`eche de restriction Z → Z H2(P4, /(cid:96)m) H2(X, /(cid:96)m) est un isomorphisme, comme on voit en utilisant la suite exacte de cohomologie ´etale `a supports, la dua- \ lit´e de Poincar´e sur U = P4 X, et le th´eor`eme de Lefschetz aﬃne. Z (cid:39) Z Ceci implique H2(X, /(cid:96)m) /(cid:96)m, et ceci implique que l’application → cycle CH1(X)/(cid:96)m H2(X, µ ) d´eﬁnie via l’application de Kummer → (cid:96)m Pic(X)/(cid:96)m H2(X, µ ) est un isomorphisme. (cid:96)m Le cup-produit sur la cohomologie ´etale Z × Z → Z Z H4(X, /(cid:96)m(2)) H2(X, /(cid:96)m(1)) H6(X, /(cid:96)m(3)) = /(cid:96)m

text: J.-L.

text: COLLIOT-THE´LE`NE

text: (ii)

text: Pour

text: n

text: et

text: d

text: l’application

text: cycle

text: Z

text: Z

text: cyc

text: X

text: est

text: surjective.

text: Q

text: Z

text: (iii)

text: Pour

text: n

text: et

text: d

text: n,

text: on

text: a

text: (X,

text: nr

text: D´emonstration.

text: E´tablissons

text: (i).

text: Pour

text: n

text: la

text: classe

text: de

text: tout

text: k-point

text: de

text: Z

text: Z

text: Z

text: X

text: engendre

text: le

text: -module

text: L’´enonc´e

text: (i)

text: est

text: donc

text: clair

text: pour

text: n

text: Supposons

text: n

text: Soit

text: U

text: Pn

text: X.

text: Pour

text: tout

text: entier

text: m

text: on

text: a

text: la

text: suite

text: k

text: exacte

text: de

text: cohomologie

text: ´etale

text: `a

text: supports

text: propres

text: Z

text: Z

text: Z

text: Z

text: c

text: c

text: Z

text: Z

text: Les

text: groupes

text: ﬁnis

text: Hi(U,

text: et

text: i(U,

text: sont

text: duaux

text: (dua-

text: c

text: lit´e

text: de

text: Poincar´e

text: Pour

text: n

text: on

text: a

text: n.

text: Le

text: th´eor`eme

text: de

text: Lefschetz

text: aﬃne

text: Z

text: Z

text: donne

text: et

text: Z

text: Z

text: La

text: ﬂ`eche

text: de

text: restriction

text: est

text: donc

text: un

text: isomorphisme

text: de

text: groupes

text: ﬁnis

text: pour

text: tout

text: m.

text: La

text: ﬂ`eche

text: de

text: restriction

text: Z

text: Z

text: Z

text: est

text: donc

text: un

text: isomorphisme.

text: Ceci

text: implique

text: que

text: l’application

text: cycle

text: Z

text: Z

text: cyc

text: X

text: est

text: surjective.

text: Ceci

text: ´etablit

text: (i)

text: pour

text: n

text: Pour

text: n

text: la

text: consid´eration

text: de

text: la

text: suite

text: exacte

text: Z

text: Z

text: Z

text: c

text: la

text: dualit´e

text: de

text: Poincar´e

text: et

text: le

text: th´eor`eme

text: de

text: Lefschetz

text: aﬃne

text: donnent

text: alors

text: Z

text: Z

text: pour

text: tout

text: m

text: et

text: donc

text: Ceci

text: sera

text: utilis´e

text: dans

text: la

text: d´emonstration

text: du

text: th´eor`eme

text: ci-apr`es.

text: E´tablissons

text: l’´enonc´e

text: (ii).

text: Soit

text: donc

text: n

text: L’argument

text: qui

text: suit

text: corrige

text: celui

text: donn´e

text: dans

text: Thm.

text: (vi)].

text: Pour

text: tout

text: degr´e

text: d,

text: et

text: tout

text: entier

text: m

text: la

text: ﬂ`eche

text: de

text: restriction

text: Z

text: Z

text: est

text: un

text: isomorphisme,

text: comme

text: on

text: voit

text: en

text: utilisant

text: la

text: suite

text: exacte

text: de

text: cohomologie

text: ´etale

text: `a

text: supports,

text: la

text: dua-

text: lit´e

text: de

text: Poincar´e

text: sur

text: U

text: X,

text: et

text: le

text: th´eor`eme

text: de

text: Lefschetz

text: aﬃne.

text: Z

text: Z

text: Ceci

text: implique

text: et

text: ceci

text: implique

text: que

text: l’application

text: cycle

text: µ

text: d´eﬁnie

text: via

text: l’application

text: de

text: Kummer

text: µ

text: est

text: un

text: isomorphisme.

text: Le

text: cup-produit

text: sur

text: la

text: cohomologie

text: ´etale

text: Z

text: Z

text: Z

text: Z


text: 8 J.-L. COLLIOT-THE´LE`NE repose de fa¸con essentielle sur [3], dont les m´ethodes g´eom´etriques sont ins- pir´ees de celles de [20] (qui cite [22]). F 5. Hypersurfaces cubiques dans P4F, corps ﬁni Pour les hypersurfaces cubiques lisses sur un corps ﬁni, le travail [18] de Parimala et Suresh permet de compl´eter le th´eor`eme 3.1 pour n = 4. ⊂ Th´eor`eme 5.1. — Soit X P4F une hypersurface cubique lisse sur un corps F ﬁni de caract´eristique diﬀ´erente de 2. F (i) Pour tout (cid:96) premier diﬀ´erent de la caract´eristique de , on a Q Z H3 (X, / (2)) = 0. nr (cid:96) F(cid:96) F F F (ii) Soit une clˆoture alg´ebrique de et G = Gal( / ). L’application naturelle → CH2(X) CH2(X)G est un isomorphisme. (iii) L’application cycle ⊗ Z → Z cyc : CH2(X) H4(X, (2)) X (cid:96) (cid:96) est surjective. (cid:54) D´emonstration. — (i) Le cas (cid:96) = 2 r´esulte d´ej`a de la proposition 1.4. Pour d´emontrer la proposition, par le lemme 1.1 et un argument de restriction- ⊂ corestriction, on peut supposer que X contient une droite L X d´eﬁnie sur F F le corps . En ´eclatant X le long de L, on trouve une -vari´et´e projective F et lisse Y -birationnelle `a X et munie d’une structure de ﬁbration en co- niques sur P2F. Le th´eor`eme de Parimala-Suresh [18, Cor. 5.6] donne alors Q Z Q Z H3 (Y, / (2)) = 0, et donc H3 (X, / (2)) = 0. nr (cid:96) (cid:96) nr (cid:96) (cid:96) Z (ii) On sait (th´eor`eme de Lefschetz faible) que H3(X, ) est sans torsion. Q Z (cid:96) La nullit´e de H3 (X, / (2)) et [7, Cor. 6.9] donnent (ii). nr (cid:96) (cid:96) (iii) Comme X est g´eom´etriquement unirationnelle de dimension 3, le co- ⊗ Z → Z noyau de l’application cycle CH2(X) H4(X, (2)) est un groupe ﬁni (cid:96) (cid:96) [7, Prop. 3.23]. D’apr`es [12, Thm. 1.1] ou [7, Thm. 2.2], la torsion du co- Q Z noyau de l’application cycle s’identiﬁe au quotient de H3 (X, / (2)) par nr (cid:96) (cid:96) (cid:3) son sous-groupe divisible maximal. De (i) r´esulte donc (iii). Remarque 5.2. — La d´emonstration du th´eor`eme de Parimala et Suresh [18] utilise un r´esultat de th´eorie du corps de classes sup´erieur, `a savoir la Q Z nullit´e de H3 (S, / (2)) pour S une surface projective et lisse sur un corps nr (cid:96) (cid:96) ﬁni ([8, Rem. 2, p. 790] ; [13, Thm. 0.7, Cor]). Elle utilise aussi beaucoup d’autres arguments d´elicats. En utilisant la th´eorie du corps de classes sup´erieur, et le lien entre la surface de Fano des droites de X et le groupe des cycles de codimension 2 de ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: J.-L.

text: COLLIOT-THE´LE`NE

text: repose

text: de

text: fa¸con

text: essentielle

text: sur

text: dont

text: les

text: m´ethodes

text: g´eom´etriques

text: sont

text: ins-

text: pir´ees

text: de

text: celles

text: de

text: (qui

text: cite

text: F

text: Hypersurfaces

text: cubiques

text: dans

text: corps

text: ﬁni

text: Pour

text: les

text: hypersurfaces

text: cubiques

text: lisses

text: sur

text: un

text: corps

text: ﬁni,

text: le

text: travail

text: de

text: Parimala

text: et

text: Suresh

text: permet

text: de

text: compl´eter

text: le

text: th´eor`eme

text: pour

text: n

text: Th´eor`eme

text: Soit

text: X

text: une

text: hypersurface

text: cubique

text: lisse

text: sur

text: un

text: corps

text: F

text: ﬁni

text: de

text: caract´eristique

text: diﬀ´erente

text: de

text: F

text: (i)

text: Pour

text: tout

text: premier

text: diﬀ´erent

text: de

text: la

text: caract´eristique

text: de

text: on

text: a

text: Q

text: Z

text: (X,

text: nr

text: F

text: F

text: F

text: (ii)

text: Soit

text: une

text: clˆoture

text: alg´ebrique

text: de

text: et

text: G

text: Gal(

text: L’application

text: naturelle

text: est

text: un

text: isomorphisme.

text: (iii)

text: L’application

text: cycle

text: Z

text: Z

text: cyc

text: X

text: est

text: surjective.

text: D´emonstration.

text: (i)

text: Le

text: cas

text: r´esulte

text: d´ej`a

text: de

text: la

text: proposition

text: Pour

text: d´emontrer

text: la

text: proposition,

text: par

text: le

text: lemme

text: et

text: un

text: argument

text: de

text: restriction-

text: corestriction,

text: on

text: peut

text: supposer

text: que

text: X

text: contient

text: une

text: droite

text: L

text: X

text: d´eﬁnie

text: sur

text: F

text: F

text: le

text: corps

text: En

text: ´eclatant

text: X

text: le

text: long

text: de

text: L,

text: on

text: trouve

text: une

text: -vari´et´e

text: projective

text: F

text: et

text: lisse

text: Y

text: -birationnelle

text: `a

text: X

text: et

text: munie

text: d’une

text: structure

text: de

text: ﬁbration

text: en

text: co-

text: niques

text: sur

text: Le

text: th´eor`eme

text: de

text: Parimala-Suresh

text: Cor.

text: donne

text: alors

text: Q

text: Z

text: Q

text: Z

text: (Y,

text: et

text: donc

text: (X,

text: nr

text: nr

text: Z

text: (ii)

text: On

text: sait

text: (th´eor`eme

text: de

text: Lefschetz

text: faible)

text: que

text: est

text: sans

text: torsion.

text: Q

text: Z

text: La

text: nullit´e

text: de

text: (X,

text: et

text: Cor.

text: donnent

text: (ii).

text: nr

text: (iii)

text: Comme

text: X

text: est

text: g´eom´etriquement

text: unirationnelle

text: de

text: dimension

text: le

text: co-

text: Z

text: Z

text: noyau

text: de

text: l’application

text: cycle

text: est

text: un

text: groupe

text: ﬁni

text: Prop.

text: D’apr`es

text: Thm.

text: ou

text: Thm.

text: la

text: torsion

text: du

text: co-

text: Q

text: Z

text: noyau

text: de

text: l’application

text: cycle

text: s’identiﬁe

text: au

text: quotient

text: de

text: (X,

text: par

text: nr

text: son

text: sous-groupe

text: divisible

text: maximal.

text: De

text: (i)

text: r´esulte

text: donc

text: (iii).

text: Remarque

text: La

text: d´emonstration

text: du

text: th´eor`eme

text: de

text: Parimala

text: et

text: Suresh

text: utilise

text: un

text: r´esultat

text: de

text: th´eorie

text: du

text: corps

text: de

text: classes

text: sup´erieur,

text: `a

text: savoir

text: la

text: Q

text: Z

text: nullit´e

text: de

text: (S,

text: pour

text: S

text: une

text: surface

text: projective

text: et

text: lisse

text: sur

text: un

text: corps

text: nr

text: ﬁni

text: Rem.

text: p.

text: Thm.

text: Cor]).

text: Elle

text: utilise

text: aussi

text: beaucoup

text: d’autres

text: arguments

text: d´elicats.

text: En

text: utilisant

text: la

text: th´eorie

text: du

text: corps

text: de

text: classes

text: sup´erieur,

text: et

text: le

text: lien

text: entre

text: la

text: surface

text: de

text: Fano

text: des

text: droites

text: de

text: X

text: et

text: le

text: groupe

text: des

text: cycles

text: de

text: codimension

text: de

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: STATISTICAL ANALYSIS OF THE FIRST PASSAGE PATH ENSEMBLE OF JUMP PROCESSES MAX VON KLEIST1 AND CHRISTOF SCHU¨TTE1,2 AND WEI ZHANG1 Abstract. The transition mechanism of jump processes between two diﬀerent subsets in state space reveals important dynamical information of the processes and therefore has attracted considerable attention in the past years. In this paper, we study the ﬁrst passage path ensemble of both discrete-time and continuous-time jump processes on a ﬁnite state space. The main approach is to divide each ﬁrst passage path into nonreactive and reactive segments and to study them separately. The analysis can be applied to jump processes which are non- ergodic, as well as continuous-time jump processes where the waiting time distributions are non-exponential. In the particular case that the jump processes are both Markovian and ergodic, our analysis elucidates the relations between the study of the ﬁrst passage paths and the study of the transition paths in transition path theory. We provide algorithms to numerically compute statistics of the ﬁrst passage path ensemble. The computational complexity of these algorithms scales with the complexity of solving a linear system, for which eﬃcient methods are available. Several examples demonstrate the wide applicability of the derived results across research areas. Key words. jump process, non-ergodic process, non-exponential distribution, ﬁrst passage path, transition path theory (Markov) jump process has been extensively studied in the past decades and nowadays it becomes a standard mathematical model that is widely applied to problems arising from physics, chemistry, biology, etc [7, 24]. Among many important topics in the study of jump processes, the ﬁrst passage paths have attracted considerable attention within diﬀerent disciplines, where the main purpose is to understand the transition mechanism of the system between diﬀerent subsets in state space and to e.e. access how much time the transitions typically take [21]. Examples include reaction networks in chemistry [38, 37], phenotypic switches in cell biology [39], confor- mational changes in molecular dynamics [17, 25], disease spreading within certain geographical areas [4, 6], as well as spread of information on social networks [16]. In these contexts, studies of the ﬁrst passage paths are often very helpful to understand the underlying processes and to foster- or prevent the transition events. A common situation that one often encounters in the study of many real-world applications is that the system exhibits metastability and transition events become very rare [27, 15, 30]. To study the transition events in these scenarios, transition path theory (TPT) has been developed both for diﬀusion processes on continuous state space [9, 8, 34] and for Markov jump processes on discrete state space [23, 5]. It provides a probabilistic framework to analyze the statistical properties of system’s reactive trajectories (following the terminology in [19]) and enables us to answer the questions which are important in order to understand the transition mechanism of the system. In the discrete state space setting, TPT can be applied to study Markov jump processes which are ergodic with a unique invariant measure [23, 35]. Meanwhile, inspired by the wide applications that have emerged due to the rapid develop- ment of network science, scientiﬁc interest has been extended to study processes which go beyond ergodic Markov jump processes [26, 29, 28]. One simple example when ergodicity is violated is 1 Institute of Mathematics, Freie Universit¨at Berlin, Arnimallee 6, 14195 Berlin, Germany 2 Zuse Institute Berlin, Takustrasse 7, 14195 Berlin, Germany Email : vkleist@zedat.fu-berlin.de, schuette@zib.de, wei.zhang@fu-berlin.de 1 ##LTLine##

text: STATISTICAL

text: ANALYSIS

text: OF

text: THE

text: FIRST

text: PASSAGE

text: PATH

text: ENSEMBLE

text: OF

text: JUMP

text: PROCESSES

text: MAX

text: VON

text: AND

text: CHRISTOF

text: AND

text: WEI

text: Abstract.

text: The

text: transition

text: mechanism

text: of

text: jump

text: processes

text: between

text: two

text: diﬀerent

text: subsets

text: in

text: state

text: space

text: reveals

text: important

text: dynamical

text: information

text: of

text: the

text: processes

text: and

text: therefore

text: has

text: attracted

text: considerable

text: attention

text: in

text: the

text: past

text: years.

text: In

text: this

text: paper,

text: we

text: study

text: the

text: ﬁrst

text: passage

text: path

text: ensemble

text: of

text: both

text: discrete-time

text: and

text: continuous-time

text: jump

text: processes

text: on

text: a

text: ﬁnite

text: state

text: space.

text: The

text: main

text: approach

text: is

text: to

text: divide

text: each

text: ﬁrst

text: passage

text: path

text: into

text: nonreactive

text: and

text: reactive

text: segments

text: and

text: to

text: study

text: them

text: separately.

text: The

text: analysis

text: can

text: be

text: applied

text: to

text: jump

text: processes

text: which

text: are

text: non-

text: ergodic,

text: as

text: well

text: as

text: continuous-time

text: jump

text: processes

text: where

text: the

text: waiting

text: time

text: distributions

text: are

text: non-exponential.

text: In

text: the

text: particular

text: case

text: that

text: the

text: jump

text: processes

text: are

text: both

text: Markovian

text: and

text: ergodic,

text: our

text: analysis

text: elucidates

text: the

text: relations

text: between

text: the

text: study

text: of

text: the

text: ﬁrst

text: passage

text: paths

text: and

text: the

text: study

text: of

text: the

text: transition

text: paths

text: in

text: transition

text: path

text: theory.

text: We

text: provide

text: algorithms

text: to

text: numerically

text: compute

text: statistics

text: of

text: the

text: ﬁrst

text: passage

text: path

text: ensemble.

text: The

text: computational

text: complexity

text: of

text: these

text: algorithms

text: scales

text: with

text: the

text: complexity

text: of

text: solving

text: a

text: linear

text: system,

text: for

text: which

text: eﬃcient

text: methods

text: are

text: available.

text: Several

text: examples

text: demonstrate

text: the

text: wide

text: applicability

text: of

text: the

text: derived

text: results

text: across

text: research

text: areas.

text: Key

text: words.

text: jump

text: process,

text: non-ergodic

text: process,

text: non-exponential

text: distribution,

text: ﬁrst

text: passage

text: path,

text: transition

text: path

text: theory

text: (Markov)

text: jump

text: process

text: has

text: been

text: extensively

text: studied

text: in

text: the

text: past

text: decades

text: and

text: nowadays

text: it

text: becomes

text: a

text: standard

text: mathematical

text: model

text: that

text: is

text: widely

text: applied

text: to

text: problems

text: arising

text: from

text: physics,

text: chemistry,

text: biology,

text: etc

text: Among

text: many

text: important

text: topics

text: in

text: the

text: study

text: of

text: jump

text: processes,

text: the

text: ﬁrst

text: passage

text: paths

text: have

text: attracted

text: considerable

text: attention

text: within

text: diﬀerent

text: disciplines,

text: where

text: the

text: main

text: purpose

text: is

text: to

text: understand

text: the

text: transition

text: mechanism

text: of

text: the

text: system

text: between

text: diﬀerent

text: subsets

text: in

text: state

text: space

text: and

text: to

text: e.e.

text: access

text: how

text: much

text: time

text: the

text: transitions

text: typically

text: take

text: Examples

text: include

text: reaction

text: networks

text: in

text: chemistry

text: phenotypic

text: switches

text: in

text: cell

text: biology

text: confor-

text: mational

text: changes

text: in

text: molecular

text: dynamics

text: disease

text: spreading

text: within

text: certain

text: geographical

text: areas

text: as

text: well

text: as

text: spread

text: of

text: information

text: on

text: social

text: networks

text: In

text: these

text: contexts,

text: studies

text: of

text: the

text: ﬁrst

text: passage

text: paths

text: are

text: often

text: very

text: helpful

text: to

text: understand

text: the

text: underlying

text: processes

text: and

text: to

text: foster-

text: or

text: prevent

text: the

text: transition

text: events.

text: A

text: common

text: situation

text: that

text: one

text: often

text: encounters

text: in

text: the

text: study

text: of

text: many

text: real-world

text: applications

text: is

text: that

text: the

text: system

text: exhibits

text: metastability

text: and

text: transition

text: events

text: become

text: very

text: rare

text: To

text: study

text: the

text: transition

text: events

text: in

text: these

text: scenarios,

text: transition

text: path

text: theory

text: (TPT)

text: has

text: been

text: developed

text: both

text: for

text: diﬀusion

text: processes

text: on

text: continuous

text: state

text: space

text: and

text: for

text: Markov

text: jump

text: processes

text: on

text: discrete

text: state

text: space

text: It

text: provides

text: a

text: probabilistic

text: framework

text: to

text: analyze

text: the

text: statistical

text: properties

text: of

text: system’s

text: reactive

text: trajectories

text: (following

text: the

text: terminology

text: in

text: and

text: enables

text: us

text: to

text: answer

text: the

text: questions

text: which

text: are

text: important

text: in

text: order

text: to

text: understand

text: the

text: transition

text: mechanism

text: of

text: the

text: system.

text: In

text: the

text: discrete

text: state

text: space

text: setting,

text: TPT

text: can

text: be

text: applied

text: to

text: study

text: Markov

text: jump

text: processes

text: which

text: are

text: ergodic

text: with

text: a

text: unique

text: invariant

text: measure

text: Meanwhile,

text: inspired

text: by

text: the

text: wide

text: applications

text: that

text: have

text: emerged

text: due

text: to

text: the

text: rapid

text: develop-

text: ment

text: of

text: network

text: science,

text: scientiﬁc

text: interest

text: has

text: been

text: extended

text: to

text: study

text: processes

text: which

text: go

text: beyond

text: ergodic

text: Markov

text: jump

text: processes

text: One

text: simple

text: example

text: when

text: ergodicity

text: is

text: violated

text: is

text: Institute

text: of

text: Mathematics,

text: Freie

text: Universit¨at

text: Berlin,

text: Arnimallee

text: Berlin,

text: Germany

text: Zuse

text: Institute

text: Berlin,

text: Takustrasse

text: Berlin,

text: Germany

text: Email

text: vkleist@zedat.fu-berlin.de,

text: schuette@zib.de,

text: wei.zhang@fu-berlin.de

text: ##LTLine##


text: (cid:80) Accordingly, if the probability distribution of the initial state on set A is µ, the average length of the ﬁrst passage path is then given by µ(x)f(x). Also, let θ(x) be the average ∈ x A number of times that node x has been visited by the ﬁrst passage path with initial distribution µ, i.e. (cid:88) (cid:104) τB (cid:105) θ(x) = 1 (x ) . (2.29) x l l=0 In analogy to Proposition 3 and Proposi(cid:88)tion 6, we know it satisﬁes | ∈ (cid:88) θ(x) = µ(x)1 (x) + θ(y)p(x y) , x V , A ∈ y Bc (2.30) θ(x) = 1 . ∈ x B (cid:101) (cid:48) Furthermore, taking into account the deﬁnitions of θ¯, θ¯ and θ in (2.8), (2.15) and (2.21), as well as relations (2.9), (2.22), we can verify the following relations (cid:101) (cid:101) (cid:88) (cid:88) (cid:88) (cid:88) (cid:88) (cid:88) − (cid:48) (cid:101) ∀ ∈ (cid:101) θ(x) = θ¯(x) + θ(x) µ (x)1 (x) = θ¯ (x) + θ(x) , x V , r A µ(x)f(x) = θ(x) = θ¯(x) + θ(x) − 1 = θ¯(cid:48)(x) + θ(x) . (2.31) ∈ ∈ ∈ ∈ ∈ ∈ x A x Bc x Bc x Bc x Bc x Bc In fact, the following explicit relations hold. ∈ Proposition 7. For any x V , (cid:101) − θ¯(x) = θ(x)(1 q(x)), θ(x) = θ(x)q(x) + µ (x)1 (x) . (2.32) r A (cid:101) Proof. Clearly θ¯(x) = θ(x) = 0 if θ(x) = 0. Hence we only need to consider the case when P θ(x) > 0. Using the Markov property, we can compute the probability (τ < τ ) from the A,x B,x · · · ﬁrst passage ensemble. Speciﬁcally, let (x , x , , x ) be a ﬁrst passage path. Conditioning on ≤ ≤ { 0 1 } τB { ≤ } x = x where 0 l τ , the event τ < τ is actually equivalent to l σ . Therefore, l B A,x B,x (cid:16) (cid:12) (cid:17) 1 − q(x) =P(τ <(cid:12)(cid:12)τ ) A,x B,x P(cid:16) ≤ ≤ ≤ (cid:17) (cid:10) (cid:80) (cid:11) = l σ x = x, 0 l τ l B P ≤(cid:16) ≤ ≤(cid:17) (cid:10) (cid:80)σ 1 (x )(cid:11) l σ , xl = x, 0 l τB x l θ¯(x) = = l=0 = , (cid:0) P xl (cid:1)= x, 0 ≤ l ≤ τB (cid:101) τB 1x(xl) θ(x) l=0 − · which implies θ¯(x) = θ(x) 1 q(x) . The expression of θ( ) follows from (2.31). → We also introduce the probability ﬂux for edge x y, which is deﬁned as 12 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Accordingly,

text: if

text: the

text: probability

text: distribution

text: of

text: the

text: initial

text: state

text: on

text: set

text: A

text: is

text: µ,

text: the

text: average

text: length

text: of

text: the

text: ﬁrst

text: passage

text: path

text: is

text: then

text: given

text: by

text: µ(x)f(x).

text: Also,

text: let

text: θ(x)

text: be

text: the

text: average

text: x

text: A

text: number

text: of

text: times

text: that

text: node

text: x

text: has

text: been

text: visited

text: by

text: the

text: ﬁrst

text: passage

text: path

text: with

text: initial

text: distribution

text: µ,

text: i.e.

label: equation

text: τB

text: θ(x)

text: (x

text: x

text: l

text: In

text: analogy

text: to

text: Proposition

text: and

text: we

text: know

text: it

text: satisﬁes

text: θ(x)

text: (x)

text: θ(y)p(x

text: y)

text: x

text: V

text: A

text: y

text: Bc

text: θ(x)

text: x

text: B

text: Furthermore,

text: taking

text: into

text: account

text: the

text: deﬁnitions

text: of

text: θ¯,

text: θ¯

text: and

text: θ

text: in

text: and

text: as

text: well

text: as

text: relations

text: we

text: can

text: verify

text: the

text: following

text: relations

text: θ(x)

text: θ¯(x)

text: θ(x)

text: µ

text: (x)

text: θ¯

text: (x)

text: θ(x)

text: x

text: V

text: r

text: A

text: µ(x)f(x)

text: θ(x)

text: θ¯(x)

text: θ(x)

text: θ(x)

text: x

text: A

text: x

text: Bc

text: x

text: Bc

text: x

text: Bc

text: x

text: Bc

text: x

text: Bc

text: In

text: fact,

text: the

text: following

text: explicit

text: relations

text: hold.

text: Proposition

text: For

text: any

text: x

text: V

text: θ¯(x)

text: q(x)),

text: θ(x)

text: θ(x)q(x)

text: µ

text: (x)

text: r

text: A

text: Proof.

text: Clearly

text: θ¯(x)

text: θ(x)

text: if

text: θ(x)

text: Hence

text: we

text: only

text: need

text: to

text: consider

text: the

text: case

text: when

text: P

text: θ(x)

text: Using

text: the

text: Markov

text: property,

text: we

text: can

text: compute

text: the

text: probability

text: (τ

text: τ

text: from

text: the

text: A,x

text: B,x

text: ﬁrst

text: passage

text: ensemble.

text: Speciﬁcally,

text: let

text: (x

text: x

text: x

text: be

text: a

text: ﬁrst

text: passage

text: path.

text: Conditioning

text: on

text: τB

text: x

text: x

text: where

text: l

text: τ

text: the

text: event

text: τ

text: τ

text: is

text: actually

text: equivalent

text: to

text: l

text: σ

text: Therefore,

text: l

text: B

text: A,x

text: B,x

text: q(x)

text: =P(τ

text: A,x

text: B,x

text: l

text: σ

text: x

text: x,

text: l

text: τ

text: l

text: B

text: P

text: (x

text: l

text: σ

text: xl

text: x,

text: l

text: τB

text: x

text: l

text: θ¯(x)

text: P

text: xl

text: x,

text: l

text: τB

text: τB

text: θ(x)

text: which

text: implies

text: θ¯(x)

text: θ(x)

text: q(x)

text: The

text: expression

text: of

text: θ(

text: follows

text: from

text: We

text: also

text: introduce

text: the

text: probability

text: ﬂux

text: for

text: edge

text: x

text: y,

text: which

text: is

text: deﬁned

text: as

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:88)−| (cid:88) −| (cid:88) (cid:88) Therefore, using q = 1, q = 0, from (4.9) we obtain A B (cid:88) − (cid:88) | (cid:88) − | m(y)q (y)p(x y) + m(y)q (y)p(x y) ∈ ∈ ∈ ∈ x A y V x B y V − − = m(y)q (y) + m(y)q (y) = m(y) , ∈ ∈ ∈ y A y B y A which implies (4.5). − 2. Using Markovianity of the discrete-time jump process and the deﬁnition of q , we can derive P | ∈ µ(x) = (x = x x − = y, coming from B, y V ) j j 1 P ∈ (cid:88)(x = x , x − = y, coming from B, y V ) = jP j 1 ∈ (x − = y, coming from B, y V ) j 1 ∝ (cid:88) P(x = x , x − = y , coming from B) j j 1 ∈ y V ∝ (cid:88) P(x = x | x − = y) P(x − = y, coming from B) j j 1 j 1 ∈ y V ∝ − − | m(y)(1 q (y))p(x y) , ∈ y V ∝ where means “equal up to a constant” and the normalization constant is given in (4.5). − 3. Using the equation of q in (4(cid:101).1) and the formula of µ in (4.6), it is easy to verify that θ deﬁned in (4.7) satisﬁes equations (2.30). 4. The expressions of θ¯, µ and θ follows from (4.7), Proposition 4 and Proposition 7. r Remark 2. Using the expression of µ in (4.8) and an argument similar to the proof of r (4.5), we could obtain (cid:88) (cid:88) (cid:88) (cid:88) (cid:88) (cid:88) − − (cid:88)| (cid:88) − | Z = m(y)(1 q (y))p(x y) = m(y)q (y)p(x y) ∈ ∈ ∈ ∈ x A y V x B y V | − | (4.10) = m(x)q(y)p(y x) = m(x)(1 q(y))p(y x) . ∈ ∈ ∈ ∈ x A y V x B y V Finally, we consider the continuous-time jump process introduced in Section and assume its associated discrete-time Markov jump process obtained in Subsection is ergodic. In this case, we consider an inﬁnitely long trajectory of the process as in (B.3) in Appendix and let ≤ M = M(N) > 0 be the integer such that τ(M) N < τ(M+1), where the stopping times τ(M) B B B are deﬁned in (4.2). Following [23], we deﬁne the reaction rate k between sets A and B by AB M k = lim . (4.11) AB N→+∞ tN We have the following result. 20 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Therefore,

text: using

text: q

text: q

text: from

text: we

text: obtain

text: A

text: B

text: m(y)q

text: (y)p(x

text: y)

text: m(y)q

text: (y)p(x

text: y)

text: x

text: A

text: y

text: V

text: x

text: B

text: y

text: V

text: m(y)q

text: (y)

text: m(y)q

text: (y)

text: m(y)

text: y

text: A

text: y

text: B

text: y

text: A

text: which

text: implies

text: Using

text: Markovianity

text: of

text: the

text: discrete-time

text: jump

text: process

text: and

text: the

text: deﬁnition

text: of

text: q

text: we

text: can

text: derive

text: P

text: µ(x)

text: (x

text: x

text: x

text: y,

text: coming

text: from

text: B,

text: y

text: V

text: j

text: j

text: P

text: x

text: x

text: y,

text: coming

text: from

text: B,

text: y

text: V

text: jP

text: j

text: (x

text: y,

text: coming

text: from

text: B,

text: y

text: V

text: j

text: P(x

text: x

text: x

text: y

text: coming

text: from

text: B)

text: j

text: j

text: y

text: V

text: P(x

text: x

text: x

text: y)

text: P(x

text: y,

text: coming

text: from

text: B)

text: j

text: j

text: j

text: y

text: V

text: q

text: (y))p(x

text: y)

text: y

text: V

text: where

text: means

text: “equal

text: up

text: to

text: a

text: constant”

text: and

text: the

text: normalization

text: constant

text: is

text: given

text: in

text: Using

text: the

text: equation

text: of

text: q

text: in

text: and

text: the

text: formula

text: of

text: µ

text: in

text: it

text: is

text: easy

text: to

text: verify

text: that

text: θ

text: deﬁned

text: in

text: satisﬁes

text: equations

text: The

text: expressions

text: of

text: θ¯,

text: µ

text: and

text: θ

text: follows

text: from

text: Proposition

text: and

text: Proposition

text: r

text: Remark

text: Using

text: the

text: expression

text: of

text: µ

text: in

text: and

text: an

text: argument

text: similar

text: to

text: the

text: proof

text: of

text: r

text: we

text: could

text: obtain

text: Z

text: q

text: (y))p(x

text: y)

text: m(y)q

text: (y)p(x

text: y)

text: x

text: A

text: y

text: V

text: x

text: B

text: y

text: V

text: m(x)q(y)p(y

text: x)

text: q(y))p(y

text: x)

text: x

text: A

text: y

text: V

text: x

text: B

text: y

text: V

text: Finally,

text: we

text: consider

text: the

text: continuous-time

text: jump

text: process

text: introduced

text: in

text: Section

text: and

text: assume

text: its

text: associated

text: discrete-time

text: Markov

text: jump

text: process

text: obtained

text: in

text: Subsection

text: is

text: ergodic.

text: In

text: this

text: case,

text: we

text: consider

text: an

text: inﬁnitely

text: long

text: trajectory

text: of

text: the

text: process

text: as

text: in

text: in

text: Appendix

text: and

text: let

text: M

text: M(N)

text: be

text: the

text: integer

text: such

text: that

text: τ(M)

text: N

text: where

text: the

text: stopping

text: times

text: τ(M)

text: B

text: B

text: B

text: are

text: deﬁned

text: in

text: Following

text: we

text: deﬁne

text: the

text: reaction

text: rate

text: k

text: between

text: sets

text: A

text: and

text: B

text: by

text: AB

text: M

text: k

text: lim

text: AB

text: N→+∞

text: tN

text: We

text: have

text: the

text: following

text: result.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: ./flux-bar-discrete-exp-eps-converted-t.o/.fpldufx-reactive-discrete-exp-eps-converted-to.pdf (cid:101) (cid:101) (cid:48) (a) θ¯ and J¯ (b) θ and J Figure 6.2: Example 1 in the case of exponential waiting time distributions. (a) average number (cid:48) of times θ¯ that each node has(cid:101) been visited by the nonreactive trajectories (in the brackets) as well as the nonreactive ﬂux J¯ for each edge related to the nonreactive trajectories (on the edges). (cid:101) (b) average number of times θ that each node has been visited by the reactive trajectories (in the brackets) as well as the reactive ﬂux J for each edge related to the reactive trajectories (on the edges). See Table 1.1 for concrete deﬁnitions of these quantities. node corresponding to the cell at the bottom left and the upper right corner (see Figure 6.3), respectively. Since the process itself is a discrete-time Markov jump process, its transition probabilities p can be directly computed from the connectivity structure of the maze. Committor function q and function θ related to the ﬁrst passage paths are shown in Figure 6.4. From the committor function q, we can observe that there are long vertical walls which roughly separate the maze into left and right parts. In the left part (dark blue cells in the left panel of Figure 6.4), the walker is likely to return to A ﬁrst before it goes to B. On the other hand, from the plot of the function θ in the right panel of Figure 6.4, we can realize that, unlike the simple pathway shown in Figure 6.3, the ﬁrst passage paths from A to B taken by the walker are very diﬀusive on average, since many cells have been repeatedly visited within a single ﬁrst passage path. The diﬀusiveness of the paths can also be observed from the very long average path lengths in Table 6.2. (cid:101) In order to further investigate the nonreactive and reactive segments of the ﬁrst passage (cid:48) paths, functions θ¯ , θ are computed and are shown in Figure 6.5. We can see that both of these two segments are diﬀusive on average, since many cells are repeatedly visited. Comparing these two segments, however, it is observed that a large number of the visits at cells in the left part of the maze actually belong to the nonreactive segments and do not contribute to the reactive pathways. Comparing the two panels in Figure 6.5, we can also conclude that the nonreactive segments are typically more diﬀusive than the reactive segments. Roughly speaking, this phenomena occurs similarly when studying the transitions of a metastable diﬀusion process 27 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: (a)

text: θ¯

text: and

text: J¯

text: (b)

text: θ

text: and

text: J

text: Figure

text: Example

label: caption

text: in

text: the

text: case

text: of

text: exponential

text: waiting

text: time

text: distributions.

text: (a)

text: average

text: number

text: of

text: times

text: θ¯

text: that

text: each

text: node

text: been

text: visited

text: by

text: the

text: nonreactive

text: trajectories

text: (in

text: the

text: brackets)

text: as

text: well

text: as

text: the

text: nonreactive

text: ﬂux

text: J¯

text: for

text: each

text: edge

text: related

text: to

text: the

text: nonreactive

text: trajectories

text: (on

text: the

text: edges).

text: (b)

text: average

text: number

text: of

text: times

text: θ

text: that

text: each

text: node

text: has

text: been

text: visited

text: by

text: the

text: reactive

text: trajectories

text: (in

text: the

text: brackets)

text: as

text: well

text: as

text: the

text: reactive

text: ﬂux

text: J

text: for

text: each

text: edge

text: related

text: to

text: the

text: reactive

text: trajectories

text: (on

text: the

text: edges).

text: See

text: Table

text: for

text: concrete

text: deﬁnitions

text: of

text: these

text: quantities.

text: node

text: corresponding

text: to

text: the

text: cell

text: at

text: the

text: bottom

text: left

text: and

text: the

text: upper

text: right

text: corner

text: (see

text: Figure

text: respectively.

text: Since

text: the

text: process

text: itself

text: is

text: a

text: discrete-time

text: Markov

text: jump

text: process,

text: its

text: transition

text: probabilities

text: p

text: can

text: be

text: directly

text: computed

text: from

text: the

text: connectivity

text: structure

text: of

text: the

text: maze.

text: Committor

text: function

text: q

text: and

text: function

text: θ

text: related

text: to

text: the

text: ﬁrst

text: passage

text: paths

text: are

text: shown

text: in

text: Figure

text: From

text: the

text: committor

text: function

text: q,

text: we

text: can

text: observe

text: that

text: there

text: are

text: long

text: vertical

text: walls

text: which

text: roughly

text: separate

text: the

text: maze

text: into

text: left

text: and

text: right

text: parts.

text: In

text: the

text: left

text: part

text: (dark

text: blue

text: cells

text: in

text: the

text: left

text: panel

text: of

text: Figure

text: the

text: walker

text: is

text: likely

text: to

text: return

text: to

text: A

text: ﬁrst

text: before

text: it

text: goes

text: to

text: B.

text: On

text: the

text: other

text: hand,

text: from

text: the

text: plot

text: of

text: the

text: function

text: θ

text: in

text: the

text: right

text: panel

text: of

text: Figure

text: we

text: can

text: realize

text: that,

text: unlike

text: the

text: simple

text: pathway

text: shown

text: in

text: Figure

text: the

text: ﬁrst

text: passage

text: paths

text: from

text: A

text: to

text: B

text: taken

text: by

text: the

text: walker

text: are

text: very

text: diﬀusive

text: on

text: average,

text: since

text: many

text: cells

text: have

text: been

text: repeatedly

text: visited

text: within

text: a

text: single

text: ﬁrst

text: passage

text: path.

text: The

text: diﬀusiveness

text: of

text: the

text: paths

text: can

text: also

text: be

text: observed

text: from

text: the

text: very

text: long

text: average

text: path

text: lengths

text: in

text: Table

text: In

text: order

text: to

text: further

text: investigate

text: the

text: nonreactive

text: and

text: reactive

text: segments

text: of

text: the

text: ﬁrst

text: passage

text: paths,

text: functions

text: θ¯

text: θ

text: are

text: computed

text: and

text: are

text: shown

text: in

text: Figure

text: We

text: can

text: see

text: that

text: both

text: of

text: these

text: two

text: segments

text: are

text: diﬀusive

text: on

text: average,

text: since

text: many

text: cells

text: are

text: repeatedly

text: visited.

text: Comparing

text: these

text: two

text: segments,

text: however,

text: it

text: is

text: observed

text: that

text: a

text: large

text: number

text: of

text: the

text: visits

text: at

text: cells

text: in

text: the

text: left

text: part

text: of

text: the

text: maze

text: actually

text: belong

text: to

text: the

text: nonreactive

text: segments

text: and

text: do

text: not

text: contribute

text: to

text: the

text: reactive

text: pathways.

text: Comparing

text: the

text: two

text: panels

text: in

text: Figure

text: we

text: can

text: also

text: conclude

text: that

text: the

text: nonreactive

text: segments

text: are

text: typically

text: more

text: diﬀusive

text: than

text: the

text: reactive

text: segments.

text: Roughly

text: speaking,

text: this

text: phenomena

text: occurs

text: similarly

text: when

text: studying

text: the

text: transitions

text: of

text: a

text: metastable

text: diﬀusion

text: process

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##