text: [52] Tian, Fei-Ran, Oscillations of the zero dispersion limit of the Korteweg de Vries equations. Comm. Pure App. Math. 46 (1993) 1093 - 1129. [53] Tian, Fei-Ran The Whitham-type equations and linear overdetermined sys- tems of Euler-Poisson-Darboux type. Duke Math. J. 74 (1994), no. 1, 203 - 221. [54] Tian, Fei-Ran , The initial value problem for the Whitham averaged system. Comm. Math. Phys. 166 (1994), no. 1, 79 - 115. [55] Trillo S.; Klein M.; Clauss G.; Onorato M., Observation of dispersive shock waves developing from initial depressions in shallow water, to appear in Physica D, http://dx.doi.org/10.1016/j.physd.2016.01.007 [56] Tsarev, S. P., Poisson brackets and one-dimensional Hamiltonian systems of hydrodynamic type. Soviet Math. Dokl. 31 (1985), 488 - 491. [57] Venakides S. Long time asymptotics of the Kortewege Vries equation Trans. Am. Math. Soc. 293 (1986), 411 - 419. [58] Venakides, V., The zero dispersion limit of the Korteweg de Vries equation for initial potential with nontrivial reﬂection coefﬁcient. Comm. Pure Appl. Math. 38 (1985), 125 - 155. [59] Venakides, S., The Korteweg de Vries equations with small dispersion: higher order Lax-Levermore theory. Comm. Pure Appl. Math. 43 (1990), 335 - 361. [60] Whitham, G. B., Linear and nonlinear waves, J.Wiley, New York, 1974. [61] Zakharov, V. E.; Manakov, S. V. Asymptotic behavior of non-linear wave systems integrated by the inverse scattering method. Soviet Physics JETP 44 (1976), no. 1, 106 - 112. 26 ##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: Tian,

text: Fei-Ran,

text: Oscillations

text: of

text: the

text: zero

text: dispersion

text: limit

text: of

text: the

text: Korteweg

text: de

text: Vries

text: equations.

text: Comm.

text: Pure

text: App.

text: Math.

text: Tian,

text: Fei-Ran

text: The

text: Whitham-type

text: equations

text: and

text: linear

text: overdetermined

text: sys-

text: tems

text: of

text: Euler-Poisson-Darboux

text: type.

text: Duke

text: Math.

text: J.

text: no.

text: Tian,

text: Fei-Ran

text: The

text: initial

text: value

text: problem

text: for

text: the

text: Whitham

text: averaged

text: system.

text: Comm.

text: Math.

text: Phys.

text: no.

text: Trillo

text: S.;

text: Klein

text: M.;

text: Clauss

text: G.;

text: Onorato

text: M.,

text: Observation

text: of

text: dispersive

text: shock

text: waves

text: developing

text: from

text: initial

text: depressions

text: in

text: shallow

text: water,

text: to

text: appear

text: in

text: Physica

text: D,

text: Tsarev,

text: S.

text: P.,

text: Poisson

text: brackets

text: and

text: one-dimensional

text: Hamiltonian

text: systems

text: of

text: hydrodynamic

text: type.

text: Soviet

text: Math.

text: Dokl.

text: Venakides

text: S.

text: Long

text: time

text: asymptotics

text: of

text: the

text: Kortewege

text: Vries

text: equation

text: Trans.

text: Am.

text: Math.

text: Soc.

text: Venakides,

text: V.,

text: The

text: zero

text: dispersion

text: limit

text: of

text: the

text: Korteweg

text: de

text: Vries

text: equation

text: for

text: initial

text: potential

text: with

text: nontrivial

text: reﬂection

text: coefﬁcient.

text: Comm.

text: Pure

text: Appl.

text: Math.

text: Venakides,

text: S.,

text: The

text: Korteweg

text: de

text: Vries

text: equations

text: with

text: small

text: dispersion:

text: higher

text: order

text: Lax-Levermore

text: theory.

text: Comm.

text: Pure

text: Appl.

text: Math.

text: Whitham,

text: G.

text: B.,

text: Linear

text: and

text: nonlinear

text: waves,

text: J.Wiley,

text: New

text: York,

text: Zakharov,

text: V.

text: E.;

text: Manakov,

text: S.

text: V.

text: Asymptotic

text: behavior

text: of

text: non-linear

text: wave

text: systems

text: integrated

text: by

text: the

text: inverse

text: scattering

text: method.

text: Soviet

text: Physics

text: JETP

text: no.

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text: ∂ − ∂ − ∂3 with formal adjoint L † = ω ∂φ 6ku0 ∂φ k3 ∂φ3 . Then at order ε0 equation (12) gives L u1 =R(u0), R(u0) := u0,t +6u0u0,x +3θx2u0,φφx +3θxxθxu0,φφ. In a similar way it is possible to get the equations for the higher order correction terms. A condition of solvability of the above equation can be obtained by ob- serving that the integral over a period of the l.h.s of the above equation against the constant function and the function u is equal to zero because 1 and u are in the (cid:90) 0 (cid:90) (cid:90) 0 kernel of L †. Therefore it follows that 2π 2π 2π 0 = R(u0)dφ = ∂t u0dφ +3∂x u20dφ 0 0 0 (cid:90) (cid:90) (cid:90) and 0 = 02π u0R(u0)dφ =∂t 0(cid:90)2π 12u20dφ +2∂x 02π u30dφ 2π +3 u0(θx2u0,φφx +θxxθxu0,φφ)dφ. 0 (cid:104) (cid:105) By denoting with the bracket . the average over a period, we rewrite the above two equations, after elementary algebra and an integration by parts, in the form (cid:104) (cid:105) (cid:104) (cid:105) ∂t u0 +3∂x u20 = 0 (15) (cid:104) (cid:105) (cid:104) (cid:105)− (cid:104) (cid:105) ∂t u20 +4∂x u30 3∂x θx2u20,φ = 0. (16) Using the identities (cid:104) (cid:105) (cid:104) (cid:105) u0u0,φφ +u20,φ = 0, u0,φφ = 0, an(cid:90)d (6), we obtained the identities for the e(cid:90)lliptic integrals (cid:112) (cid:112) − − − − e2 5η−3 4Vη2 3Bη 2Adη = 0(cid:112), e2 − 3η2 +2Vη +B dη = 0. e1 η3 +Vη2 +Bη +A (cid:82) e1 η3 +Vη2 +Bη +A √ Introducing the integral W := π2 ee2 −η3 +Vη2 +Bη +Adη and using the 1 (cid:104) (cid:105) (cid:104) (cid:105) above two identities and the relations kWA = 1, u0 = 2πkWB and u20 = 2πkWV where W , W and W are the partial derivatives of W with respect to A, B and V A B V respectively, we can reduce (14), (15) and (16) to the form ∂ ∂ − ∂ ∂tWA +2V ∂xWA 2WA ∂xV = 0 (17) ∂ ∂ ∂ ∂tWB +2V ∂xWB +WA ∂xB = 0 (18) ∂ ∂ − ∂ ∂tWV +2V ∂xWV 2WA ∂xA = 0. (19) 5 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: ∂3

text: with

text: formal

text: adjoint

text: L

text: ω

text: ∂φ

text: ∂φ

text: ∂φ3

text: Then

text: at

text: order

text: ε0

text: equation

text: gives

text: L

text: In

text: a

text: similar

text: way

text: it

text: is

text: possible

text: to

text: get

text: the

text: equations

text: for

text: the

text: higher

text: order

text: correction

text: terms.

text: A

text: condition

text: of

text: solvability

text: of

text: the

text: above

text: equation

text: can

text: be

text: obtained

text: by

text: ob-

text: serving

text: that

text: the

text: integral

text: over

text: a

text: period

text: of

text: the

text: l.h.s

text: of

text: the

text: above

text: equation

text: against

text: the

text: constant

text: function

text: and

text: the

text: function

text: u

text: is

text: equal

text: to

text: zero

text: because

text: and

text: u

text: are

text: in

text: the

text: kernel

text: of

text: L

text: Therefore

text: it

text: follows

text: that

text: 2π

text: 2π

text: 2π

text: ∂t

text: and

text: =∂t

text: 2π

text: By

text: denoting

text: with

text: the

text: bracket

text: the

text: average

text: over

text: a

text: period,

text: we

text: rewrite

text: the

text: above

text: two

text: equations,

text: after

text: elementary

text: algebra

text: and

text: an

text: integration

text: by

text: parts,

text: in

text: the

text: form

text: ∂t

text: ∂t

text: Using

text: the

text: identities

text: we

text: obtained

text: the

text: identities

text: for

text: the

text: integrals

text: 5η−3

text: +B

text: dη

text: η3

text: +Bη

text: +A

text: η3

text: +Bη

text: +A

text: Introducing

text: the

text: integral

text: W

text: π2

text: −η3

text: +Bη

text: +Adη

text: and

text: using

text: the

text: above

text: two

text: identities

text: and

text: the

text: relations

text: kWA

text: and

text: where

text: W

text: W

text: and

text: W

text: are

text: the

text: partial

text: derivatives

text: of

text: W

text: with

text: respect

text: to

text: A,

text: B

text: and

text: V

text: A

text: B

text: V

text: respectively,

text: we

text: can

text: reduce

text: and

text: to

text: the

text: form

text: ∂tWA

text: ∂xWA

text: ∂xV

text: ∂tWB

text: ∂xWB

text: +WA

text: ∂xB

text: ∂tWV

text: ∂xWV

text: ∂xA

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text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: form (25) is quite general and easily adapts to other modulation equations ( see for example the book [37]). The equations (25) gives another expression for the speed k λi = 2(β1 +β2 +β3)+2∂ k which was obtained in [33]. βi × The Whitham equations are a systems of 3 3 quasi-linear hyperbolic equa- tions namely for β1 > β2 > β3 one has [45] λ1 > λ2 > λ3. → Using the e(cid:18)xpansion of the elliptic int(cid:19)egrals as m 0(cid:18)(see e.g. [43]) (cid:19) π m 9 π − m − 3 K(m)= 1+ + m2 +O(m3) , E(m)= 1 m2 +O(m3) , 2 4 64 2 4 64 (26) → and m 1 (cid:20) (cid:21) √ (cid:39) 1 − 16 − (cid:39) 1 16 E(m) 1+ (1 m) log − 1 , K(m) log − , (27) 2 1 m 2 1 m one can verify that the speeds λ have the following limiting behaviour respec- i tively • at β2 = β1 λ1(β1,β1,β3) = λ2(β1,β1,β3) = 4β1 +2β3 (28) λ3(β1,β1,β3) = 6β3; • at β2 = β3 one has λ1(β1,β3,β3) = 6β1 − (29) λ2(β1,β3,β3) = λ3(β1,β3,β3) = 12β3 6β1. ∂ Namely, when β1 = β2, the equation for β3 reduces to the Hopf equation ∂t β3 + ∂ 6β3 ∂xβ3 = 0. In the same way when β2 = β3 the equation for β1 reduces to the Hopf equation. In the coordinates βi, i = 1,2,3 the travelling(cid:18)wave solution (10) ta(cid:19)kes the form − − Ω u(x,t;ε) = β1 +β3 β2 +2(β2 β3)cn2 K(m)πε +K(m);m , (30) where (cid:112) − − Ω := kx−ωt +φ0 = π Kβ1(m)β3 (x−2t(β1 +β2 +β3))+φ0, m = ββ21 −ββ33 . (31) 7 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: form

text: is

text: quite

text: general

text: and

text: easily

text: adapts

text: to

text: other

text: modulation

text: equations

text: see

text: for

text: example

text: the

text: book

text: The

text: equations

text: gives

text: another

text: expression

text: for

text: the

text: speed

text: k

text: λi

text: k

text: which

text: was

text: obtained

text: in

text: βi

text: The

text: Whitham

text: equations

text: are

text: a

text: systems

text: of

text: quasi-linear

text: hyperbolic

text: equa-

text: tions

text: namely

text: for

text: β1

text: β2

text: β3

text: one

text: has

text: λ1

text: λ2

text: Using

text: the

label: equation

text: of

text: the

text: elliptic

text: as

text: m

text: e.g.

text: π

text: m

text: π

text: m

text: K(m)=

text: E(m)=

text: and

text: m

text: E(m)

text: m)

text: log

text: K(m)

text: log

text: m

text: m

text: one

text: can

text: verify

text: that

text: the

text: speeds

text: λ

text: have

text: the

text: following

text: limiting

text: behaviour

text: respec-

text: i

text: tively

text: at

text: β2

text: β1

text: at

text: β2

text: β3

text: one

text: has

text: Namely,

text: when

text: β1

text: the

text: equation

text: for

text: β3

text: reduces

text: to

text: the

text: Hopf

text: equation

text: ∂t

text: β3

text: ∂xβ3

text: In

text: the

text: same

text: way

text: when

text: β2

text: β3

text: the

text: equation

text: for

text: β1

text: reduces

text: to

text: the

text: Hopf

text: equation.

text: In

text: the

text: coordinates

text: βi,

text: i

text: the

text: solution

text: the

text: form

text: Ω

text: u(x,t;ε)

text: β1

text: β2

text: K(m)πε

text: +K(m);m

text: where

text: Ω

text: kx−ωt

text: π

text: m

text: ββ21

text: −ββ33

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: Cyclone III clock buffer x, ps 64 ν0 ∆ ~ 400 –> 200 32 x 10–15 drift 400 --> 200 200 –> 100 16 MHz 50 –> 25 100 –> 50 environment drift 8 1 2 4 8 16 time constant 4 ∆𝝼 , AU 0 effect of ∆P 25 –> 12.5 2 File: Cyclone-III-Thermal-effect-log 10–15 drift 10–15 drift 10–15 drift 10–15 drift File: Cyclone-III-Thermal-effect C.Calosso, E.Rubiola Figure 15: Thermal eﬀects measured on a Cyclone III FPGA. Each curve represents the thermal transient when the clock frequency is divided by two. in Fig. 15. In the main body, all the curves show an exponential behavior plus a linear drift (cid:0) (cid:1) x(t) = k(cid:48)∆T 1 − e−t/τ˜ + k(cid:48)(cid:48)t , (31) where ∆T = T − T results from setting ν in powers of two, and τ˜ is the J A 0 time constant. For reference, we observed P = 1 W at 400 MHz, which means ∆T ≈ 10 K with Θ ≈ 10 K/W (including the thermal pad on the pcb), and JA neglecting the dissipation at ν = 0. 0 The linear drift (1 fs/s, or 10−15 fractional frequency) does not scale with power. This behavior is typical of the environment temperature, slowly drifting during the measurement (a fraction of a Kelvin over 1 hour). Extrapolating the drift to t = 0, we get the asymptotic eﬀect of the ∆P transient alone. The time constant τ˜ is found as the intercept of the tangent at t = 0 and the linear drift (dashed lines). This graphical process removes the drift. The value τ˜ = 400 s is the same for all the transients. The inset of Fig. 15 shows the delay versus the carrier frequency (dissipated power). As expected, the delay is proportional to T , set through ν . Account- J 0 ing for P and Θ , the thermal coeﬃcient of the delay is 10 ps/K. JA 20 s 004 ##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##

text: Cyclone

font: default

text: III

text: clock

text: buffer

text: x,

text: ps

text: ν0

text: x

text: drift

text: MHz

text: environment

text: drift

text: time

text: constant

text: ∆𝝼

text: AU

text: effect

text: of

text: ∆P

text: File:

text: Cyclone-III-Thermal-effect-log

text: drift

text: drift

text: drift

text: drift

text: File:

text: Cyclone-III-Thermal-effect

text: C.Calosso,

text: E.Rubiola

text: Figure

text: Thermal

text: eﬀects

text: measured

text: on

text: a

text: Cyclone

text: III

text: FPGA.

text: Each

text: curve

text: represents

text: the

text: thermal

text: transient

text: when

text: the

text: clock

text: frequency

text: is

text: divided

text: by

text: two.

text: in

text: Fig.

text: In

text: the

text: main

text: body,

text: all

text: the

text: curves

text: show

text: an

text: exponential

text: behavior

text: plus

text: a

text: linear

text: drift

text: x(t)

text: e−t/τ˜

text: where

text: ∆T

text: T

text: T

text: results

text: from

text: setting

text: ν

text: in

text: powers

text: of

text: two,

text: and

text: τ˜

text: is

text: the

text: J

text: A

text: time

text: constant.

text: For

text: reference,

text: we

text: observed

text: P

text: W

text: at

text: MHz,

text: which

text: means

text: ∆T

text: K

text: with

text: Θ

text: K/W

text: (including

text: the

text: thermal

text: pad

text: on

text: the

text: pcb),

text: and

text: JA

text: neglecting

text: the

text: dissipation

text: at

text: ν

text: The

text: linear

text: drift

text: fs/s,

text: or

text: fractional

text: frequency)

text: does

text: not

text: scale

text: with

text: power.

text: This

text: behavior

text: is

text: typical

text: of

text: the

text: environment

text: temperature,

text: slowly

text: drifting

text: during

text: the

text: measurement

text: (a

text: fraction

text: of

text: a

text: Kelvin

text: over

text: hour).

text: Extrapolating

text: the

text: drift

text: to

text: t

text: we

text: get

text: the

text: asymptotic

text: eﬀect

text: of

text: the

text: ∆P

text: transient

text: alone.

text: The

text: time

text: constant

text: τ˜

text: is

text: found

text: as

text: the

text: intercept

text: of

text: the

text: tangent

text: at

text: t

text: and

text: the

text: linear

text: drift

text: (dashed

text: lines).

text: This

text: graphical

text: process

text: removes

text: the

text: drift.

text: The

text: value

text: τ˜

text: s

text: is

text: the

text: same

text: for

text: all

text: the

text: transients.

text: The

text: inset

text: of

text: Fig.

text: shows

text: the

text: delay

text: versus

text: the

text: carrier

text: frequency

text: (dissipated

text: power).

text: As

text: expected,

text: the

text: delay

text: is

text: proportional

text: to

text: T

text: set

text: through

text: ν

text: Account-

text: J

text: ing

text: for

text: P

text: and

text: Θ

text: the

text: thermal

text: coeﬃcient

text: of

text: the

text: delay

text: is

text: ps/K.

text: JA

text: s

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

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text: A) Clock distribution B) Output buﬀer in in DUT DUT ref ref Figure 5: The digital phase meter is either a Symmetricom (now Microsemi) 5125 or 5120. The two outputs may have diﬀerent frequency. h = V 2 b (ν (cid:28) ν ) (21) −1 0 −1 0 c b k = −1 (ν (cid:29) ν ) . (22) −1 4π2ν2 0 c 0 The white phase noise S (f) = b is described by (10) at low ν , and by ϕ 0 0 (18) at high ν , separated by the cutoﬀ 0 √ B h 0 ν = . (white) . (23) c 2πV J 0 This is shown on Fig. 4 B. At low ν , (10) enables to calculate the noise (cid:10) (cid:11) 0 power n2(t) = h B of the input threshold 0 h B = V 2 b ν (ν (cid:28) ν ) . (24) 0 0 0 0 0 c Assuming that B is equal to 3–4 times the maximum ν , we can infer h √ 0 0 and the noise voltage e = h . Conversely, at high ν we can extract the n 0 0 ﬂuctuation (cid:114) 1 b J = 0 (ν (cid:29) ν ) . (25) 0 c 2π ν 0 This can be compared to the rms jitter, if available in the specs. 4 Selected Noise Measurements We measured the phase noise of several devices routinely used in our labs. This is a necessary step, before considering an unbound search for the best. 10 latigid retem esahp latigid retem esahp ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: A)

text: Clock

text: distribution

text: B)

text: Output

text: buﬀer

text: in

text: in

text: DUT

text: DUT

text: ref

text: ref

text: Figure

text: The

label: caption

text: digital

text: phase

text: meter

text: is

text: either

text: a

text: Symmetricom

text: (now

text: Microsemi)

text: or

text: The

text: two

text: outputs

text: may

text: have

text: diﬀerent

text: frequency.

text: h

text: V

text: b

text: (ν

text: ν

text: −1

text: −1

text: c

text: b

text: k

text: −1

text: (ν

text: ν

text: −1

text: c

text: The

text: white

text: phase

text: noise

text: S

text: (f)

text: b

text: is

text: described

text: by

text: at

text: low

text: ν

text: and

text: by

text: ϕ

text: at

text: high

text: ν

text: separated

text: by

text: the

text: cutoﬀ

text: B

text: h

text: ν

text: (white)

text: c

text: J

text: This

text: is

text: shown

text: on

text: Fig.

text: B.

text: At

text: low

text: ν

text: enables

text: to

text: calculate

text: the

text: noise

text: power

text: h

text: B

text: of

text: the

text: input

text: threshold

text: h

text: B

text: V

text: b

text: ν

text: (ν

text: ν

text: c

text: Assuming

text: that

text: B

text: is

text: equal

text: to

text: times

text: the

text: maximum

text: ν

text: we

text: can

text: infer

text: h

text: and

text: the

text: noise

text: voltage

text: e

text: h

text: Conversely,

text: at

text: high

text: ν

text: we

text: can

text: extract

text: the

text: n

text: ﬂuctuation

text: b

text: J

text: (ν

text: ν

text: c

text: 2π

text: ν

text: This

text: can

text: be

text: compared

text: to

text: the

text: rms

text: jitter,

text: if

text: available

text: in

text: the

text: specs.

text: Selected

text: Noise

text: Measurements

text: We

text: measured

text: the

text: phase

text: noise

text: of

text: several

text: devices

text: routinely

text: used

text: in

text: our

text: labs.

text: This

text: is

text: a

text: necessary

text: step,

text: before

text: considering

text: an

text: unbound

text: search

text: for

text: the

text: best.

text: latigid

text: retem

text: esahp

text: latigid

text: retem

text: esahp

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text: 2 fbc printed on May 29, 2020 As is well known, the long-range rapidity correlations in hadronic colli- sion experiments reveal information on the dynamics and evolution of the system in its earliest partonic phase. Experimentally, the multiplicity corre- lations in early pp and pp¯ collisions [4–9] and nuclear collisions [10, 11] were followed by the relativistic heavy-ion and pp experiments at RHIC [12–14] and the LHC [1, 15–18]. Physical pictures, models, and theoretical methods have been constructed along the quest to understand the data [19–47]. The basic assumptions of the applied superposition framework are fol- lowing [2, 3]: (a) Particle emission occurs independently from longitudinally extended sources. (b) The forward (F) and backward (B) bins are suﬃciently well separated in pseudorapidity, such that the transition from the initial state to the ﬁnal hadron distribution does not cause mixing between particles belonging to the F and B bins. Actually, our approach takes into account three stages typically distin- guished in the evolution of the system: 1) early production of initial particles (forming an entropy density) from sources, 2) hydrodynamic or transport evolution in the intermediate phase, and ﬁnally 3) production of hadrons and their subsequent registration in detectors. Our derivation assumes for simplicity a single type of sources. In Ap- pendix we show how and under what conditions the model may be gener- alized to a case with multiple types of sources 2. Formulas As explained in detail in Refs. [2, 3] (cf. also Appendix in the present work), the above-mentioned stages 1) and 3) involve, from the statistical point of view, folding of statistical distributions, whereas stage 2) results in a linear transformation of the particle (ﬂuid) density. The three stages may be combined to yield a very simple “pocket” formula involving only one free parameter, relating the correlation of the initial sources s and s F B in the F and B bins in spatial rapidity, denoted as ρ(s , s ), to statistical F B quantities accessible experimentally. These quantities are the correlation of the numbers of charged hadrons n and n in the experimental F and B F B bins in pseudorapidity, denoted as ρ(n , n ) (a.k.a. the b coeﬃcient), and F B the scaled variances of multiplicities in the F and B bins, ω(n ) and ω(n ). F B For symmetric collisions and for symmetrically arranged pseudorapidity bins ≡ ω(n ) = ω(n ) ω(n ), and we have (see Appendix ) F B A

text: fbc

text: printed

text: on

text: May

text: As

text: is

text: well

text: known,

text: the

text: long-range

text: rapidity

text: correlations

text: in

text: hadronic

text: colli-

text: sion

text: experiments

text: reveal

text: information

text: on

text: the

text: dynamics

text: and

text: evolution

text: of

text: the

text: system

text: in

text: its

text: earliest

text: partonic

text: phase.

text: Experimentally,

text: the

text: multiplicity

text: corre-

text: lations

text: in

text: early

text: pp

text: and

text: pp¯

text: collisions

text: and

text: nuclear

text: collisions

text: were

text: followed

text: by

text: the

text: relativistic

text: heavy-ion

text: and

text: pp

text: experiments

text: at

text: RHIC

text: and

text: the

text: LHC

text: Physical

text: pictures,

text: models,

text: and

text: theoretical

text: methods

text: have

text: been

text: constructed

text: along

text: the

text: quest

text: to

text: understand

text: the

text: data

text: The

text: basic

text: assumptions

text: of

text: the

text: applied

text: superposition

text: framework

text: are

text: fol-

text: lowing

text: (a)

label: list

text: Particle

text: emission

text: occurs

text: independently

text: from

text: longitudinally

text: extended

text: sources.

text: (b)

text: The

text: forward

text: (F)

text: and

text: backward

text: (B)

text: bins

text: are

text: suﬃciently

text: well

text: separated

text: in

text: pseudorapidity,

text: such

text: that

text: the

text: transition

text: from

text: the

text: initial

text: state

text: to

text: the

text: ﬁnal

text: hadron

text: distribution

text: does

text: not

text: cause

text: mixing

text: between

text: particles

text: belonging

text: to

text: the

text: F

text: and

text: B

text: bins.

text: Actually,

text: our

text: approach

text: takes

text: into

text: account

text: three

text: stages

text: typically

text: distin-

text: guished

text: in

text: the

text: evolution

text: of

text: the

text: system:

text: early

text: production

text: of

text: initial

text: particles

text: (forming

text: an

text: entropy

text: density)

text: from

text: sources,

text: hydrodynamic

text: or

text: transport

text: evolution

text: in

text: the

text: intermediate

text: phase,

text: and

text: ﬁnally

text: production

text: of

text: hadrons

text: and

text: their

text: subsequent

text: registration

text: in

text: detectors.

text: Our

text: derivation

text: assumes

text: for

text: simplicity

text: a

text: single

text: type

text: of

text: sources.

text: In

text: Ap-

text: pendix

text: we

text: show

text: how

text: and

text: under

text: what

text: conditions

text: the

text: model

text: may

text: be

text: gener-

text: alized

text: to

text: a

text: case

text: with

text: multiple

text: types

text: of

text: sources

text: Formulas

text: As

text: explained

text: in

text: detail

text: in

text: Refs.

text: (cf.

text: also

text: Appendix

text: in

text: the

text: present

text: work),

text: the

text: above-mentioned

text: stages

text: and

text: involve,

text: from

text: the

text: statistical

text: point

text: of

text: view,

text: folding

text: of

text: statistical

text: distributions,

text: whereas

text: stage

text: results

text: in

text: a

text: linear

text: transformation

text: of

text: the

text: particle

text: (ﬂuid)

text: density.

text: The

text: three

text: stages

text: may

text: be

text: combined

text: to

text: yield

text: a

text: very

text: simple

text: “pocket”

text: formula

text: involving

text: only

text: one

text: free

text: parameter,

text: relating

text: the

text: correlation

text: of

text: the

text: initial

text: sources

text: s

text: and

text: s

text: F

text: B

text: in

text: the

text: F

text: and

text: B

text: bins

text: in

text: spatial

text: rapidity,

text: denoted

text: as

text: ρ(s

text: s

text: to

text: statistical

text: F

text: B

text: quantities

text: accessible

text: experimentally.

text: These

text: quantities

text: are

text: the

text: correlation

text: of

text: the

text: numbers

text: of

text: charged

text: hadrons

text: n

text: and

text: n

text: in

text: the

text: experimental

text: F

text: and

text: B

text: F

text: B

text: bins

text: in

text: pseudorapidity,

text: denoted

text: as

text: ρ(n

text: n

text: (a.k.a.

text: the

text: b

text: coeﬃcient),

text: and

text: F

text: B

text: the

text: scaled

text: variances

text: of

text: multiplicities

text: in

text: the

text: F

text: and

text: B

text: bins,

text: ω(n

text: and

text: ω(n

text: F

text: B

text: For

text: symmetric

text: collisions

text: and

text: for

text: symmetrically

text: arranged

text: pseudorapidity

text: bins

text: ω(n

text: ω(n

text: ω(n

text: and

text: we

text: have

text: (see

text: Appendix

text: F

text: B

text: A


text: 10 fbc printed on May 29, 2020 ρ(n , n ) F B ρ(uF , uB) = (cid:104) (cid:105) (cid:104) (cid:48) (cid:105) (cid:48) , (C.2) − SA(cid:104) va(cid:105)r(cid:104)(m(cid:105))+(cid:104)SA(cid:48) (cid:105)v(cid:104)ar(cid:48)((cid:105)m ) 1 SA m + SA m ω(nA) where (cid:104) (cid:105) (cid:48) (cid:104) (cid:48)(cid:105) u = S m + S m . (C.3) A A A We note the same structure as in Eq. (1), with δ replaced with the combination (cid:104) (cid:105) (cid:104) (cid:48) (cid:105) (cid:48) S var(m) + S var(m ) δ = A(cid:104) (cid:105)(cid:104) (cid:105) (cid:104) A(cid:48) (cid:105)(cid:104) (cid:48)(cid:105) . (C.4) S m + S m A A This combination is constant in two interesting cases: (cid:104) (cid:48) (cid:105) (cid:104) (cid:105) 1. S = λ S , A A (cid:104) (cid:105) (cid:48) (cid:104) (cid:48)(cid:105) 2. var(m) = κ m , var(m ) = κ m , where constants λ or κ do not depend on centrality or the pseudorapidity (cid:48) (cid:104) (cid:105) (cid:104) (cid:48)(cid:105) separation. In the ﬁrst case δ = (var(m)+λvar(m ))/( m +λ m ) = const., whereas in the second case δ = κ = const. The correlation ρ(u , u ) is a more complicated object which now plays F B the role of ρ(S , S ) from Eq. (1,2). In a more general analysis with sources F B of multiple types we should keep it as is. A simpliﬁcation occurs, however, (cid:48) (cid:39) when in each event S λS , i.e., the relative ﬂuctuations are not too A A(cid:39) (cid:39) (cid:48) (cid:48) large. Then we have ρ(u , u ) ρ(S , S ) ρ(S , S ). F B F B F B A physical realization of scenario 1) is the quark-diquark model of Ref. [57] for the A-A collisions, where we expect that (event-by-event) the numbers of wounded quarks and diquarks are proportional to each other. Scenario (cid:48) 2) occurs where the scaled variances of m and m are equal. This is, e.g., the case of the Poisson distributions, or more general negative binomial distributions with the same parameters controlling the scaled variance. A generalization of the discussion of this Appendix to more than two types of sources is straightforward, with the sums showing up in the formulas extending from 2 to n kinds. In conclusion, the analysis of this paper may be extended to the case where the superposition model involves more types of sources under the con- dition that the combination (C.4) is (approximately) constant. Conversely, the constant value of δ (as to a good approximation occurs in Fig. (1)), does not require the assumption of a single type of sources. ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: fbc

text: printed

text: on

text: May

text: ρ(n

text: n

text: F

text: B

text: ρ(uF

text: uB)

text: SA

text: m

text: SA

text: m

text: ω(nA)

text: where

text: u

text: S

text: m

text: S

text: m

text: A

text: A

text: A

text: We

text: note

text: the

text: same

text: structure

text: as

text: in

text: Eq.

text: with

text: δ

text: replaced

text: with

text: the

text: combination

text: S

text: var(m)

text: S

text: var(m

text: δ

text: S

text: m

text: S

text: m

text: A

text: A

text: This

text: combination

text: is

text: constant

text: in

text: two

text: interesting

text: cases:

text: S

text: λ

text: S

text: A

text: A

text: var(m)

text: κ

text: m

text: var(m

text: κ

text: m

text: where

text: constants

text: λ

text: or

text: κ

text: do

text: not

text: depend

text: on

text: centrality

text: or

text: the

text: pseudorapidity

text: separation.

text: In

text: the

text: ﬁrst

text: case

text: δ

text: (var(m)+λvar(m

text: m

text: +λ

text: m

text: const.,

text: whereas

text: in

text: the

text: second

text: case

text: δ

text: κ

text: const.

text: The

text: correlation

text: ρ(u

text: u

text: is

text: a

text: more

text: complicated

text: object

text: which

text: now

text: plays

text: F

text: B

text: the

text: role

text: of

text: ρ(S

text: S

text: from

text: Eq.

text: In

text: a

text: more

text: general

text: analysis

text: with

text: sources

text: F

text: B

text: of

text: multiple

text: types

text: we

text: should

text: keep

text: it

text: as

text: is.

text: A

text: simpliﬁcation

text: occurs,

text: however,

text: when

text: in

text: each

text: event

text: S

text: λS

text: i.e.,

text: the

text: relative

text: ﬂuctuations

text: are

text: not

text: too

text: A

text: large.

text: Then

text: we

text: have

text: ρ(u

text: u

text: ρ(S

text: S

text: ρ(S

text: S

text: F

text: B

text: F

text: B

text: F

text: B

text: A

text: physical

text: realization

text: of

text: scenario

text: is

text: the

text: quark-diquark

text: model

text: of

text: Ref.

text: for

text: the

text: A-A

text: collisions,

text: where

text: we

text: expect

text: that

text: (event-by-event)

text: the

text: numbers

text: of

text: wounded

text: quarks

text: and

text: diquarks

text: are

text: proportional

text: to

text: each

text: other.

text: Scenario

text: occurs

text: where

text: the

text: scaled

text: variances

text: of

text: m

text: and

text: m

text: are

text: equal.

text: This

text: is,

text: e.g.,

text: the

text: case

text: of

text: the

text: Poisson

text: distributions,

text: or

text: more

text: general

text: negative

text: binomial

text: distributions

text: with

text: the

text: same

text: parameters

text: controlling

text: the

text: scaled

text: variance.

text: A

text: generalization

text: of

text: the

text: discussion

text: of

text: this

text: Appendix

text: to

text: more

text: than

text: two

text: types

text: of

text: sources

text: is

text: straightforward,

text: with

text: the

text: sums

text: showing

text: up

text: in

text: the

text: formulas

text: extending

text: from

text: to

text: n

text: kinds.

text: In

text: conclusion,

text: the

text: analysis

text: of

text: this

text: paper

text: may

text: be

text: extended

text: to

text: the

text: case

text: where

text: the

text: superposition

text: model

text: involves

text: more

text: types

text: of

text: sources

text: under

text: the

text: con-

text: dition

text: that

text: the

text: combination

text: is

text: (approximately)

text: constant.

text: Conversely,

text: the

text: constant

text: value

text: of

text: δ

text: (as

text: to

text: a

text: good

text: approximation

text: occurs

text: in

text: Fig.

text: does

text: not

text: require

text: the

text: assumption

text: of

text: a

text: single

text: type

text: of

text: sources.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 6 A.Sezer, T.Ergin and R.Yamazaki − Figure 3. The gamma-ray TS map of the neighborhood of G306.3 0.9 and SourceA. Left Panel: SourceA is not included in the − background model and G306.3 0.9 is left in the model as a point-like source. On both panels, Suzaku X-ray counts (50, 200, 500) are shown in yellow color, white lines show the CO intensity contours of 12.5 (dashed), 30 (solid), 55 (solid), 78 (solid), 101 (solid) K km − s 1, and the red circle and red cross represent the extension and best-ﬁt location of SourceA, respectively. Fermi-LAT sources are shown − in magenta color. Right Panel: Both G306.3 0.9 and SourceA are included in the background model. The black contours on the left panel and green contours on the right panel are for the gamma-ray TS values (25, 30, 40, 45, 49). − Figure 4. The SED of G306.3 0.9 created assuming the SNR as Figure 5. The monthly gamma-ray variability for SourceA as- − a Disk like extended source with a PL-type spectra. The shaded suming a PL type spectrum in the energy range of 0.2 300 region represents the model ﬂux and its statistical errors obtained GeV. from ﬁtting a PL-type spectrum to the given spectral data. of the clouds that coincide with SourceA are in the veloc- − − − − − − − duced at diﬀerent velocity ranges of [ 70, 50], [ 50, 35], ity range of [ 35, 10] km s 1, where the distance range − − − − ∼ − [ 35, 10] km s 1 from bottom right to left and [ 10,0], calculated is 1 9 kpc. The distance range found here is − [0,15], [15,40] km s 1 from top right to left. The white con- consistent with other measurements (Reynolds et al. 2013; tours represent the TS values of SourceA gamma-ray data at Combi et al. 2016), but it is not much constraining. So, we 25, 30, 40, and 45, and yellow contours are the X-ray counts will use the distance to the SNR as 8 kpc assuming that the at 50, 200 and 500. The color scale for W is set to the SNR lies at a ﬁducial distance of the Galactic center. CO same range for all plots, which is between 0.0 and 76.2 K km The total W value found for the regions overlapping − CO − − s 1, with the W values peaking in the velocity range of with SourceA and G306.3 0.9 is about 44 and 3 K km s 1, − − −CO [ 50, 35] km s 1. Calculating the distance for this velocity respectively. Using the CO-to-H conversion factor of X = ∼ − × − − − 2 range, we obtain a distance range of 3 6 kpc. The distance 1.8 1020 cm 2 K 1 km s 1 (Dame, Hartmann & Thaddeus (cid:13) c 2016 RAS, MNRAS 000, 1–?? ##LTFigure## ##LTFigure## ##LTFigure##

text: A.Sezer,

text: T.Ergin

text: and

text: R.Yamazaki

text: Figure

text: The

text: gamma-ray

text: TS

text: map

text: of

text: the

text: neighborhood

text: of

text: and

text: SourceA.

text: Left

text: Panel:

text: SourceA

text: is

text: not

text: included

text: in

text: the

text: background

text: model

text: and

text: is

text: left

text: in

text: the

text: model

text: as

text: a

text: point-like

text: source.

text: On

text: both

text: panels,

text: Suzaku

text: X-ray

text: counts

text: are

text: shown

text: in

text: yellow

text: color,

text: white

text: lines

text: show

text: the

text: CO

text: intensity

text: contours

text: of

text: (dashed),

text: (solid),

text: (solid),

text: (solid),

text: (solid)

text: K

text: km

text: s

text: and

text: the

text: red

text: circle

text: and

text: red

text: cross

text: represent

text: the

text: extension

text: and

text: best-ﬁt

text: location

text: of

text: SourceA,

text: respectively.

text: Fermi-LAT

text: sources

text: are

text: shown

text: in

text: magenta

text: color.

text: Right

text: Panel:

text: Both

text: and

text: SourceA

text: are

text: included

text: in

text: the

text: background

text: model.

text: The

text: black

text: contours

text: on

text: the

text: left

text: panel

text: and

text: green

text: contours

text: on

text: the

text: right

text: panel

text: are

text: for

text: the

text: gamma-ray

text: TS

text: values

text: Figure

text: The

text: SED

text: of

text: created

text: assuming

text: the

text: SNR

text: as

text: Figure

text: The

text: monthly

text: gamma-ray

text: variability

text: for

text: SourceA

text: as-

text: a

text: Disk

text: like

text: extended

text: source

text: with

text: a

text: PL-type

text: spectra.

text: The

text: shaded

text: suming

text: a

text: PL

text: type

text: spectrum

text: in

text: the

text: energy

text: range

text: of

text: region

text: represents

text: the

text: model

text: ﬂux

text: and

text: its

text: statistical

text: errors

text: obtained

text: GeV.

text: from

text: ﬁtting

text: a

text: PL-type

text: spectrum

text: to

text: the

text: given

text: spectral

text: data.

text: of

text: the

text: clouds

text: that

text: coincide

text: with

text: SourceA

text: are

text: in

text: the

text: veloc-

text: duced

text: at

text: diﬀerent

text: velocity

text: ranges

text: of

text: ity

text: range

text: of

text: km

text: s

text: where

text: the

text: distance

text: range

text: km

text: s

text: from

text: bottom

text: right

text: to

text: left

text: and

text: calculated

text: is

text: kpc.

text: The

text: distance

text: range

text: found

text: here

text: is

text: km

text: s

text: from

text: top

text: right

text: to

text: left.

text: The

text: white

text: con-

text: consistent

text: with

text: other

text: measurements

text: (Reynolds

text: et

text: al.

text: tours

text: represent

text: the

text: TS

text: values

text: of

text: SourceA

text: gamma-ray

text: data

text: at

text: Combi

text: et

text: al.

text: but

text: it

text: is

text: not

text: much

text: constraining.

text: So,

text: we

text: and

text: and

text: yellow

text: contours

text: are

text: the

text: X-ray

text: counts

text: will

text: use

text: the

text: distance

text: to

text: the

text: SNR

text: as

text: kpc

text: assuming

text: that

text: the

text: at

text: and

text: The

text: color

text: scale

text: for

text: W

text: is

text: set

text: to

text: the

text: SNR

text: lies

text: at

text: a

text: ﬁducial

text: distance

text: of

text: the

text: Galactic

text: center.

text: CO

text: same

text: range

text: for

text: all

text: plots,

text: which

text: is

text: between

text: and

text: K

text: km

text: The

text: total

text: W

text: value

text: found

text: for

text: the

text: regions

text: overlapping

text: CO

text: s

text: with

text: the

text: W

text: values

text: peaking

text: in

text: the

text: velocity

text: range

text: of

text: with

text: SourceA

text: and

text: is

text: about

text: and

text: K

text: km

text: s

text: −CO

text: km

text: s

text: Calculating

text: the

text: distance

text: for

text: this

text: velocity

text: respectively.

text: Using

text: the

text: CO-to-H

text: conversion

text: factor

text: of

text: X

text: range,

text: we

text: obtain

text: a

text: distance

text: range

text: of

text: kpc.

text: The

text: distance

text: cm

text: K

text: km

text: s

text: (Dame,

text: Hartmann

text: Thaddeus

text: c

text: RAS,

text: MNRAS

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##


text: XXV European Cosmic Ray Symposium, Turin, Sept. 4-9 2016 1 Electron Acceleration Mechanisms in Thunderstorms S. Celestin LPC2E, University of Orleans, CNRS, Orleans, France Thunderstorms produce strong electric ﬁelds over regions on the order of kilometer. The corre- sponding electric potential diﬀerences are on the order of 100 MV. Secondary cosmic rays reaching these regions may be signiﬁcantly accelerated and even ampliﬁed in relativistic runaway avalanche processes. These phenomena lead to enhancements of the high-energy background radiation ob- served by detectors on the ground and on board aircraft. Moreover, intense submillisecond gamma- ray bursts named terrestrial gamma-ray ﬂashes (TGFs) produced in thunderstorms are detected from low Earth orbit satellites. When passing through the atmosphere, these gamma-rays are rec- ognized to produce secondary relativistic electrons and positrons rapidly trapped in the geomagnetic ﬁeld and injected into the near-Earth space environment. In the present work, we attempt to give an overview of the current state of research on high-energy phenomena associated with thunderstorms. I. INTRODUCTION tracloud lightning that transports negative charges upward (+IC) [10–14]. Recently, studies focusing on observation and theory of radio emissions believed to A. High-Energy Radiation Bursts be associated with the acceleration processes of elec- trons producing TGF have provided a new insight in Terrestrial gamma-ray ﬂashes (TGFs) are bursts this phenomenon (e.g., see [15–18]). of high-energy photons originating from the Earth’s X-ray and gamma-ray bursts have also been re- atmosphere in association with thunderstorm activ- cently observed during natural and rocket-triggered ity. TGFs were serendipitously discovered in the lightning discharges usually from the ground (e.g., 1990s by the instrument BATSE on board the Comp- [19–27]) but also using airborne detectors [28]. The ton Gamma-Ray Observatory (CGRO), which had energy of photons measured in lightning-produced X- been originally launched to perform observations of ray bursts are typically lower than that in TGFs. celestial gamma-ray sources [1]. Later on, the de- B. Enhancement of Background Radiation by tection of these events has been reported using the Thunderstorms Reuven Ramaty High Energy Solar Spectroscopic Imager (RHESSI) satellite [2], the Astrorivelatore In addition to radiation bursts, another type of Gamma a Immagini Leggero (AGILE) satellite [3], high-energy emission has been observed inside thun- and the Fermi Gamma-ray Space Telescope [4]. An il- derstorms using detectors on board balloons [29, 30] lustration of a TGF is given in Figure 1. A few events and airplanes (e.g., [31–33]). These events, so-called closely resembling TGFs have also been detected from gamma-ray glows, correspond to signiﬁcant enhance- the ground [5–8] and from an aircraft [9]. ments of background radiation that last for more than Moreover, measurements have correlated TGFs a few seconds up until a lightning discharge occurs and with initial development stages of normal polarity in- presumably depletes the charges responsible for the large-scale electric ﬁeld present in the thunderstorm, which abruptly terminates the gamma-ray glow event. In fact, gamma-ray glows are often observed over a time corresponding to the duration of the presence of the detector in the active region, which is lim- ited by the speed of the aircraft [33]. Similar emis- sions have been measured from ground-based detec- tors (e.g., [34–39]) and are usually referred to as Thun- derstorm Ground Enhancements. C. Objectives The objectives of the present work are: • To present the recent and rapidly growing cross- FIG. 1: Representation of a TGF (pink points) produced disciplinary ﬁeld of research named High-Energy by a terrestrial thunderstorm and its associated electron Atmospheric Physics [40]. (yellow points) and positron (green points) beams. Credit: • NASA/Goddard Space Flight Center/J. Dwyer, Florida To give insight into the main physical mecha- Inst. of Technology. nisms and concepts believed to be at play in the eConf C16-09-04.3 ##LTFigure## ##LTFigure##

text: XXV

text: European

text: Cosmic

text: Ray

text: Symposium,

text: Turin,

text: Sept.

text: Electron

label: title

text: Acceleration

text: Mechanisms

text: in

text: Thunderstorms

text: S.

label: author

text: Celestin

text: University

text: of

text: Orleans,

text: CNRS,

text: Orleans,

text: France

text: Thunderstorms

label: abstract

text: produce

text: strong

text: electric

text: ﬁelds

text: over

text: regions

text: on

text: the

text: order

text: of

text: kilometer.

text: The

text: corre-

text: sponding

text: electric

text: potential

text: diﬀerences

text: are

text: on

text: the

text: order

text: of

text: MV.

text: Secondary

text: cosmic

text: rays

text: reaching

text: these

text: regions

text: may

text: be

text: signiﬁcantly

text: accelerated

text: and

text: even

text: ampliﬁed

text: in

text: relativistic

text: runaway

text: avalanche

text: processes.

text: These

text: phenomena

text: lead

text: to

text: enhancements

text: of

text: the

text: high-energy

text: background

text: radiation

text: ob-

text: served

text: by

text: detectors

text: on

text: the

text: ground

text: and

text: on

text: board

text: aircraft.

text: Moreover,

text: intense

text: submillisecond

text: gamma-

text: ray

text: bursts

text: named

text: terrestrial

text: gamma-ray

text: ﬂashes

text: (TGFs)

text: produced

text: in

text: thunderstorms

text: are

text: detected

text: from

text: low

text: Earth

text: orbit

text: satellites.

text: When

text: passing

text: through

text: the

text: atmosphere,

text: these

text: gamma-rays

text: are

text: rec-

text: ognized

text: to

text: produce

text: secondary

text: relativistic

text: electrons

text: and

text: positrons

text: rapidly

text: trapped

text: in

text: the

text: geomagnetic

text: ﬁeld

text: and

text: injected

text: into

text: the

text: near-Earth

text: space

text: environment.

text: In

text: the

text: present

text: work,

text: we

text: attempt

text: to

text: give

text: an

text: overview

text: of

text: the

text: current

text: state

text: of

text: research

text: on

text: high-energy

text: phenomena

text: associated

text: with

text: thunderstorms.

text: I.

text: INTRODUCTION

text: tracloud

text: lightning

text: that

text: transports

text: negative

text: charges

text: upward

text: (+IC)

text: Recently,

text: studies

text: focusing

text: on

text: observation

text: and

text: theory

text: of

text: radio

text: emissions

text: believed

text: to

text: A.

text: High-Energy

text: Radiation

text: Bursts

text: be

text: associated

text: with

text: the

text: acceleration

text: processes

text: of

text: elec-

text: trons

text: producing

text: TGF

text: have

text: provided

text: a

text: new

text: insight

text: in

text: Terrestrial

text: gamma-ray

text: ﬂashes

text: (TGFs)

text: are

text: bursts

text: this

text: phenomenon

text: (e.g.,

text: see

text: of

text: high-energy

text: photons

text: originating

text: from

text: the

text: Earth’s

text: X-ray

text: and

text: gamma-ray

text: bursts

text: have

text: also

text: been

text: re-

text: atmosphere

text: in

text: association

text: with

text: thunderstorm

text: activ-

text: cently

text: observed

text: during

text: natural

text: and

text: rocket-triggered

text: ity.

text: TGFs

text: were

text: serendipitously

text: discovered

text: in

text: the

text: lightning

text: discharges

text: usually

text: from

text: the

text: ground

text: (e.g.,

text: by

text: the

text: instrument

text: BATSE

text: on

text: board

text: the

text: Comp-

text: but

text: also

text: using

text: airborne

text: detectors

text: The

text: ton

text: Gamma-Ray

text: Observatory

text: (CGRO),

text: which

text: had

text: energy

text: of

text: photons

text: measured

text: in

text: lightning-produced

text: X-

text: been

text: originally

text: launched

text: to

text: perform

text: observations

text: of

text: ray

text: bursts

text: are

text: typically

text: lower

text: than

text: that

text: in

text: TGFs.

text: celestial

text: gamma-ray

text: sources

text: Later

text: on,

text: the

text: de-

text: B.

text: Enhancement

text: of

text: Background

text: Radiation

text: by

text: tection

text: of

text: these

text: events

text: has

text: been

text: reported

text: using

text: the

text: Thunderstorms

text: Reuven

text: Ramaty

text: High

text: Energy

text: Solar

text: Spectroscopic

text: Imager

text: (RHESSI)

text: satellite

text: the

text: Astrorivelatore

text: In

text: addition

text: to

text: radiation

text: bursts,

text: another

text: type

text: of

text: Gamma

text: a

text: Immagini

text: Leggero

text: (AGILE)

text: satellite

text: high-energy

text: emission

text: has

text: been

text: observed

text: inside

text: thun-

text: and

text: the

text: Fermi

text: Gamma-ray

text: Space

text: Telescope

text: An

text: il-

text: derstorms

text: using

text: detectors

text: on

text: board

text: balloons

text: lustration

text: of

text: a

text: TGF

text: is

text: given

text: in

text: Figure

text: A

text: few

text: events

text: and

text: airplanes

text: (e.g.,

text: These

text: events,

text: so-called

text: closely

text: resembling

text: TGFs

text: have

text: also

text: been

text: detected

text: from

text: gamma-ray

text: glows,

text: correspond

text: to

text: signiﬁcant

text: enhance-

text: the

text: ground

text: and

text: from

text: an

text: aircraft

text: ments

text: of

text: background

text: radiation

text: that

text: last

text: for

text: more

text: than

text: Moreover,

text: measurements

text: have

text: correlated

text: TGFs

text: a

text: few

text: seconds

text: up

text: until

text: a

text: lightning

text: discharge

text: occurs

text: and

text: with

text: initial

text: development

text: stages

text: of

text: normal

text: polarity

text: in-

text: presumably

text: depletes

text: the

text: charges

text: responsible

text: for

text: the

text: large-scale

text: electric

text: ﬁeld

text: present

text: in

text: the

text: thunderstorm,

text: which

text: abruptly

text: terminates

text: the

text: gamma-ray

text: glow

text: event.

text: In

text: fact,

text: gamma-ray

text: glows

text: are

text: often

text: observed

text: over

text: a

text: time

text: corresponding

text: to

text: the

text: duration

text: of

text: the

text: presence

text: of

text: the

text: detector

text: in

text: the

text: active

text: region,

text: which

text: is

text: lim-

text: ited

text: by

text: the

text: speed

text: of

text: the

text: aircraft

text: Similar

text: emis-

text: sions

text: have

text: been

text: measured

text: from

text: ground-based

text: detec-

text: tors

text: (e.g.,

text: and

text: are

text: usually

text: referred

text: to

text: as

text: Thun-

text: derstorm

text: Ground

text: Enhancements.

text: C.

text: Objectives

text: The

text: objectives

text: of

text: the

text: present

text: work

text: are:

text: To

text: present

text: the

text: recent

text: and

text: rapidly

text: growing

text: cross-

text: FIG.

text: Representation

text: of

text: a

text: TGF

text: (pink

text: points)

text: produced

text: disciplinary

text: ﬁeld

text: of

text: research

text: named

text: High-Energy

text: by

text: a

text: terrestrial

text: thunderstorm

text: and

text: its

text: associated

text: electron

text: Atmospheric

text: Physics

text: (yellow

text: points)

text: and

text: positron

text: (green

text: points)

text: beams.

text: Credit:

text: NASA/Goddard

text: Space

text: Flight

text: Center/J.

text: Dwyer,

text: Florida

text: To

text: give

text: insight

text: into

text: the

text: main

text: physical

text: mecha-

text: Inst.

text: of

text: Technology.

text: nisms

text: and

text: concepts

text: believed

text: to

text: be

text: at

text: play

text: in

text: the

text: eConf

text: ##LTFigure##

text: ##LTFigure##


text: Multinomial method for option pricing under Variance Gamma ∗ Nicola Cantarutti , Jo˜ao Guerra CEMAPRE - Centre for Applied Mathematics and Economics ISEG - University of Lisbon May 29, 2020 Abstract This paper presents a multinomial method for option pricing when the underlying asset follows an exponential Variance Gamma process. The continuous time Variance Gamma process is approximated by a discrete time Markov chain with the same ﬁrst four cumulants. This approach is particularly convenient for pricing American and Bermudan options, which can be exercised at any time up to expiration date. Numerical computations of European and American options are presented, and com- pared with results obtained with ﬁnite diﬀerences methods and with the Black Scholes model. Keywords: American option, L´evy processes, Moment Matching, Multi- nomial tree, Variance Gamma. 1 Introduction Since the early nineties, a lot of research has been done on the topic of pure jump L´evy processes to describe the dynamics of the asset returns. The main contributions are due to Madan and Seneta (1990), Eberlein and Keller (1995), Geman et al. (1998), Barndorﬀ-Nielsen (1998). L´evy processes are stochastic processes with independent and stationary increments that have nice analytical properties and reproduce quite well the statistical features of the ﬁnancial data. For example, in Figure 1 we show four histograms of the daily log-returns of four indices: the S&P 500 Stock Index, the KOSPI (Korea Composite Stock Price Index), XAO (All Ordinaries Aus- tralian Index) and TAIEX (Taiwan Capitalization weighted Stock Index). The histograms are plotted together with the ﬁtted Normal and Variance Gamma (VG) densities. It is straightforward to check that the VG density reproduces much better the high peaks near the origin, and the heavy tails of the empirical data. ∗ Corresponding author. Email: nicolacantarutti@gmail.com 1 ##LTLine##

text: Multinomial

text: method

text: for

text: option

text: pricing

text: under

text: Variance

text: Gamma

text: Nicola

text: Cantarutti

text: Jo˜ao

text: Guerra

text: CEMAPRE

text: Centre

text: for

text: Applied

text: Mathematics

text: and

text: Economics

text: ISEG

text: University

text: of

text: Lisbon

text: May

text: Abstract

text: This

text: paper

text: presents

text: a

text: multinomial

text: method

text: for

text: option

text: pricing

text: when

text: the

text: underlying

text: asset

text: follows

text: an

text: exponential

text: Variance

text: Gamma

text: process.

text: The

text: continuous

text: time

text: Variance

text: Gamma

text: process

text: is

text: approximated

text: by

text: a

text: discrete

text: time

text: Markov

text: chain

text: with

text: the

text: same

text: ﬁrst

text: four

text: cumulants.

text: This

text: approach

text: is

text: particularly

text: convenient

text: for

text: pricing

text: American

text: and

text: Bermudan

text: options,

text: which

text: can

text: be

text: exercised

text: at

text: any

text: time

text: up

text: to

text: expiration

text: date.

text: Numerical

text: computations

text: of

text: European

text: and

text: American

text: options

text: are

text: presented,

text: and

text: com-

text: pared

text: with

text: results

text: obtained

text: with

text: ﬁnite

text: diﬀerences

text: methods

text: and

text: with

text: the

text: Black

text: Scholes

text: model.

text: Keywords:

text: American

text: option,

text: L´evy

text: processes,

text: Moment

text: Matching,

text: Multi-

text: nomial

text: tree,

text: Variance

text: Gamma.

text: Introduction

text: Since

text: the

text: early

text: nineties,

text: a

text: lot

text: of

text: research

text: has

text: been

text: done

text: on

text: the

text: topic

text: of

text: pure

text: jump

text: L´evy

text: processes

text: to

text: describe

text: the

text: dynamics

text: of

text: the

text: asset

text: returns.

text: The

text: main

text: contributions

text: are

text: due

text: to

text: Madan

text: and

text: Seneta

text: Eberlein

text: and

text: Keller

text: Geman

text: et

text: al.

text: Barndorﬀ-Nielsen

text: L´evy

text: processes

text: are

text: stochastic

text: processes

text: with

text: independent

text: and

text: stationary

text: increments

text: that

text: have

text: nice

text: analytical

text: properties

text: and

text: reproduce

text: quite

text: well

text: the

text: statistical

text: features

text: of

text: the

text: ﬁnancial

text: data.

text: For

text: example,

text: in

text: Figure

text: we

text: show

text: four

text: histograms

text: of

text: the

text: daily

text: log-returns

text: of

text: four

text: indices:

text: the

text: S&P

text: Stock

text: Index,

text: the

text: KOSPI

text: (Korea

text: Composite

text: Stock

text: Price

text: Index),

text: XAO

text: (All

text: Ordinaries

text: Aus-

text: tralian

text: Index)

text: and

text: TAIEX

text: (Taiwan

text: Capitalization

text: weighted

text: Stock

text: Index).

text: The

text: histograms

text: are

text: plotted

text: together

text: with

text: the

text: ﬁtted

text: Normal

text: and

text: Variance

text: Gamma

text: (VG)

text: densities.

text: It

text: is

text: straightforward

text: to

text: check

text: that

text: the

text: VG

text: density

text: reproduces

text: much

text: better

text: the

text: high

text: peaks

text: near

text: the

text: origin,

text: and

text: the

text: heavy

text: tails

text: of

text: the

text: empirical

text: data.

text: Corresponding

text: author.

text: Email:

text: nicolacantarutti@gmail.com

text: ##LTLine##


text: choice of the parameters. This relation can be further investigated. Another possible topic of further research is the comparison of our results for the Amer- ican options with other numerical methods such as the Least Square Monte Carlo (Longstaﬀ and Schwartz (2001)) and ﬁnite diﬀerences (Almendral and Oosterlee (2007)). Acknowledgements Our sincere thanks are for the Department of Mathematics of ISEG and CEMAPRE, University of Lisbon, http://cemapre.iseg.ulisboa.pt/. This research was supported by the European Union in the FP7-PEOPLE-2012-ITN project STRIKE - Novel Methods in Computational Finance (304617), and by CEMAPRE MULTI/00491, ﬁnanced by FCT/MEC through Portuguese na- tional funds. We wish also to acknowledge all the members of the STRIKE network, http://www.itn-strike.eu/. A Poisson integration A convenient tool to analyse the jumps of a L´evy process is the random mea- sure of jumps. Within this formalism it is possible to describe jump processes with inﬁnite activity, as the VG process. The jump process associated to the ≤ ≤ L´evy process X is deﬁned, for each 0 t T , by: t − ∆X = X X − (57) t t t ∈ B R\{ } where X − = lim ↑ X . Consider the set A ( 0 ) , the random t s t s measure of the jumps of X is deﬁned by: t (cid:88) { ∈ ≤ ≤ } N(t, A)(ω) = # ∆X (ω) A : 0 s t (58) s = 1 (∆X (ω)). A s ≤ s t This measure counts the number of jumps of size in A, up to time t. We say that A ∈ B(R\{0}) is bounded below if 0 (cid:54)∈ A¯ (zero does not belong to the ∞ closure of A). If A is bounded below, then N(t, A) < and is a Poisson process with intensity E ν(A) = [N(1, A)], (59) (see Applebaum (2009) theorem 2.3.4 and 2.3.5). If A is not bounded below, ∞ it is possible to have ν(A) = and N(t, A) is not a Poisson process because of the accumulation of inﬁnite numbers of small jumps. The process N(t, A) is called Poisson random measure. The L´evy measure corresponds to the intensity of the Poisson measure. The Compensated Poisson random measure is deﬁned by − N˜(t, A) = N(t, A) tν(A), (60) 19

text: choice

text: of

text: the

text: parameters.

text: This

text: relation

text: can

text: be

text: further

text: investigated.

text: Another

text: possible

text: topic

text: of

text: further

text: research

text: is

text: the

text: comparison

text: of

text: our

text: results

text: for

text: the

text: Amer-

text: ican

text: options

text: with

text: other

text: numerical

text: methods

text: such

text: as

text: the

text: Least

text: Square

text: Monte

text: Carlo

text: (Longstaﬀ

text: and

text: Schwartz

text: and

text: ﬁnite

text: diﬀerences

text: (Almendral

text: and

text: Oosterlee

text: Acknowledgements

label: section

text: Our

text: sincere

text: thanks

text: are

text: for

text: the

text: Department

text: of

text: Mathematics

text: of

text: ISEG

text: and

text: CEMAPRE,

text: University

text: of

text: Lisbon,

text: This

text: research

text: was

text: supported

text: by

text: the

text: European

text: Union

text: in

text: the

text: project

text: STRIKE

text: Novel

text: Methods

text: in

text: Computational

text: Finance

text: and

text: by

text: CEMAPRE

text: ﬁnanced

text: by

text: FCT/MEC

text: through

text: Portuguese

text: na-

text: tional

text: funds.

text: We

text: wish

text: also

text: to

text: acknowledge

text: all

text: the

text: members

text: of

text: the

text: STRIKE

text: network,

text: A

text: Poisson

text: integration

text: A

text: convenient

text: tool

text: to

text: analyse

text: the

text: jumps

text: of

text: a

text: L´evy

text: process

text: is

text: the

text: random

text: mea-

text: sure

text: of

text: jumps.

text: Within

text: this

text: formalism

text: it

text: is

text: possible

text: to

text: describe

text: jump

text: processes

text: with

text: inﬁnite

text: activity,

text: as

text: the

text: VG

text: process.

text: The

text: jump

text: process

text: associated

text: to

text: the

text: L´evy

text: process

text: X

text: is

text: deﬁned,

text: for

text: each

text: t

text: T

text: by:

text: t

text: ∆X

text: X

text: X

text: t

text: t

text: t

text: B

text: R\{

text: where

text: X

text: lim

text: X

text: Consider

text: the

text: set

text: A

text: the

text: random

text: t

text: s

text: t

text: s

text: measure

text: of

text: the

text: jumps

text: of

text: X

text: is

text: deﬁned

text: by:

text: t

text: N(t,

text: A)(ω)

text: ∆X

text: (ω)

text: A

text: s

text: t

text: s

text: (∆X

text: (ω)).

text: A

text: s

text: s

text: t

text: This

text: measure

text: counts

text: the

text: number

text: of

text: jumps

text: of

text: size

text: in

text: A,

text: up

text: to

text: time

text: t.

text: We

text: say

text: that

text: A

text: is

text: bounded

text: below

text: if

text: A¯

text: (zero

text: does

text: not

text: belong

text: to

text: the

text: closure

text: of

text: A).

text: If

text: A

text: is

text: bounded

text: below,

text: then

text: N(t,

text: A)

text: and

text: is

text: a

text: Poisson

text: process

text: with

text: intensity

text: E

text: ν(A)

text: A)],

text: (see

text: Applebaum

text: theorem

text: and

text: If

text: A

text: is

text: not

text: bounded

text: below,

text: it

text: is

text: possible

text: to

text: have

text: ν(A)

text: and

text: N(t,

text: A)

text: is

text: not

text: a

text: Poisson

text: process

text: because

text: of

text: the

text: accumulation

text: of

text: inﬁnite

text: numbers

text: of

text: small

text: jumps.

text: The

text: process

text: N(t,

text: A)

text: is

text: called

text: Poisson

text: random

text: measure.

text: The

text: L´evy

text: measure

text: corresponds

text: to

text: the

text: intensity

text: of

text: the

text: Poisson

text: measure.

text: The

text: Compensated

text: Poisson

text: random

text: measure

text: is

text: deﬁned

text: by

text: N˜(t,

text: A)

text: N(t,

text: A)

text: tν(A),


text: Black, F. and Scholes, M. (1973). The pricing of options and corporate liabilities. The Journal of Political Economy, 81(3):637–654. Carr, P. and Madan, D. (1998). Option valuation using the Fast Fourier trans- form. Journal of Computational Finance, 2:61–73. Cont, R. and Tankov, P. (2003). Financial Modelling with Jump Processes. Chapman and Hall/CRC; 1 edition. Cont, R. and Voltchkova, E. (2005). A ﬁnite diﬀerence scheme for option pricing in jump diﬀusion and exponential L´evy models. SIAM Journal of numerical analysis, 43(4):1596–1626. Cox, J., Ross, S., and Rubinstein, M. (1979). Option pricing: A simpliﬁed approach. Journal of ﬁnancial economics, 7:229–263. Eberlein, E. and Keller, U. (1995). Hyperbolic distributions in ﬁnance. Bernoulli, 1(3):281–299. Fu, M. C. (2000). Variance-Gamma and Monte Carlo. Advances in Mathematical Finance., pages 21–34. Geman, H., Madan, D., and Yor, M. (1998). Asset prices are Brownian motion: only in business time. Quantitative Analysis in Financial Markets: Collected Papers of the New York University Mathematical Finance Seminar, 2(1):103– 146. Hirsa, A. and Madan, D. (2001). Pricing American options under Variance Gamma. Journal of Computational Finance, 7. Longstaﬀ, F. A. and Schwartz, E. S. (2001). Valuing American options by simulation: A simple Least-Squares approach. Review of Financial Studies, pages 113–147. Madan, D., Carr, P., and Chang, E. (1998). The Variance Gamma process and option pricing. European Finance Review, 2:79–105. Madan, D. and Milne, F. (1991). Option pricing with V.G. martingale compo- nents. Mathematical Finance, 1:39–55. Madan, D. and Seneta, E. (1990). The Variance Gamma (V.G.) model for share market returns. The journal of Business, 63(4):511–524. Sato, K. I. (1999). L´evy processes and inﬁnitely divisible distributions. Cam- bridge University Press. Seneta, E. (2004). Fitting the Variance-Gamma model to ﬁnancial data. Journal of Applied Probability, 41:177–187. Ssebugenyi, C.S. Mwaniki, I. and Konlack, V. (2013). On the minimal entropy martingale measure and multinomial lattices with cumulants. Applied Math- ematical Finance, 20(4):359–379. Yamada, Y. and Primbs, J. (2001). Construction of multinomial lattice ran- dom walks for optimal hedges. Proc. of the 2001 international conference of Computational Science, pages 579–588. 22

text: Black,

label: reference

text: F.

text: and

text: Scholes,

text: M.

text: The

text: pricing

text: of

text: options

text: and

text: corporate

text: liabilities.

text: The

text: Journal

text: of

text: Political

text: Economy,

text: Carr,

text: P.

text: and

text: Madan,

text: D.

text: Option

text: valuation

text: using

text: the

text: Fast

text: Fourier

text: trans-

text: form.

text: Journal

text: of

text: Computational

text: Finance,

text: Cont,

text: R.

text: and

text: Tankov,

text: P.

text: Financial

text: Modelling

text: with

text: Jump

text: Processes.

text: Chapman

text: and

text: Hall/CRC;

text: edition.

text: Cont,

text: R.

text: and

text: Voltchkova,

text: E.

text: A

text: ﬁnite

text: diﬀerence

text: scheme

text: for

text: option

text: pricing

text: in

text: jump

text: diﬀusion

text: and

text: exponential

text: L´evy

text: models.

text: SIAM

text: Journal

text: of

text: numerical

text: analysis,

text: Cox,

text: J.,

text: Ross,

text: S.,

text: and

text: Rubinstein,

text: M.

text: Option

text: pricing:

text: A

text: simpliﬁed

text: approach.

text: Journal

text: of

text: ﬁnancial

text: economics,

text: Eberlein,

text: E.

text: and

text: Keller,

text: U.

text: Hyperbolic

text: distributions

text: in

text: ﬁnance.

text: Bernoulli,

text: Fu,

text: M.

text: C.

text: Variance-Gamma

text: and

text: Monte

text: Carlo.

text: Advances

text: in

text: Mathematical

text: Finance.,

text: pages

text: Geman,

text: H.,

text: Madan,

text: D.,

text: and

text: Yor,

text: M.

text: Asset

text: prices

text: are

text: Brownian

text: motion:

text: only

text: in

text: business

text: time.

text: Quantitative

text: Analysis

text: in

text: Financial

text: Markets:

text: Collected

text: Papers

text: of

text: the

text: New

text: York

text: University

text: Mathematical

text: Finance

text: Seminar,

text: Hirsa,

text: A.

text: and

text: Madan,

text: D.

text: Pricing

text: American

text: options

text: under

text: Variance

text: Gamma.

text: Journal

text: of

text: Computational

text: Finance,

text: Longstaﬀ,

text: F.

text: A.

text: and

text: Schwartz,

text: E.

text: S.

text: Valuing

text: American

text: options

text: by

text: simulation:

text: A

text: simple

text: Least-Squares

text: approach.

text: Review

text: of

text: Financial

text: Studies,

text: pages

text: Madan,

text: D.,

text: Carr,

text: P.,

text: and

text: Chang,

text: E.

text: The

text: Variance

text: Gamma

text: process

text: and

text: option

text: pricing.

text: European

text: Finance

text: Review,

text: Madan,

text: D.

text: and

text: Milne,

text: F.

text: Option

text: pricing

text: with

text: V.G.

text: martingale

text: compo-

text: nents.

text: Mathematical

text: Finance,

text: Madan,

text: D.

text: and

text: Seneta,

text: E.

text: The

text: Variance

text: Gamma

text: (V.G.)

text: model

text: for

text: share

text: market

text: returns.

text: The

text: journal

text: of

text: Business,

text: Sato,

text: K.

text: I.

text: L´evy

text: processes

text: and

text: inﬁnitely

text: divisible

text: distributions.

text: Cam-

text: bridge

text: University

text: Press.

text: Seneta,

text: E.

text: Fitting

text: the

text: Variance-Gamma

text: model

text: to

text: ﬁnancial

text: data.

text: Journal

text: of

text: Applied

text: Probability,

text: Ssebugenyi,

text: C.S.

text: Mwaniki,

text: I.

text: and

text: Konlack,

text: V.

text: On

text: the

text: minimal

text: entropy

text: martingale

text: measure

text: and

text: multinomial

text: lattices

text: with

text: cumulants.

text: Applied

text: Math-

text: ematical

text: Finance,

text: Yamada,

text: Y.

text: and

text: Primbs,

text: J.

text: Construction

text: of

text: multinomial

text: lattice

text: ran-

text: dom

text: walks

text: for

text: optimal

text: hedges.

text: Proc.

text: of

text: the

text: international

text: conference

text: of

text: Computational

text: Science,

text: pages


text: By the martingale pricing theory, the discounted price of the option is a martin- gale. From t(cid:20)his it is possible(cid:21)to derive the partial integro-diﬀerential equation (cid:0) (cid:1) (PIDE) for the price of the option EQ − ∂V (t, y) L − d e rtV (t, Y ) = + YtV (t, y) rV (t, y) = 0, (23) t ∂t L where YtV (t, Yt) is the inﬁnitesimal generator of the log-price process (22). The resulting PIDE is (cid:90) (cid:2) (cid:3) ∂V (t, y) ∂V (t, y) − + (r + ω) + V (t, y + x) V (t, y) ν(dx) = rV (t, y). (24) ∂t ∂y R 3 The multinomial method In this section we introduce the multinomial method proposed in Yamada and Primbs (2004). The stock price is considered as a Markov chain with L ∈ possible future states at each time. In this setting, the time t [t , T] is − 0 discretized as t = t + n∆t for n = 0, ..., N and ∆t = (T t )/N. We denote n 0 0 the stock price at time t as S(t ) = S . n n n Consider the up/down factors u > d > 0 and write the discrete evolution of the stock price S as: n − − S = uL ldl 1S l = 1, ..., L (cid:80) (25) n+1 n where each future state has transition probability p , satisfying L p = 1. ∈ l− l=1 l The value of the stock at time t can assume j [1, n(L 1)+1] possible values: n − − − S(j) = un(L l)+1 jdj 1S . (26) n 0 The multinomial tree is recombining if for a constant c > 1, u/d = c. Regard- ing our work, we only consider ﬁve branches, L = 5. As explained in Yamada and Primbs (2004), this number of branches is enough to model the features of a stochastic process up to its fourth moment. 3.1 Moment matching To determine the parameters of the Markov chain we ask for the local mo- ments to be equal to that of the continuous process. First, rewrite the continuous process (22) as the sum of a drift term an(cid:90)d a martingale term: − (cid:90) Y Y = (r + ω)∆t + xN(∆t, dx) (27) t+∆t t R\{ } 0 (cid:82) = (r + ω + θ)∆t + xN˜(∆t, dx) R\{ } 0 where θ = R xν(dx) is the mean of the Poisson process (for ∆t = 1), and the compensated Poisson integral term is a martingale (see Appendix ). 8 ##LTLine## ##LTLine## ##LTLine##

text: By

text: the

text: martingale

text: pricing

text: theory,

text: the

text: discounted

text: price

text: of

text: the

text: option

text: is

text: a

text: martin-

text: gale.

text: From

text: it

text: is

text: derive

text: the

text: partial

text: integro-diﬀerential

text: equation

text: (PIDE)

text: for

text: the

text: price

text: of

text: the

text: option

text: EQ

text: ∂V

text: (t,

text: y)

text: L

text: d

text: e

text: rtV

text: (t,

text: Y

text: YtV

text: (t,

text: y)

text: rV

text: (t,

text: y)

text: t

text: ∂t

text: L

text: where

text: YtV

text: (t,

text: Yt)

text: is

text: the

text: inﬁnitesimal

text: generator

text: of

text: the

text: log-price

text: process

text: The

text: resulting

text: PIDE

text: is

text: ∂V

text: (t,

text: y)

text: ∂V

text: (t,

text: y)

text: (r

text: ω)

text: V

text: (t,

text: y

text: x)

text: V

text: (t,

text: y)

text: ν(dx)

text: rV

text: (t,

text: y).

text: ∂t

text: ∂y

text: R

text: The

text: multinomial

text: method

text: In

text: this

text: section

text: we

text: introduce

text: the

text: multinomial

text: method

text: proposed

text: in

text: Yamada

text: and

text: Primbs

text: The

text: stock

text: price

text: is

text: considered

text: as

text: a

text: Markov

text: chain

text: with

text: L

text: possible

text: future

text: states

text: at

text: each

text: time.

text: In

text: this

text: setting,

text: the

text: time

text: t

text: [t

text: T]

text: is

text: discretized

text: as

text: t

text: t

text: n∆t

text: for

text: n

text: N

text: and

text: ∆t

text: (T

text: t

text: )/N.

text: We

text: denote

text: n

text: the

text: stock

text: price

text: at

text: time

text: t

text: as

text: S(t

text: S

text: n

text: n

text: n

text: Consider

text: the

text: up/down

text: factors

text: u

text: d

text: and

text: write

text: the

text: discrete

text: evolution

text: of

text: the

text: stock

text: price

text: S

text: as:

text: n

text: S

text: uL

text: ldl

text: l

text: L

text: n

text: where

text: each

text: future

text: state

text: has

text: transition

text: probability

text: p

text: satisfying

text: L

text: p

text: l−

text: l

text: The

text: value

text: of

text: the

text: stock

text: at

text: time

text: t

text: can

text: assume

text: j

text: n(L

text: possible

text: values:

text: n

text: S(j)

text: un(L

text: jdj

text: n

text: The

text: multinomial

text: tree

text: is

text: recombining

text: if

text: for

text: a

text: constant

text: c

text: u/d

text: c.

text: Regard-

text: ing

text: our

text: work,

text: we

text: only

text: consider

text: ﬁve

text: branches,

text: L

text: As

text: explained

text: in

text: Yamada

text: and

text: Primbs

text: this

text: number

text: of

text: branches

text: is

text: enough

text: to

text: model

text: the

text: features

text: of

text: a

text: stochastic

text: process

text: up

text: to

text: its

text: fourth

text: moment.

text: Moment

text: matching

text: To

text: determine

text: the

text: parameters

text: of

text: the

text: Markov

text: chain

text: we

text: ask

text: for

text: the

text: local

text: mo-

text: ments

text: to

text: be

text: equal

text: to

text: that

text: of

text: the

text: continuous

text: process.

text: First,

text: rewrite

text: the

text: continuous

text: process

text: as

text: the

text: sum

text: of

text: a

text: drift

text: term

text: a

text: martingale

text: term:

text: Y

text: Y

text: (r

text: ω)∆t

text: xN(∆t,

text: dx)

text: t+∆t

text: t

text: R\{

text: (r

text: ω

text: θ)∆t

text: xN˜(∆t,

text: dx)

text: R\{

text: where

text: θ

text: R

text: xν(dx)

text: is

text: the

text: mean

text: of

text: the

text: Poisson

text: process

text: (for

text: ∆t

text: and

text: the

text: compensated

text: Poisson

text: integral

text: term

text: is

text: a

text: martingale

text: (see

text: Appendix

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text: ##LTLine##

text: ##LTLine##


text: The drift parameter (for ∆t = 1) can be easily computed as b = r + ω + θ. The only missing parameter to determine is α(∆t). This is a function of the time increment ∆t and can be determined using the higher order in the moment matching condition together with the condition of positive probabilities. √ Recall that in the well known binomial model for a diﬀusion process, it takes the value α(∆t) = σ˜ ∆t, and represents the volatility of the increments √ in ∆t, see Cox et al. (1979). In the trinomial model, it takes the well known value α(∆t) = 3σ˜ ∆t, see for instance Y(cid:114)amada and Primbs (2001). For the 4 multinomial method a good representation for the parameter is √ 3 + κ¯ α(∆t) = c , (33) 2 12 where κ¯ = c /c2 is the excess of kurtosis6. We refer to the paper of Yamada 4 2 and Primbs (2004) for the derivation. This choice guarantees that the probabil- ities p for i = 1...5 are always positive and sum to one. We can use the formula i (33), together with (32), to ob(cid:20)tain the simpler form: √ √ − 3 + κ¯ + s 9 + 3κ¯ 3 + κ¯ (cid:21)s 9 + 3κ¯ [p , p , p , p , p ] = , , (34) 1 2 3 4 5 √ 4(3 + κ¯)2 √ 2(3 + κ¯)2 − 3 + 2κ¯ 3 + κ¯ + s 9 + 3κ¯ 3 + κ¯ s 9 + 3κ¯ (cid:112), , , 2(3 + κ¯) 2(3 + κ¯)2 4(3 + κ¯)2 where s = c / c3 is the skewness. 3 2 Remark 1. The standard deviation of a L´evy process with ﬁnite moments fol- lows the square root rule. This means that the term α(∆t) has to be proportional to the square root of ∆t. In the binomial and trinomial models, the proportion- ality constant is explicit, while for the pentanomial method it is implicit in the formula (33). Expanding the formula using the expression (14) for the cumu- √ lants, it is possible to check that the square root rule is satisﬁed at ﬁrst order in ∆t. 3.2 Convergence In this section we show that the multinomial method converges to a com- pound Poisson process when the time step goes to zero. We call a generic jump process (6), with the same ﬁrst four cumulants (14) of the original VG process (22), the approximated VG process XA. The cumulant generating function of the increment ∆XA has the following series representa- tion (see Appendix ()): − c u2 − ic u3 c u4 O H (u) = ic u 2 3 + 4 + (u5). (35) ∆XA 1 2 3! 4! We can check that this expression holds for the VG increments as well, simply by using a Taylor expansion up to the fourth order on the VG L´evy exponent (13), and adding the addidional drift term c = b∆t = (r + ω + θ)∆t. 1 6We use the bar over κ, to distinguish the kurtosis from the variance of the gamma process κ. 10 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: The

text: drift

text: parameter

text: (for

text: ∆t

text: can

text: be

text: easily

text: computed

text: as

text: b

text: r

text: ω

text: θ.

text: The

text: only

text: missing

text: parameter

text: to

text: determine

text: is

text: α(∆t).

text: This

text: is

text: a

text: function

text: of

text: the

text: time

text: increment

text: ∆t

text: and

text: can

text: be

text: determined

text: using

text: the

text: higher

text: order

text: in

text: the

text: moment

text: matching

text: condition

text: together

text: with

text: the

text: condition

text: of

text: positive

text: probabilities.

text: Recall

text: that

text: in

text: the

text: well

text: known

text: binomial

text: model

text: for

text: a

text: diﬀusion

text: process,

text: it

text: takes

text: the

text: value

text: α(∆t)

text: σ˜

text: ∆t,

text: and

text: represents

text: the

text: volatility

text: of

text: the

text: increments

text: in

text: ∆t,

text: see

text: Cox

text: et

text: al.

text: In

text: the

text: trinomial

text: model,

text: it

text: takes

text: the

text: well

text: known

text: value

text: α(∆t)

text: 3σ˜

text: ∆t,

text: see

text: for

text: instance

text: and

text: Primbs

text: For

text: the

text: multinomial

text: method

text: a

text: good

text: representation

text: for

text: the

text: parameter

text: is

text: κ¯

text: α(∆t)

text: c

text: where

text: κ¯

text: c

text: is

text: the

text: excess

text: of

text: We

text: refer

text: to

text: the

text: paper

text: of

text: Yamada

text: and

text: Primbs

text: for

text: the

text: derivation.

text: This

text: choice

text: guarantees

text: that

text: the

text: probabil-

text: ities

text: p

text: for

text: i

text: are

text: always

text: positive

text: and

text: sum

text: to

text: one.

text: We

text: can

text: use

text: the

text: formula

text: i

text: together

text: with

text: to

text: the

text: simpler

text: form:

text: κ¯

text: s

text: 3κ¯

text: κ¯

text: 3κ¯

text: [p

text: p

text: p

text: p

text: p

text: κ¯)2

text: κ¯)2

text: 2κ¯

text: κ¯

text: s

text: 3κ¯

text: κ¯

text: s

text: 3κ¯

text: κ¯)

text: κ¯)2

text: κ¯)2

text: where

text: s

text: c

text: is

text: the

text: skewness.

text: Remark

text: The

text: standard

text: deviation

text: of

text: a

text: L´evy

text: process

text: with

text: ﬁnite

text: moments

text: fol-

text: lows

text: the

text: square

text: root

text: rule.

text: This

text: means

text: that

text: the

text: term

text: α(∆t)

text: has

text: to

text: be

text: proportional

text: to

text: the

text: square

text: root

text: of

text: ∆t.

text: In

text: the

text: binomial

text: and

text: trinomial

text: models,

text: the

text: proportion-

text: ality

text: constant

text: is

text: explicit,

text: while

text: for

text: the

text: pentanomial

text: method

text: it

text: is

text: implicit

text: in

text: the

text: formula

text: Expanding

text: the

text: formula

text: using

text: the

text: expression

text: for

text: the

text: cumu-

text: lants,

text: it

text: is

text: possible

text: to

text: check

text: that

text: the

text: square

text: root

text: rule

text: is

text: satisﬁed

text: at

text: ﬁrst

text: order

text: in

text: ∆t.

text: Convergence

text: In

text: this

text: section

text: we

text: show

text: that

text: the

text: multinomial

text: method

text: converges

text: to

text: a

text: com-

text: pound

text: Poisson

text: process

text: when

text: the

text: time

text: step

text: goes

text: to

text: zero.

text: We

text: call

text: a

text: generic

text: jump

text: process

text: with

text: the

text: same

text: ﬁrst

text: four

text: cumulants

text: of

text: the

text: original

text: VG

text: process

text: the

text: approximated

text: VG

text: process

text: XA.

text: The

text: cumulant

text: generating

text: function

text: of

text: the

text: increment

text: ∆XA

text: has

text: the

text: following

text: series

text: representa-

text: tion

text: (see

text: Appendix

text: c

text: ic

text: c

text: O

text: H

text: (u)

text: ic

text: u

text: ∆XA

text: We

text: can

text: check

text: that

text: this

text: expression

text: holds

text: for

text: the

text: VG

text: increments

text: as

text: well,

text: simply

text: by

text: using

text: a

text: Taylor

text: expansion

text: up

text: to

text: the

text: fourth

text: order

text: on

text: the

text: VG

text: L´evy

text: exponent

text: and

text: adding

text: the

text: addidional

text: drift

text: term

text: c

text: b∆t

text: (r

text: ω

text: θ)∆t.

text: use

text: the

text: bar

text: over

text: κ,

text: to

text: distinguish

text: the

text: kurtosis

text: from

text: the

text: variance

text: of

text: the

text: gamma

text: process

text: κ.

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text: The convolution algebra of an absolutely locally compact topos Simon Henry May 29, 2020 Abstract We introduce a class of toposes called “absolutely locally compact” toposes and of “admissible” sheaf of rings over such toposes. To any such T C T ringed topos ( , A) we attach an involutive convolution algebra ( , A) c which is well deﬁned up to Morita equivalence and characterized by the C T fact that the category of non-degenerate modules over ( , A) is equiv- T c alent to the category of sheaf of A-module over . In the case where A is the sheaf of real or complex Dedekind numbers, we construct several norms on this involutive algebra that allows to complete it in various Ba- ∗ T ∗ T ∗ T nach and C -algebras: L1( , A), C ( , A) and C ( , A). We also red max give some examples where this construction corresponds to well known constructions of involutive algebras, like groupoids convolution algebra and Leavitt path algebras. Contents 1 Introduction 1 2 Locally compact locales and compactly supported functions 3 3 Absolutely separated and absolutely locally compact toposes 8 4 Admissible sheaf of rings and the algebra of compactly sup- ported functions 17 5 Examples 31 1 Introduction Both topos theory and non-commutative geometry are theories that are con- cerned with the study of certain “generalized topological spaces” that are too singular to be treated by ordinary topology, like the space of leaves of a foliation ∗ or the space of orbits of a group action or of a groupoid. Indeed C -algebras ∗ Keywords. topos, convolution algebra, C -algebra 2010 Mathematics Subject Classiﬁcation. 18B25, 03G30, 46L05 email: simon.henry@college-de-france.fr 1 ##LTLine##

text: The

text: convolution

text: algebra

text: of

text: an

text: absolutely

text: locally

text: compact

text: topos

text: Simon

text: Henry

text: May

text: Abstract

text: We

text: introduce

text: a

text: class

text: of

text: toposes

text: called

text: “absolutely

text: locally

text: compact”

text: toposes

text: and

text: of

text: “admissible”

text: sheaf

text: of

text: rings

text: over

text: such

text: toposes.

text: To

text: any

text: such

text: T

text: C

text: T

text: ringed

text: topos

text: A)

text: we

text: attach

text: an

text: involutive

text: convolution

text: algebra

text: A)

text: c

text: which

text: is

text: well

text: deﬁned

text: up

text: to

text: Morita

text: equivalence

text: and

text: characterized

text: by

text: the

text: C

text: T

text: fact

text: that

text: the

text: category

text: of

text: non-degenerate

text: modules

text: over

text: A)

text: is

text: equiv-

text: T

text: c

text: alent

text: to

text: the

text: category

text: of

text: sheaf

text: of

text: A-module

text: over

text: In

text: the

text: case

text: where

text: A

text: is

text: the

text: sheaf

text: of

text: real

text: or

text: complex

text: Dedekind

text: numbers,

text: we

text: construct

text: several

text: norms

text: on

text: this

text: involutive

text: algebra

text: that

text: allows

text: to

text: complete

text: it

text: in

text: various

text: Ba-

text: T

text: T

text: T

text: nach

text: and

text: C

text: -algebras:

text: A),

text: C

text: A)

text: and

text: C

text: A).

text: We

text: also

text: red

text: max

text: give

text: some

text: examples

text: where

text: this

text: construction

text: corresponds

text: to

text: well

text: known

text: constructions

text: of

text: involutive

text: algebras,

text: like

text: groupoids

text: convolution

text: algebra

text: and

text: Leavitt

text: path

text: algebras.

text: Contents

text: Introduction

text: Locally

text: compact

text: locales

text: and

text: compactly

text: supported

text: functions

text: Absolutely

text: separated

text: and

text: absolutely

text: locally

text: compact

text: toposes

text: Admissible

text: sheaf

text: of

text: rings

text: and

text: the

text: algebra

text: of

text: compactly

text: sup-

text: ported

text: functions

text: Examples

text: Introduction

text: Both

text: topos

text: theory

text: and

text: non-commutative

text: geometry

text: are

text: theories

text: that

text: are

text: con-

text: cerned

text: with

text: the

text: study

text: of

text: certain

text: “generalized

text: topological

text: spaces”

text: that

text: are

text: too

text: singular

text: to

text: be

text: treated

text: by

text: ordinary

text: topology,

text: like

text: the

text: space

text: of

text: leaves

text: of

text: a

text: foliation

text: or

text: the

text: space

text: of

text: orbits

text: of

text: a

text: group

text: action

text: or

text: of

text: a

text: groupoid.

text: Indeed

text: C

text: -algebras

text: Keywords.

text: topos,

text: convolution

text: algebra,

text: C

text: -algebra

text: Mathematics

text: Subject

text: Classiﬁcation.

text: email:

text: simon.henry@college-de-france.fr

text: ##LTLine##