text: A SURVEY ON REVERSE CARLESON MEASURES 27 Furthermore, the norms on these spaces are the same. In fact, these are the only harmonically weighted Dirichlet spaces which are equal to an H (b) space with equal norm [12]. In [15] it was shown that if (cid:88) n ∈ T (8.2) µ = c δ , c > 0, ζ j ζj j j j=1 T is a ﬁnite linear combination of point masses on and a is the unique polynomial with a(0) > 0 and with simple zeros at ζ (and no other zeros) and b is the Pythagorean mate for a (which j must also be a polynomial), then H (b) = D(µ) with equivalent norms. In this case we can use Theorem 7.7 to obtain a characterization of the Carleson measures for D(µ): Theorem 8.3. For µ as in (8.2) and ν ∈ M (D−), the following assertions are equivalent: (cid:81) + (i) ν is a Carleson measure for D(µ); | − | (ii) n z ζ 2 dν is a Carleson measure for H2. i=1 i This result appeared in [10] (see also [11]). In fact, Theorem 6.1 from [10] shows that the above conditions are equivalent to (cid:107)kD(µ)(cid:107) (cid:46) (cid:107)kD(µ)(cid:107) ∀λ ∈ D. λ ν λ D(µ) In other words, at least when µ is a linear combination of point masses, the reproducing kernel thesis characterizes the Carleson measures for D(µ). The discussion of reverse Carleson measures for D(µ) is dramatically simpler since they do not ∈ D− (cid:107) (cid:107) (cid:46) (cid:107) (cid:107) ∈ exist! Indeed, suppose that ν M ( ) and f f for all f D(µ). In particular, this (cid:62)+ (cid:107) (cid:107) µ(cid:46) ν(cid:107) (cid:107) T is true for the monomials zn, n 0. But zn 1 and zn 2 = 1 + nµ( ), which gives a ∞ ν µ contradiction when n tends to . We point out some related results from [11] which discuss a type of reverse Carleson measure for D(µ) spaces except that the deﬁnitions of “reverse Carleson measures” and “sets of domination” (dominating sets) are quite different, and not equivalent, to ours. 9. APPENDIX ∞ We reproduce here an adaption to 1 < p < + of Baranov’s proof as presented in [8, Section 7] and which is based on the Bernstein-type inequalities in model spaces he obtained in [3, 5]. It T \ ∈ uses a Whitney type decomposition of σ(Θ). Let ε > 0, let δ (0, 1/2) and let (cid:90) d (ζ) = d(ζ, Ω(Θ, ε)), ε { ∈ D | | } where we recall that Ω(Θ, ε) = z : Θ(z) < ε . Since − ∞ d 1(ζ) dm(ζ) = , T\ ε σ(Θ)

text: A

text: SURVEY

text: ON

text: REVERSE

text: CARLESON

text: MEASURES

text: Furthermore,

text: the

text: norms

text: on

text: these

text: spaces

text: are

text: the

text: same.

text: In

text: fact,

text: these

text: are

text: the

text: only

text: harmonically

text: weighted

text: Dirichlet

text: spaces

text: which

text: are

text: equal

text: to

text: an

text: H

text: (b)

text: space

text: with

text: equal

text: norm

text: In

text: it

text: was

text: shown

text: that

text: if

text: n

text: T

text: µ

text: c

text: δ

text: c

text: ζ

text: j

text: ζj

text: j

text: j

text: T

text: is

text: a

text: ﬁnite

text: linear

text: combination

text: of

text: point

text: masses

text: on

text: and

text: a

text: is

text: the

text: unique

text: polynomial

text: with

text: and

text: with

text: simple

text: zeros

text: at

text: ζ

text: (and

text: no

text: other

text: zeros)

text: and

text: b

text: is

text: the

text: Pythagorean

text: mate

text: for

text: a

text: (which

text: j

text: must

text: also

text: be

text: a

text: polynomial),

text: then

text: H

text: (b)

text: D(µ)

text: with

text: equivalent

text: norms.

text: In

text: this

text: case

text: we

text: can

text: use

text: Theorem

text: to

text: obtain

text: a

text: characterization

text: of

text: the

text: Carleson

text: measures

text: for

text: D(µ):

text: Theorem

text: For

text: µ

text: as

text: in

text: and

text: ν

text: M

text: (D−),

text: the

text: following

text: assertions

text: are

text: equivalent:

text: (i)

text: ν

text: is

text: a

text: Carleson

text: measure

text: for

text: D(µ);

text: (ii)

text: n

text: z

text: ζ

text: dν

text: is

text: a

text: Carleson

text: measure

text: for

text: i

text: This

text: result

text: appeared

text: in

text: (see

text: also

text: In

text: fact,

text: Theorem

text: from

text: shows

text: that

text: the

text: above

text: conditions

text: are

text: equivalent

text: to

text: ∀λ

text: D.

text: λ

text: ν

text: λ

text: D(µ)

text: In

text: other

text: words,

text: at

text: least

text: when

text: µ

text: is

text: a

text: linear

text: combination

text: of

text: point

text: masses,

text: the

text: reproducing

text: kernel

text: thesis

text: characterizes

text: the

text: Carleson

text: measures

text: for

text: D(µ).

text: The

text: discussion

text: of

text: reverse

text: Carleson

text: measures

text: for

text: D(µ)

text: is

text: dramatically

text: simpler

text: since

text: they

text: do

text: not

text: D−

text: exist!

text: Indeed,

text: suppose

text: that

text: ν

text: M

text: and

text: f

text: f

text: for

text: all

text: f

text: D(µ).

text: In

text: particular,

text: this

text: T

text: is

text: true

text: for

text: the

text: monomials

text: zn,

text: n

text: But

text: zn

text: and

text: zn

text: nµ(

text: which

text: gives

text: a

text: ν

text: µ

text: contradiction

text: when

text: n

text: tends

text: to

text: We

text: point

text: out

text: some

text: related

text: results

text: from

text: which

text: discuss

text: a

text: type

text: of

text: reverse

text: Carleson

text: measure

text: for

text: D(µ)

text: spaces

text: except

text: that

text: the

text: deﬁnitions

text: of

text: “reverse

text: Carleson

text: measures”

text: and

text: “sets

text: of

text: domination”

text: (dominating

text: sets)

text: are

text: quite

text: different,

text: and

text: not

text: equivalent,

text: to

text: ours.

text: APPENDIX

text: We

text: reproduce

text: here

text: an

text: adaption

text: to

text: p

text: of

text: Baranov’s

text: proof

text: as

text: presented

text: in

text: Section

text: and

text: which

text: is

text: based

text: on

text: the

text: Bernstein-type

text: inequalities

text: in

text: model

text: spaces

text: he

text: obtained

text: in

text: It

text: T

text: uses

text: a

text: Whitney

text: type

text: decomposition

text: of

text: σ(Θ).

text: Let

text: ε

text: let

text: δ

text: and

text: let

text: d

text: (ζ)

text: d(ζ,

text: Ω(Θ,

text: ε)),

text: ε

text: D

text: where

text: we

text: recall

text: that

text: Ω(Θ,

text: ε)

text: z

text: Θ(z)

text: ε

text: Since

text: d

text: dm(ζ)

text: T\

text: ε

text: σ(Θ)


text: 30 FRICAIN, HARTMANN, AND ROSS (cid:107) (cid:107) ≤ (cid:107) (cid:107) (9.4) kΘ C(q) kΘ z˜ α z α (which we use h(cid:32)ere for α = 2q). Usi(cid:33)ng (9.4), (9.2) and the fact that the angle2 between [s(3), u] T (cid:90) k and is separated from π, we conclude that 2 p/q − | | ≤ − − δp (cid:88) wr q(v(cid:88)) dv (cid:90) Cδ(cid:90)p(m(Ik) 1/(p 1))p/q = C m(I ), [s(3),u] k k Hence ≤ 1 | (cid:48) | | | J1,3 Cδp f (v) pwp(v) dv dm(u). k m(I ) r (cid:90) (cid:90) k I(1) [s(3),u] (cid:90) (cid:88) k k k k (cid:88) ∈ Again just by the mean value property, there exists u I(1) such that k k 1 (cid:80)| (cid:48) | | | 1 | (cid:48) | | | f (v) pwp(v) dv dm(u) = f (v) pwp(v) dv . m(I ) r 4 r k I(1) [s(3),u] [s(3),u ] k k k k k k Now note that the measure m (sum of Lebesgue measures on the intervals) is a Car- k [s(3),u ] leson measure with a uniform (cid:90)bounkd okn the Carleson constant independent of the location of (cid:88) ∈ ∈ u I(1) and s(3) S(3) (and of δ). Then by the Bernstein inequality [3, Theorem 1.1], we have k k k k | (cid:48) | | | ≤ (cid:107) (cid:107) f (v) pwp(v) dv C f p, (cid:88) r p [s(3),u ] k k k which gives (cid:80)≤ (cid:80)(cid:107) (cid:107) (cid:80) J1,3 Cδp f p. k p (cid:90) (cid:88) k Using similar estimates for the other terms J2,4, J3,1 and J4,2, we obtain k k k k k k | | ≤ (cid:107) (cid:83)(cid:107) − (cid:107) (cid:107) f p dm Cδp f p + c A 1 f p. p p µ I k k T \ Finally we consider the integral over σ(Θ) = I . For this, as indicated in [8], we use k k an argument from [31] which we would like to make more explicit here, thereby showing that ∈ the direct Carleson measure condition is indeed not required in the argument. Recall that f K ∩ T (cid:107) (cid:107) (cid:54) p C( ). Also, it is clear that we can assume f √ = 0. By uniform continuity there exists Θ (cid:48) ∈ D | − (cid:48)| Lp(µ) ρ > 0 such that for every z, z with z z < 2ρ we have (cid:107) (cid:107) | − (cid:48) | ≤ f Lp(µ) f(z) (cid:83)f(z ) . A1/p Now there exists a sequence of arcs (J ) (not necessarily open) with pairwise disjoint interiors ⊂k ∩ (cid:54) ∅ ∈ such that m(J ) < ρ, and with σ(Θ) J and S(J ) Ω(Θ, ε) = . Let z S(J ) such k k k k k k (cid:54) ∈ 2That explains why we choose a decomposition with Ji,j, i = j, since in this case the interval [s(j), u], u I(i), k k k will never be orthogonal to the boundary. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: FRICAIN,

text: HARTMANN,

text: AND

text: ROSS

label: equation

text: kΘ

text: C(q)

text: kΘ

text: z˜

text: α

text: z

text: α

text: (which

text: we

text: use

text: for

text: α

text: and

text: the

text: fact

text: that

text: the

text: between

text: u]

text: T

text: k

text: and

text: is

text: separated

text: from

text: π,

text: we

text: conclude

text: that

text: p/q

text: δp

text: wr

text: dv

text: C

text: m(I

text: k

text: k

text: Hence

text: Cδp

text: f

text: (v)

text: pwp(v)

text: dv

text: dm(u).

text: k

text: m(I

text: r

text: k

text: k

text: k

text: k

text: k

text: Again

text: just

text: by

text: the

text: mean

text: value

text: property,

text: there

text: exists

text: u

text: such

text: that

text: k

text: k

text: f

text: (v)

text: pwp(v)

text: dv

text: dm(u)

text: f

text: (v)

text: pwp(v)

text: dv

text: m(I

text: r

text: r

text: k

text: k

text: k

text: k

text: k

text: k

text: k

text: Now

text: note

text: that

text: the

text: measure

text: m

text: (sum

text: of

text: Lebesgue

text: measures

text: on

text: the

text: intervals)

text: is

text: a

text: Car-

text: k

text: leson

text: measure

text: with

text: a

text: uniform

text: okn

text: the

text: Carleson

text: constant

text: independent

text: of

text: the

text: location

text: of

text: u

text: and

text: (and

text: of

text: δ).

text: Then

text: by

text: the

text: Bernstein

text: inequality

text: Theorem

text: we

text: have

text: k

text: k

text: k

text: k

text: f

text: (v)

text: pwp(v)

text: dv

text: C

text: f

text: p,

text: r

text: p

text: k

text: k

text: k

text: which

text: gives

text: Cδp

text: f

text: p.

text: k

text: p

text: k

text: Using

text: similar

text: estimates

text: for

text: the

text: other

text: terms

text: and

text: we

text: obtain

text: k

text: k

text: k

text: k

text: k

text: k

text: f

text: p

text: dm

text: Cδp

text: f

text: p

text: c

text: A

text: f

text: p.

text: p

text: p

text: µ

text: I

text: k

text: k

text: T

text: Finally

text: we

text: consider

text: the

text: integral

text: over

text: σ(Θ)

text: I

text: For

text: this,

text: as

text: indicated

text: in

text: we

text: use

text: k

text: k

text: an

text: argument

text: from

text: which

text: we

text: would

text: like

text: to

text: make

text: more

text: explicit

text: here,

text: thereby

text: showing

text: that

text: the

text: direct

text: Carleson

text: measure

text: condition

text: is

text: indeed

text: not

text: required

text: in

text: the

text: argument.

text: Recall

text: that

text: f

text: K

text: T

text: p

text: C(

text: Also,

text: it

text: is

text: clear

text: that

text: we

text: can

text: assume

text: f

text: By

text: uniform

text: continuity

text: there

text: exists

text: Θ

text: D

text: Lp(µ)

text: ρ

text: such

text: that

text: for

text: every

text: z,

text: z

text: with

text: z

text: z

text: 2ρ

text: we

text: have

text: f

text: Lp(µ)

text: f(z)

text: Now

text: there

text: exists

text: a

text: sequence

text: of

text: arcs

text: (J

text: (not

text: necessarily

text: open)

text: with

text: pairwise

text: disjoint

text: interiors

text: ⊂k

text: such

text: that

text: m(J

text: ρ,

text: and

text: with

text: σ(Θ)

text: J

text: and

text: S(J

text: Ω(Θ,

text: ε)

text: Let

text: z

text: S(J

text: such

text: k

text: k

text: k

text: k

text: k

text: k

text: explains

text: why

text: we

text: choose

text: a

text: decomposition

text: with

text: Ji,j,

text: i

text: j,

text: since

text: in

text: this

text: case

text: the

text: interval

text: [s(j),

text: u],

text: u

text: I(i),

text: k

text: k

text: k

text: will

text: never

text: be

text: orthogonal

text: to

text: the

text: boundary.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: A detailed heterogeneous agent model for a single asset ﬁnancial market with trading via an order book Roberto Mota Navarro and Herna´n Larralde Instituto de Ciencias F´ısicas, Universidad Nacional Auto´noma de Me´xico, Cuernavaca, Morelos, C.P. 62210, Me´xico E-mail: mvr@fis.unam.mx Abstract. We present an agent based model of a single asset ﬁnancial market that is capable of replicating most of the non-trivial statistical properties observed in real ﬁnancial markets, generically referred to as stylized facts. In our model agents employ strategies inspired on those used in real markets, and a realistic trade mechanism based on a double auction order book. We study the role of the distinct types of trader on the return statistics: speciﬁcally, correlation properties (or lack thereof), volatility clustering, heavy tails, and the degree to which the distribution can be described by a log-normal. Further, by introducing the practice of “proﬁt taking”, our model is also capable of replicating the stylized fact related to an asymmetry in the distribution of losses and gains.

text: A

label: title

font: VOINKX+URWPalladioL-Bold

text: detailed

text: heterogeneous

text: agent

text: model

text: for

text: a

text: single

text: asset

text: ﬁnancial

text: market

text: with

text: trading

text: via

text: an

text: order

text: book

text: Roberto

label: author

text: Mota

text: Navarro

text: and

text: Herna´n

text: Larralde

text: Instituto

font: BIFPVV+URWPalladioL-Roma

text: de

text: Ciencias

text: F´ısicas,

text: Universidad

text: Nacional

text: Auto´noma

text: de

text: Me´xico,

text: Cuernavaca,

text: Morelos,

text: C.P.

text: Me´xico

text: E-mail:

text: mvr@fis.unam.mx

label: footer

text: Abstract.

text: We

label: abstract

text: present

text: an

text: agent

text: based

text: model

text: of

text: a

text: single

text: asset

text: ﬁnancial

text: market

text: that

text: is

text: capable

text: of

text: replicating

text: most

text: of

text: the

text: non-trivial

text: statistical

text: properties

text: observed

text: in

text: real

text: ﬁnancial

text: markets,

text: generically

text: referred

text: to

text: as

text: stylized

text: facts.

text: In

text: our

text: model

text: agents

text: employ

text: strategies

text: inspired

text: on

text: those

text: used

text: in

text: real

text: markets,

text: and

text: a

text: realistic

text: trade

text: mechanism

text: based

text: on

text: a

text: double

text: auction

text: order

text: book.

text: We

text: study

text: the

text: role

text: of

text: the

text: distinct

text: types

text: of

text: trader

text: on

text: the

text: return

text: statistics:

text: speciﬁcally,

text: correlation

text: properties

text: (or

text: lack

text: thereof),

text: volatility

text: clustering,

text: heavy

text: tails,

text: and

text: the

text: degree

text: to

text: which

text: the

text: distribution

text: can

text: be

text: described

text: by

text: a

text: log-normal.

text: Further,

text: by

text: introducing

text: the

text: practice

text: of

text: “proﬁt

text: taking”,

text: our

text: model

text: is

text: also

text: capable

text: of

text: replicating

text: the

text: stylized

text: fact

text: related

text: to

text: an

text: asymmetry

text: in

text: the

text: distribution

text: of

text: losses

text: and

text: gains.


text: A detailed heterogeneous agent model for a single asset ﬁnancial market with trading via an order book12 (a) Returns corresponding to the simulation. (b) Returns from Consol Energy Inc. Data obtained from QuantQuote[35]. Figure 6: Returns time series for the simulation with a time lag τ = 1 (a) and comparison with empirical data from Consol Energy Inc (b). The blue shaded regions show the times in which technical agents were active, as can be seen, these times coincide with the periods with the largest changes of price. line). We observe that the auto-correlation function of the absolute returns remains positive over a long time interval, and that it decays slowly to zero. Figure 8b illustrates the same auto-correlation functions for a representative company listed in the Standard & Poor’s 500. Figure 9a shows the distribution function of returns from our model. This distribution shows heavier tails than a normal distribution with the same mean and standard deviation and it is possible to observe that the left tail is heavier than the right one. For comparison, ﬁgure 9b illustrates the distribution function of returns for a representative company listed in the Standard & Poor’s 500. Figure 10a shows the cumulative complementary distribution of positive and negative returns, highlighting the asymmetry between losses and gains. The tail of the distribution of negative price changes is signiﬁcantly heavier than the ##LTFigure## ##LTFigure##

text: A

font: DJYUCA+URWPalladioL-Ital

text: detailed

text: heterogeneous

text: agent

text: model

text: for

text: a

text: single

text: asset

text: ﬁnancial

text: market

text: with

text: trading

text: via

text: an

text: order

text: (a)

text: Returns

label: caption

text: corresponding

text: to

text: the

text: simulation.

text: (b)

text: Returns

text: from

text: Consol

text: Energy

text: Inc.

text: Data

text: obtained

text: from

text: Figure

text: Returns

text: time

text: series

text: for

text: the

text: simulation

text: with

text: a

text: time

text: lag

text: τ

text: (a)

text: and

text: comparison

text: with

text: empirical

text: data

text: from

text: Consol

text: Energy

text: Inc

text: (b).

text: The

text: blue

text: shaded

text: regions

text: show

text: the

text: times

text: in

text: which

text: technical

text: agents

text: were

text: active,

text: as

text: can

text: be

text: seen,

text: these

text: times

text: coincide

text: with

text: the

text: periods

text: with

text: the

text: largest

text: changes

text: of

text: price.

text: line).

text: We

text: observe

text: that

text: the

text: auto-correlation

text: function

text: of

text: the

text: absolute

text: returns

text: remains

text: positive

text: over

text: a

text: long

text: time

text: interval,

text: and

text: that

text: it

text: decays

text: slowly

text: to

text: zero.

text: Figure

text: illustrates

text: the

text: same

text: auto-correlation

text: functions

text: for

text: a

text: representative

text: company

text: listed

text: in

text: the

text: Standard

text: Poor’s

text: Figure

text: shows

text: the

text: distribution

text: function

text: of

text: returns

text: from

text: our

text: model.

text: This

text: distribution

text: shows

text: heavier

text: tails

text: than

text: a

text: normal

text: distribution

text: with

text: the

text: same

text: mean

text: and

text: standard

text: deviation

text: and

text: it

text: is

text: possible

text: to

text: observe

text: that

text: the

text: left

text: tail

text: is

text: heavier

text: than

text: the

text: right

text: one.

text: For

text: comparison,

text: ﬁgure

text: illustrates

text: the

text: distribution

text: function

text: of

text: returns

text: for

text: a

text: representative

text: company

text: listed

text: in

text: the

text: Standard

text: Poor’s

text: Figure

text: shows

text: the

text: cumulative

text: complementary

text: distribution

text: of

text: positive

text: and

text: negative

text: returns,

text: highlighting

text: the

text: asymmetry

text: between

text: losses

text: and

text: gains.

text: The

text: tail

text: of

text: the

text: distribution

text: of

text: negative

text: price

text: changes

text: is

text: signiﬁcantly

text: heavier

text: than

text: the

text: ##LTFigure##

text: ##LTFigure##


text: A detailed heterogeneous agent model for a single asset ﬁnancial market with trading via an order book14 (a) Direct (blue) and absolute (red) auto-correlation functions of returns from our the simulation. (b) Corresponding auto-correlation functions from Airgas Inc. Data obtained from QuantQuote[35]. Figure 8: Returns auto-correlation function for the simulation (a) and comparison with empirical data from Airgas Inc (b). While the auto-correlation of the direct returns (blue lines) is zero, the auto-correlation of the absolute value of the returns (red lines) remains positive for a long period of time, and decays slowly to zero. < R− > < p− > < R > < p > + + γ = 0.0025 -0.006 0.595 -0.032 0.057 γ = 0.0225 0.004 0.597 -0.053 0.623 γ = 0.0400 -0.050 0.609 0.203 0.489 Table 2: Values of the mean log-likelihood ratios < R > between the powerlaw and lognormal ﬁts and of the mean signiﬁcance values < p >. The values are presented for empirical data and for three representative cases of our model with different values of γ. Here < R− > and < R+ > stand for the log-likelihoods of the left and right tails, correspondingly. Similarly, < p− > and < p > stand for the mean + signiﬁcance values for the left and right tails. ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: A

text: detailed

text: heterogeneous

text: agent

text: model

text: for

text: a

text: single

text: asset

text: ﬁnancial

text: market

text: with

text: trading

text: via

text: an

text: order

text: (a)

text: Direct

text: (blue)

text: and

text: absolute

text: (red)

text: auto-correlation

text: functions

text: of

text: returns

text: from

text: our

text: the

text: simulation.

text: (b)

text: Corresponding

text: auto-correlation

text: functions

text: from

text: Airgas

text: Inc.

text: Data

text: obtained

text: from

text: Figure

text: Returns

text: auto-correlation

text: function

text: for

text: the

text: simulation

text: (a)

text: and

text: comparison

text: with

text: empirical

text: data

text: from

text: Airgas

text: Inc

text: (b).

text: While

text: the

text: auto-correlation

text: of

text: the

text: direct

text: returns

text: (blue

text: lines)

text: is

text: zero,

text: the

text: auto-correlation

text: of

text: the

text: absolute

text: value

text: of

text: the

text: returns

text: (red

text: lines)

text: remains

text: positive

text: for

text: a

text: long

text: period

text: of

text: time,

text: and

text: decays

text: slowly

text: to

text: zero.

text: R−

text: p−

text: R

text: p

text: γ

text: γ

text: γ

text: Table

text: Values

text: of

text: the

text: mean

text: log-likelihood

text: ratios

text: R

text: between

text: the

text: powerlaw

text: and

text: lognormal

text: ﬁts

text: and

text: of

text: the

text: mean

text: signiﬁcance

text: values

text: p

text: The

text: values

text: are

text: presented

text: for

text: empirical

text: data

text: and

text: for

text: three

text: representative

text: cases

text: of

text: our

text: model

text: with

text: different

text: values

text: of

text: γ.

text: Here

text: R−

text: and

text: R+

text: stand

text: for

text: the

text: log-likelihoods

text: of

text: the

text: left

text: and

text: right

text: tails,

text: correspondingly.

text: Similarly,

text: p−

text: and

text: p

text: stand

text: for

text: the

text: mean

text: signiﬁcance

text: values

text: for

text: the

text: left

text: and

text: right

text: tails.

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: A detailed heterogeneous agent model for a single asset ﬁnancial market with trading via an order book15 (a) Returns PDF corresponding from the simulation. (b) Returns PDF from United States Steel Corporation. Data obtained from QuantQuote[35]. Figure 9: Returns PDF from the simulation (a) and comparison with empirical data from United States Steel Corporation, Inc. The tails of the distribution (red line) are clearly heavier than those of a normal distribution (blue line). < R− > < p− > < R > < p > + + 0.258 0.399 0.403 0.338 Table 3: Values of the mean log-likelihood ratios < R > between the powerlaw and lognormal ﬁts and of the mean signiﬁcance values < p > for the empirical data. In ﬁgure 14a we present the distribution of volatilities measured as the average | (cid:12) | (cid:12) of the absolute value of returns r(cid:12)(cid:12)(t) o(cid:12)(cid:12)ver a time window T = n∆t, i.e. − 1 t+∑n 1 (cid:48) VT(t) = n (cid:48) r(t ) t =t For the present result we took values of n = 30 and ∆t = 1 time steps. The distribution of volatilities is not well described by a log-normal distribution, however, the central part of the distribution may be approximated by one[31]. On the other hand, when we remove the technical agents from the simulation, the volatilities are remarkably well described by a log-normal distribution as shown in Figure 15, which corresponds to a run with the same parameter values described in Table 1 without technical agents. ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: A

text: detailed

text: heterogeneous

text: agent

text: model

text: for

text: a

text: single

text: asset

text: ﬁnancial

text: market

text: with

text: trading

text: via

text: an

text: order

text: (a)

text: Returns

text: PDF

text: corresponding

text: from

text: the

text: simulation.

text: (b)

text: Returns

text: PDF

text: from

text: United

text: States

text: Steel

text: Corporation.

text: Data

text: obtained

text: from

text: Figure

text: Returns

text: PDF

text: from

text: the

text: simulation

text: (a)

text: and

text: comparison

text: with

text: empirical

text: data

text: from

text: United

text: States

text: Steel

text: Corporation,

text: Inc.

text: The

text: tails

text: of

text: the

text: distribution

text: (red

text: line)

text: are

text: clearly

text: heavier

text: than

text: those

text: of

text: a

text: normal

text: distribution

text: (blue

text: line).

text: R−

text: p−

text: R

text: p

text: Table

text: Values

text: of

text: the

text: mean

text: log-likelihood

text: ratios

text: R

text: between

text: the

text: powerlaw

text: and

text: lognormal

text: ﬁts

text: and

text: of

text: the

text: mean

text: signiﬁcance

text: values

text: p

text: for

text: the

text: empirical

text: data.

text: In

text: ﬁgure

text: we

text: present

text: the

text: distribution

text: of

text: volatilities

text: measured

text: as

text: the

text: average

text: of

text: the

text: absolute

text: value

text: of

text: returns

text: a

text: time

text: window

text: T

text: n∆t,

text: i.e.

text: t+∑n

text: VT(t)

text: n

text: r(t

text: t

text: =t

text: For

text: the

text: present

text: result

text: we

text: took

text: values

text: of

text: n

text: and

text: ∆t

text: time

text: steps.

text: The

text: distribution

text: of

text: volatilities

text: is

text: not

text: well

text: described

text: by

text: a

text: log-normal

text: distribution,

text: however,

text: the

text: central

text: part

text: of

text: the

text: distribution

text: may

text: be

text: approximated

text: by

text: On

text: the

text: other

text: hand,

text: when

text: we

text: remove

text: the

text: technical

text: agents

text: from

text: the

text: simulation,

text: the

text: volatilities

text: are

text: remarkably

text: well

text: described

text: by

text: a

text: log-normal

text: distribution

text: as

text: shown

text: in

text: Figure

text: which

text: corresponds

text: to

text: a

text: run

text: with

text: the

text: same

text: parameter

text: values

text: described

text: in

text: Table

text: without

text: technical

text: agents.

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: A detailed heterogeneous agent model for a single asset ﬁnancial market with trading via an order book24 References [1] Adrian Pagan. The econometrics of ﬁnancial markets. Journal of empirical ﬁnance, 3(1):15–102, 1996. [2] Rama Cont. Empirical properties of asset returns: stylized facts and statistical issues. Quantitative Finance, 1:223–236, 2001. [3] Rosario N Mantegna and H Eugene Stanley. Introduction to econophysics: correlations and complexity in ﬁnance. Cambridge university press, 1999. [4] Benoit Mandelbrot. The variation of certain speculative prices. The Journal of Business, pages 394–419, 1963. [5] Benoit Mandelbrot and Howard M Taylor. On the distribution of stock price differences. Operations research, 15(6):1057–1062, 1967. [6] Christian T Brownlees, Robert F Engle, and Bryan T Kelly. A practical guide to volatility forecasting through calm and storm. The Journal of Risk, 14(2):3–22, 2011. [7] Argia M Sbordone, Andrea Tambalotti, Krishna Rao, and Kieran James Walsh. Policy analysis using dsge models: an introduction. Economic policy review, 16(2), 2010. [8] J Doyne Farmer and Duncan Foley. The economy needs agent-based modelling. Nature, 460(7256):685–686, 2009. [9] Paul De Grauwe. The scientiﬁc foundation of dynamic stochastic general equilibrium (dsge) models. Public choice, 144(3-4):413–443, 2010. [10] Eric Bonabeau. Predicting the unpredictable. Harvard Business Review, 80(3):109–116, 2002. [11] Leigh Tesfatsion and Kenneth L Judd. Handbook of computational economics: agent-based computational economics, volume 2. Elsevier, 2006. [12] Tobias Preis, Sebastian Golke, Wolfgang Paul, and Johannes J Schneider. Multi-agent-based order book model of ﬁnancial markets. EPL (Europhysics Letters), 75(3):510, 2006. [13] J Doyne Farmer, Paolo Patelli, and Ilija I Zovko. The predictive power of zero intelligence in ﬁnancial markets. Proceedings of the National Academy of Sciences of the United States of America, 102(6):2254– 2259, 2005. [14] Sergei Maslov. Simple model of a limit order-driven market. Physica A: Statistical Mechanics and its Applications, 278(3):571–578, 2000. [15] Damien Challet, Robin Stinchcombe, et al. Non-constant rates and over-diffusive prices in a simple model of limit order markets. Quantitative ﬁnance, 3(3):155–162, 2003. [16] Thilo A. Schmitt, Rudi Scha¨fer, Michael C. Mu¨nnix, and Thomas Guhr. Microscopic understanding of heavy-tailed return distributions in an agent-based model. EPL (Europhysics Letters), 100, 2012. [17] Lux T. and Marchesi M. Volatility clustering in ﬁnancial markets: A micro-simulation of interacting agents. International Journal of Theoretical and Applied Finance, 2000. [18] Rama Cont. Volatility clustering in ﬁnancial markets: Empirical facts and agent-based models. In Springer, editor, Long Memory in Economics. 2007. [19] Mario A Bertella, Felipe R Pires, Ling Feng, and Harry Eugene Stanley. Conﬁdence and the stock market: An agent-based approach. PloS one, 9(1):e83488, 2014. [20] V Alﬁ, M Cristelli, L Pietronero, and A Zaccaria. Minimal agent based model for ﬁnancial markets i. The European Physical Journal B, 67(3):385–397, 2009. [21] Carl Chiarella, Giulia Iori, et al. A simulation analysis of the microstructure of double auction markets*. Quantitative Finance, 2(5):346–353, 2002. [22] Carl Chiarella, Giulia Iori, and Josep Perello´. The impact of heterogeneous trading rules on the limit order book and order ﬂows. Journal of Economic Dynamics and Control, 33(3):525–537, 2009. [23] Andrea Consiglio*, Valerio Lacagnina, and Annalisa Russino. A simulation analysis of the microstructure of an order driven ﬁnancial market with multiple securities and portfolio choices. Quantitative Finance, 5(1):71–87, 2005. [24] Marco LiCalzi, Paolo Pellizzari, et al. Fundamentalists clashing over the book: a study of order- driven stock markets. Quantitative ﬁnance, 3(6):470–480, 2003. [25] William Brock, Josef Lakonishok, and Blake LeBaron. Simple technical trading rules and the stochastic properties of stock returns. The Journal of Finance, 47(5):1731–1764, 1992. [26] Anil K Bera, Sergey Ivliev, and Fabrizio Lillo. Financial Econometrics and Empirical Market Microstructure. Springer, 2015. [27] Matthieu Cristelli. Complexity in Financial Markets. Springer, 2014. [28] Marc Potters and Jean-Philippe Bouchaud. More stylized facts of ﬁnancial markets: Leverage effect and downside correlations. Physica A, 2001. [29] Vasiliki Plerou, Parameswaran Gopikrishnan, Luis A Nunes Amaral, Martin Meyer, and H Eugene Stanley. Scaling of the distribution of price ﬂuctuations of individual companies. Physical Review E, 60(6):6519, 1999. [30] J.P. Bouchaud Rama Cont, M. Potters. Scale Invariance and Beyond (Proc. CNRS Workshop on Scale

text: A

text: detailed

text: heterogeneous

text: agent

text: model

text: for

text: a

text: single

text: asset

text: ﬁnancial

text: market

text: with

text: trading

text: via

text: an

text: order

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text: Empirical

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text: of

text: asset

text: returns:

text: stylized

text: facts

text: and

text: statistical

text: issues.

text: Quantitative

text: Finance,

text: Rosario

text: N

text: Mantegna

text: and

text: H

text: Eugene

text: Stanley.

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text: econophysics:

text: correlations

text: and

text: complexity

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text: Cambridge

text: university

text: press,

text: Benoit

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text: variation

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text: Brownlees,

text: Robert

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text: Engle,

text: and

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text: practical

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text: forecasting

text: through

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text: storm.

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text: Journal

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text: Risk,

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text: Letters),

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text: Paolo

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text: predictive

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text: of

text: zero

text: intelligence

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text: ﬁnancial

text: markets.

text: Proceedings

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text: the

text: National

text: Academy

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text: United

text: States

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text: Sergei

text: Maslov.

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text: Statistical

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text: Damien

text: Challet,

text: Robin

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text: et

text: al.

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text: rates

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text: over-diffusive

text: prices

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text: simple

text: model

text: of

text: limit

text: order

text: markets.

text: Quantitative

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text: Thilo

text: A.

text: Schmitt,

text: Rudi

text: Scha¨fer,

text: Michael

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text: Mu¨nnix,

text: and

text: Thomas

text: Guhr.

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text: understanding

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text: heavy-tailed

text: return

text: distributions

text: in

text: an

text: agent-based

text: model.

text: EPL

text: (Europhysics

text: Letters),

text: Lux

text: T.

text: and

text: Marchesi

text: M.

text: Volatility

text: clustering

text: in

text: ﬁnancial

text: markets:

text: A

text: micro-simulation

text: of

text: interacting

text: agents.

text: International

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text: of

text: Theoretical

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text: clustering

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text: markets:

text: Empirical

text: facts

text: and

text: agent-based

text: models.

text: In

text: Springer,

text: editor,

text: Long

text: Memory

text: in

text: Economics.

text: Mario

text: A

text: Bertella,

text: Felipe

text: R

text: Pires,

text: Ling

text: Feng,

text: and

text: Harry

text: Eugene

text: Stanley.

text: Conﬁdence

text: and

text: the

text: stock

text: market:

text: An

text: agent-based

text: approach.

text: PloS

text: one,

text: V

text: Alﬁ,

text: M

text: Cristelli,

text: L

text: Pietronero,

text: and

text: A

text: Zaccaria.

text: Minimal

text: agent

text: based

text: model

text: for

text: ﬁnancial

text: markets

text: i.

text: The

text: European

text: Physical

text: Journal

text: B,

text: Carl

text: Chiarella,

text: Giulia

text: Iori,

text: et

text: al.

text: A

text: simulation

text: analysis

text: of

text: the

text: microstructure

text: of

text: double

text: auction

text: markets*.

text: Quantitative

text: Finance,

text: Carl

text: Chiarella,

text: Giulia

text: Iori,

text: and

text: Josep

text: Perello´.

text: The

text: impact

text: of

text: heterogeneous

text: trading

text: rules

text: on

text: the

text: limit

text: order

text: book

text: and

text: order

text: ﬂows.

text: Journal

text: of

text: Economic

text: Dynamics

text: and

text: Control,

text: Andrea

text: Consiglio*,

text: Valerio

text: Lacagnina,

text: and

text: Annalisa

text: Russino.

text: A

text: simulation

text: analysis

text: of

text: the

text: microstructure

text: of

text: an

text: order

text: driven

text: ﬁnancial

text: market

text: with

text: multiple

text: securities

text: and

text: portfolio

text: choices.

text: Quantitative

text: Finance,

text: Marco

text: LiCalzi,

text: Paolo

text: Pellizzari,

text: et

text: al.

text: Fundamentalists

text: clashing

text: over

text: the

text: book:

text: a

text: study

text: of

text: order-

text: driven

text: stock

text: markets.

text: Quantitative

text: ﬁnance,

text: William

text: Brock,

text: Josef

text: Lakonishok,

text: and

text: Blake

text: LeBaron.

text: Simple

text: technical

text: trading

text: rules

text: and

text: the

text: stochastic

text: properties

text: of

text: stock

text: returns.

text: The

text: Journal

text: of

text: Finance,

text: Anil

text: K

text: Bera,

text: Sergey

text: Ivliev,

text: and

text: Fabrizio

text: Lillo.

text: Financial

text: Econometrics

text: and

text: Empirical

text: Market

text: Microstructure.

text: Springer,

text: Matthieu

text: Cristelli.

text: Complexity

text: in

text: Financial

text: Markets.

text: Springer,

text: Marc

text: Potters

text: and

text: Jean-Philippe

text: Bouchaud.

text: More

text: stylized

text: facts

text: of

text: ﬁnancial

text: markets:

text: Leverage

text: effect

text: and

text: downside

text: correlations.

text: Physica

text: A,

text: Vasiliki

text: Plerou,

text: Parameswaran

text: Gopikrishnan,

text: Luis

text: A

text: Nunes

text: Amaral,

text: Martin

text: Meyer,

text: and

text: H

text: Eugene

text: Stanley.

text: Scaling

text: of

text: the

text: distribution

text: of

text: price

text: ﬂuctuations

text: of

text: individual

text: companies.

text: Physical

text: Review

text: E,

text: J.P.

text: Bouchaud

text: Rama

text: Cont,

text: M.

text: Potters.

text: Scale

text: Invariance

text: and

text: Beyond

text: (Proc.

text: CNRS

text: Workshop

text: on

text: Scale


text: A detailed heterogeneous agent model for a single asset ﬁnancial market with trading via an order book4 In the next section we present a detailed discussion of the agent based model we propose. The paper continues with a section in which we present the results obtained in simulations of the model and we focus on the stylized facts listed above, comparing the behavior of the model with representative empirical data. We also study the effect of varying the relative populations of agents as well as the parameters that control the practice of proﬁt taking by the technical agents in the system. We end with a section of concluding remarks and perspectives. 2. Model 2.1. General Aspects The model represents a ﬁnancial market in which N agents trade a single asset through a double auction order book in which the standing orders are registered until executed. In the model we only consider market and limit orders [32] of unit volume. Like in actual ﬁnancial markets, in the model, the population of agents is divided into two different sub-populations, with each sub-population employing one of two basic trading strategies: fundamental analysis -by which a “fundamental price” p is f estimated, and then the traders attempt to take advantage of the deviations between p and its present trading price P -; or technical analysis -by which the trader tries to f t identify and exploit trends in the price time series-. These two types of strategies are representative of the main strategies used in real life trading and were ﬁrst introduced in the Lux-Marchesi (LM) model [33]. The effects of these strategies on the dynamics of the price are opposed: while fundamental agents tend to stabilize the prices around the average value of their fundamental prices, technical agents tend to create periods of violent price changes. The parameters controlling the behavior of each agent are assigned at the beginning of each simulation, and even if two agents belong to the same group (fundamental or technical) the difference in the values of their controlling parameters will generate different ”personalities” within each strategy. We make time run in discrete units corresponding to simulation steps and on each simulation step, each fundamental agent will engage in trading with a probability p while technical agents will be active when they observe a favorable active trend or when they can obtain a high immediate proﬁt, as will be explained later. In our system, every agent is assigned unlimited credit, and short selling is allowed. These two liberties are meant to ensure that an agent is able to engage in trading whenever it becomes active, thus providing the market with enough liquidity. Although the model we propose includes the main components of the Lux- Marchesi model, there are important differences in the way in which we designed both the agents and the market environment. Of central importance is the fact that in our model the process of price formation is directly governed by the demand and supply provided by the agents and all the transactions are mediated through an order book. Another important difference is the fact that by assigning different parameters, we include heterogeneity within each strategy. Further, in our model the only sources of “exogenous” randomness are the entry times of the fundamental agents; and the time of arrival and nature of the news in the system, which in turn ellicit randomly distributed reactions from fundamental agents. 2.2. Types of Agents

text: A

text: detailed

text: heterogeneous

text: agent

text: model

text: for

text: a

text: single

text: asset

text: ﬁnancial

text: market

text: with

text: trading

text: via

text: an

text: order

text: In

text: the

text: next

text: section

text: we

text: present

text: a

text: detailed

text: discussion

text: of

text: the

text: agent

text: based

text: model

text: we

text: propose.

text: The

text: paper

text: continues

text: with

text: a

text: section

text: in

text: which

text: we

text: present

text: the

text: results

text: obtained

text: in

text: simulations

text: of

text: the

text: model

text: and

text: we

text: focus

text: on

text: the

text: stylized

text: facts

text: listed

text: above,

text: comparing

text: the

text: behavior

text: of

text: the

text: model

text: with

text: representative

text: empirical

text: data.

text: We

text: also

text: study

text: the

text: effect

text: of

text: varying

text: the

text: relative

text: populations

text: of

text: agents

text: as

text: well

text: as

text: the

text: parameters

text: that

text: control

text: the

text: practice

text: of

text: proﬁt

text: taking

text: by

text: the

text: technical

text: agents

text: in

text: the

text: system.

text: We

text: end

text: with

text: a

text: section

text: of

text: concluding

text: remarks

text: and

text: perspectives.

text: Model

text: General

text: Aspects

text: The

text: model

text: represents

text: a

text: ﬁnancial

text: market

text: in

text: which

text: N

text: agents

text: trade

text: a

text: single

text: asset

text: through

text: a

text: double

text: auction

text: order

text: book

text: in

text: which

text: the

text: standing

text: orders

text: are

text: registered

text: until

text: executed.

text: In

text: the

text: model

text: we

text: only

text: consider

text: market

text: and

text: limit

text: orders

text: of

text: unit

text: volume.

text: Like

text: in

text: actual

text: ﬁnancial

text: markets,

text: in

text: the

text: model,

text: the

text: population

text: of

text: agents

text: is

text: divided

text: into

text: two

text: different

text: sub-populations,

text: with

text: each

text: sub-population

text: employing

text: one

text: of

text: two

text: basic

text: trading

text: strategies:

text: fundamental

text: analysis

text: -by

text: which

text: a

text: “fundamental

text: price”

text: p

text: is

text: f

text: estimated,

text: and

text: then

text: the

text: traders

text: attempt

text: to

text: take

text: advantage

text: of

text: the

text: deviations

text: between

text: p

text: and

text: its

text: present

text: trading

text: price

text: P

text: or

text: technical

text: analysis

text: -by

text: which

text: the

text: trader

text: tries

text: to

text: f

text: t

text: identify

text: and

text: exploit

text: trends

text: in

text: the

text: price

text: time

text: series-.

text: These

text: two

text: types

text: of

text: strategies

text: are

text: representative

text: of

text: the

text: main

text: strategies

text: used

text: in

text: real

text: life

text: trading

text: and

text: were

text: ﬁrst

text: introduced

text: in

text: the

text: Lux-Marchesi

text: (LM)

text: model

text: The

text: effects

text: of

text: these

text: strategies

text: on

text: the

text: dynamics

text: of

text: the

text: price

text: are

text: opposed:

text: while

text: fundamental

text: agents

text: tend

text: to

text: stabilize

text: the

text: prices

text: around

text: the

text: average

text: value

text: of

text: their

text: fundamental

text: prices,

text: technical

text: agents

text: tend

text: to

text: create

text: periods

text: of

text: violent

text: price

text: changes.

text: The

text: parameters

text: controlling

text: the

text: behavior

text: of

text: each

text: agent

text: are

text: assigned

text: at

text: the

text: beginning

text: of

text: each

text: simulation,

text: and

text: even

text: if

text: two

text: agents

text: belong

text: to

text: the

text: same

text: group

text: (fundamental

text: or

text: technical)

text: the

text: difference

text: in

text: the

text: values

text: of

text: their

text: controlling

text: parameters

text: will

text: generate

text: different

text: ”personalities”

text: within

text: each

text: strategy.

text: We

text: make

text: time

text: run

text: in

text: discrete

text: units

text: corresponding

text: to

text: simulation

text: steps

text: and

text: on

text: each

text: simulation

text: step,

text: each

text: fundamental

text: agent

text: will

text: engage

text: in

text: trading

text: with

text: a

text: probability

text: p

text: while

text: technical

text: agents

text: will

text: be

text: active

text: when

text: they

text: observe

text: a

text: favorable

text: active

text: trend

text: or

text: when

text: they

text: can

text: obtain

text: a

text: high

text: immediate

text: proﬁt,

text: as

text: will

text: be

text: explained

text: later.

text: In

text: our

text: system,

text: every

text: agent

text: is

text: assigned

text: unlimited

text: credit,

text: and

text: short

text: selling

text: is

text: allowed.

text: These

text: two

text: liberties

text: are

text: meant

text: to

text: ensure

text: that

text: an

text: agent

text: is

text: able

text: to

text: engage

text: in

text: trading

text: whenever

text: it

text: becomes

text: active,

text: thus

text: providing

text: the

text: market

text: with

text: enough

text: liquidity.

text: Although

text: the

text: model

text: we

text: propose

text: includes

text: the

text: main

text: components

text: of

text: the

text: Lux-

text: Marchesi

text: model,

text: there

text: are

text: important

text: differences

text: in

text: the

text: way

text: in

text: which

text: we

text: designed

text: both

text: the

text: agents

text: and

text: the

text: market

text: environment.

text: Of

text: central

text: importance

text: is

text: the

text: fact

text: that

text: in

text: our

text: model

text: the

text: process

text: of

text: price

text: formation

text: is

text: directly

text: governed

text: by

text: the

text: demand

text: and

text: supply

text: provided

text: by

text: the

text: agents

text: and

text: all

text: the

text: transactions

text: are

text: mediated

text: through

text: an

text: order

text: book.

text: Another

text: important

text: difference

text: is

text: the

text: fact

text: that

text: by

text: assigning

text: different

text: parameters,

text: we

text: include

text: heterogeneity

text: within

text: each

text: strategy.

text: Further,

text: in

text: our

text: model

text: the

text: only

text: sources

text: of

text: “exogenous”

text: randomness

text: are

text: the

text: entry

text: times

text: of

text: the

text: fundamental

text: agents;

text: and

text: the

text: time

text: of

text: arrival

text: and

text: nature

text: of

text: the

text: news

text: in

text: the

text: system,

text: which

text: in

text: turn

text: ellicit

text: randomly

text: distributed

text: reactions

text: from

text: fundamental

text: agents.

text: Types

text: of

text: Agents


text: A detailed heterogeneous agent model for a single asset ﬁnancial market with trading via an order book5 As mentioned above, technical agents employ “technical analysis” in an attempt to predict the future behavior of the price time series with the purpose of exploiting the knowledge of that future behavior. In our model, technical agents utilize a technique used in real life called Moving Average-Oscillator (MAO) [25], which consists of a pair of moving averages with different window sizes: a long period average called the slow moving average, and a short period average aptly called the fast moving average. The fast moving average is intended to capture the tendency of the price movements in a short term while the slow moving average has the purpose of capturing the long term trend. Figure 1 shows an example of this technical indicator. Figure 1: Moving average-oscillator (MAO). This is a common technical indicator which is formed by two moving averages of different window sizes that are constantly observed. The moving average with the largest window size is called fast moving average and the one with the smallest window is called slow moving average. When the fast moving average crosses the slow one from below, a signal to buy is generated; conversely, when the fast moving average crosses the slow from above, a signal to sell is generated. When the fast average crosses the slow one from above, the MAO strategy suggests that this is a “signal to sell”, since the prices show a short term tendency to fall below the long term trend captured by the slow moving average. Similarly, a “signal to buy” occurs when the fast moving average crosses the slow one from below, since this can be interpreted as the prices having a short time tendency to rise above the long term trend. We employ the MAO indicator in our model because while it is very simple and easy to implement, it is representative of the plethora of technical analysis tools and it is widely used in real markets[34]. In our model we use MAO indicators that differ in the window sizes of the two averages which compose them. For each of these indicators there is a population of technical agents following its evolution over time and engaging in trading as a result of the signals that the indicator generates. Further, when an indicator generates a signal to either a buy or sell, each technical agent following that particular indicator ##LTFigure##

text: A

text: detailed

text: heterogeneous

text: agent

text: model

text: for

text: a

text: single

text: asset

text: ﬁnancial

text: market

text: with

text: trading

text: via

text: an

text: order

text: As

text: mentioned

text: above,

text: technical

text: agents

text: employ

text: “technical

text: analysis”

text: in

text: an

text: attempt

text: to

text: predict

text: the

text: future

text: behavior

text: of

text: the

text: price

text: time

text: series

text: with

text: the

text: purpose

text: of

text: exploiting

text: the

text: knowledge

text: of

text: that

text: future

text: behavior.

text: In

text: our

text: model,

text: technical

text: agents

text: utilize

text: a

text: technique

text: used

text: in

text: real

text: life

text: called

text: Moving

text: Average-Oscillator

text: (MAO)

text: which

text: consists

text: of

text: a

text: pair

text: of

text: moving

text: averages

text: with

text: different

text: window

text: sizes:

text: a

text: long

text: period

text: average

text: called

text: the

text: slow

text: moving

text: average,

text: and

text: a

text: short

text: period

text: average

text: aptly

text: called

text: the

text: fast

text: moving

text: average.

text: The

text: fast

text: moving

text: average

text: is

text: intended

text: to

text: capture

text: the

text: tendency

text: of

text: the

text: price

text: movements

text: in

text: a

text: short

text: term

text: while

text: the

text: slow

text: moving

text: average

text: has

text: the

text: purpose

text: of

text: capturing

text: the

text: long

text: term

text: trend.

text: Figure

text: shows

text: an

text: example

text: of

text: this

text: technical

text: indicator.

text: Figure

text: Moving

text: average-oscillator

text: (MAO).

text: This

text: is

text: a

text: common

text: technical

text: indicator

text: which

text: is

text: formed

text: by

text: two

text: moving

text: averages

text: of

text: different

text: window

text: sizes

text: that

text: are

text: constantly

text: observed.

text: The

text: moving

text: average

text: with

text: the

text: largest

text: window

text: size

text: is

text: called

text: fast

text: moving

text: average

text: and

text: the

text: one

text: with

text: the

text: smallest

text: window

text: is

text: called

text: slow

text: moving

text: average.

text: When

text: the

text: fast

text: moving

text: average

text: crosses

text: the

text: slow

text: one

text: from

text: below,

text: a

text: signal

text: to

text: buy

text: is

text: generated;

text: conversely,

text: when

text: the

text: fast

text: moving

text: average

text: crosses

text: the

text: slow

text: from

text: above,

text: a

text: signal

text: to

text: sell

text: is

text: generated.

text: When

text: the

text: fast

text: average

text: crosses

text: the

text: slow

text: one

text: from

text: above,

text: the

text: MAO

text: strategy

text: suggests

text: that

text: this

text: is

text: a

text: “signal

text: to

text: sell”,

text: since

text: the

text: prices

text: show

text: a

text: short

text: term

text: tendency

text: to

text: fall

text: below

text: the

text: long

text: term

text: trend

text: captured

text: by

text: the

text: slow

text: moving

text: average.

text: Similarly,

text: a

text: “signal

text: to

text: buy”

text: occurs

text: when

text: the

text: fast

text: moving

text: average

text: crosses

text: the

text: slow

text: one

text: from

text: below,

text: since

text: this

text: can

text: be

text: interpreted

text: as

text: the

text: prices

text: having

text: a

text: short

text: time

text: tendency

text: to

text: rise

text: above

text: the

text: long

text: term

text: trend.

text: We

text: employ

text: the

text: MAO

text: indicator

text: in

text: our

text: model

text: because

text: while

text: it

text: is

text: very

text: simple

text: and

text: easy

text: to

text: implement,

text: it

text: is

text: representative

text: of

text: the

text: plethora

text: of

text: technical

text: analysis

text: tools

text: and

text: it

text: is

text: widely

text: used

text: in

text: real

text: In

text: our

text: model

text: we

text: use

text: MAO

text: indicators

text: that

text: differ

text: in

text: the

text: window

text: sizes

text: of

text: the

text: two

text: averages

text: which

text: compose

text: them.

text: For

text: each

text: of

text: these

text: indicators

text: there

text: is

text: a

text: population

text: of

text: technical

text: agents

text: following

text: its

text: evolution

text: over

text: time

text: and

text: engaging

text: in

text: trading

text: as

text: a

text: result

text: of

text: the

text: signals

text: that

text: the

text: indicator

text: generates.

text: Further,

text: when

text: an

text: indicator

text: generates

text: a

text: signal

text: to

text: either

text: a

text: buy

text: or

text: sell,

text: each

text: technical

text: agent

text: following

text: that

text: particular

text: indicator

text: ##LTFigure##


text: Fluctuations in a Cosmology with a Space-Like Singularity and their Gauge Theory Dual Description ∗ † ‡ Robert H. Brandenberger , Elisa G. M. Ferreira , and Ian A. Morrison Department of Physics, McGill University, Montr´eal, QC, H3A 2T8, Canada § Yi-Fu Cai CAS Key Laboratory for Researches in Galaxies and Cosmology, Department of Astronomy, University of Science and Technology of China, Chinese Academy of Sciences, Hefei, Anhui 230026, China and Department of Physics, McGill University, Montr´eal, QC, H3A 2T8, Canada ¶ Sumit R. Das Department of Physics and Astronomy, University of Kentucky, Lexington, KY, 40506, USA ∗∗ Yi Wang Department of Physics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, P.R.China We consider a time-dependent deformation of anti-de-Sitter (AdS) space-time which contains a spacelike “singularity” - a space-like region of high curvature. Making use of the AdS/CFT corre- spondence we can map the bulk dynamics onto the boundary. The boundary theory has a time de- pendent coupling constant which becomes small at times when the bulk space-time is highly curved. We investigate the propagation of small ﬂuctuations of a test scalar ﬁeld from early times before the bulk singularity to late times after the singularity. Under the assumption that the AdS/CFT correspondence extends to deformed AdS space-times, we can map the bulk evolution of the scalar ﬁeld onto the evolution of the boundary gauge ﬁeld. The time evolution of linearized ﬂuctuations is well deﬁned in the boundary theory as long as the coupling remains ﬁnite, so that we can extend the boundary perturbations to late times after the singularity. Assuming that the spacetime in the future of the singularity has a weakly coupled region near the boundary, we reconstruct the bulk ﬂuctuations after the singularity crossing making use of generic properties of boundary-to-bulk propagators. Finally, we extract the spectrum of the ﬂuctuations at late times given some initial spectrum. We ﬁnd that the spectral index is unchanged, but the amplitude increases due to the squeezing of the ﬂuctuations during the course of the evolution. This investigation can teach us im- portant lessons on how the spectrum of cosmological perturbations passes through a bounce which is singular from the bulk point of view but which is resolved using an ultraviolet complete theory of quantum gravity. PACS numbers: 98.80.Cq I. INTRODUCTION The AdS/CFT correspondence [1] is a most promising proposal for a non-perturbative deﬁnition of string theory. Thus, this correspondence should also have important consequences for early universe cosmology. In fact, over the years there have been several proposals which address the meaning of cosmological singularities in the dual ﬁeld theory [2–5]. The general idea is the following: consider an asymptotically AdS space-time which is contracting towards a curvature singularity. According to the AdS/CFT dictionary, this may correspond to a dual conformal ﬁeld theory which lives on the boundary which is in a nontrivial unstable state [2, 5] or which has a time dependent coupling [3, 4] which becomes small when the bulk singularity is reached. While the bulk theory cannot be used to evolve further ∗ Email: rhb@physics.mcgill.ca † Email: elisa.ferreira@mail.mcgill.ca ‡ Email: imorrison@physics.mcgill.ca § Email: yifucai@ustc.edu.cn ¶ ∗∗ Email: das@pa.uky.edu Email: phyw@ust.hk ##LTLine##

text: Fluctuations

text: in

text: a

text: Cosmology

text: with

text: a

text: Space-Like

text: Singularity

text: and

text: their

text: Gauge

text: Theory

text: Dual

text: Description

text: Robert

text: H.

text: Brandenberger

text: Elisa

text: G.

text: M.

text: Ferreira

text: and

text: Ian

text: A.

text: Morrison

text: Department

text: of

text: Physics,

text: McGill

text: University,

text: Montr´eal,

text: QC,

text: Canada

text: Yi-Fu

text: Cai

text: CAS

text: Key

text: Laboratory

text: for

text: Researches

text: in

text: Galaxies

text: and

text: Cosmology,

text: Department

text: of

text: Astronomy,

text: University

text: of

text: Science

text: and

text: Technology

text: of

text: China,

text: Chinese

text: Academy

text: of

text: Sciences,

text: Hefei,

text: Anhui

text: China

text: and

text: Department

text: of

text: Physics,

text: McGill

text: University,

text: Montr´eal,

text: QC,

text: Canada

text: Sumit

text: R.

text: Das

text: Department

text: of

text: Physics

text: and

text: Astronomy,

text: University

text: of

text: Kentucky,

text: Lexington,

text: KY,

text: USA

text: Yi

text: Wang

text: Department

text: of

text: Physics,

text: The

text: Hong

text: Kong

text: University

text: of

text: Science

text: and

text: Technology,

text: Clear

text: Water

text: Bay,

text: Kowloon,

text: Hong

text: Kong,

text: P.R.China

text: We

text: consider

text: a

text: time-dependent

text: deformation

text: of

text: anti-de-Sitter

text: (AdS)

text: space-time

text: which

text: contains

text: a

text: spacelike

text: “singularity”

text: a

text: space-like

text: region

text: of

text: high

text: curvature.

text: Making

text: use

text: of

text: the

text: AdS/CFT

text: corre-

text: spondence

text: we

text: can

text: map

text: the

text: bulk

text: dynamics

text: onto

text: the

text: boundary.

text: The

text: boundary

text: theory

text: has

text: a

text: time

text: de-

text: pendent

text: coupling

text: constant

text: which

text: becomes

text: small

text: at

text: times

text: when

text: the

text: bulk

text: space-time

text: is

text: highly

text: curved.

text: We

text: investigate

text: the

text: propagation

text: of

text: small

text: ﬂuctuations

text: of

text: a

text: test

text: scalar

text: ﬁeld

text: from

text: early

text: times

text: before

text: the

text: bulk

text: singularity

text: to

text: late

text: times

text: after

text: the

text: singularity.

text: Under

text: the

text: assumption

text: that

text: the

text: AdS/CFT

text: correspondence

text: extends

text: to

text: deformed

text: AdS

text: space-times,

text: we

text: can

text: map

text: the

text: bulk

text: evolution

text: of

text: the

text: scalar

text: ﬁeld

text: onto

text: the

text: evolution

text: of

text: the

text: boundary

text: gauge

text: ﬁeld.

text: The

text: time

text: evolution

text: of

text: linearized

text: ﬂuctuations

text: is

text: well

text: deﬁned

text: in

text: the

text: boundary

text: theory

text: as

text: long

text: as

text: the

text: coupling

text: remains

text: ﬁnite,

text: so

text: that

text: we

text: can

text: extend

text: the

text: boundary

text: perturbations

text: to

text: late

text: times

text: after

text: the

text: singularity.

text: Assuming

text: that

text: the

text: spacetime

text: in

text: the

text: future

text: of

text: the

text: singularity

text: has

text: a

text: weakly

text: coupled

text: region

text: near

text: the

text: boundary,

text: we

text: reconstruct

text: the

text: bulk

text: ﬂuctuations

text: after

text: the

text: singularity

text: crossing

text: making

text: use

text: of

text: generic

text: properties

text: of

text: boundary-to-bulk

text: propagators.

text: Finally,

text: we

text: extract

text: the

text: spectrum

text: of

text: the

text: ﬂuctuations

text: at

text: late

text: times

text: given

text: some

text: initial

text: spectrum.

text: We

text: ﬁnd

text: that

text: the

text: spectral

text: index

text: is

text: unchanged,

text: but

text: the

text: amplitude

text: increases

text: due

text: to

text: the

text: squeezing

text: of

text: the

text: ﬂuctuations

text: during

text: the

text: course

text: of

text: the

text: evolution.

text: This

text: investigation

text: can

text: teach

text: us

text: im-

text: portant

text: lessons

text: on

text: how

text: the

text: spectrum

text: of

text: cosmological

text: perturbations

text: passes

text: through

text: a

text: bounce

text: which

text: is

text: singular

text: from

text: the

text: bulk

text: point

text: of

text: view

text: but

text: which

text: is

text: resolved

text: using

text: an

text: ultraviolet

text: complete

text: theory

text: of

text: quantum

text: gravity.

text: PACS

text: numbers:

text: I.

text: INTRODUCTION

text: The

text: AdS/CFT

text: correspondence

text: is

text: a

text: most

text: promising

text: proposal

text: for

text: a

text: non-perturbative

text: deﬁnition

text: of

text: string

text: theory.

text: Thus,

text: this

text: correspondence

text: should

text: also

text: have

text: important

text: consequences

text: for

text: early

text: universe

text: cosmology.

text: In

text: fact,

text: over

text: the

text: years

text: there

text: have

text: been

text: several

text: proposals

text: which

text: address

text: the

text: meaning

text: of

text: cosmological

text: singularities

text: in

text: the

text: dual

text: ﬁeld

text: theory

text: The

text: general

text: idea

text: is

text: the

text: following:

text: consider

text: an

text: asymptotically

text: AdS

text: space-time

text: which

text: is

text: contracting

text: towards

text: a

text: curvature

text: singularity.

text: According

text: to

text: the

text: AdS/CFT

text: dictionary,

text: this

text: may

text: correspond

text: to

text: a

text: dual

text: conformal

text: ﬁeld

text: theory

text: which

text: lives

text: on

text: the

text: boundary

text: which

text: is

text: in

text: a

text: nontrivial

text: unstable

text: state

text: or

text: which

text: has

text: a

text: time

text: dependent

text: coupling

text: which

text: becomes

text: small

text: when

text: the

text: bulk

text: singularity

text: is

text: reached.

text: While

text: the

text: bulk

text: theory

text: cannot

text: be

text: used

text: to

text: evolve

text: further

text: Email:

text: rhb@physics.mcgill.ca

text: Email:

text: elisa.ferreira@mail.mcgill.ca

text: Email:

text: imorrison@physics.mcgill.ca

text: Email:

text: yifucai@ustc.edu.cn

text: Email:

text: das@pa.uky.edu

text: Email:

text: phyw@ust.hk

text: ##LTLine##


text: 12 thus non-trivial to infer the gauge ﬁeld ﬂuctuations from the bulk dilaton. The approach we will take is to look for a power law form of the gauge ﬁeld Fourier modes A (k) which yields the spectrum of the bulk dilaton ﬂuctuations we µ are starting with. (cid:90) The dominant mode for infrared modes in Tr[F2(ξ)] is the term A˙2. Inserting the Fourier expansion of A (x) yields i i A˙2 = d3k d3k A˙ (k )A˙ (k )ei(k1+k2)xV , (44) i 1 2 i 1 i 2 where V is the normalization volume used to deﬁne the Fourier transform. We introduce new momenta 1 (cid:48) k = (k + k ) 1 2 1 (cid:48) k = (k − k ) . (45) 2 2 We write down the Fourier expansion of the dilat(cid:90)on ﬁeld ϕ, and insert into (30) and (31) to identify Fourier coeﬃcients. This leads to (cid:48) (cid:48) 1 (cid:48) k + k k − k ϕ(k) = d3k A˙ ( )A˙ ( )V 1/2 (46) 4 i 2 i 2 where ϕ is deﬁned in (31). − One way to ﬁnd a consistent A (k) to give rise to a power law bulk spectrum ϕ(k) ∼ k γ is as follows. We can i (cid:48) (cid:48) divide this integral into a region R with k < k and a region R with k > k. In the integral over region R we can (cid:48) 1 2 1 set k = 0 to ﬁnd the approximate result − k3A˙2(k) ∼ k γ (47) µ and hence (making use of the fact that on super-Hubble scales A˙ ∼ HA) − γ+3 Ai(k) ∼ k 2 . (48) (cid:90) (cid:90) It is straightforward to check that this is a consistent solution for any γ > 0. Substituting (48) into (46) we get (cid:48) (cid:48) − − − ϕ(k) ∼ d3k (k2 − (k )2) (γ+3) = k γ d4q(1 − q2) γ+23 (49) The integral over q is convergent when γ > 0. In particular, for a scale invariant spectrum γ = 3/2. For cosmological scales we are interested in, the modes are outside of the Hubble radius at the time −t . Hence, we b can use the small argument limit of the Bessel functions. We will assume that at t = −t both modes have the same b amplitude. Let us denote the total amplitude of A(k) (we are dropping the index i) by 2A˜ (which is k-dependent). − (cid:0)− (cid:1) Then we ﬁnd the following values of the coeﬃcients D and D before the bounce Y J D− = A˜| − t |−1(cid:0)/2 | − kt(cid:1)b| νg Y b 2 D− = A˜| − t |−1/2 | − ktb| −νg . (50) J b 2 At this point we know the equation of motion and the initial conditions for the boundary gauge ﬁeld at the time −t when the boundary theory becomes weakly coupled and when we begin the evolution of the ﬂuctuations on the b boundary. The evolution can be followed by determining the coeﬃcients A, B, D+, D+ by standard matching of the J Y function A˜ (t) and its time derivative at t = ±ξ, as detailed in the next section. k D. Evolution of Boundary Fluctuations through the Singularity  It is useful to discuss matching of solutions of the general form A−(k)F (t) + B−(k)G (t) if t ≤ −ξ, 1 1 A˜ (t) = A exp[βt] + B exp[−βt] if − ξ ≤ t ≤ ξ, k A (k)F (t) + B (k)G (t) if t ≥ ξ. + 2 + 2 Matching A˜ and its time derivative across t = ±ξ then leads to ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: thus

text: non-trivial

text: to

text: infer

text: the

text: gauge

text: ﬁeld

text: ﬂuctuations

text: from

text: the

text: bulk

text: dilaton.

text: The

text: approach

text: we

text: will

text: take

text: is

text: to

text: look

text: for

text: a

text: power

text: law

text: form

text: of

text: the

text: gauge

text: ﬁeld

text: Fourier

text: modes

text: A

text: (k)

text: which

text: yields

text: the

text: spectrum

text: of

text: the

text: bulk

text: dilaton

text: ﬂuctuations

text: we

text: µ

text: are

text: starting

text: with.

text: The

text: dominant

text: mode

text: for

text: infrared

text: modes

text: in

text: is

text: the

text: term

text: Inserting

text: the

text: Fourier

text: expansion

text: of

text: A

text: (x)

text: yields

text: i

text: i

text: A˙

text: (k

text: )A˙

text: (k

text: i

text: i

text: i

text: where

text: V

text: is

text: the

text: normalization

text: volume

text: used

text: to

text: deﬁne

text: the

text: Fourier

text: transform.

text: We

text: introduce

text: new

text: momenta

text: k

text: (k

text: k

text: k

text: (k

text: k

text: We

text: write

text: down

text: the

text: Fourier

text: expansion

text: of

text: the

text: ﬁeld

text: ϕ,

text: and

text: insert

text: into

text: and

text: to

text: identify

text: Fourier

text: coeﬃcients.

text: This

text: leads

text: to

text: k

text: k

text: k

text: k

text: ϕ(k)

text: A˙

text: )A˙

text: )V

text: i

text: i

text: where

text: ϕ

text: is

text: deﬁned

text: in

text: One

text: way

text: to

text: ﬁnd

text: a

text: consistent

text: A

text: (k)

text: to

text: give

text: rise

text: to

text: a

text: power

text: law

text: bulk

text: spectrum

text: ϕ(k)

text: k

text: γ

text: is

text: as

text: follows.

text: We

text: can

text: i

text: divide

text: this

text: integral

text: into

text: a

text: region

text: R

text: with

text: k

text: k

text: and

text: a

text: region

text: R

text: with

text: k

text: k.

text: In

text: the

text: integral

text: over

text: region

text: R

text: we

text: can

text: set

text: k

text: to

text: ﬁnd

text: the

text: approximate

text: result

text: k

text: γ

text: µ

text: and

text: hence

text: (making

text: use

text: of

text: the

text: fact

text: that

text: on

text: super-Hubble

text: scales

text: A˙

text: HA)

text: Ai(k)

text: k

text: It

text: is

text: straightforward

text: to

text: check

text: that

text: this

text: is

text: a

text: consistent

text: solution

text: for

text: any

text: γ

text: Substituting

text: into

text: we

text: get

text: ϕ(k)

text: (k

text: k

text: γ

text: The

text: integral

text: over

text: q

text: is

text: convergent

text: when

text: γ

text: In

text: particular,

text: for

text: a

text: scale

text: invariant

text: spectrum

text: γ

text: For

text: cosmological

text: scales

text: we

text: are

text: interested

text: in,

text: the

text: modes

text: are

text: outside

text: of

text: the

text: Hubble

text: radius

text: at

text: the

text: time

text: −t

text: Hence,

text: we

text: b

text: can

text: use

text: the

text: small

text: argument

text: limit

text: of

text: the

text: Bessel

text: functions.

text: We

text: will

text: assume

text: that

text: at

text: t

text: −t

text: both

text: modes

text: have

text: the

text: same

text: b

text: amplitude.

text: Let

text: us

text: denote

text: the

text: total

text: amplitude

text: of

text: A(k)

text: (we

text: are

text: dropping

text: the

text: index

text: i)

text: by

text: (which

text: is

text: k-dependent).

text: Then

text: we

text: ﬁnd

text: the

text: following

text: values

text: of

text: the

text: coeﬃcients

text: D

text: and

text: D

text: before

text: the

text: bounce

text: Y

text: J

text: D−

text: A˜|

text: t

text: νg

text: Y

text: b

text: D−

text: A˜|

text: t

text: ktb|

text: −νg

text: J

text: b

text: At

text: this

text: point

text: we

text: know

text: the

text: equation

text: of

text: motion

text: and

text: the

text: initial

text: conditions

text: for

text: the

text: boundary

text: gauge

text: ﬁeld

text: at

text: the

text: time

text: −t

text: when

text: the

text: boundary

text: theory

text: becomes

text: weakly

text: coupled

text: and

text: when

text: we

text: begin

text: the

text: evolution

text: of

text: the

text: ﬂuctuations

text: on

text: the

text: b

text: boundary.

text: The

text: evolution

text: can

text: be

text: followed

text: by

text: determining

text: the

text: coeﬃcients

text: A,

text: B,

text: D+,

text: D+

text: by

text: standard

text: matching

text: of

text: the

text: J

text: Y

text: function

text: A˜

text: (t)

text: and

text: its

text: time

text: derivative

text: at

text: t

text: ±ξ,

text: as

text: detailed

text: in

text: the

text: next

text: section.

text: k

text: D.

text: Evolution

text: of

text: Boundary

text: Fluctuations

text: through

text: the

text: Singularity

text: It

text: is

text: useful

text: to

text: discuss

text: matching

text: of

text: solutions

text: of

text: the

text: general

text: form

text: A−(k)F

text: (t)

text: B−(k)G

text: (t)

text: if

text: t

text: −ξ,

text: A˜

text: (t)

text: A

text: exp[βt]

text: B

text: exp[−βt]

text: if

text: ξ

text: t

text: ξ,

text: k

text: A

text: (k)F

text: (t)

text: B

text: (k)G

text: (t)

text: if

text: t

text: ξ.

text: Matching

text: A˜

text: and

text: its

text: time

text: derivative

text: across

text: t

text: ±ξ

text: then

text: leads

text: to

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 9 (cid:87) (cid:87) B+ (cid:91) x -(cid:91) c (cid:87) B- k-1 |H|-1 FIG. 2: Space-time sketch of the relevant times and length scales in our deformed AdS . The horizontal axis is comoving 5 spatial coordinate (in direction perpendicular to the radial direction. The vertical direction is conformal time. τ and τ − ± B+ B are the times when the coupling constant is 1, the times ξ occur when the bulk curvature reaches string scale. The vertical line represents the wavelength of a mode which we are interested in. Note that the length is larger than the Hubble radius at τ and τ −. The blue lines at 45o indicate the Hubble radius. B+ B We are interested in evolving linearized bulk ﬂuctuations. This means we will turn on normalizable bulk modes. Hence, we will be evolving the linear ﬂuctuations in the gauge ﬁeld A on the boundary. The initial condiitions for µ the gauge ﬁeld ﬂuctuations on the boundary are set by the dilaton perturbations via the linearized version of the correspondence 1 ϕ(y) → (cid:104)Tr[F2(y)](cid:105) , (30) g2 YM where the normalizable dilaton ﬂuctuation behaves in the standard fashion as ϕ(z, y) → z4ϕ(y) (31) as one approaches the boundary z = 0. The eﬀects of linear dilaton ﬂuctuations on the evolution of the gauge ﬁeld ﬂuctuations would be a second order eﬀect in the amplitude of ﬂuctuations. Hence, at linear level in perturbation theory there is no such coupling and the only eﬀect of the dilaton on the gauge ﬁeld ﬂuctuations is via the dilaton-dependence of the gauge coupling constant at background level. Given a spectrum of dilaton ﬂuctuations at early times in the bulk, we will evolve them in the bulk until bulk per- turbation theory breaks down at time −t . At that point, we compute the boundary values of the dilaton ﬂuctuations b and use them to determine the initial values of the gauge ﬁeld ﬂuctuations δA at that time. mu B. Evolution of the Boundary Fluctuations Before the Singularity As we have mentioned, the dilaton ﬁeld evolves as a function of cosmic time and initially takes a large value, and thus the Yang Mills coupling is also very large initially which implies that the boundary theory is strongly coupled ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

font: default

text: B+

text: x

text: c

text: B-

text: FIG.

text: Space-time

text: sketch

text: of

text: the

text: relevant

text: times

text: and

text: length

text: scales

text: in

text: our

text: deformed

text: AdS

text: The

text: horizontal

text: axis

text: is

text: comoving

text: spatial

text: coordinate

text: (in

text: direction

text: perpendicular

text: to

text: the

text: radial

text: direction.

text: The

text: vertical

text: direction

text: is

text: conformal

text: time.

text: τ

text: and

text: τ

text: B+

text: B

text: are

text: the

text: times

text: when

text: the

text: coupling

text: constant

text: is

text: the

text: times

text: ξ

text: occur

text: when

text: the

text: bulk

text: curvature

text: reaches

text: string

text: scale.

text: The

text: vertical

text: line

text: represents

text: the

text: wavelength

text: of

text: a

text: mode

text: which

text: we

text: are

text: interested

text: in.

text: Note

text: that

text: the

text: length

text: is

text: larger

text: than

text: the

text: Hubble

text: radius

text: at

text: τ

text: and

text: τ

text: The

text: blue

text: lines

text: at

text: indicate

text: the

text: Hubble

text: radius.

text: B+

text: B

text: We

text: are

text: interested

text: in

text: evolving

text: linearized

text: bulk

text: ﬂuctuations.

text: This

text: means

text: we

text: will

text: turn

text: on

text: normalizable

text: bulk

text: modes.

text: Hence,

text: we

text: will

text: be

text: evolving

text: the

text: linear

text: ﬂuctuations

text: in

text: the

text: gauge

text: ﬁeld

text: A

text: on

text: the

text: boundary.

text: The

text: initial

text: condiitions

text: for

text: µ

text: the

text: gauge

text: ﬁeld

text: ﬂuctuations

text: on

text: the

text: boundary

text: are

text: set

text: by

text: the

text: dilaton

text: perturbations

text: via

text: the

text: linearized

text: version

text: of

text: the

text: correspondence

text: ϕ(y)

text: YM

text: where

text: the

text: normalizable

text: dilaton

text: ﬂuctuation

text: behaves

text: in

text: the

text: standard

text: fashion

text: as

text: ϕ(z,

text: y)

text: as

text: one

text: approaches

text: the

text: boundary

text: z

text: The

text: eﬀects

text: of

text: linear

text: dilaton

text: ﬂuctuations

text: on

text: the

text: evolution

text: of

text: the

text: gauge

text: ﬁeld

text: ﬂuctuations

text: would

text: be

text: a

text: second

text: order

text: eﬀect

text: in

text: the

text: amplitude

text: of

text: ﬂuctuations.

text: Hence,

text: at

text: linear

text: level

text: in

text: perturbation

text: theory

text: there

text: is

text: no

text: such

text: coupling

text: and

text: the

text: only

text: eﬀect

text: of

text: the

text: dilaton

text: on

text: the

text: gauge

text: ﬁeld

text: ﬂuctuations

text: is

text: via

text: the

text: dilaton-dependence

text: of

text: the

text: gauge

text: coupling

text: constant

text: at

text: background

text: level.

text: Given

text: a

text: spectrum

text: of

text: dilaton

text: ﬂuctuations

text: at

text: early

text: times

text: in

text: the

text: bulk,

text: we

text: will

text: evolve

text: them

text: in

text: the

text: bulk

text: until

text: bulk

text: per-

text: turbation

text: theory

text: breaks

text: down

text: at

text: time

text: −t

text: At

text: that

text: point,

text: we

text: compute

text: the

text: boundary

text: values

text: of

text: the

text: dilaton

text: ﬂuctuations

text: b

text: and

text: use

text: them

text: to

text: determine

text: the

text: initial

text: values

text: of

text: the

text: gauge

text: ﬁeld

text: ﬂuctuations

text: δA

text: at

text: that

text: time.

text: mu

text: B.

text: Evolution

text: of

text: the

text: Boundary

text: Fluctuations

text: Before

text: the

text: Singularity

text: As

text: we

text: have

text: mentioned,

text: the

text: dilaton

text: ﬁeld

text: evolves

text: as

text: a

text: function

text: of

text: cosmic

text: time

text: and

text: initially

text: takes

text: a

text: large

text: value,

text: and

text: thus

text: the

text: Yang

text: Mills

text: coupling

text: is

text: also

text: very

text: large

text: initially

text: which

text: implies

text: that

text: the

text: boundary

text: theory

text: is

text: strongly

text: coupled

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##