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Continuous Time Random Walks in the Continuum Limit Simulation of Atmospheric Wind Speeds David Kleinhans, Joachim Peinke, Rudolf Friedrich kleinhan@uni-muenster.de FORWIND Institut f ur Physik Institut f ur Theoretische Physik

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Continuous Time Random Walks in the Continuum Limit

Simulation of Atmospheric Wind Speeds

David Kleinhans, Joachim Peinke, Rudolf Friedrich

kleinhan@uni-muenster.de

FORWIND Zentrum f¨ ur Windenergieforschung D-26129 Oldenburg, Germany Institut f¨ ur Physik Carl-von-Ossietzky-Universit¨ at Oldenburg D-26111 Oldenburg, Germany Institut f¨ ur Theoretische Physik Westf¨ alische Wilhelms-Universit¨ at M¨ unster D-48149 M¨ unster, Germany

Version vom 31. Januar 2008 David Kleinhans — Continuous Time Random Walks in the Continuum Limit — Stochastics in Turbulence and Finance, January 2008 1

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SLIDE 2

Turbulence and Finance: Instationary Processes

David Kleinhans — Continuous Time Random Walks in the Continuum Limit — Stochastics in Turbulence and Finance, January 2008 2

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SLIDE 3

Motivation

[Böttcher, Bath & Peinke 2007]

–0.04 0.00 0.04 –4 4 8 y(τ) log p(y(τ))

[Nawroth & Peinke 2003] u Log(P(u)) [arb. units] [Friedrich 2003] [Mordant, Metz, Michel & Pinton 2001]

David Kleinhans — Continuous Time Random Walks in the Continuum Limit — Stochastics in Turbulence and Finance, January 2008 3

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SLIDE 4

Motivation

[Böttcher, Bath & Peinke 2007]

–0.04 0.00 0.04 –4 4 8 y(τ) log p(y(τ))

[Nawroth & Peinke 2003] u Log(P(u)) [arb. units] [Friedrich 2003] [Mordant, Metz, Michel & Pinton 2001]

CTRWs are potential generators of Lagrangian tracer dynamics Similar structure of increment statistis in Finance and Atmospheric turbulence

David Kleinhans — Continuous Time Random Walks in the Continuum Limit — Stochastics in Turbulence and Finance, January 2008 3

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SLIDE 5

Motivation

[Böttcher, Bath & Peinke 2007]

–0.04 0.00 0.04 –4 4 8 y(τ) log p(y(τ))

[Nawroth & Peinke 2003] u Log(P(u)) [arb. units] [Friedrich 2003] [Mordant, Metz, Michel & Pinton 2001]

Outline: Introduction to Continuous Time Random Walks (CTRWs) Initial definition Continuous sample paths, application to finance CTRW model for atmospheric turbulence

David Kleinhans — Continuous Time Random Walks in the Continuum Limit — Stochastics in Turbulence and Finance, January 2008 3

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SLIDE 6

Continuous time random walks (CTRWs)

David Kleinhans — Continuous Time Random Walks in the Continuum Limit — Stochastics in Turbulence and Finance, January 2008 4

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SLIDE 7

Discrete Random Walks

[Metzler & Klafter 2000]

Continuous time random walk: xi+1 = xi + ηi ti+1 = ti + τi PDFs of jumping times are continuous, paths are discontinuous

David Kleinhans — Continuous Time Random Walks in the Continuum Limit — Stochastics in Turbulence and Finance, January 2008 5

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Diffusion Limit of CTRWs

Evolution of CTRWs Sums of random variables ⇒ Fourier / Laplace-Representation Montroll-Weiss equation: ˆ ˜ P(k, u) = 1 − ˆ Pt(u) u h 1 − ˜ Px(k) ˆ Pt(u) i .

(1)

David Kleinhans — Continuous Time Random Walks in the Continuum Limit — Stochastics in Turbulence and Finance, January 2008 6

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Diffusion Limit of CTRWs

Evolution of CTRWs Sums of random variables ⇒ Fourier / Laplace-Representation Montroll-Weiss equation: ˆ ˜ P(k, u) = 1 − ˆ Pt(u) u h 1 − ˜ Px(k) ˆ Pt(u) i .

(2)

Diffusion Limit (⇒ Long Time) Assumptions: Jump PDF has finite variance Asymptotical Power-Law decay ∼ x−(1+α) of waiting time PDF (heavy tailed) Fractional Diffusion equation ∂ ∂tW(x, t) = δ(k)δ(t) + σ2 T α 0 D1−α

t

X

i,j

∂2 ∂xi∂xj W(x, t)

(3)

David Kleinhans — Continuous Time Random Walks in the Continuum Limit — Stochastics in Turbulence and Finance, January 2008 6

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Diffusion Limit of CTRWs

With 0D1−α

t

f(t), 0 < α ≤ 1: Riemann-Liouville integro-differential fractional operator

0D1−α t

f(t) := 1 Γ(α) ∂ ∂t

∞

Z dt′ f(t) (t − t′)1−α

(4)

Connection to integer PDEs: Memory Kernel [Metzler & Klafter 2000, Barkai 2001] W(x, t) =

∞

Z s A(s|t)W1(x, (Kα/K1)1/α s)

(5)

with A(s|t) = 1 α t s1+1/α Lα,1 „ t s1/α «

(6)

David Kleinhans — Continuous Time Random Walks in the Continuum Limit — Stochastics in Turbulence and Finance, January 2008 6

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Long Time Limit: t ≫ 1

Waiting time PDF P(τ) =

8 > < > : τ ≥ 0 : q

2 πσ2 exp

„

τ2 2σ2

« τ < 0 : 8 < : τ ≤ 100 : N (0.8, 100)L0.8,1(τ) τ > 100 : L0.8,1(τ)

  • 6
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2 4 6 8 25 50 75 100 125 150 x1(t) t

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  • 2

2 4 6 8 25 50 75 100 125 150 x2(t) t

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  • 2

2 4 6 8 25 50 75 100 125 150 x3(t) t

  • 40
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20 40 60 80 100 120 140 160 2500 5000 7500 10000 12500 15000 x1(t) t

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20 40 60 80 100 120 140 160 2500 5000 7500 10000 12500 15000 x2(t) t

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  • 20

20 40 60 80 100 120 140 160 2500 5000 7500 10000 12500 15000 x3(t) t David Kleinhans — Continuous Time Random Walks in the Continuum Limit — Stochastics in Turbulence and Finance, January 2008 7

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Continuous Trajectories at Finite Time

Adequate scaling of processes: Continuous (fractional) trajectories at finite time ⇒ Monte-Carlo simulation of fractional PDEs

David Kleinhans — Continuous Time Random Walks in the Continuum Limit — Stochastics in Turbulence and Finance, January 2008 8

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Continuous Trajectories at Finite Time

Adequate scaling of processes: Continuous (fractional) trajectories at finite time ⇒ Monte-Carlo simulation of fractional PDEs Initial work: Heinsalu, Patriarca, Goychuk, Schmid & Hänggi 2006, Fractional Fokker-Planck dynamics: Numerical algorithm and simulations

David Kleinhans — Continuous Time Random Walks in the Continuum Limit — Stochastics in Turbulence and Finance, January 2008 8

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Continuous Trajectories at Finite Time

Adequate scaling of processes: Continuous (fractional) trajectories at finite time ⇒ Monte-Carlo simulation of fractional PDEs Initial work: Heinsalu, Patriarca, Goychuk, Schmid & Hänggi 2006, Fractional Fokker-Planck dynamics: Numerical algorithm and simulations Recent advancements: Magdziarz & Weron 2007, Fractional FP dynamics: Stochastic representation and computer simulation Gorenflo, Mainardi & Vivoli 2007, Continuous-time random walk and parametric subordination in fractional diffusion Kleinhans & Friedrich 2007, Continuous time random walks: Simulation of continuous trajectories

David Kleinhans — Continuous Time Random Walks in the Continuum Limit — Stochastics in Turbulence and Finance, January 2008 8

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Continuum limit [Fogedby 1994]

According to Fogedby: Continuum limit of discrete equations, xi+1 = xi + ηi ti+1 = ti + τi 9 > = > ;

i→s

⇒ 8 > < > :

∂ ∂s x(s)

= η(s)

∂ ∂s t(s)

= τ(s)

David Kleinhans — Continuous Time Random Walks in the Continuum Limit — Stochastics in Turbulence and Finance, January 2008 9

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SLIDE 16

Continuum limit [Fogedby 1994]

According to Fogedby: Continuum limit of discrete equations, xi+1 = xi + ηi ti+1 = ti + τi 9 > = > ;

i→s

⇒ 8 > < > :

∂ ∂s x(s)

= η(s)

∂ ∂s t(s)

= τ(s) Notation: dxs = dWs dts = dLα

s

David Kleinhans — Continuous Time Random Walks in the Continuum Limit — Stochastics in Turbulence and Finance, January 2008 9

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SLIDE 17

Continuum limit [Fogedby 1994]

According to Fogedby: Continuum limit of discrete equations, xi+1 = xi + ηi ti+1 = ti + τi 9 > = > ;

i→s

⇒ 8 > < > :

∂ ∂s x(s)

= η(s)

∂ ∂s t(s)

= τ(s)

David Kleinhans — Continuous Time Random Walks in the Continuum Limit — Stochastics in Turbulence and Finance, January 2008 9

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SLIDE 18

Continuum limit [Fogedby 1994]

According to Fogedby: Continuum limit of discrete equations, xi+1 = xi + ηi ti+1 = ti + τi 9 > = > ;

i→s

⇒ 8 > < > :

∂ ∂s x(s)

= η(s)

∂ ∂s t(s)

= τ(s) Here: η(s) and τ(s) have to obey stable distribution Lα,1(x) = 1 π Re 8 < :

∞

Z dk exp h −ikx − |k|α exp “ −iπα 2 ”i 9 = ;

0.5 1 1.5 2 2.5 0.5 1 1.5 2 Lα(x) x α=0.9 α=0.8 α=0.7 α=0.6 α=0.5 α=0.4

Result: Fractional dynamics at finite time

David Kleinhans — Continuous Time Random Walks in the Continuum Limit — Stochastics in Turbulence and Finance, January 2008 9

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Intrinsic Time in Finance [Müller 1993]

David Kleinhans — Continuous Time Random Walks in the Continuum Limit — Stochastics in Turbulence and Finance, January 2008 10

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Intrinsic Time in Finance [Müller 1993]

David Kleinhans — Continuous Time Random Walks in the Continuum Limit — Stochastics in Turbulence and Finance, January 2008 10

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Numerical algorithm

Numerical Integration Scheme: (Itô) x(s + ∆s) = x(s) + ∆sF(x(s)) + (∆s)1/2 η(s) t(s + ∆s) = t(s) + (∆s)1/α τα(s) .

David Kleinhans — Continuous Time Random Walks in the Continuum Limit — Stochastics in Turbulence and Finance, January 2008 11

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Numerical algorithm

Numerical Integration Scheme: (Itô) x(s + ∆s) = x(s) + ∆sF(x(s)) + (∆s)1/2 η(s) t(s + ∆s) = t(s) + (∆s)1/α τα(s) . Discontinuous character of t(s):

0.5 1 1.5 2 2.5 3 3.5 4 0.5 1 1.5 2 2.5 3 t(s) s

David Kleinhans — Continuous Time Random Walks in the Continuum Limit — Stochastics in Turbulence and Finance, January 2008 11

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Numerical algorithm

Numerical Integration Scheme: (Itô) x(s + ∆s) = x(s) + ∆sF(x(s)) + (∆s)1/2 η(s) t(s + ∆s) = t(s) + (∆s)1/α τα(s) . Algorithm for simulation of x(t): Initialisation of xs(0) and ts(0), set s = 0 for every j = 0 to N:

  • 1. while (ts(s) < tj):

(a) calculate xs(s + ∆s) and ts(s + ∆s) from discrete equations (see above) (b) increase s by ∆s

  • 2. set x(tj) := xs(s)

David Kleinhans — Continuous Time Random Walks in the Continuum Limit — Stochastics in Turbulence and Finance, January 2008 11

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Numerical algorithm

Numerical Integration Scheme: (Itô) x(s + ∆s) = x(s) + ∆sF(x(s)) + (∆s)1/2 η(s) t(s + ∆s) = t(s) + (∆s)1/α τα(s) . Appropriate ∆s can be determined:

10-2 10-1 100 10-5 10-4 10-3 10-2 10-1 100 ∆s √〈∆x2〉

David Kleinhans — Continuous Time Random Walks in the Continuum Limit — Stochastics in Turbulence and Finance, January 2008 11

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Numerical results

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  • 1.5
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  • 0.5

0.5 1 1.5 2 2.5 1 2 3 4 5 6 7 8 9 10 2 4 6 8 10 x(t) s(t) t α=1.00

  • 2
  • 1.5
  • 1
  • 0.5

0.5 1 1.5 2 2.5 1 2 3 4 5 6 7 8 9 10 2 4 6 8 10 x(t) s(t) t α=0.95

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  • 1.5
  • 1
  • 0.5

0.5 1 1.5 2 2.5 1 2 3 4 5 6 7 8 9 10 2 4 6 8 10 x(t) s(t) t α=0.90

  • 2
  • 1.5
  • 1
  • 0.5

0.5 1 1.5 2 2.5 1 2 3 4 5 6 7 8 9 10 2 4 6 8 10 x(t) s(t) t α=0.85

David Kleinhans — Continuous Time Random Walks in the Continuum Limit — Stochastics in Turbulence and Finance, January 2008 12

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Validation: Fractional statistics

0.05 0.1 0.15 0.2 0.25

  • 4
  • 2

2 4 6 P(x) x

David Kleinhans — Continuous Time Random Walks in the Continuum Limit — Stochastics in Turbulence and Finance, January 2008 13

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Validation: Fractional statistics

0.2 0.4 0.6 0.8 1 1 2 3 4 5 6 7 8 9 10 <x(t1=1)x(t2)> t2

[Kleinhans & Friedrich 2007, Baule & Friedrich 2007]

David Kleinhans — Continuous Time Random Walks in the Continuum Limit — Stochastics in Turbulence and Finance, January 2008 13

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SLIDE 28

Simulation of Atmospheric Wind Fields

David Kleinhans — Continuous Time Random Walks in the Continuum Limit — Stochastics in Turbulence and Finance, January 2008 14

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Atmospheric Wind Fields

[NASA visible earth]

20 40 60 80 100 2 4 6 8 10 wind speed (m/s) height (m)

logarithmic profile: atmosphere unstable / neutral atmosphere stable

[Berg (2004)] [R. Gasch, Windkraftanlagen, Teubner, 1993.]

David Kleinhans — Continuous Time Random Walks in the Continuum Limit — Stochastics in Turbulence and Finance, January 2008 15

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Turbulence intensity

Engineer: Character of wind field fully specified by (here: T = 600s) Mean wind speed u(t)T Turbulence intensity TI := u2T −[u(t)T ]2

u(t)T

David Kleinhans — Continuous Time Random Walks in the Continuum Limit — Stochastics in Turbulence and Finance, January 2008 16

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Turbulence intensity

Engineer: Character of wind field fully specified by (here: T = 600s) Mean wind speed u(t)T Turbulence intensity TI := u2T −[u(t)T ]2

u(t)T

Example: Temporal drift of wind velocity u(t) = u0 + at + u′(t) with

˙ u′(t) ¸ = ˙ tu′(t) ¸ = 0 D u′2(t) E = σ2

TI = 100 q a2 T 2

12 + σ2/u0

⇒ Turbulence intensity very weak parameter

7 8 9 10 11 12 13

  • 300
  • 150

150 300 u(t) [m/s] t [s] 7 8 9 10 11 12 13

  • 300
  • 150

150 300 u(t) [m/s] t [s]

Two sample time series with turbulence intensity TI= 10%

David Kleinhans — Continuous Time Random Walks in the Continuum Limit — Stochastics in Turbulence and Finance, January 2008 16

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Windfield models: Overview

DNS, Large Eddy Simulation (LES), Reynolds Averaged Navier Stokes (RANS) Spectral Models (IEC compliant) Veers: Three-dimensional wind fiels simulation (1984) Mann: Wind field simulation (1998) some further works e.g. by Bierbooms, Nielsen, Larsen, Hansen . . . (Fung: Kinematic Simulation (KS) (1992)) Fractal models Schertzer, Lovejoy: . . . anisotropic scaling multiplicative processes (1987) Cleve: Fractional Brownian motion (2005) Stochastic / Probabilistic Nawroth: Reconstruction of processes with Markov properties in scale Schmiegel / Barndorff-Nielsen: Delay kernel Cleve: Cascade model

David Kleinhans — Continuous Time Random Walks in the Continuum Limit — Stochastics in Turbulence and Finance, January 2008 17

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Spectral surrogate data (1D)[Shinozuka, Deodatis 1991]

Aim: Simulation of u′(t) with u′ = 0 and proper energy spectrum. ’Fourier-Stieltjes-Integral’: u′(t) =

∞

R [cos(ωt)du(ω) + sin(ωt)dv(ω)] Properties of the coefficients (for ω, ω′ ≥ 0): Mean du(ω) = dv(ω) = 0 Autocorrelation du(ω)du(ω′) = 2δ(ω′ − ω)S(ω)dω dv(ω)dv(ω′) = 2δ(ω′ − ω)S(ω)dω Crosscorrelation du(ω)dv(ω′) = 0

David Kleinhans — Continuous Time Random Walks in the Continuum Limit — Stochastics in Turbulence and Finance, January 2008 18

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Intermittent velocity increments

Atmospheric measurement: Spectral surrogates: [Shinozuka, Deodatis 1991; Veers 1988; Mann 1998]

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2 4 6 8 10 12 14

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  • 3
  • 2
  • 1

1 2 3 4 5 Log[P(∆x)] ∆x [σ m/s]

Spectral surrogates do not reproduce intermittent statistics How could they? Advanced methods for fast simulation of atmospheric winds required

David Kleinhans — Continuous Time Random Walks in the Continuum Limit — Stochastics in Turbulence and Finance, January 2008 19

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Motivation: Starting point

Requirements Intermittent statistics of atmospheric wind Simulation of a 2D (3D?) field in time Adaptability to measured wind data Recent work: [Friedrich (2003)] Motivation of the applicability of CTRW’s for Lagrangian tracers [Baule (2005)] Several tools for the application of CTRW processes. [Castaing (1990), Beck (2003), Böttcher (2005)] Analysis of turbulent data sets: Generation of intermittency by superposition of gaussian processes.

David Kleinhans — Continuous Time Random Walks in the Continuum Limit — Stochastics in Turbulence and Finance, January 2008 20

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Model

Coupled Langevin equations ∂ ∂txref(s) = −γref ˆ xref(s) − x0 ˜ + q DrefΓref(s) ∂ ∂sxi(s) = −γ ˆ xi(s) − αixref(s) ˜ + X

j

p DijΓj(s) ∀ i ∂ ∂st(s) = τ(s) Properties x0: Mean wind speed at reference height Reference process xref: Models wind variations on larger timescales (γref ≪ γ)

David Kleinhans — Continuous Time Random Walks in the Continuum Limit — Stochastics in Turbulence and Finance, January 2008 21

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Model

Coupled Langevin equations ∂ ∂txref(s) = −γref ˆ xref(s) − x0 ˜ + q DrefΓref(s) ∂ ∂sxi(s) = −γ ˆ xi(s) − αixref(s) ˜ + X

j

p DijΓj(s) ∀ i ∂ ∂st(s) = τ(s) Properties x0: Mean wind speed at reference height Reference process xref: Models wind variations on larger timescales (γref ≪ γ) αi incorporate (logarithmic) wind profile Interaction of fluctuations decays exponentially, Dij ∼ exp(− ˛ ˛ri − rj ˛ ˛) Kramers-like equations (11a,11a) can be treates analytically Stochastic process t(s) naturally introduces intermittency

David Kleinhans — Continuous Time Random Walks in the Continuum Limit — Stochastics in Turbulence and Finance, January 2008 21

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Fokker-Planck Equation

Evolution equation for PDF: ∂ ∂s P(x, xref, s) = h ∇xγ ˆ x + αxref ˜ + ∇x∇T

x D

(13)

+ ∂ ∂xref γ ˆ xref + x0 ˜ + ∂2 ∂x2

ref

Dref # P(x, xref, s) Time evolution of expectation values Estimation of stationary results Here: Mean and Variance: xi = αix0

(14)

D (xi − xi)2E = D γ + α2

i

γ γ + γref Dref γref

(15)

David Kleinhans — Continuous Time Random Walks in the Continuum Limit — Stochastics in Turbulence and Finance, January 2008 22

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Spectrum

Spectrum of (hidden) x(s): (s ≫ γ, γref und i ≡ k) f(ω) =  D + α2

i

γ3 (γ + γref)(γ − γref) Dref γref ff 1 γ2 + ω2 +α2

i

γ2 (γ + γref)(γ − γref)Dref 1 γ2

ref + ω2

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  • 1.4
  • 1.2
  • 1
  • 0.8
  • 0.6
  • 0.4
  • 0.2
  • 10
  • 8
  • 6
  • 4
  • 2
2 4

d d(log[ω]) log

  • f(log[ω])
  • log[ω]

David Kleinhans — Continuous Time Random Walks in the Continuum Limit — Stochastics in Turbulence and Finance, January 2008 23

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SLIDE 40

Spectrum

Spectrum of (hidden) x(s): (s ≫ γ, γref und i ≡ k) f(ω) =  D + α2

i

γ3 (γ + γref)(γ − γref) Dref γref ff 1 γ2 + ω2 +α2

i

γ2 (γ + γref)(γ − γref)Dref 1 γ2

ref + ω2

  • 2
  • 1.8
  • 1.6
  • 1.4
  • 1.2
  • 1
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  • 0.6
  • 0.4
  • 0.2
  • 10
  • 8
  • 6
  • 4
  • 2
2 4

d d(log[ω]) log

  • f(log[ω])
  • log[ω]

David Kleinhans — Continuous Time Random Walks in the Continuum Limit — Stochastics in Turbulence and Finance, January 2008 23

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SLIDE 41

Comparisson: Measurement ↔ Model

7 8 9 10 11 12 13 100 200 300 400 500 600 x(t) t [s]

  • 5

5 10

  • 6
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  • 2

2 4 6 Log[P(∆x)] ∆x [σ m/s] 4 6 8 10 12 14 16 18 100 200 300 400 500 600 x(t) t [s]

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  • 2

2 4 6 8 10 12 14

  • 4
  • 3
  • 2
  • 1

1 2 3 4 5 Log[P(∆x)] ∆x [σ m/s] 2 4 6 8 10 12 14 16 18 100 200 300 400 500 600 x(t) t [s]

  • 5

5 10

  • 6
  • 4
  • 2

2 4 6 Log[P(∆x)] ∆x [σ m/s]

David Kleinhans — Continuous Time Random Walks in the Continuum Limit — Stochastics in Turbulence and Finance, January 2008 24

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SLIDE 42

Results: Loads on Wind Turbine Blade

[H. Gontier, A.P . Schaffarczyk, Fachhochschule Kiel, Germany]

David Kleinhans — Continuous Time Random Walks in the Continuum Limit — Stochastics in Turbulence and Finance, January 2008 25

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Results: Loads on Wind Turbine Blade

[H. Gontier, A.P . Schaffarczyk, Fachhochschule Kiel, Germany]

David Kleinhans — Continuous Time Random Walks in the Continuum Limit — Stochastics in Turbulence and Finance, January 2008 25

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SLIDE 44

Results: Loads on Wind Turbine Blade

[H. Gontier, A.P . Schaffarczyk, Fachhochschule Kiel, Germany]

David Kleinhans — Continuous Time Random Walks in the Continuum Limit — Stochastics in Turbulence and Finance, January 2008 25

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SLIDE 45

Conclusion

David Kleinhans — Continuous Time Random Walks in the Continuum Limit — Stochastics in Turbulence and Finance, January 2008 26

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SLIDE 46

Conclusion

CTRWs Powerfull tool for numerical simulation of fractional processes Simulation of continuous sample paths ⇒ Accurate reproduction of fractional dynamics at finite time Multivariate joint statistics? Wind field model Modells currently applied: Poor reproduction of intermittent statistics Stochastic approach based on CTRW ⇒ Intermittency can be controlled Infinite waiting times unphysical: Truncation of power laws ( ) Multiplicative character of atmospheric turbulence?

David Kleinhans — Continuous Time Random Walks in the Continuum Limit — Stochastics in Turbulence and Finance, January 2008 27

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SLIDE 47

Conclusion

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 5 10 15 20 25 30 35 sqrt[<(u-<u>)2>] [m/s] u [m/s] Fit: 0.027 u + 0.009 Analysis of FINO data

David Kleinhans — Continuous Time Random Walks in the Continuum Limit — Stochastics in Turbulence and Finance, January 2008 27

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SLIDE 48

Thank you for your attention!

David Kleinhans — Continuous Time Random Walks in the Continuum Limit — Stochastics in Turbulence and Finance, January 2008 28