Continuous Time Random Walks in the Continuum Limit Simulation of - PowerPoint PPT Presentation

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

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

Motivation Log(P(u)) [arb. units] 8 log p(y( τ )) 4 0 u –4 –0.04 0.00 0.04 [Friedrich 2003] y( τ ) [Böttcher, Bath & Peinke 2007] [Mordant, Metz, Michel & Pinton 2001] [Nawroth & Peinke 2003] David Kleinhans — Continuous Time Random Walks in the Continuum Limit — Stochastics in Turbulence and Finance, January 2008 3

Motivation Log(P(u)) [arb. units] 8 log p(y( τ )) 4 0 u –4 –0.04 0.00 0.04 [Friedrich 2003] y( τ ) [Böttcher, Bath & Peinke 2007] [Mordant, Metz, Michel & Pinton 2001] [Nawroth & Peinke 2003] 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

Motivation Log(P(u)) [arb. units] 8 log p(y( τ )) 4 0 u –4 –0.04 0.00 0.04 [Friedrich 2003] y( τ ) [Böttcher, Bath & Peinke 2007] [Mordant, Metz, Michel & Pinton 2001] [Nawroth & Peinke 2003] 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

Continuous time random walks (CTRWs) David Kleinhans — Continuous Time Random Walks in the Continuum Limit — Stochastics in Turbulence and Finance, January 2008 4

Discrete Random Walks [Metzler & Klafter 2000] Continuous time random walk: x i +1 = x i + η i t i +1 = t i + τ i 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

Diffusion Limit of CTRWs Evolution of CTRWs Sums of random variables ⇒ Fourier / Laplace-Representation Montroll-Weiss equation: 1 − ˆ P t ( u ) ˆ ˜ P ( k , u ) = . (1) h i 1 − ˜ P x ( k ) ˆ u P t ( u ) David Kleinhans — Continuous Time Random Walks in the Continuum Limit — Stochastics in Turbulence and Finance, January 2008 6

Diffusion Limit of CTRWs Evolution of CTRWs Sums of random variables ⇒ Fourier / Laplace-Representation Montroll-Weiss equation: 1 − ˆ P t ( u ) ˆ ˜ P ( k , u ) = . (2) h i 1 − ˜ P x ( k ) ˆ u P t ( u ) 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 ) + σ 2 ∂ 2 ∂ D 1 − α X W ( x , t ) (3) t T α 0 ∂x i ∂x j i,j David Kleinhans — Continuous Time Random Walks in the Continuum Limit — Stochastics in Turbulence and Finance, January 2008 6

Diffusion Limit of CTRWs With 0 D 1 − α f ( t ) , 0 < α ≤ 1 : Riemann-Liouville integro-differential fractional operator t ∞ 1 ∂ Z f ( t ) 0 D 1 − α dt ′ f ( t ) := (4) t ( t − t ′ ) 1 − α Γ( α ) ∂t 0 Connection to integer PDEs: Memory Kernel [Metzler & Klafter 2000, Barkai 2001] ∞ Z s A ( s | t ) W 1 ( x , ( K α /K 1 ) 1 /α s ) W ( x , t ) = (5) 0 with A ( s | t ) = 1 t „ t « s 1+1 /α L α, 1 (6) s 1 /α α David Kleinhans — Continuous Time Random Walks in the Continuum Limit — Stochastics in Turbulence and Finance, January 2008 6

Long Time Limit: t ≫ 1 Waiting time PDF P ( τ ) = 8 160 140 6 120 4 100 8 „ « τ 2 q 2 τ ≥ 0 : πσ 2 exp > 80 2 < 2 σ 2 x 1 (t) x 1 (t) 60 0 40 > τ < 0 : 0 : 20 -2 0 -4 -20 -6 -40 0 25 50 75 100 125 150 0 2500 5000 7500 10000 12500 15000 t t 8 160 140 6 120 8 4 100 τ ≤ 100 N (0 . 8 , 100) L 0 . 8 , 1 ( τ ) : < 80 2 x 2 (t) x 2 (t) 60 τ > 100 : 0 : 0 40 20 -2 0 -4 -20 -6 -40 0 25 50 75 100 125 150 0 2500 5000 7500 10000 12500 15000 t t 8 160 140 6 120 4 100 80 L 0 . 8 , 1 ( τ ) 2 x 3 (t) x 3 (t) 60 0 40 20 -2 0 -4 -20 -6 -40 0 25 50 75 100 125 150 0 2500 5000 7500 10000 12500 15000 t t David Kleinhans — Continuous Time Random Walks in the Continuum Limit — Stochastics in Turbulence and Finance, January 2008 7

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

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

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

Continuum limit [Fogedby 1994] According to Fogedby: Continuum limit of discrete equations, 9 8 ∂ x i +1 = x i + η i ∂s x ( s ) = η ( s ) > > = < i → s ⇒ ∂ t i +1 = t i + τ i ∂s t ( s ) = τ ( s ) > > ; : David Kleinhans — Continuous Time Random Walks in the Continuum Limit — Stochastics in Turbulence and Finance, January 2008 9

Continuum limit [Fogedby 1994] According to Fogedby: Continuum limit of discrete equations, 9 8 ∂ x i +1 = x i + η i ∂s x ( s ) = η ( s ) > > = < i → s ⇒ ∂ t i +1 = t i + τ i ∂s t ( s ) = τ ( s ) > > ; : Notation: dx s = dW s dL α dt s = s David Kleinhans — Continuous Time Random Walks in the Continuum Limit — Stochastics in Turbulence and Finance, January 2008 9

Continuum limit [Fogedby 1994] According to Fogedby: Continuum limit of discrete equations, 9 8 ∂ x i +1 = x i + η i ∂s x ( s ) = η ( s ) > > = < i → s ⇒ ∂ t i +1 = t i + τ i ∂s t ( s ) = τ ( s ) > > ; : David Kleinhans — Continuous Time Random Walks in the Continuum Limit — Stochastics in Turbulence and Finance, January 2008 9

Continuum limit [Fogedby 1994] According to Fogedby: Continuum limit of discrete equations, 9 8 ∂ x i +1 = x i + η i ∂s x ( s ) = η ( s ) > > = < i → s ⇒ ∂ t i +1 = t i + τ i ∂s t ( s ) = τ ( s ) > > ; : 2.5 α =0.9 α =0.8 α =0.7 Here: η ( s ) and τ ( s ) have to obey stable distribution α =0.6 2 α =0.5 α =0.4 1.5 ∞ 8 9 L α (x) L α, 1 ( x ) = 1 − iπα Z < = h − ikx − | k | α exp “ ”i π Re dk exp 1 2 : ; 0 0.5 0 0 0.5 1 1.5 2 x Result: Fractional dynamics at finite time David Kleinhans — Continuous Time Random Walks in the Continuum Limit — Stochastics in Turbulence and Finance, January 2008 9

Intrinsic Time in Finance [Müller 1993] David Kleinhans — Continuous Time Random Walks in the Continuum Limit — Stochastics in Turbulence and Finance, January 2008 10

Intrinsic Time in Finance [Müller 1993] David Kleinhans — Continuous Time Random Walks in the Continuum Limit — Stochastics in Turbulence and Finance, January 2008 10

Numerical algorithm Numerical Integration Scheme: (Itô) x ( s ) + ∆ sF ( x ( s )) + (∆ s ) 1 / 2 η ( s ) x ( s + ∆ s ) = t ( s ) + (∆ s ) 1 /α τ α ( s ) t ( s + ∆ s ) = . David Kleinhans — Continuous Time Random Walks in the Continuum Limit — Stochastics in Turbulence and Finance, January 2008 11