Power Variation of α-stable Lévy-Processes

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Power Variation of α-stable Lévy-Processes and application to paleoclimatic modelling Jan Gairing C. Hein, P. Imkeller Department of Mathematics Humboldt Universität zu Berlin Weather Derivatives and Risk Workshop 27.-28. January, 2010

Table of Contents Introduction Stable Distributions Stable Lévy-Processes

Table of Contents Introduction Stable Distributions Stable Lévy-Processes Theoretical results Results link to SDE s

Table of Contents Introduction Stable Distributions Stable Lévy-Processes Theoretical results Results link to SDE s Statistics Simulations Paleoclimatic Modelling

Paleoclimatic Model (Ditlevsen 99)

Paleoclimatic Model (Ditlevsen 99) Temperature proxi: log Calcium concentration 4 2 0 2 4 10.000 30.000 50.000 70.000 90.000 Time B.P.

Paleoclimatic Model (Ditlevsen 99) Temperature proxi: double-well potential: log Calcium concentration 4 2 0 2 4 Frequency 0 500 1000 1500 2000 2500 3000 3500 10.000 30.000 50.000 70.000 90.000 Histogram of log(ca) with pseudo potential Time B.P. 2 0 2 4

Paleoclimatic Model (Ditlevsen 99) Temperature proxi: double-well potential: log Calcium concentration 4 2 0 2 4 Frequency 0 500 1000 1500 2000 2500 3000 3500 10.000 30.000 50.000 70.000 90.000 Histogram of log(ca) with pseudo potential Time B.P. model X t = x 0 t 0 U (X s )ds + L t 2 0 2 4

Paleoclimatic Model (Ditlevsen 99) Temperature proxi: log Calcium concentration 4 2 0 2 4 10.000 30.000 50.000 70.000 90.000 Time B.P. model X t = x 0 t 0 U (X s )ds + L t U some adequate potential (double-welled) double-well potential: Frequency 0 500 1000 1500 2000 2500 3000 3500 Histogram of log(ca) with pseudo potential 2 0 2 4

Paleoclimatic Model (Ditlevsen 99) Temperature proxi: log Calcium concentration 4 2 0 2 4 10.000 30.000 50.000 70.000 90.000 Time B.P. model X t = x 0 t 0 U (X s )ds + L t U some adequate potential (double-welled) L heavy tailed noise (stable) double-well potential: Frequency 0 500 1000 1500 2000 2500 3000 3500 Histogram of log(ca) with pseudo potential 2 0 2 4

Paleoclimatic Model X t = x 0 t 0 U (X s )ds + L t How to obtain information on L without knowledge of U?

stable Ditstributions A distribution having second characteristic: ln E e i λy c α λ α 1 i β sign(λ) tan πα 2, α 1, = c λ 1 i β π 2 sign(λ) log λ, α = 1 is called (strictly) stable, write Y S α (c, β, 0), for 0 <α 2.

stable Ditstributions A distribution having second characteristic: ln E e i λy c α λ α 1 i β sign(λ) tan πα 2, α 1, = c λ 1 i β π 2 sign(λ) log λ, α = 1 is called (strictly) stable, write Y S α (c, β, 0), for 0 <α 2. heavy tails: P(Y > x) x α, x

stable Ditstributions A distribution having second characteristic: ln E e i λy c α λ α 1 i β sign(λ) tan πα 2, α 1, = c λ 1 i β π 2 sign(λ) log λ, α = 1 is called (strictly) stable, write Y S α (c, β, 0), for 0 <α 2. heavy tails: P(Y > x) x α, x moments: E Y p < p <α

stable Ditstributions Examples: Normal distribution S 2 (c, 0, 0): ln E e i λy = c 2 λ 2, P(Y x) =Φ(x)

stable Ditstributions Examples: Normal distribution S 2 (c, 0, 0): ln E e i λy = c 2 λ 2, P(Y x) =Φ(x) Cauchy distribution S 1 (c, 0, 0): ln E e i λy = c λ, P(Y x) = 1 2 + 1 π arctan(c 1 x)

stable Ditstributions Examples: Normal distribution S 2 (c, 0, 0): ln E e i λy = c 2 λ 2, P(Y x) =Φ(x) Cauchy distribution S 1 (c, 0, 0): ln E e i λy = c λ, P(Y x) = 1 2 + 1 π arctan(c 1 x) Lévy distribution S 1/2 (c, 1, 0): ln E e i λy = c 1/2 λ 1/2 (1 i sign(λ)), P(Y x) = 2(1 Φ( c/x)), x 0

stable Ditstributions Theorem (Stable limit distribution) Let (Y n ) n be a sequence of i.i.d. rv s s.t. there are a rv Y, a n, b n R s.t. then Y is stable. a n k=1 n Y n b n Y,

Brownian Motion Let W be a Brownian Motion, then ln E e i λw t = 1 2 tλ2 = tc 2 λ 2

Brownian Motion Let W be a Brownian Motion, then ln E e i λw t = 1 2 tλ2 = tc 2 λ 2 scaling: W t d = t W1

Brownian Motion Let W be a Brownian Motion, then ln E e i λw t = 1 2 tλ2 = tc 2 λ 2 scaling: continuous paths W t d = t W1

Brownian Motion Let W be a Brownian Motion, then ln E e i λw t = 1 2 tλ2 = tc 2 λ 2 scaling: d W t = t W1 continuous paths infinite variation

Brownian Motion Let W be a Brownian Motion, then ln E e i λw t = 1 2 tλ2 = tc 2 λ 2 scaling: d W t = t W1 continuous paths infinite variation finite quadratic variation

stable Lévy Processes Let L be a Lévy process with ln E e i λl t = tcα λ α 1 i β sign(λ) tan πα, α 1, 2 tc λ 1 i β π 2 sign(λ) log λ, α = 1

stable Lévy Processes Let L be a Lévy process with ln E e i λl tc α λ α, α 1, t = tc λ, α = 1

stable Lévy Processes Let L be a Lévy process with ln E e i λl tc α λ α, α 1, t = tc λ, α = 1 scaling: L t d = t 1/α L 1

stable Lévy Processes Let L be a Lévy process with ln E e i λl tc α λ α, α 1, t = tc λ, α = 1 scaling: discontinuous paths for α<2 L t d = t 1/α L 1

stable Lévy Processes Let L be a Lévy process with ln E e i λl tc α λ α, α 1, t = tc λ, α = 1 scaling: d L t = t 1/α L 1 discontinuous paths for α<2 finite variation for α<1

stable Lévy Processes Let L be a Lévy process with ln E e i λl tc α λ α, α 1, t = tc λ, α = 1 scaling: d L t = t 1/α L 1 discontinuous paths for α<2 finite variation for α<1 infinite variation for 1 α<2

stable Lévy Processes Let L be a Lévy process with ln E e i λl tc α λ α, α 1, t = tc λ, α = 1 scaling: d L t = t 1/α L 1 discontinuous paths for α<2 finite variation for α<1 infinite variation for 1 α<2 finite quadratic variation

stable sample paths 50 0 50 100 150 200 α=1 0 2000 4000 6000 8000 10000 Time

stable sample paths 50 0 50 100 150 200 α=1 α=1.5 0 2000 4000 6000 8000 10000 Time

stable sample paths 50 0 50 100 150 200 α=1 α=1.5 α=2 0 2000 4000 6000 8000 10000 Time

Definition Let X be a stochastic process. For p > 0 the power variation process of X is, if it exists, defined as V p (X) t := lim n V n p(x) t := lim n [nt] X i/n X (i 1)/n p. i=1

Definition Let X be a stochastic process. For p > 0 the power variation process of X is, if it exists, defined as V p (X) t := lim n V n p(x) t := lim n [nt] X i/n X (i 1)/n p. The alternating power variation process of X is defined as Ṽ p (X) t := lim n Ṽ n p(x) t := lim n i=1 [nt] ( 1) i X i/n X (i 1)/n p. i=1

Remark Let L S α a stable Lévy process with index 0 <α 2, then for any t V p (L) t (resp. Ṽ p (L) t ) < p α

Hein, Imkeller, Pavlyukevich (2008) Theorem Let L S α be an α-stable Lévy process. If p > α/2 then V n p (L) t ntb n (α, p) t 0 D (L t) t 0 (n ), where L is an α/p-stable process, L 1 S α/p(c, 1, 0)

Hein, Imkeller, Pavlyukevich (2008) Theorem Let L S α be an α-stable Lévy process. If p > α/2 then V n p (L) t ntb n (α, p) t 0 D (L t) t 0 (n ), where L is an α/p-stable process, L 1 S α/p(c, 1, 0) where: B n (α, p) is deterministic converging to 0.

Hein, Imkeller, Pavlyukevich (2008) Theorem Let L S α be an α-stable Lévy process. If p > α/2 then V n p (L) t ntb n (α, p) t 0 D (L t) t 0 (n ), where L is an α/p-stable process, L 1 S α/p(c, 1, 0) where: B n (α, p) is deterministic converging to 0. B n (α, p) = 0 for p >α

Hein, Imkeller, Pavlyukevich (2008) Theorem Let L S α be an α-stable Lévy process. If p > α/2 then V n p (L) t ntb n (α, p) t 0 D (L t) t 0 (n ), where L is an α/p-stable process, L 1 S α/p(c, 1, 0) where: B n (α, p) is deterministic converging to 0. B n (α, p) = 0 for p >α c = c (α, p)

Hein, Imkeller, Pavlyukevich (2008) Theorem Let L S α be an α-stable Lévy process. If p > α/2 then (Ṽ n p(l) t ) t 0 D (L t ) t 0 (n ), where L is a symmetric α/p-stable process, i.e. L 1 S α/p(c, 0, 0).

Hein, Imkeller, Pavlyukevich (2008) Theorem Let L S α be an α-stable Lévy process. If p > α/2 then (Ṽ n p(l) t ) t 0 D (L t ) t 0 (n ), where L is a symmetric α/p-stable process, i.e. L 1 S α/p(c, 0, 0). Theorem Both Theorems hold for L t + Y t whenever V n p(y) t P 0 t 0 the sum converging to the same limit.

link to SDE s Look at perturbations Y such that: V n p(y) t P 0 t 0

link to SDE s Look at perturbations Y such that: V n p(y) t P 0 t 0 for p > 1: Y = 0 f (s)ds for any integrable f.

link to SDE s Look at perturbations Y such that: V n p(y) t P 0 t 0 for p > 1: Y = f (s)ds for any integrable f. 0 for p > 2: Y = W, W a brownian motion.

link to SDE s Look at perturbations Y such that: V n p(y) t P 0 t 0 for p > 1: Y = f (s)ds for any integrable f. 0 for p > 2: Y = W, W a brownian motion. Thus for p > 2 V n p x 0 0 U (X s )ds + W + L D L

Wasserstein-bounds Definition (Wasserstein-metric) F X and F Y and 0 p < we define the L q -Wasserstein metric by W q (F X, F Y ):= inf E µ X Y q µ M(F X,F Y ) 1 1/q M(µ, ν) is the class of all laws having marginals µ and ν.

Wasserstein-bounds Definition (Wasserstein-metric) F X and F Y and 0 p < we define the L q -Wasserstein metric by W q (F X, F Y ):= inf E µ X Y q µ M(F X,F Y ) 1 1/q M(µ, ν) is the class of all laws having marginals µ and ν. W q is a metric on F : x q df(x) < W 2 is equivalent to weak convergence + second moments

Wasserstein-bounds Theorem Let L be an α-stable Lévy process with α ]0, 2[, i.e. L 1 S α (c, β, 0) 1. the 2α-variation: For 1 > q > 1/2: W q V n 2α (L) 1, Y = O(n 2+1/q ) Y S 1/2 (c, 1, 0) has a Lévy distribution.

Wasserstein-bounds Theorem Let L be an α-stable Lévy process with α ]0, 2[, i.e. L 1 S α (c, β, 0) 1. the 2α-variation: For 1 > q > 1/2: W q V n 2α (L) 1, Y = O(n 2+1/q ) Y S 1/2 (c, 1, 0) has a Lévy distribution. 2. the alternating α-variation: For 1 < q 2 W q Ṽn α (L) 1, Y = O(n 1+1/q ) Y S 1 (c, 0, 0) is Cauchy distributed.

Statistics Obtain information on stable Noise?

Statistics Obtain information on stable Noise? Fit V n p(x) t or Ṽ n p(x) t to stable reference distribution

Statistics Obtain information on stable Noise? Fit Vp(X) n t or Ṽp(X) n t to stable reference distribution Regain Noise parameter from fitted reference

Statistics Obtain information on stable Noise? Fit Vp(X) n t or Ṽp(X) n t to stable reference distribution Regain Noise parameter from fitted reference To do so: Split the path L 0, L 1,...L T into m subparts L 0,...L t1 to L tm 1,...L T

Statistics Obtain information on stable Noise? Fit Vp(X) n t or Ṽp(X) n t to stable reference distribution Regain Noise parameter from fitted reference To do so: Split the path L 0, L 1,...L T into m subparts L 0,...L t1 to L tm 1,...L T Compute a sample of discrete power variations V p n (L) 0<t<t1,...V p n (L) tm 1 <t<t

Statistics Obtain information on stable Noise? Fit Vp(X) n t or Ṽp(X) n t to stable reference distribution Regain Noise parameter from fitted reference To do so: Split the path L 0, L 1,...L T into m subparts L 0,...L t1 to L tm 1,...L T Compute a sample of discrete power variations V p n (L) 0<t<t1,...V p n (L) tm 1 <t<t Fit to stable reference distribution to the sample choosing p and the scale C

The Test Statistics The Kolmogorov-Smirnoff distance D KS (F n, G) := sup F n (x) G(x) x R

The Test Statistics The Kolmogorov-Smirnoff distance D KS (F n, G) := sup F n (x) G(x) x R For the empirical distribution function F n (x) := 1 n n 1{y j x} j=1 to some stable reference distribution G. (if p = 2α then G is a Lévy distribution)

The Test Statistics A weighted L 2 -distance (Gürtler and Henze, 2000) to the Cauchy distribution (p = α): D n,κ := n Φ n (u) e u 2 w(u)du, w(u) := e κ u,κ>0

The Test Statistics A weighted L 2 -distance (Gürtler and Henze, 2000) to the Cauchy distribution (p = α): D n,κ := n = 2 n j,k=1 Φ n (u) e u 2 w(u)du, w(u) := e κ u,κ>0 n κ n κ 2 + (y j y k ) 4 1 + κ 2 (1 + κ) 2 + y 2 j j=1 + 2n 2 + κ

The Test Statistics A weighted L 2 -distance (Gürtler and Henze, 2000) to the Cauchy distribution (p = α): D n,κ := n = 2 n j,k=1 Φ n (u) e u 2 w(u)du, w(u) := e κ u,κ>0 n κ n κ 2 + (y j y k ) 4 1 + κ 2 (1 + κ) 2 + y 2 j For the empirical characteristic function j=1 + 2n 2 + κ Φ n (u) := 1 n n exp(i uy j ) j=1

L Introduction Theoretical results Statistics References Simulations L S 0.6 (5, 0, 0) 80000 60000 40000 20000 0 0 10000 20000 30000 40000 Time

L d d Introduction Theoretical results Statistics References Simulations 1.0 α = 0.6; C = 5; d = 0.05854243 L S 0.6 (5, 0, 0) 0.8 0.6 0.4 80000 60000 40000 20000 0 0 10000 20000 30000 40000 0.2 8 10 2.0 100 80 60 40 C 2 4 6 1.5 1.0 0.5 a α = 0.6; C = 5; d = 0.520927 Time 20 C 8 2 4 6 10 2.0 1.5 1.0 0.5 a

Simulations L S 1.5 (5, 0, 0) 150 100 50 0 50 L 0 10000 20000 30000 40000 Time

L d d Introduction Theoretical results Statistics References Simulations 1.0 α = 1.5; C = 5; d = 0.05035901 L S 1.5 (5, 0, 0) 0.8 0.6 0.4 150 100 50 0 50 0.2 8 10 2.0 100 80 60 C 2 4 6 1.5 1.0 a α = 1.4; C = 5; d = 0.2386760 0.5 0 10000 20000 30000 40000 40 Time 20 C 8 2 4 6 10 2.0 1.5 1.0 0.5 a

Ice-core data The log-calcium signal as temperature proxi: log Calcium concentration 4 2 0 2 4 10.000 30.000 50.000 70.000 90.000 Time B.P.

Ice-core data Fitting the Model : X t = x 0 t 0 U (X s )ds + L t

d 0.9 Introduction Theoretical results Statistics References Ice-core data Fitting the Model : X t = x 0 t 0 U (X s )ds + L t α = 1.85; C = 1.5; d = 0.09321583 α = 1.85; C = 1.5; d = 0.09321583 1.0 0.8 0.6 0.4 C 1.0 0.2 0.5 3.0 2.5 2.0 1.5 0.5 1.0 a 1.5 2.0 C 0.5 1.0 1.5 2.0 2.5 3.0 0.85 0.8 0.75 0.65 0.7 0.55 0.25 0.25 0.6 0.35 0.35 0.55 0.2 0.6 0.15 0.7 0.55 0.65 0.5 0.3 0.15 0.3 0.5 0.45 0.75 0.15 0.5 1.0 1.5 2.0 0.45 0.65 a

Outlook Berry-Esseen type bounds for the convergence.

Outlook Berry-Esseen type bounds for the convergence. Analyze the behaviour of the estimator.

Outlook Berry-Esseen type bounds for the convergence. Analyze the behaviour of the estimator. Generalize the algorithm to allow larger values of p.

Outlook Berry-Esseen type bounds for the convergence. Analyze the behaviour of the estimator. Generalize the algorithm to allow larger values of p. Reduce the number of datapoints required.

References I [1] Nualart David Corcuera, José Manuel and Jeanette H.C. Woerner. A functional central limit theorem for the realized power variation of integrated stable processes. Stochastic Analysis and Apllications, 25(1):169 186, January 2007. [2] Peter D. Ditlevsen. Observation of α-stable noise induced millennial climate changes from an ice-core record. Geophysical Researcher Letters, 26(10):1441 1444, May 1999. [3] Imkeller P. Hein, C. and I. Pavlyukevich. Limit theorems for p-variations of solutions of sde s driven by additive non-gaussian stable levy noise. arxiv:0811.3769v1[math.pr], 2008.

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