Power Variation of α-stable Lévy-Processes

Transcription

1 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 January, 2010

2 Table of Contents Introduction Stable Distributions Stable Lévy-Processes

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

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

5 Paleoclimatic Model (Ditlevsen 99)

6 Paleoclimatic Model (Ditlevsen 99) Temperature proxi: log Calcium concentration Time B.P.

7 Paleoclimatic Model (Ditlevsen 99) Temperature proxi: double-well potential: log Calcium concentration Frequency Histogram of log(ca) with pseudo potential Time B.P

8 Paleoclimatic Model (Ditlevsen 99) Temperature proxi: double-well potential: log Calcium concentration Frequency Histogram of log(ca) with pseudo potential Time B.P. model X t = x 0 t 0 U (X s )ds + L t

9 Paleoclimatic Model (Ditlevsen 99) Temperature proxi: log Calcium concentration 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 Histogram of log(ca) with pseudo potential

10 Paleoclimatic Model (Ditlevsen 99) Temperature proxi: log Calcium concentration 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 Histogram of log(ca) with pseudo potential

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

12 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.

13 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

14 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

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

16 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) = π arctan(c 1 x)

17 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) = 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

18 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,

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

20 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

21 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

22 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

23 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

24 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

25 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

26 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

27 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

28 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

29 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

30 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

31 stable sample paths α= Time

32 stable sample paths α=1 α= Time

33 stable sample paths α=1 α=1.5 α= Time

34 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

35 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

36 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

37 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)

38 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.

39 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

40 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)

41 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).

42 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.

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

44 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.

45 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.

46 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

47 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.

48 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

49 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.

50 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.

51 Statistics Obtain information on stable Noise?

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

53 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

54 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

55 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

56 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

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

58 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)

59 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

60 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 ) κ 2 (1 + κ) 2 + y 2 j j=1 + 2n 2 + κ

61 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 ) 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

62 L Introduction Theoretical results Statistics References Simulations L S 0.6 (5, 0, 0) Time

63 L d d Introduction Theoretical results Statistics References Simulations 1.0 α = 0.6; C = 5; d = L S 0.6 (5, 0, 0) C a α = 0.6; C = 5; d = Time 20 C a

64 Simulations L S 1.5 (5, 0, 0) L Time

65 L d d Introduction Theoretical results Statistics References Simulations 1.0 α = 1.5; C = 5; d = L S 1.5 (5, 0, 0) C a α = 1.4; C = 5; d = Time 20 C a

66 Ice-core data The log-calcium signal as temperature proxi: log Calcium concentration Time B.P.

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

68 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 = α = 1.85; C = 1.5; d = C a C a

69 Outlook Berry-Esseen type bounds for the convergence.

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

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

72 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.

73 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): , January [2] Peter D. Ditlevsen. Observation of α-stable noise induced millennial climate changes from an ice-core record. Geophysical Researcher Letters, 26(10): , May [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: v1[math.pr], 2008.