Claudia Klüppelberg Technische Universität München Zentrum Mathematik. Oslo, September 2012

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1 Claudia Klüppelberg Technische Universität München Zentrum Mathematik Oslo, September 2012

2 Collaborators Fred E. Benth, Oslo U Christine Bernhard, former Diploma student TUM Peter Brockwell Vicky Fasen, ETH Zurich Vincenzo Ferrazzano, TUM Isabel García, Bochum U. Alexander Lindner, Braunschweig U. Ross Maller, ANU Canberra Thilo Meyer-Brandis, LMU Gernot Müller, TUM Linda Vos, former PhD student Oslo U 2

3 Outline today Lévy processes 3

4 Outline today Lévy processes Lévy-driven CARMA processes 4

5 Outline today Lévy processes Lévy-driven CARMA processes Causal CARMA Processes 5

6 Outline today Lévy processes Lévy-driven CARMA processes Causal CARMA Processes Strictly Lévy-driven CARMA processes 6

7 Outline today Lévy processes Lévy-driven CARMA processes Causal CARMA Processes Strictly Lévy-driven CARMA processes Connections with discrete-time ARMAs 7

8 Outline today Lévy processes Lévy-driven CARMA processes Causal CARMA Processes Strictly Lévy-driven CARMA processes Connections with discrete-time ARMAs The discretely sampled ARMA process 8

9 Outline tomorrow 9

10 Outline tomorrow Invoking CARMA processes for statistical wind speed modelling 10

11 Outline tomorrow Invoking CARMA processes for statistical wind speed modelling Modelling electricity market data by CARMA processes 11

12 Outline tomorrow Invoking CARMA processes for statistical wind speed modelling Modelling electricity market data by CARMA processes Continuous-time GARCH models as self-exciting CARMA processes 12

13 Motivation Interest in continuous-time models stems from continuous-time modelling in physics pricing models in financial applications statistical modelling of irregularly spaced data availability of high-frequency data Stylized facts in financial/turbulence data: heavy-tailed modelling dependence without correlation stochastic volatility/intermittency effects 13

14 Lévy processes A Lévy process L = {L(t), t 0} is a stochastic process with the following properties: 1. L(0) = 0 a.s. 2. L has stationary and independent increments, 3. L is continuous in probability: for every t 0 and ε > 0 we have lim s t P( L s L t > ε) = 0 4. L has càdlàg sample paths. The increments of L on disjoint intervals of equal length are then independent and identically distributed random variables with some infinitely divisible distribution which could, for example, be Gaussian, compound Poisson, gamma, (normal) inverse Gaussian or one of many other possibilities. 14

15 The characteristic function of L(t) has for every t 0 the form where E [ exp(iθl(t)) ] = exp(tψ(θ)), θ R, ψ(θ) = iθγ 1 ( ) 2 θ2 τ 2 1 e iθx iθx1 { x <1} ν(dx), R for some γ R, τ 0, and measure ν on the Borel subsets of R. The measure ν is called the Lévy measure of the process L and has the property, ν({0}) = 0 and (x 2 1)ν(dx) <. R 15

16 Examples ν = 0: Brownian motion. γ = τ 2 = 0, ν(du) = λf(du); i.e. ν(r) = λ < : compound Poisson process with drift. ψ(θ) = 0 (eiθx 1)ν(dx) = (1 iθ β ) α, ν (u) = αu 1 e βu 1 (0, ) (u): gamma process (standardized if α = β 2 ). ν (u) = 1 2 α u 1 e β u : symmetrized gamma process (L 1 L 2 ). ψ(θ) = exp( c θ α ), 0 < α 2, symmetric α-stable process. Then ν (u) = c u 1 α for u R \ {0}. 16

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20 Extension to (, ) By introducing a second Lévy process {M(t), 0 t < }, independent of L and with the same distribution, we can extend {L(t), t 0} to a process with index set (, ) by defining L (t) = L(t)I [0, ) (t) M( t )I (, 0] (t). Then L has stationary independent increments, cadlag paths and satisfies L (0) = 0. In what follows we shall refer to L as L. 20

21 Second order Lévy processes We can sometimes work with a weaker version of a Lévy process (provided it has finite variance): we call {L(t), t 0} or its extension to R a second order Lévy process, if its increments have stationary mean and variance: E[L(1)] = E[L(t + 1) L(t)] and Var[L(1)] = Var[L(t + 1) L(t)] and if increments are uncorrelated. If not said explicitly otherwise, we shall use the standardized version: E[L(1)] = 0 and Var[L(1)] = 1. 21

22 Second order Lévy-driven CARMA(p, q) processes Formally, a CARMA(p, q) process Y is a (weakly) stationary solution of the pth order linear differential equation, a(d)y(t) = σb(d)dl(t), (1) where D denotes differentiation with respect to t, a(z) = z p + a 1 z p a p and b(z) = b 0 + b 1 z + + b p 1 z p 1, b q = 1, b j := 0 for j > q, and a( ) and b( ) have no common zeros. Moreover, {L(t)} is a second order Lévy process with Var[L(1)] = 1. We call L the Driving Lévy Process (DLP). 22

23 Interpret (1) as equivalent to the observation and state equations Y(t) = b X(t) (2) dx(t) = AX(t)dt + e p dl(t), (3) with X(t) = (X(t), X (1) (t),..., X (p 2) (t), X (p 1) (t)) b = b 0 b 1. b q = , e p = , A = a p a p 1 a p 2. a 1. a 1,..., a p, b 0,..., b p 1 are such that b q = 1 and b j = 0 for j q. Note that q < p. 23

24 The solution of (3) is a multivariate OU process, which satisfies X(t) = e A(t s) X(s) + t s e A(t u) e p dl(u), t 0. (4) Proposition There exists a covariance stationary solution X of (4) with the property that X(t) is uncorrelated with {L(s) L(t), s > t} for all t R if and only if the eigenvalues λ 1,..., λ p of A satisfy Re(λ i ) < 0, i = 1,..., p. (5) If the condition (5) is satisfied then the solution with the specified properties is X(t) = t e A(t u) e p dl(u), (6) which has the distribution of e Au e 0 p dl(u). 24

25 Proof The eigenvalues of A (which are the roots of the autoregressive polynomial a( )) must have negative real parts for the sum of the covariance matrices of the terms on the right of (4) to be bounded in t for every fixed s. If this condition is satisfied then letting s in (4) with t fixed gives (6). Conversely, {X(t)} as defined by (6) is clearly a covariance stationary solution of (4) with the property that {X(t)} is independent of {L(s) L(t), s > t} for all t R. We call the resulting CARMA process a causal CARMA process. 25

26 Causal CARMA Processes Definition. If Re(λ j ) < 0, j = 1,..., p, then the causal CARMA(p, q) process with standardized second order DLP {L(t), < t < } and coefficients {σ, a 1,..., a p, b 0,..., b q = 1} is the covariance and strictly stationary process, Y(t) = σb X(t), t R, where i.e. X(t) = t e A(t u) e p dl(u), (7) t Y(t) = σ b e A(t u) e p dl(u). (8) 26

27 Causality and Non-causality Under (5) {Y(t), < t < } is a causal function of {L(t), < t < }, where g(t) = Y(t) = g(t u) dl(u), (9) { σb e At e p if t > 0, 0 otherwise. The function g is the kernel of the CARMA process {Y(t)}. Under condition (5) (cf. Lemma 2.3 of Brockwell and Lindner (2009)), (10) g(t) = σ itω b(iω) e dω. (11) 2π a(iω) Under the less restrictive conditions, Re(λ j ) 0, j = 1,..., p, (12) (9) and (11) define a stationary, not-necessarily causal CARMA process. 27

28 Second order Properties From (8) we find for E[L(1)] = µ R that and for µ = 0, EY(t) = σb EX(t) = µσ b 0 a p Cov[Y(t + h), Y(t)] = σ 2 b e A h Σb, where Σ = E[X(t)X(t) ] = 0 e Ay e p e p e A y dy. 28

29 From (9) and the convolution theorem for Fourier transforms the ACVF of Y is γ(h) = g(u + h )g(u)du = σ2 e iωh b(iω) 2 2π a(iω) dω. (13) and, consequently, the spectral density of Y is f (ω) = σ2 b(iω) 2 2π a(iω). (14) 29

30 Figure: ACVF of the CARMA(3,2) process 30

31 Explicit Representations: Causal Case Under the causality condition (5), changing the variable of integration in (11) to z = iω, and evaluating the contour integral gives D m(λ) 1 [ g(h) = σ (z λ) m(λ) zh b(z) e (m(λ) 1)! a(z) λ ], h 0, (15) z=λ where D := d/dz, the sum is over the distinct roots λ of a( ), and m(λ) is the multiplicity of the root λ. The same argument applied to (13) gives, for the autocovariance function, γ(h) = σ 2 λ D m(λ) 1 [ (z λ) m(λ) z h b(z)b( z) e (m(λ) 1)! a(z)a( z) ]. z=λ 31

32 Distinct Autoregressive Roots When the zeroes λ 1,..., λ p of a( ) are distinct and satisfy the causality condition (5), these expressions for the kernel g and the autocovariance function γ reduce to g(h) = σ p r=1 b(λ r ) a (λ r ) eλ rh I [0, ) (h) (16) and γ(h) = σ 2 p r=1 b(λ r )b( λ r ) a (λ r )a( λ r ) eλ r h. (17) 32

33 Canonical representation of Y When the autoregressive roots are distinct we obtain a very useful representation of Y from (16). Defining α r = σ b(λ r) a, r = 1,..., p, (18) (λ r ) we can write where Y(t) = p Y r (t), (19) r=1 Y r (t) = t α r e λ r(t u) dl(u). (20) 33

34 Equivalently, where Y is the solution of Y(t) = [1,..., 1]Y(t), t 0, (21) dy(t) = diag[λ i ] p i=1 Y(t)dt + σbr 1 e p dl(t) (22) with matrices B = diag[b(λ i )] p i=1 and R = [λi 1 j ] p i,j=1. We shall refer to Y(t) = σbr 1 X(t) (23) as the canonical state vector. The canonical representation of the process Y reduces the problem of simulating CARMA(p, q) processes with distinct autoregressive roots to the much simpler problem of simulating the (possibly complex-valued) component CAR(1) processes and adding them. 34

35 Strictly Lévy-driven CARMA processes E[L(1) 2 ] < can be relaxed to the following. Theorem [Brockwell and Lindner (2009)] Let L be the DLP and suppose that a( ) and b( ) have no common zeroes. Then the CARMA equations (2) and (3) have a strictly stationary solution Y on R if and only if E[ln + L(1) ] < and a( ) is non-zero on the imaginary axis. In this case the solution Y is unique and given by (9), and g can be given explicitly. Moreover, the corresponding state vector (3) can be chosen to be strictly stationary. 35

36 If the conditions of the Theorem hold and, additionally, all eigenvalues have negative real parts, then (8) still defines a causal (but not necessarily second-order) strictly stationary process with joint distributions satisfying 0 t2 t 1 ln E[exp(iθ 1 Y(t 1 ) + + iθ n Y(t n ))] = n t1 n ψ θ i g(t i + u) du + ψ θ i g(t i u) du + i=1 0 n tn ψ θ i g(t i u) du + + ψ (θ n g(t n u)) du, t n 1 i=2 where g(u) = b e Au e p. i=1 36

37 ν = 0: Gaussian CARMA(p, q). {L(t)} compound Poisson with bilateral exponential jumps CAR(1) {Y(t)} has marginal cgf κ(θ) = λ 2a 1 ln (1 + θ2 i.e., Y(t) has symmetrized gamma distribution, bilateral exponential if λ = 2a 1. Interesting examples for DLP are α-stable Lévy processes. β 2 ), 37

38 History Gaussian processes with the rational spectral density (14) have been of interest for many years. In particular, there was a very extensive study of such processes by Doob (1944). A very nice paper, also based heavily on the spectral density is that of Pham Din Tuan (1977). The SDE approach to such processes has also been used widely in the engineering literature and was employed by Jones (1978) in his approach to the modelling of time series with irregularly-spaced data. Since the 1990ies Brockwell extended the Gaussian model to Lévy-driven models. The CAR(1) model, driven by a subordinator, was suggested by Barndorff-Nielsen and Shephard as stochastic volatility model. It was also used previously in storage modelling. 38

39 The SDE approach outlined above suggests a number of interesting generalizations such as replacement of the SDE by a non-linear SDE to obtain, in particular, continuous-time generalised Ornstein-Uhlenbeck processes of the form dv(t) = AdM(t) + V(t)dL(t) (and its multivariate versions) for a bivariate Lévy process (M, L). Such models go back to Yor and collaborators in the 1990ies, cf. Lindner and Maller (2005) and Anita Behme (PhD Thesis, 2011). Special cases are continuous-time GARCH models, which we will present later. 39

40 An important issue for statistical inference is the parameter estimation, the extraction of the driving process L and the state-vector X from discrete (or continuous) observations of Y. For this goal some knowledge on the connection between continuous- and discrete-time ARMA models is needed. 40

41 Discrete-time ARMA processes The discrete-time ARMA(p, q) process {Y n, n Z} is a (weakly) stationary solution of the p th order linear difference equations, φ(b)y n = σ d θ(b)z n, n Z (24) where B is the backward shift operator, {Z n } WN(0, 1), σ d > 0 and φ(z) := 1 φ 1 z φ p z p and θ(z) := 1 + θ 1 z + + θ q z q, with θ q 0 and φ p 0. We shall assume that the polynomials φ( ) and θ( ) have no common zeroes. 41

42 Proposition There exists a covariance stationary solution Y of (24) with the property that Y n is uncorrelated with Z n+j for all j 1 if and only if φ(z) = 1 φ 1 z φ p z p is non-zero for all complex z such that z 1. In this case the solution with the specified properties is causal with representation n Y n = ψ n j Z j = j= ψ j Z n j, n Z j=0 42

43 Comparison with the Discrete Time Case I For the causal discrete-time ARMA(p, q) process, φ(b)y n = σ d θ(b)z n, n Z, {Z n } WN(0, 1), with roots λ r of φ( ) we have the analogues to (15) σ d D m(λ) 1 [ ] ψ j = (z λ) m(λ) z j 1 θ(z 1 ) + I (m(λ) 1)! φ(z 1 [0,q p] (j) σ dd q p j [ z q p θ(z 1 ] ) ) z=λ (q p j)! φ(z 1 ) and λ σ 2 d γ(h) = Dm(λ) 1 (m(λ) 1)! λ +I [0,q p] (h σ2 d Dq p h (q p h)! [ (z λ) m(λ) z h 1 θ(z 1 )θ(z) φ(z 1 )φ(z) [ z q p θ(z 1 )θ(z) φ(z 1 )φ(z) ] ] z=λ. z=0 The sums are over the distinct roots λ of φ(1/z) = 0 and m(λ) is the multiplicity of λ. z=0 43

44 If also q < p and m(λ) = 1 for each λ, we get analogues to (16) and (17): ψ j = σ d p r=1 λ j 1 r θ(λ r ) φ (λ r ), j 0, and γ(h) = σ 2 d p r=1 λ h 1 r θ(λ r )θ(λ 1 r ) φ (λ r )φ(λ 1 r ). 44

45 There is also a corresponding canonical representation analogous to that in (18). It takes the form, Y n = p Y r,n, r=1 where and Y r,n = n k= β r λ k n r Z k, r = 1,..., p, β r = σ d λ 1 r θ(λ r )φ (λ r ), r = 1,..., p. 45

46 Remark Thus when q < p and the autoregressive roots are distinct, both the CARMA and ARMA processes can be represented as a sum of autoregressive processes of order 1. Note however that in both cases the component processes are not independent and are in general complex valued. Example 3: ARMA(2,1) If φ(z) = (1 ρ 1 z)(1 ρ 2 z), where ρ 1 ρ 2 and θ(z) = 1 + θ 1 z, then β r = σ d ρ r + θ 1 ρ r ρ 3 r, r = 1, 2. The canonical representation of the ARMA(2,1) process is thus Y n = Y 1,n + Y 2,n, where Y r,n = β r n k= ρ n k r Z k, r = 1, 2, 46

47 Sampling from a CARMA process Let Y be a causal CARMA(p, q) process as in (1) and > 0. Then the sampled process Y := {Y n } n Z satisfies the ARMA(p, p 1) equations φ (B)Y n = θ (B)Z n, n Z, {Z n } WN(0, σ 2 ). B is the backshift operator on the grid; i.e. BYn = Yn 1, φ (z) = p i=1 (1 e λ i z) and θ ( ) is a polynomial of order p 1, chosen such that it has no roots inside the complex unit circle. 47

48 If Y is a second order stationary CARMA(p, q) process, then it is well-known (since Doob (1944), cf. Bloomfield (1976)) that the sampled process (Y(n )) n Z is an ARMA(p, p 1) process with spectral density f (ω) = 1 2π γ Y (k )e ikω = 1 k= k= ( ω + 2kπ ) f Y, π ω π, where f Y (ω), < ω <, is the spectral density of Y. 48

49 Aliasing problem 49

50 Recall that for the CARMA(p, q) process γ(h) = σ2 b(z)b( z) 2π ρ a(z)a( z) e h z dz, h R which also holds for h = k for k Z. We write a(λ) = p i=1 (λ λ i) and b(λ) = q i=1 (1 µ iλ). Then with the above, = σ2 4π 2 i ρ f (ω) = 1 2π h= γ(h )e ihω b(z)b( z) sinh( z) dz, π ω π. a(z)a( z) cosh( z) cos(ω) 50

51 Recall p. 45. We apply the filter φ (B) := p j=1 (1 eλj B) to Y {λ j, j = 1,..., p} are again the eigenvalues of A, equivalently the roots of a( ). With φ (z) = 1 d 1 z d p z p for z C we obtain φ(b)y n = Y n d 1 Y n 1 d py n p = θ(b)z n, n Z, where {Z n } n Z WN(0, σ 2 ); in particular, the rhs is a MA(p 1). For 0 the AR polynomial degenerates, but we can study the MA polynomial on the rhs. 51

52 We write now a(z) = p i=1 (z λ i) and b(z) = q i=1 (z µ i). Theorem The MA(p 1) process X n = θ(b)zn for n Z has representation X n = where (1 + η(ξ i )B) p 1 q i=1 q (1 ζ k B)Zn, {Zn } n Z WN(0, σ 2 ), k=1 σ 2 = 2(p q) 1 e a1 σ 2 [2(p q) 1]! p q 1 i=1 η(ξ i ) q k=1 ζ, k with (as 0) ζ k = 1 ± µ k + o( ) and η(ξ i ) = ξ i 1 ± (ξ i 1) o(1), (25) and the signs are chosen so that lim 0 ζ k < 1 and lim 0 η(ξ i ) 1. 52

53 Wold and Cramér-Wold representations Every second order stationary causal stochastic process with mean 0 has representation either, for {L t } t R weak Lévy Y(t) = t or, for {Z n } n Z WN(0, σ 2 ) g(t s)dl(s), t R, Y n = ψ j Z n j, n Z. j=0 53

54 Wold representation of the discretely observed CARMA Y n = ( ) ψ j Z n j = σ ψ j Zn j, n Z. (26) σ j=0 j=0 Consequently, we approximate the kernel g(t) by g (t) = j=0 σ ψ j 1 (j,(j+1) ](t). (27) 54

55 Example: CAR(1) g(t) = e λt 1 (0, ) (t), where λ < 0. Then Yn = e λ Yn 1 + Z n, where Z n = n (n 1) e λ(n u) dl(u) are i.i.d. Then ψ j = e jλ for j N 0, and σ 2 = σ2 2λ (e2λ 1). Hence, e g (t) = σ 2λ 1 2λ ejλ 1 (j,(j+1) ] (t), j=0 which converges pointwise to σg as 0. 55

56 Theorem Let Y be the causal CARMA(p, q) with different λ j, j = 1,..., p. Then with ζ k and η(ξ i ) as in (25): and ψ j = p r=1 p 1 q i=1 (1 + η(ξ i )e λr ) q k=1 (1 ζ ke λr ) m r(1 e (λ m λ r ) e jλr, ) σ 2 = 2(p q) 1 e a1 σ 2 [2(p q) 1]! p q 1 i=1 η(ξ i ) q k=1 ζ. j Moreover, for every t 0, g (t) = Hence, j=0 σ ψ j 1 (j,(j+1) ](t) σg(t) = σ p j=1 σ lim ψ 0 [t/ ] = σg(t), t 0. b(λ i ) a (λ i ) eλ it 1 (0, ) (t). 56

57 Noise Recovery: Invertible CARMA Processes A CARMA(p, q) process Y is said to be invertible if the roots of the moving average polynomial b( ) have negative real parts; i.e., Re(µ i ) > 0 for all i = 1,..., q. We need the following Assumption 1. (i) The roots of the polynomial a( ) satisfy Re(λ j ) < 0 for all j = 1,..., p (Y is causal), and (ii) the roots of the moving average polynomial b( ) have negative real parts; i.e., Re(µ i ) > 0 for all i = 1,..., q (Y is invertible). 57

58 Theorem [Ferrazzano and Fuchs (2012)] Let Y be a second order CARMA(p, q) process and Z the noise of the sampled process Y. If Assumption 1 holds, then as 0, σ [t/ ] n=1 Z n L 2 L t, t (0, ). (28) CK s guess: convergence is a.s. [Jurek and Vervaat (1983)] Ferrazzano and Ueltzhöfer (2012, in preparation) suggest a model for L based on non-parametric estimation of the Lévy measure. They allow for a weak Lévy process, so that they can also fit the dependence structure of the increments by a time-change. More statistics later! 58

59 [1] Brockwell, P. J. (1995) A note on the embedding of discrete-time ARMA processes. J. Time Series Anal. 16(5), [2] Brockwell, P. J. (2001) Lévy-driven CARMA processes. Ann. Inst. Statist. Math. 53(1), [3] Brockwell, P. J. (2001) Continuous-time ARMA processes. In: C. R. Rao and D. N. Shanbhag (eds), Handbook of Statistics: Stochastic Processes, Theory and Methods, pp Elsevier, Amsterdam. [4] Brockwell, P. J. (2004) Representations of continuous-time ARMA processes. J. Appl. Prob. 41A, [5] Brockwell, P. and Lindner, A. (2009) Existence and uniqueness of stationary Lévy-driven CARMA processes. Stoch. Proc. Appl. 119, [6] Brockwell, P. J., Ferrazzano, V. and Klüppelberg, C. (2012) High frequency sampling of a continuous-time ARMA process. J. Time Series Analysis 33(1), [7] Brockwell, P. J., Ferrazzano, V. and Klüppelberg, C. (2011) Kernel estimation in continuous-time moving average processes. Submitted for publication. [8] Ferrazzano, V. and Fuchs, F. (2012) Noise recovery for Lévy-driven CARMA processes and high-frequency behaviour of approximating Riemann sums. In preparation. 59