Detecting instants of jumps and estimating intensity of jumps from continuous or discrete data

Transcription

1 Detecting instants of jumps and estimating intensity of jumps from continuous or discrete data Denis Bosq 1 Delphine Blanke 2 1 LSTA, Université Pierre et Marie Curie - Paris 6 2 LMA, Université d'avignon et des Pays de Vaucluse SAPS X, 2015, Le Mans D. Bosq, D. Blanke Detecting and estimating jumps 1 / 32

2 Outline 1 Examples 2 Properties of the space D 3 ARMAD processes 4 Prediction of ARMAD processes 5 Observation in continuous time 6 High frequency data (HFD) D. Bosq, D. Blanke Detecting and estimating jumps 2 / 32

3 Outline 1 Examples 2 Properties of the space D 3 ARMAD processes 4 Prediction of ARMAD processes 5 Observation in continuous time 6 High frequency data (HFD) D. Bosq, D. Blanke Detecting and estimating jumps 2 / 32

4 Outline 1 Examples 2 Properties of the space D 3 ARMAD processes 4 Prediction of ARMAD processes 5 Observation in continuous time 6 High frequency data (HFD) D. Bosq, D. Blanke Detecting and estimating jumps 2 / 32

5 Outline 1 Examples 2 Properties of the space D 3 ARMAD processes 4 Prediction of ARMAD processes 5 Observation in continuous time 6 High frequency data (HFD) D. Bosq, D. Blanke Detecting and estimating jumps 2 / 32

6 Outline 1 Examples 2 Properties of the space D 3 ARMAD processes 4 Prediction of ARMAD processes 5 Observation in continuous time 6 High frequency data (HFD) D. Bosq, D. Blanke Detecting and estimating jumps 2 / 32

7 Outline 1 Examples 2 Properties of the space D 3 ARMAD processes 4 Prediction of ARMAD processes 5 Observation in continuous time 6 High frequency data (HFD) D. Bosq, D. Blanke Detecting and estimating jumps 2 / 32

8 The mistral Examples Source: Jacq, V, Albert P., Delorme R. (2005) Le mistral quelques aspects des connaissances actuelles. La Météorologie, n 50, p D. Bosq, D. Blanke Detecting and estimating jumps 3 / 32

9 The mistral Examples Some facts about the mistral: Maximum wind speed: 140 km/h (38.9 m/s) 115 days of mistral per year data from 2005 to 2014 (forthcoming work) Source: Jacq, V, Albert P., Delorme R. (2005) Le mistral quelques aspects des connaissances actuelles. La Météorologie, n 50, p D. Bosq, D. Blanke Detecting and estimating jumps 3 / 32

10 Examples Other examples Magnetic eld Stock prices Pollution cf Horváth-Kokoszka (2012) Questions At what instant one may say that there exists a jump? Detect the instants of jumps Estimate intensity of jumps Estimate the greatest jump Predict the mistral next day (or next week). D. Bosq, D. Blanke Detecting and estimating jumps 4 / 32

11 Examples Other examples Magnetic eld Stock prices Pollution cf Horváth-Kokoszka (2012) Questions At what instant one may say that there exists a jump? Detect the instants of jumps Estimate intensity of jumps Estimate the greatest jump Predict the mistral next day (or next week). D. Bosq, D. Blanke Detecting and estimating jumps 4 / 32

12 Examples Other examples Magnetic eld Stock prices Pollution cf Horváth-Kokoszka (2012) Questions At what instant one may say that there exists a jump? Detect the instants of jumps Estimate intensity of jumps Estimate the greatest jump Predict the mistral next day (or next week). D. Bosq, D. Blanke Detecting and estimating jumps 4 / 32

13 Examples Other examples Magnetic eld Stock prices Pollution cf Horváth-Kokoszka (2012) Questions At what instant one may say that there exists a jump? Detect the instants of jumps Estimate intensity of jumps Estimate the greatest jump Predict the mistral next day (or next week). D. Bosq, D. Blanke Detecting and estimating jumps 4 / 32

14 Examples Other examples Magnetic eld Stock prices Pollution cf Horváth-Kokoszka (2012) Questions At what instant one may say that there exists a jump? Detect the instants of jumps Estimate intensity of jumps Estimate the greatest jump Predict the mistral next day (or next week). D. Bosq, D. Blanke Detecting and estimating jumps 4 / 32

15 Examples Other examples Magnetic eld Stock prices Pollution cf Horváth-Kokoszka (2012) Questions At what instant one may say that there exists a jump? Detect the instants of jumps Estimate intensity of jumps Estimate the greatest jump Predict the mistral next day (or next week). D. Bosq, D. Blanke Detecting and estimating jumps 4 / 32

16 Properties of the space D Properties of the space D D = D[0,1] space of càdlàg real functions dened on [0,1]. (D, ) equipped with the supnorm is not separable. It is better to use the modied Skorohod metric dened as { } d(x,y) = inf λ 0 x y λ ; x,y D λ Λ where Λ is the class of nondecreasing mapping of [0,1] onto [0,1], with λ 0 = sup λ(t) λ(s) log s<t t s, then, D becomes a separable complete metric space. See Billingsley (1999). D. Bosq, D. Blanke Detecting and estimating jumps 5 / 32

17 Properties of the space D Properties of the space D D = D[0,1] space of càdlàg real functions dened on [0,1]. (D, ) equipped with the supnorm is not separable. It is better to use the modied Skorohod metric dened as { } d(x,y) = inf λ 0 x y λ ; x,y D λ Λ where Λ is the class of nondecreasing mapping of [0,1] onto [0,1], with λ 0 = sup λ(t) λ(s) log s<t t s, then, D becomes a separable complete metric space. See Billingsley (1999). D. Bosq, D. Blanke Detecting and estimating jumps 5 / 32

18 Properties of the space D We recall three useful properties of D: If x C = C [0,1] then d (x n,x) 0 x n x 0 If ρ is a bounded linear operator (for ) then it is measurable (for the σ algebra associated with d), x (x) = 1 0 x dµ + a j (x(t j ) x(t j )) denes a continuous linear j 1 form on (D,. ), where µ is a bounded signed measure and (a j ) is a sequence of real numbers such that j 1 a j (x(t j ) x(t j )) <. D. Bosq, D. Blanke Detecting and estimating jumps 6 / 32

19 Properties of the space D We recall three useful properties of D: If x C = C [0,1] then d (x n,x) 0 x n x 0 If ρ is a bounded linear operator (for ) then it is measurable (for the σ algebra associated with d), x (x) = 1 0 x dµ + a j (x(t j ) x(t j )) denes a continuous linear j 1 form on (D,. ), where µ is a bounded signed measure and (a j ) is a sequence of real numbers such that j 1 a j (x(t j ) x(t j )) <. D. Bosq, D. Blanke Detecting and estimating jumps 6 / 32

20 Properties of the space D We recall three useful properties of D: If x C = C [0,1] then d (x n,x) 0 x n x 0 If ρ is a bounded linear operator (for ) then it is measurable (for the σ algebra associated with d), x (x) = 1 0 x dµ + a j (x(t j ) x(t j )) denes a continuous linear j 1 form on (D,. ), where µ is a bounded signed measure and (a j ) is a sequence of real numbers such that j 1 a j (x(t j ) x(t j )) <. D. Bosq, D. Blanke Detecting and estimating jumps 6 / 32

21 ARMAD processes ARMAD processes Let (ξ t, t R) be a continuous-time process (possibly with jumps). In order to modelize a sequence (X n, n Z) of D valued random variables, one may set: X n (t) = ξ n+t, 0 t 1, n Z. Now, consider the ARMAD (1,1) process dened as X n ρ(x n 1 ) = Z n ρ (Z n 1 ), n Z where (Z n ) is a strong white noise in D, and ρ and ρ are bounded linear operators. Assumption 1 ρ j 1 L < 1 and ρ j 2 L < 1 for some integers j 1 1 and j 2 1. Proposition 2 Assumption 1 implies existence and uniqueness of the ARMAD (1,1) process a.s. and in L 2 (in sup norm) and (Z n ) is innovation. D. Bosq, D. Blanke Detecting and estimating jumps 7 / 32

22 ARMAD processes ARMAD processes Let (ξ t, t R) be a continuous-time process (possibly with jumps). In order to modelize a sequence (X n, n Z) of D valued random variables, one may set: X n (t) = ξ n+t, 0 t 1, n Z. Now, consider the ARMAD (1,1) process dened as X n ρ(x n 1 ) = Z n ρ (Z n 1 ), n Z where (Z n ) is a strong white noise in D, and ρ and ρ are bounded linear operators. Assumption 1 ρ j 1 L < 1 and ρ j 2 L < 1 for some integers j 1 1 and j 2 1. Proposition 2 Assumption 1 implies existence and uniqueness of the ARMAD (1,1) process a.s. and in L 2 (in sup norm) and (Z n ) is innovation. D. Bosq, D. Blanke Detecting and estimating jumps 7 / 32

23 ARMAD processes ARMAD processes Let (ξ t, t R) be a continuous-time process (possibly with jumps). In order to modelize a sequence (X n, n Z) of D valued random variables, one may set: X n (t) = ξ n+t, 0 t 1, n Z. Now, consider the ARMAD (1,1) process dened as X n ρ(x n 1 ) = Z n ρ (Z n 1 ), n Z where (Z n ) is a strong white noise in D, and ρ and ρ are bounded linear operators. Assumption 1 ρ j 1 L < 1 and ρ j 2 L < 1 for some integers j 1 1 and j 2 1. Proposition 2 Assumption 1 implies existence and uniqueness of the ARMAD (1,1) process a.s. and in L 2 (in sup norm) and (Z n ) is innovation. D. Bosq, D. Blanke Detecting and estimating jumps 7 / 32

24 ARMAD processes ARMAD processes Let (ξ t, t R) be a continuous-time process (possibly with jumps). In order to modelize a sequence (X n, n Z) of D valued random variables, one may set: X n (t) = ξ n+t, 0 t 1, n Z. Now, consider the ARMAD (1,1) process dened as X n ρ(x n 1 ) = Z n ρ (Z n 1 ), n Z where (Z n ) is a strong white noise in D, and ρ and ρ are bounded linear operators. Assumption 1 ρ j 1 L < 1 and ρ j 2 L < 1 for some integers j 1 1 and j 2 1. Proposition 2 Assumption 1 implies existence and uniqueness of the ARMAD (1,1) process a.s. and in L 2 (in sup norm) and (Z n ) is innovation. D. Bosq, D. Blanke Detecting and estimating jumps 7 / 32

25 ARMAD processes Examples of linear operators in D l(x)(t) = a(t)x(1), 0 t 1, x D where a is continuous: then l is linear and continuous. l m (x) = x( i 1 2 m ) I J m,i 2 m i=1 where J m,i = [ [ i 1 2, 1 i 2 m m 1, and J m,2m = [ 2 m 1 2 m,1 ]. That operator is in general not continuous 2 m, i A typical example of regular linear operator in D is given by 1 ρ r (x)(t) = r(s,t)x(s)ds, 0 t 1, x D 0 where r is continuous (uniformly with respect to s) and sup 0 s,t 1 r(s,t) < 1, then one has ρ r L < 1. In addition, one gets ρ r (D) C = C [0,1]. D. Bosq, D. Blanke Detecting and estimating jumps 8 / 32

26 ARMAD processes Examples of linear operators in D l(x)(t) = a(t)x(1), 0 t 1, x D where a is continuous: then l is linear and continuous. l m (x) = x( i 1 2 m ) I J m,i 2 m i=1 where J m,i = [ [ i 1 2, 1 i 2 m m 1, and J m,2m = [ 2 m 1 2 m,1 ]. That operator is in general not continuous 2 m, i A typical example of regular linear operator in D is given by 1 ρ r (x)(t) = r(s,t)x(s)ds, 0 t 1, x D 0 where r is continuous (uniformly with respect to s) and sup 0 s,t 1 r(s,t) < 1, then one has ρ r L < 1. In addition, one gets ρ r (D) C = C [0,1]. D. Bosq, D. Blanke Detecting and estimating jumps 8 / 32

27 ARMAD processes Examples of linear operators in D l(x)(t) = a(t)x(1), 0 t 1, x D where a is continuous: then l is linear and continuous. l m (x) = x( i 1 2 m ) I J m,i 2 m i=1 where J m,i = [ [ i 1 2, 1 i 2 m m 1, and J m,2m = [ 2 m 1 2 m,1 ]. That operator is in general not continuous 2 m, i A typical example of regular linear operator in D is given by 1 ρ r (x)(t) = r(s,t)x(s)ds, 0 t 1, x D 0 where r is continuous (uniformly with respect to s) and sup 0 s,t 1 r(s,t) < 1, then one has ρ r L < 1. In addition, one gets ρ r (D) C = C [0,1]. D. Bosq, D. Blanke Detecting and estimating jumps 8 / 32

28 ARMAD processes Ornstein-Uhlenbeck process Consider the process n+t ξ n+t = exp(θ(n + t s))dw (s), 0 t 1, n Z (θ > 0). Then: X n (t) = ρ θ (X n 1 )(t) + Z n (t), n Z with ρ θ (x)(t) = exp( θt) x(1), and n+t Z n (t) = exp( θ(n + t s)) dw (s), n Z, n cf Bosq (2000). D. Bosq, D. Blanke Detecting and estimating jumps 9 / 32

29 ARMAD processes Ornstein-Uhlenbeck process Consider the process n+t ξ n+t = exp(θ(n + t s))dw (s), 0 t 1, n Z (θ > 0). Then: X n (t) = ρ θ (X n 1 )(t) + Z n (t), n Z with ρ θ (x)(t) = exp( θt) x(1), and n+t Z n (t) = exp( θ(n + t s)) dw (s), n Z, n cf Bosq (2000). D. Bosq, D. Blanke Detecting and estimating jumps 9 / 32

30 ARMAD processes Modied O.-U. process with jumps Processes One fixed jump t Innovation (in gray), Proc. O.U. with jumps (in blue), Intensity of jumps (in red) `O.-U.' processes with jumps can be constructed with modied innovation (Z n ). For example, one may choose Z n (t) = W 1n (t)i t<t0n + W 2n (t)i t T0n where (W 1n,W 2n ) are independent Wiener, and one has one xed time T 0n t 0 of jump or one random time of jump : i.i.d. T 0n U [0,1] independent from (W 1n,W 2n ). D. Bosq, D. Blanke Detecting and estimating jumps 10 / 32

31 ARMAD processes Modied O.-U. process with jumps Processes One random jump `O.-U.' processes with jumps can be constructed with modied innovation (Z n ). For example, one may choose Z n (t) = W 1n (t)i t<t0n + W 2n (t)i t T0n where (W 1n,W 2n ) are independent Wiener, and one has one xed time T 0n t 0 of jump or one random time of jump : i.i.d. T 0n U [0,1] independent from (W 1n,W 2n ) t Innovation (in gray), Proc. O.U. with jumps (in blue), Intensity of jumps (in red) D. Bosq, D. Blanke Detecting and estimating jumps 10 / 32

32 ARMAD processes Pointwise convergence of an ARD(1) process: an example Set where X n (t) = j=0 ρ j r (Z n j )(t), 0 t 1, n Z Z n (t) = N n+t N n λt, 0 t 1, n Z. Then, (Z n ) is a D-white noise not-separably valued. But, at least, we have almost sure and L 2 consistency for each t [0,1]. D. Bosq, D. Blanke Detecting and estimating jumps 11 / 32

33 ARMAD processes Pointwise convergence of an ARD(1) process: an example Set where X n (t) = j=0 ρ j r (Z n j )(t), 0 t 1, n Z Z n (t) = N n+t N n λt, 0 t 1, n Z. Then, (Z n ) is a D-white noise not-separably valued. But, at least, we have almost sure and L 2 consistency for each t [0,1]. D. Bosq, D. Blanke Detecting and estimating jumps 11 / 32

34 ARMAD processes Limit theorems for ARMAD processes Proposition 3 Strong law of large numbers (SLLN) d(0, Z n ) 0, a.s. X n 0, a.s. Example 1: If (Z n ) is convex tight, the SLLN holds. Example 2: If (Z n ) takes its values in the cone of non-decreasing functions, the SLLN holds. In particular the SLLN can be applied to the a.s. empirical distribution function, namely : F n F 0 (with clear n notation), despite the fact that I Zi t is not a.s. separably valued! (cf Daer and Taylor (1979)). Proposition 4 Central limit theorem (CLT): (Z n ) satises the CLT if and only if (X n ) satises it. Conditions for the CLT in the i.i.d. case appear in Bloznelis-Paulauskas. D. Bosq, D. Blanke Detecting and estimating jumps 12 / 32

35 ARMAD processes Limit theorems for ARMAD processes Proposition 3 Strong law of large numbers (SLLN) d(0, Z n ) 0, a.s. X n 0, a.s. Example 1: If (Z n ) is convex tight, the SLLN holds. Example 2: If (Z n ) takes its values in the cone of non-decreasing functions, the SLLN holds. In particular the SLLN can be applied to the a.s. empirical distribution function, namely : F n F 0 (with clear n notation), despite the fact that I Zi t is not a.s. separably valued! (cf Daer and Taylor (1979)). Proposition 4 Central limit theorem (CLT): (Z n ) satises the CLT if and only if (X n ) satises it. Conditions for the CLT in the i.i.d. case appear in Bloznelis-Paulauskas. D. Bosq, D. Blanke Detecting and estimating jumps 12 / 32

36 ARMAD processes Limit theorems for ARMAD processes Proposition 3 Strong law of large numbers (SLLN) d(0, Z n ) 0, a.s. X n 0, a.s. Example 1: If (Z n ) is convex tight, the SLLN holds. Example 2: If (Z n ) takes its values in the cone of non-decreasing functions, the SLLN holds. In particular the SLLN can be applied to the a.s. empirical distribution function, namely : F n F 0 (with clear n notation), despite the fact that I Zi t is not a.s. separably valued! (cf Daer and Taylor (1979)). Proposition 4 Central limit theorem (CLT): (Z n ) satises the CLT if and only if (X n ) satises it. Conditions for the CLT in the i.i.d. case appear in Bloznelis-Paulauskas. D. Bosq, D. Blanke Detecting and estimating jumps 12 / 32

37 ARMAD processes Limit theorems for ARMAD processes Proposition 3 Strong law of large numbers (SLLN) d(0, Z n ) 0, a.s. X n 0, a.s. Example 1: If (Z n ) is convex tight, the SLLN holds. Example 2: If (Z n ) takes its values in the cone of non-decreasing functions, the SLLN holds. In particular the SLLN can be applied to the a.s. empirical distribution function, namely : F n F 0 (with clear n notation), despite the fact that I Zi t is not a.s. separably valued! (cf Daer and Taylor (1979)). Proposition 4 Central limit theorem (CLT): (Z n ) satises the CLT if and only if (X n ) satises it. Conditions for the CLT in the i.i.d. case appear in Bloznelis-Paulauskas. D. Bosq, D. Blanke Detecting and estimating jumps 12 / 32

38 ARMAD processes Limit theorems for ARMAD processes Proposition 3 Strong law of large numbers (SLLN) d(0, Z n ) 0, a.s. X n 0, a.s. Example 1: If (Z n ) is convex tight, the SLLN holds. Example 2: If (Z n ) takes its values in the cone of non-decreasing functions, the SLLN holds. In particular the SLLN can be applied to the a.s. empirical distribution function, namely : F n F 0 (with clear n notation), despite the fact that I Zi t is not a.s. separably valued! (cf Daer and Taylor (1979)). Proposition 4 Central limit theorem (CLT): (Z n ) satises the CLT if and only if (X n ) satises it. Conditions for the CLT in the i.i.d. case appear in Bloznelis-Paulauskas. D. Bosq, D. Blanke Detecting and estimating jumps 12 / 32

39 Prediction of ARMAD processes Prediction of ARMAD processes For convenience we suppose that ρ = 0. Now, one observes the D valued random variables X 1,...X n, (n 2) and want to predict X n+1. The main problem is to estimate ρ. For this purpose we consider the empirical covariance operator: C n = 1 n n i=1 X i X i and the empirical cross covariance operator D n = 1 n 1 n 1 i=1 X i X i+1. D. Bosq, D. Blanke Detecting and estimating jumps 13 / 32

40 Prediction of ARMAD processes Prediction of ARMAD processes For convenience we suppose that ρ = 0. Now, one observes the D valued random variables X 1,...X n, (n 2) and want to predict X n+1. The main problem is to estimate ρ. For this purpose we consider the empirical covariance operator: C n = 1 n n i=1 X i X i and the empirical cross covariance operator D n = 1 n 1 n 1 i=1 X i X i+1. D. Bosq, D. Blanke Detecting and estimating jumps 13 / 32

41 Prediction of ARMAD processes Now, one denes empirical eigenelements by: where k n = o(n). C n (v jn ) = λ jn v jn, 1 j k n Finally the predictor is given by ρ n (X n )(t) = 1 n 1 0 t 1. k n j=1 1 λjn 1 v jn (s)x n (s)ds 0 n 1 1 v jn (s)x i (s)ds. X i+1 (t), i=1 0 For consistency, cf El Hajj (2014), in Annales de l'isup. D. Bosq, D. Blanke Detecting and estimating jumps 14 / 32

42 Prediction of ARMAD processes Now, one denes empirical eigenelements by: where k n = o(n). C n (v jn ) = λ jn v jn, 1 j k n Finally the predictor is given by ρ n (X n )(t) = 1 n 1 0 t 1. k n j=1 1 λjn 1 v jn (s)x n (s)ds 0 n 1 1 v jn (s)x i (s)ds. X i+1 (t), i=1 0 For consistency, cf El Hajj (2014), in Annales de l'isup. D. Bosq, D. Blanke Detecting and estimating jumps 14 / 32

43 Observation in continuous time Observation in continuous time Assumption 5 ρ(d) C and ρ (D) C. Proposition 6 If Assumption 5 holds, then for every t xed (or random), we have X n (t) X n (t ) = Z n (t) Z n (t ), 0 t 1, n Z, hence, X n and Z n have the same i.i.d. jumps! Proof. Clear from Assumption 5 and X n (t) X n (t ) [(ρ(x n 1 )(t) ρ(x n 1 )(t ))] = Z n (t) Z n (t ) [ ρ (Z n 1 )(t) ρ (Z n 1 )(t ) ], n Z. D. Bosq, D. Blanke Detecting and estimating jumps 15 / 32

44 Observation in continuous time Observation in continuous time Assumption 5 ρ(d) C and ρ (D) C. Proposition 6 If Assumption 5 holds, then for every t xed (or random), we have X n (t) X n (t ) = Z n (t) Z n (t ), 0 t 1, n Z, hence, X n and Z n have the same i.i.d. jumps! Proof. Clear from Assumption 5 and X n (t) X n (t ) [(ρ(x n 1 )(t) ρ(x n 1 )(t ))] = Z n (t) Z n (t ) [ ρ (Z n 1 )(t) ρ (Z n 1 )(t ) ], n Z. D. Bosq, D. Blanke Detecting and estimating jumps 15 / 32

45 Observation in continuous time Observation in continuous time Assumption 5 ρ(d) C and ρ (D) C. Proposition 6 If Assumption 5 holds, then for every t xed (or random), we have X n (t) X n (t ) = Z n (t) Z n (t ), 0 t 1, n Z, hence, X n and Z n have the same i.i.d. jumps! Proof. Clear from Assumption 5 and X n (t) X n (t ) [(ρ(x n 1 )(t) ρ(x n 1 )(t ))] = Z n (t) Z n (t ) [ ρ (Z n 1 )(t) ρ (Z n 1 )(t ) ], n Z. D. Bosq, D. Blanke Detecting and estimating jumps 15 / 32

46 Observation in continuous time The case of xed jumps - If E X <, we have jn = 1 n n i=1 X i(t j ) X i (t j ) a.s. n E j - If E X 2 < : n ( jn E j, 1 j k) L n N k(0,γ k ) Finally, if E(expc X ) <, one may estimate the greatest jump: P( max jn E ( max j) 1 j k 1 j k > ε) = O(exp( nγ(c,ε)). D. Bosq, D. Blanke Detecting and estimating jumps 16 / 32

47 Observation in continuous time The case of xed jumps - If E X <, we have jn = 1 n n i=1 X i(t j ) X i (t j ) a.s. n E j - If E X 2 < : n ( jn E j, 1 j k) L n N k(0,γ k ) Finally, if E(expc X ) <, one may estimate the greatest jump: P( max jn E ( max j) 1 j k 1 j k > ε) = O(exp( nγ(c,ε)). D. Bosq, D. Blanke Detecting and estimating jumps 16 / 32

48 Observation in continuous time The case of xed jumps - If E X <, we have jn = 1 n n i=1 X i(t j ) X i (t j ) a.s. n E j - If E X 2 < : n ( jn E j, 1 j k) L n N k(0,γ k ) Finally, if E(expc X ) <, one may estimate the greatest jump: P( max jn E ( max j) 1 j k 1 j k > ε) = O(exp( nγ(c,ε)). D. Bosq, D. Blanke Detecting and estimating jumps 16 / 32

49 Observation in continuous time The case of ordered random instants X admits k jumps at ordered random instants: 0 < T 1 <... < T k < 1 (a.s.) with random intensity j = X (T j ) X (T j ) 1 j k. One observes n i.i.d. copies, ji, 1 i n, with jumps 0 < T 1i <... < T ki < 1 (a.s.), 1 i n. Set jn = 1 n n i=1 X i (T ji ) X i (T ji ), 1 j k. D. Bosq, D. Blanke Detecting and estimating jumps 17 / 32

50 Observation in continuous time The case of ordered random instants X admits k jumps at ordered random instants: 0 < T 1 <... < T k < 1 (a.s.) with random intensity j = X (T j ) X (T j ) 1 j k. One observes n i.i.d. copies, ji, 1 i n, with jumps 0 < T 1i <... < T ki < 1 (a.s.), 1 i n. Set jn = 1 n n i=1 X i (T ji ) X i (T ji ), 1 j k. D. Bosq, D. Blanke Detecting and estimating jumps 17 / 32

51 Observation in continuous time The case of ordered random instants X admits k jumps at ordered random instants: 0 < T 1 <... < T k < 1 (a.s.) with random intensity j = X (T j ) X (T j ) 1 j k. One observes n i.i.d. copies, ji, 1 i n, with jumps 0 < T 1i <... < T ki < 1 (a.s.), 1 i n. Set jn = 1 n n i=1 X i (T ji ) X i (T ji ), 1 j k. D. Bosq, D. Blanke Detecting and estimating jumps 17 / 32

52 Observation in continuous time Thus, we have jn a.s. n E j, 1 j k. Now, if E (exp( c X ) <, c > 0, then P ( jn E j ε ) a exp( b nε 2 ), a > 0, b > 0, 1 j k. Also, if E X 2 < n ( jn E j ) N (0,σ 2 ), and, Berry-Esseen inequality applies provided E X 3 <. D. Bosq, D. Blanke Detecting and estimating jumps 18 / 32

53 Observation in continuous time Thus, we have jn a.s. n E j, 1 j k. Now, if E (exp( c X ) <, c > 0, then P ( jn E j ε ) a exp( b nε 2 ), a > 0, b > 0, 1 j k. Also, if E X 2 < n ( jn E j ) N (0,σ 2 ), and, Berry-Esseen inequality applies provided E X 3 <. D. Bosq, D. Blanke Detecting and estimating jumps 18 / 32

54 Observation in continuous time Thus, we have jn a.s. n E j, 1 j k. Now, if E (exp( c X ) <, c > 0, then P ( jn E j ε ) a exp( b nε 2 ), a > 0, b > 0, 1 j k. Also, if E X 2 < n ( jn E j ) N (0,σ 2 ), and, Berry-Esseen inequality applies provided E X 3 <. D. Bosq, D. Blanke Detecting and estimating jumps 18 / 32

55 Observation in continuous time Non-ordered random instants Suppose that X admits k non-ordered random instants of jumps T 1,,T k with independent intensity 1,..., k, and E 1 > > E k > 0. Their respective positions are unknown but we have k j=1 (x E j ) = 0 = x k + a k 1 x k a 0. (P) We estimate (a j ) by using X 1,, X n and the ordered jumps i1 ik,,1 i n. D. Bosq, D. Blanke Detecting and estimating jumps 19 / 32

56 Observation in continuous time Non-ordered random instants Suppose that X admits k non-ordered random instants of jumps T 1,,T k with independent intensity 1,..., k, and E 1 > > E k > 0. Their respective positions are unknown but we have k j=1 (x E j ) = 0 = x k + a k 1 x k a 0. (P) We estimate (a j ) by using X 1,, X n and the ordered jumps i1 ik,,1 i n. D. Bosq, D. Blanke Detecting and estimating jumps 19 / 32

57 Observation in continuous time Non-ordered random instants Suppose that X admits k non-ordered random instants of jumps T 1,,T k with independent intensity 1,..., k, and E 1 > > E k > 0. Their respective positions are unknown but we have k j=1 (x E j ) = 0 = x k + a k 1 x k a 0. (P) We estimate (a j ) by using X 1,, X n and the ordered jumps i1 ik,,1 i n. D. Bosq, D. Blanke Detecting and estimating jumps 19 / 32

58 Observation in continuous time Now, we set and we obtain 1 n n ( 1) k 1 n 1 n n i=1 i=1 1 j<j k k almost surely for j = 0,...,k. j=1 ij = 1 n ij ij = n n k i=1 j=1 i=1 1 j<j k n i=1 i1 ik = ( 1)k 1 n â j,n a j n i=1 j := â k 1,n ij ij := â k 2,n i1 ik := â 0,n Now, replacing in equation (P) one may solve it (at least) numerically. Of course, if k 4 the situation is better... D. Bosq, D. Blanke Detecting and estimating jumps 20 / 32

59 Observation in continuous time Now, we set and we obtain 1 n n ( 1) k 1 n 1 n n i=1 i=1 1 j<j k k almost surely for j = 0,...,k. j=1 ij = 1 n ij ij = n n k i=1 j=1 i=1 1 j<j k n i=1 i1 ik = ( 1)k 1 n â j,n a j n i=1 j := â k 1,n ij ij := â k 2,n i1 ik := â 0,n Now, replacing in equation (P) one may solve it (at least) numerically. Of course, if k 4 the situation is better... D. Bosq, D. Blanke Detecting and estimating jumps 20 / 32

60 Observation in continuous time Now, we set and we obtain 1 n n ( 1) k 1 n 1 n n i=1 i=1 1 j<j k k almost surely for j = 0,...,k. j=1 ij = 1 n ij ij = n n k i=1 j=1 i=1 1 j<j k n i=1 i1 ik = ( 1)k 1 n â j,n a j n i=1 j := â k 1,n ij ij := â k 2,n i1 ik := â 0,n Now, replacing in equation (P) one may solve it (at least) numerically. Of course, if k 4 the situation is better... D. Bosq, D. Blanke Detecting and estimating jumps 20 / 32

61 High frequency data (HFD) High frequency data Observation in continuous time is somewhat unrealistic since the data are very irregular! In practice data are often observed in discrete time (cf the mistral). Then we consider observations of the form X ( l ), 0 l q, q where l and q are integers with jumps T 0 = 0 < T 1 < < T k < 1 and we set I k = k [T j 1, T j [ 2 [T k,1] 2 j=1 D. Bosq, D. Blanke Detecting and estimating jumps 21 / 32

62 High frequency data (HFD) Assumption 7 Put EX n = m, then for some α ]0,1], we have the following: ρ and ρ are hölderian: ρ(x)(t) ρ(x)(s) a(x) t s α ρ (x)(t) ρ (x)(s) b(x) t s α m(t) m(s) c m t s α For i.i.d. M n : Z n (t) Z n (s) M n t s α, (s,t) I nk where I nk = [0,T n1 [ 2 [T nk,1[ 2, n Z. E(a(X o )) p <, E(b(Z 0 ) p <, EM p 0 <, p 1 If Z n is a fractional brownian motion or an Ornstein Uhlenbeck process (with jumps) it satises the above condition for some α > 0. D. Bosq, D. Blanke Detecting and estimating jumps 22 / 32

63 High frequency data (HFD) Assumption 7 Put EX n = m, then for some α ]0,1], we have the following: ρ and ρ are hölderian: ρ(x)(t) ρ(x)(s) a(x) t s α ρ (x)(t) ρ (x)(s) b(x) t s α m(t) m(s) c m t s α For i.i.d. M n : Z n (t) Z n (s) M n t s α, (s,t) I nk where I nk = [0,T n1 [ 2 [T nk,1[ 2, n Z. E(a(X o )) p <, E(b(Z 0 ) p <, EM p 0 <, p 1 If Z n is a fractional brownian motion or an Ornstein Uhlenbeck process (with jumps) it satises the above condition for some α > 0. D. Bosq, D. Blanke Detecting and estimating jumps 22 / 32

64 High frequency data (HFD) Now, we use the following result, since it gives a measure of proximity between increments of X and Z : Lemma 8 If Assumption 7 holds we have: X i (t) X i (s) Z i (t) Z i (s) (a(x i ) + b(z i 1 ) + c m I ρ L ) t s α (s,t) [0,1] 2, 1 i n. Corollary 9 For (s,t) I ik = [0,T i1 [ 2 [T ik,1[ 2 : X i (t) X i (s) (a(x i ) + b(z i ) + M i + c m I ρ L ) t s α. D. Bosq, D. Blanke Detecting and estimating jumps 23 / 32

65 High frequency data (HFD) Now, we use the following result, since it gives a measure of proximity between increments of X and Z : Lemma 8 If Assumption 7 holds we have: X i (t) X i (s) Z i (t) Z i (s) (a(x i ) + b(z i 1 ) + c m I ρ L ) t s α (s,t) [0,1] 2, 1 i n. Corollary 9 For (s,t) I ik = [0,T i1 [ 2 [T ik,1[ 2 : X i (t) X i (s) (a(x i ) + b(z i ) + M i + c m I ρ L ) t s α. D. Bosq, D. Blanke Detecting and estimating jumps 23 / 32

66 High frequency data (HFD) The case of k xed jumps Suppose that E X 4 < and that X has k xed jumps at non-ordered distinct instants t 1,..., t k ]0,1[ with E X (t 1 ) X (t 1 ) > > E X (t k ) X (t k ) > 0, k 2. If lim n q n =, then, for q n large enough, there exist integers l j,n, 1 j k such that Now set l j,n 1 q n l,n = 1 n n i=1 < t j l j,n q n := t j,n, 1 j k. X i( l ) X i ( l 1 ) q n q n, 1 l q n. D. Bosq, D. Blanke Detecting and estimating jumps 24 / 32

67 High frequency data (HFD) The case of k xed jumps Suppose that E X 4 < and that X has k xed jumps at non-ordered distinct instants t 1,..., t k ]0,1[ with E X (t 1 ) X (t 1 ) > > E X (t k ) X (t k ) > 0, k 2. If lim n q n =, then, for q n large enough, there exist integers l j,n, 1 j k such that Now set l j,n 1 q n l,n = 1 n n i=1 < t j l j,n q n := t j,n, 1 j k. X i( l ) X i ( l 1 ) q n q n, 1 l q n. D. Bosq, D. Blanke Detecting and estimating jumps 24 / 32

68 High frequency data (HFD) Detecting the instants of jumps Set ˆt 1,n = 1 q n arg max 1 l q n l,n := ˆl 1,n q n, ˆt 2,n = 1 arg max l,n := ˆl 2,n, q n 1 l q n,l ˆl 1,n q n et cetera... Then it can be shown that, hence, if q α n P( k j=1 } 1 {ˆt j,n t j,n ) = O( n ) + O ( q α ) 2 n, < +, almost surely for n large enough, ˆt j,n = t j,n, then ˆt j,n t j 1 q n, 1 j k. Consequently, ˆt j,n t j, j = 1,...,k almost surely. D. Bosq, D. Blanke Detecting and estimating jumps 25 / 32

69 High frequency data (HFD) Detecting the instants of jumps Set ˆt 1,n = 1 q n arg max 1 l q n l,n := ˆl 1,n q n, ˆt 2,n = 1 arg max l,n := ˆl 2,n, q n 1 l q n,l ˆl 1,n q n et cetera... Then it can be shown that, hence, if q α n P( k j=1 } 1 {ˆt j,n t j,n ) = O( n ) + O ( q α ) 2 n, < +, almost surely for n large enough, ˆt j,n = t j,n, then ˆt j,n t j 1 q n, 1 j k. Consequently, ˆt j,n t j, j = 1,...,k almost surely. D. Bosq, D. Blanke Detecting and estimating jumps 25 / 32

70 High frequency data (HFD) Estimating the intensity of jumps By inserting ˆt j,n in the empirical means one may construct estimators of the jumps intensity: ˆ j,n = 1 n n X i(ˆt j,n ) X i (ˆt j,n 1 ) q n, then i=1 ˆ j,n E j for 1 j k, with an almost complete rate. D. Bosq, D. Blanke Detecting and estimating jumps 26 / 32

71 High frequency data (HFD) Exponential rates With more stringent conditions, one obtains P( k } ( {ˆt j,n t j,n ) = O exp( c1 n) ) + O ( exp( c 2 nqn α ) ) j=1 and almost surely for n large enough, ˆ ( lnn j,n E j = O n ). D. Bosq, D. Blanke Detecting and estimating jumps 27 / 32

72 High frequency data (HFD) The pseudo Poisson process (PPP) For some xed k, suppose that Z i has random instants of jumps given by 0 < T i,1 < < T i,k < 1 (a.s.), 1 i n. Recall that i,j := X i (T i,j ) X i (T i,j ) = Z i (T i,j ) Z i (T i,j ), 1 i n, 1 j k. As above, the sample paths are hölderian between jumps. Assumption 10 i,j δ 0 > 0, 1 i n, 1 j k, δ 0 xed. Assumption 11 T i,j T i,j 1 δ 1 > 0, 1 j k with T i,0 = 0 and T i,k+1 = 1, δ 1 xed. These assumptions are somewhat strong but they are compatible with study of the mistral. D. Bosq, D. Blanke Detecting and estimating jumps 28 / 32

73 High frequency data (HFD) The pseudo Poisson process (PPP) For some xed k, suppose that Z i has random instants of jumps given by 0 < T i,1 < < T i,k < 1 (a.s.), 1 i n. Recall that i,j := X i (T i,j ) X i (T i,j ) = Z i (T i,j ) Z i (T i,j ), 1 i n, 1 j k. As above, the sample paths are hölderian between jumps. Assumption 10 i,j δ 0 > 0, 1 i n, 1 j k, δ 0 xed. Assumption 11 T i,j T i,j 1 δ 1 > 0, 1 j k with T i,0 = 0 and T i,k+1 = 1, δ 1 xed. These assumptions are somewhat strong but they are compatible with study of the mistral. D. Bosq, D. Blanke Detecting and estimating jumps 28 / 32

74 High frequency data (HFD) Consider the integer-valued variables L ijn dened as L ijn 1 q n ( and set iln = X l i q n ) < T i,j L ijn q n, 1 j k, 1 i n X i ( l 1 q n ), l = 1,...,qn. Theorem 12 Consider the random set L in = {L i1n,...,l ikn }, 1 i n. Under the above conditions, we obtain ( P n i=1 q n l=1,l/ L in k { } ) j=1 iln ilj n = O(nqn α p ). Thus if n nqn α p <, almost surely for n large enough, we get iln < ilj n for all 1 i n, 1 l q n with l / L in. D. Bosq, D. Blanke Detecting and estimating jumps 29 / 32

75 High frequency data (HFD) Consider the integer-valued variables L ijn dened as L ijn 1 q n ( and set iln = X l i q n ) < T i,j L ijn q n, 1 j k, 1 i n X i ( l 1 q n ), l = 1,...,qn. Theorem 12 Consider the random set L in = {L i1n,...,l ikn }, 1 i n. Under the above conditions, we obtain ( P n i=1 q n l=1,l/ L in k { } ) j=1 iln ilj n = O(nqn α p ). Thus if n nqn α p <, almost surely for n large enough, we get iln < ilj n for all 1 i n, 1 l q n with l / L in. D. Bosq, D. Blanke Detecting and estimating jumps 29 / 32

76 High frequency data (HFD) The previous theorem implies that, looking at the k greatest jumps, we may exactly identify the k intervals containing a jump. By this way, we obtain a set of k values, say L ij that can be ordered to get the estimates L i1n,..., L ikn that are equal to L i1n,...,l ikn almost surely for n large enough. Now, one may show that, for 1 j k: ( ) ( ) n 1 Lijn Lijn n X 1 a.s. i X i q n q n E X 1(T 1j ) X 1 (T 1j ). n i=1 Finally, it is possible to detect jumps in derivative and to estimate their intensity (work in progress...). D. Bosq, D. Blanke Detecting and estimating jumps 30 / 32

77 High frequency data (HFD) The previous theorem implies that, looking at the k greatest jumps, we may exactly identify the k intervals containing a jump. By this way, we obtain a set of k values, say L ij that can be ordered to get the estimates L i1n,..., L ikn that are equal to L i1n,...,l ikn almost surely for n large enough. Now, one may show that, for 1 j k: ( ) ( ) n 1 Lijn Lijn n X 1 a.s. i X i q n q n E X 1(T 1j ) X 1 (T 1j ). n i=1 Finally, it is possible to detect jumps in derivative and to estimate their intensity (work in progress...). D. Bosq, D. Blanke Detecting and estimating jumps 30 / 32

78 High frequency data (HFD) The previous theorem implies that, looking at the k greatest jumps, we may exactly identify the k intervals containing a jump. By this way, we obtain a set of k values, say L ij that can be ordered to get the estimates L i1n,..., L ikn that are equal to L i1n,...,l ikn almost surely for n large enough. Now, one may show that, for 1 j k: ( ) ( ) n 1 Lijn Lijn n X 1 a.s. i X i q n q n E X 1(T 1j ) X 1 (T 1j ). n i=1 Finally, it is possible to detect jumps in derivative and to estimate their intensity (work in progress...). D. Bosq, D. Blanke Detecting and estimating jumps 30 / 32

79 References I High frequency data (HFD) Billingsley, P. Convergence of probability measures. Wiley, Blanke, D. and Bosq, D. Exponential bounds for intensity of jumps. Math. Methods Statist., 23(4), Blanke, D. and Bosq, D. Detecting jumps for discretely observed ARMAD processes, submitted, Bloznelis, M. and Paulauskas, V. On the central limit theorem in D[0,1]. Stat Prob Let, 17(2), , Bosq, D. Estimating and detecting jumps. Applications to D[0,1]-valued linear processes. Festschrift in honour of P. Deheuvels, Springer, Daer, P. and Taylor, R. Laws of large numbers for D[0,1]. Ann. Probab. 7(1), 8595, D. Bosq, D. Blanke Detecting and estimating jumps 31 / 32

80 Chiopu-Kratina, I. and Daer, P. Tightness and the law of large numbers in D[0,1]. Rev. Roumaine Math. Pures Appl., 44(3), , D. Bosq, D. Blanke Detecting and estimating jumps 32 / 32 References II High frequency data (HFD) El Hajj, L. Estimation et prévision des processus autorégressifs à valeurs dans D[0,1], Ann. ISUP 58(2), Horváth, L. and Kokoszka, P. Inference for functional data with applications. Springer Series in Statistics. Springer, New York, Janson, S. and Kaijser, S. Higher Moments of Banach space valued random variables. Memoirs Amer. Math. Soc., to appear. arxiv: , Pestman, W.R. Mesurability of linear operators in the Skorohod topology. Bull. Belg. Math. Soc. Simon Stevin, 2(4), , 1995.