Locally stationary Hawkes processes

Transcription

1 Locally stationary Hawkes processes François Roue LTCI, CNRS, Télécom ParisTech, Université Paris-Saclay Talk based on Roue, von Sachs, and Sansonnet [2016] 1 / 40

2 Outline Introduction Non-stationary Hawkes processes Locally stationary Hawkes processes Numerical experiments References 2 / 40

3 Introduction Non-stationary Hawkes processes Locally stationary Hawkes processes Numerical experiments References 3 / 40

4 Hawkes processes and applications A Hawkes process is a self-exciting point process with linear conditional intensity (see Hawkes [1971]). 4 / 40

5 Hawkes processes and applications A Hawkes process is a self-exciting point process with linear conditional intensity (see Hawkes [1971]). Applications of this model include: seismology Ogata [1988], genomics Reynaud-Bouret and Schbath [2010], neuroscience Reynaud-Bouret et al. [2013], nance: microstructure dynamics for high-frequency data Bowsher [2007], Bacry et al. [2015, 2013]... 4 / 40

6 Preliminary remarks and conventions A point process N is identied with a random measure with discrete support: N = k δ T k, where δ t is the Dirac measure at point t. 5 / 40

7 Preliminary remarks and conventions A point process N is identied with a random measure with discrete support: N = k δ T k, where δ t is the Dirac measure at point t. We will consider point processes on R l (often with l = 1). 5 / 40

8 Preliminary remarks and conventions A point process N is identied with a random measure with discrete support: N = k δ T k, where δ t is the Dirac measure at point t. We will consider point processes on R l (often with l = 1). We do not consider marked or multivariate point processes. 5 / 40

9 Preliminary remarks and conventions A point process N is identied with a random measure with discrete support: N = k δ T k, where δ t is the Dirac measure at point t. We will consider point processes on R l (often with l = 1). We do not consider marked or multivariate point processes. For a test function g, we denote N(g) = g dn = g(t k ). k 5 / 40

10 Preliminary remarks and conventions A point process N is identied with a random measure with discrete support: N = k δ T k, where δ t is the Dirac measure at point t. We will consider point processes on R l (often with l = 1). We do not consider marked or multivariate point processes. For a test function g, we denote N(g) = g dn = g(t k ). k (Some) notation on functional norms: The L q -norm of g is denoted by g q for q [1; ], and 5 / 40

11 Preliminary remarks and conventions A point process N is identied with a random measure with discrete support: N = k δ T k, where δ t is the Dirac measure at point t. We will consider point processes on R l (often with l = 1). We do not consider marked or multivariate point processes. For a test function g, we denote N(g) = g dn = g(t k ). k (Some) notation on functional norms: The L q -norm of g is denoted by g q for q [1; ], and We will use a polynomial weighted L 1 norm depending on some β > 0, h (β) := h β 1 = h(s) s β ds. 5 / 40

12 Stationary Hawkes processes A stationary Hawkes process on R is often dened as a point process N = k δ T k on the real line with conditional intensity t λ(t) = λ c + p(t s) N(ds) = λ c + p(t T i ), (1) T i <t where λ c > 0 and p : R + R + satises p < 1. 6 / 40

13 Stationary Hawkes processes A stationary Hawkes process on R is often dened as a point process N = k δ T k on the real line with conditional intensity λ(t) = λ c + t p(t s) N(ds) = λ c + T i <t p(t T i ), (1) where λ c > 0 and p : R + R + satises p < 1. A description as a Cluster process (Hawkes and Oakes [1974]) can be used to dene a more general class of spatial stationary Hawkes processes in R l : 6 / 40

14 Stationary Hawkes processes A stationary Hawkes process on R is often dened as a point process N = k δ T k on the real line with conditional intensity λ(t) = λ c + t p(t s) N(ds) = λ c + T i <t p(t T i ), (1) where λ c > 0 and p : R + R + satises p < 1. A description as a Cluster process (Hawkes and Oakes [1974]) can be used to dene a more general class of spatial stationary Hawkes processes in R l : Start with a center process N c = k δ tk c, taken as a Poisson point process (PPP) with intensity λ c. 6 / 40

15 Stationary Hawkes processes A stationary Hawkes process on R is often dened as a point process N = k δ T k on the real line with conditional intensity λ(t) = λ c + t p(t s) N(ds) = λ c + T i <t p(t T i ), (1) where λ c > 0 and p : R + R + satises p < 1. A description as a Cluster process (Hawkes and Oakes [1974]) can be used to dene a more general class of spatial stationary Hawkes processes in R l : Start with a center process N c = k δ tk c, taken as a Poisson point process (PPP) with intensity λ c. Given N c, generate independent component processes or clusters N( tk c ) with descendants propagating through a fertility measure µ, 6 / 40

16 Stationary Hawkes processes A stationary Hawkes process on R is often dened as a point process N = k δ T k on the real line with conditional intensity λ(t) = λ c + t p(t s) N(ds) = λ c + T i <t p(t T i ), (1) where λ c > 0 and p : R + R + satises p < 1. A description as a Cluster process (Hawkes and Oakes [1974]) can be used to dene a more general class of spatial stationary Hawkes processes in R l : Start with a center process N c = k δ tk c, taken as a Poisson point process (PPP) with intensity λ c. Given N c, generate independent component processes or clusters N( tk c ) with descendants propagating through a fertility measure µ, The superposition N = N( t c ) N c (dt c ) = N( tk c ) k denes a spatial Hawkes process. 6 / 40

17 Stationary Hawkes processes : generation of clusters Ospring process: given a location t R l, we let M( t) be a PPP with control measure µ( t). 7 / 40

18 Stationary Hawkes processes : generation of clusters Ospring process: given a location t R l, we let M( t) be a PPP with control measure µ( t). Dene successive generations of descendants as follows : 7 / 40

19 Stationary Hawkes processes : generation of clusters Ospring process: given a location t R l, we let M( t) be a PPP with control measure µ( t). Dene successive generations of descendants as follows : Generation 0 : N (0) ( t c ) = δ t c. 7 / 40

20 Stationary Hawkes processes : generation of clusters Ospring process: given a location t R l, we let M( t) be a PPP with control measure µ( t). Dene successive generations of descendants as follows : Generation 0 : N (0) ( t c ) = δ t c. Generation 1 : N (1) ( t c ) = M( t c ). 7 / 40

21 Stationary Hawkes processes : generation of clusters Ospring process: given a location t R l, we let M( t) be a PPP with control measure µ( t). Dene successive generations of descendants as follows : Generation 0 : N (0) ( t c ) = δ t c. Generation 1 : N (1) ( t c ) = M( t c ). Generation n+1 : Denoting N (n) ( t c ) = i δ t (n), i N (n+1) ( t c ) = M( s) N (n) (ds t c ) = M( t (n) i ) i.. 7 / 40

22 Stationary Hawkes processes : generation of clusters Ospring process: given a location t R l, we let M( t) be a PPP with control measure µ( t). Dene successive generations of descendants as follows : Generation 0 : N (0) ( t c ) = δ t c. Generation 1 : N (1) ( t c ) = M( t c ). Generation n+1 : Denoting N (n) ( t c ) = i δ t (n), i N (n+1) ( t c ) = M( s) N (n) (ds t c ) = M( t (n) i ) i. and nally set N( t c ) = k 0 N (k) ( t c ).. 7 / 40

23 Stationary Hawkes processes Each cluster N( t c ) has a nite mean number of points if and only if dµ < 1. The distribution of N( t c t c ) does not depend on t c. Since N c is a homogeneous PPP, the resulting process N is a stationary nite intensity point process. The conditional intensity λ(t) in (1) is obtained by setting µ(dt) = p(t)dt. 8 / 40

24 First and second order moments of stationary Hawkes processes The mean intensity m 1 of N is given by m 1 = λ c 1 dµ. The Bartlett spectrum of N dened by Cov (N(g 1 ), N(g 2 )) = ĝ 1 (ω)ĝ 2 (ω) Γ(dω) with ĝ j (ω) = g j (t) e iωt dt, j = 1, 2, is given by Γ(dω) = m 1 2π 1 e itω µ(dt) 2 dω 9 / 40

25 A parametric example Consider a specic class of Gamma-shaped fertility functions p, depending on parameters ζ, δ, η, θ > 0 p(t) = ζ θη (t δ)η 1 + Γ(η) e θ(t δ). Then the stationarity assumption simply reads p = ζ < 1. Moreover, ˆp(ω) = p(t) e itω dt = ζe iδω (1 + iω/θ) η. 10 / 40

26 Bartlett spectrum of a Hawkes process with Gamma shape fertility A positive δ induces a periodic phenomenon in the self-excitation. Here we take ζ = 0.6, δ = 1, η = 2, θ = / 40

27 Non-stationary modeling In many applications, it is interesting to investigate whether the statistical behavior evolves in time or space. 12 / 40

28 Non-stationary modeling In many applications, it is interesting to investigate whether the statistical behavior evolves in time or space. For instance, one can think of a Gamma shape fertility with time-varying parameters. 12 / 40

29 Non-stationary modeling In many applications, it is interesting to investigate whether the statistical behavior evolves in time or space. For instance, one can think of a Gamma shape fertility with time-varying parameters. In the latter case, the Bartlett spectrum should also evolves along the time, resulting in a time-frequency analysis. 12 / 40

30 Non-stationary modeling In many applications, it is interesting to investigate whether the statistical behavior evolves in time or space. For instance, one can think of a Gamma shape fertility with time-varying parameters. In the latter case, the Bartlett spectrum should also evolves along the time, resulting in a time-frequency analysis. Our goal is to provide a reasonable modeling approach for a comprehensive statistical analysis in this non-stationary context. 12 / 40

31 Introduction Non-stationary Hawkes processes Locally stationary Hawkes processes Numerical experiments References 13 / 40

32 Non-stationary Hawkes processes, using conditional intensity Non-stationary Hawkes processes often refer to a process initiated in the empty state at the origin, t λ(t) = λ c + p(t s) N(ds), t 0. 0 In Chen and Hall [2013], additional non-stationarity is introduced by letting λ c depend on time, t λ(t) = λ c (t) + p(t s) N(ds), t 0. 0 We propose the more general form λ(t) = λ c (t) + t p(t s; t) N(ds). 14 / 40

33 Non-stationary Hawkes processes, using clusters Similarly, in the cluster construction, one can Replace the constant baseline intensity λ c of N c by a baseline intensity function t λ c (t). Replace the control measure µ( s) of the ospring process M( s) by any measure µ( s). 15 / 40

34 Non-stationary Hawkes processes, using clusters Similarly, in the cluster construction, one can Replace the constant baseline intensity λ c of N c by a baseline intensity function t λ c (t). Replace the control measure µ( s) of the ospring process M( s) by any measure µ( s). Question What should replace the condition µ < 1 used to guaranty the existence of a nite mean intensity stationary Hawkes process? 15 / 40

35 Non-stationary Hawkes processes, stability Consider the two following conditions : (C-1) ζ 1 = sup µ(dt s) < 1. s (C-2) µ( s)ds admits a density bounded by ζ 1 < 1. Then if µ( s) = µ( s), they both are equivalent to dµ < / 40

36 Non-stationary Hawkes processes, stability Consider the two following conditions : (C-1) ζ 1 = sup µ(dt s) < 1. s (C-2) µ( s)ds admits a density bounded by ζ 1 < 1. Then if µ( s) = µ( s), they both are equivalent to dµ < 1. It turns out that (C-1) implies that E N(ds t c ) (1 ζ 1 ) 1 (each cluster has a uniformly bounded nite mean number of points). 16 / 40

37 Non-stationary Hawkes processes, stability Consider the two following conditions : (C-1) ζ 1 = sup µ(dt s) < 1. s (C-2) µ( s)ds admits a density bounded by ζ 1 < 1. Then if µ( s) = µ( s), they both are equivalent to dµ < 1. It turns out that (C-1) implies that E N(ds t c ) (1 ζ 1 ) 1 (each cluster has a uniformly bounded nite mean number of points). (C-2) and λ c < imply that N admits a nite mean intensity bounded by λ c /(1 ζ 1 ). 16 / 40

38 Non-stationary Hawkes processes, density assumption Suppose that the measure µ( s) admits a density, written as t d(t s; s). Then (C-1) reads ζ 1 := sup s d(r; s)dr < / 40

39 Non-stationary Hawkes processes, density assumption Suppose that the measure µ( s) admits a density, written as t d(t s; s) =: p(t s; t). Then (C-1) reads ζ 1 := sup s While (C-2) reads ζ 1 := sup t d(r; s)dr < 1. p(r; t)dr < / 40

40 Non-stationary Hawkes processes, density assumption Suppose that the measure µ( s) admits a density, written as t d(t s; s) =: p(t s; t). Then (C-1) reads ζ 1 := sup s While (C-2) reads ζ 1 := sup t d(r; s)dr < 1. p(r; t)dr < 1. Denition (Non-stationary Hawkes processes) We call a non-stationary Hawkes process a process dened as N above under the condition (NS-1) ζ 1 := sup p(r; t)dr < 1 and λ c <, t which implies that N has nite mean intensity bounded by λ c /(1 ζ 1 ). 17 / 40

41 Introduction Non-stationary Hawkes processes Locally stationary Hawkes processes Numerical experiments References 18 / 40

42 Parallel with time series A Hawkes process with fertility function p is often paralleled with an autoregressive (AR) time series with parameter θ, since in both cases, conditional intensity/expectation are linear functions of the past, resulting in similar Bartlett/spectral density formula m 1 2π 1 p(t) e itω 2 / dt σ 2 2π 1 k (here expressed as functions of ω and λ, resp.) θ k e iλk 2 19 / 40

43 Parallel with time series A Hawkes process with fertility function p is often paralleled with an autoregressive (AR) time series with parameter θ, since in both cases, conditional intensity/expectation are linear functions of the past, resulting in similar Bartlett/spectral density formula m 1 2π 1 p(t) e itω 2 / dt σ 2 2π 1 k (here expressed as functions of ω and λ, resp.) θ k e iλk 2 We can draw the same parallel between the previously introduced non-stationary Hawkes process and time-varying AR (TVAR) processes dened by the equation X t = µ(t) + k θ k (t)x t k + σ(t)ɛ t. 19 / 40

44 Asymptotic setting: locally stationary TVAR processes We propose to follow the approach used in Dahlhaus [1996] for time series, using rescaled time-varying parameters σ, µ and θ, X t = µ(t ) + k θ k (t )X t k + σ(t )ɛ t. 20 / 40

45 Asymptotic setting: locally stationary TVAR processes We propose to follow the approach used in Dahlhaus [1996] for time series, using rescaled time-varying parameters σ, µ and θ, X t,t = µ(t/t ) + k θ k (t/t )X t k,t + σ(t/t )ɛ t. We thus get a collection of processes (X t,t ) t Z, T > / 40

46 Asymptotic setting: locally stationary TVAR processes We propose to follow the approach used in Dahlhaus [1996] for time series, using rescaled time-varying parameters σ, µ and θ, X t,t = µ(t/t ) + k θ k (t/t )X t k,t + σ(t/t )ɛ t. We thus get a collection of processes (X t,t ) t Z, T > 0. Two important consequences as T A T -sample X 1,T,..., X T,T basically involves the parameter function u (σ(u), µ(u), θ(u)) on a xed interval u [0, 1], allowing us for a consistent estimation of these parameters. If u (σ(u), µ(u), θ(u)) is smooth, a subsample X t,t with indices t such that t/t u (0, 1), can be approximated by a stationary AR process with parameter (σ(u), µ(u), θ(u)). 20 / 40

47 Asymptotic setting: locally stationary Hawkes processes Denition (Locally stationary Hawkes processes) A locally stationary Hawkes process with local baseline intensity λ c <LS> : R l R + and local fertility function p <LS> : R l R l R + is a collection (N T ) T >0 such that, for all T > 0, N T is a non-stationary Hawkes processes with baseline intensity t λ c <LS> (t/t ) and fertility function (r, t) p <LS> (r; t/t ). 21 / 40

48 Asymptotic setting: locally stationary Hawkes processes Denition (Locally stationary Hawkes processes) A locally stationary Hawkes process with local baseline intensity λ c <LS> : R l R + and local fertility function p <LS> : R l R l R + is a collection (N T ) T >0 such that, for all T > 0, N T is a non-stationary Hawkes processes with baseline intensity t λ c <LS> (t/t ) and fertility function (r, t) p <LS> (r; t/t ). For a given real location t, the scaled location u = t/t is typically called an absolute location. At each u, we denote by N( ; u) a stationary Hawakes process with baseline intensity λ c <LS> (u) and fertility function p <LS> ( ; u). 21 / 40

49 Assumptions on local baseline intensity and local fertility function Condition (NS-1) becomes (LS-1) ζ <LS> 1 := sup p <LS> (r; u) dr < 1 u Note that it does not involve T. and λ c <LS> <. 22 / 40

50 Assumptions on local baseline intensity and local fertility function Condition (NS-1) becomes (LS-1) ζ <LS> 1 := sup p <LS> (r; u) dr < 1 u Note that it does not involve T. and λ c <LS> <. More regularity assumptions: (LS-2): β-hölder type smoothness condition on λ c <LS> (u) (LS-3): β-hölder type smoothness conditions on p <LS> (t, u) w.r.t. its second argument u. (LS-4): some β-power decay condition on p <LS> ( ; u) uniformly bounded w.r.t. u. 22 / 40

51 Laplace functional We use Laplace functionals to show that, for a given absolute location u, as T, in the neighborhood of Tu, N T can be approximated by N( ; u). For all T > 0 and u R l, letting g denote some test function, the Laplace functional of N T is denoted by L T (g) = E [exp N T (g)]. the Laplace functional of N( ; u) is denoted by L(g; u) = E [exp N(g; u)]. Hence we compare L T S Tu with L( ; u), where S Tu g : t g(t Tu). 23 / 40

52 Approximation results A typical result is that if g 1, g, g (β) 1, then L T S Tu (g) L(g; u) = O ( T β), where constants in O-terms only depend on λ c <LS> and p <LS>. 24 / 40

53 Approximation results A typical result is that if g 1, g, g (β) 1, then L T S Tu (g) L(g; u) = O ( T β), where constants in O-terms only depend on λ c <LS> and p <LS>. In fact, under suitable conditions, the result holds uniformly in z for functions g = g( ; z) which are holomorphic w.r.t. z, see [Roue et al., 2016, Theorem 2]. 24 / 40

54 Approximation results A typical result is that if g 1, g, g (β) 1, then L T S Tu (g) L(g; u) = O ( T β), where constants in O-terms only depend on λ c <LS> and p <LS>. In fact, under suitable conditions, the result holds uniformly in z for functions g = g( ; z) which are holomorphic w.r.t. z, see [Roue et al., 2016, Theorem 2]. Consequently, we obtain, if g j 1, g j, g j (β) 1 for j = 1,..., m, then ) ( Cum ({N T S Tu (g j )} 1 j m Cum ({N(g j ; u)} 1 j m ) = O T β). 24 / 40

55 Example of application: approximation of local mean density For all T > 0 and t R l, let m 1T (t) denote the mean density function of N T. For all u R l Let m 1 <LS> (u) denote the local mean intensity, that is, the mean intensity of N( ; u), which is given by m 1 <LS> (u) = λ c <LS> (u) 1 p <LS> ( ; u). The previous result with m = 1 implies: sup m 1T (t) m <LS> 1 (u) = O t : t Tu b ((1 + b β )T β) 25 / 40

56 Another application : time-frequency analysis Suppose that l = 1. We can dene and approximate the local Bartlett spectrum as follows : For all u R l, let γ <LS> (ω; u) denote the local Bartlett density, that is, the Bartlett density of N( ; u) which is where γ <LS> (ω; u) = m 1 <LS> (u) 2π p <LS> (ω; u) = 1 p <LS> (ω; u) p <LS> (t; u) e iωt dt The cumulent approximation result with m = 2 implies: ( ) Var N T (S Tu g) = ĝ(ω) 2 γ <LS> (ω; u) dω + O(T β ) 2 Kernel estimation of γ <LS> (ω; u) can thus be achieved by an empirical estimate of Var ( N T (S Tu g) ) for a well chosen g. 26 / 40

57 Introduction Non-stationary Hawkes processes Locally stationary Hawkes processes Numerical experiments References 27 / 40

58 Simulation of a locally stationary Hawkes process If l = 1 and p <LS> ( ; u) is supported on R + for all u, we can use that N T has conditional intensity given by λ T (t) = λ <LS> c (t/t ) + p <LS> (t s; t/t ) N T (ds). Use Ogata's modied thinning algorithm Ogata [1981] to simulate the non-stationary Hawkes process N T on the interval [0, T ]. 28 / 40

59 Simulated examples We take a time-constant baseline intensity λ c <LS> (u) = 1/2. The local fertility function p <LS> ( ; u) is set as a Gamma-shaped function with time varying parameters. 29 / 40

60 Simulated examples We take a time-constant baseline intensity λ c <LS> (u) = 1/2. The local fertility function p <LS> ( ; u) is set as a Gamma-shaped function with time varying parameters. Example 1 [Time varying scale θ(u) and overall fertility ζ(u)]: p <LS> (s; u) = ζ(u) θ(u)e θ(u)s 1 s>0, with ζ(u), θ(u) of cosine form. 29 / 40

61 Simulated examples We take a time-constant baseline intensity λ c <LS> (u) = 1/2. The local fertility function p <LS> ( ; u) is set as a Gamma-shaped function with time varying parameters. Example 1 [Time varying scale θ(u) and overall fertility ζ(u)]: p <LS> (s; u) = ζ(u) θ(u)e θ(u)s 1 s>0, with ζ(u), θ(u) of cosine form. Example 2 [Time varying delay δ(u)]: p <LS> (s; u) = 1 2 (s δ(u)) +e (s δ(u)) with δ(u) = (6 10u) 1 [0;1/2] (u) + (10u 4) 1 (1/2;1] (u) inducing a periodic phenomenon in the self-excitation. 29 / 40

62 Figure: Theoretical local mean density (top) and Bartlett spectrum (bottom) for Example / 40

63 Figure: Conditional intensity function of a simulated Hawkes process following Example 1, with T = / 40

64 Figure: Estimation of the local mean density (top) and of the local Bartlett spectrum (bottom) for Example / 40

65 Figure: Theoretical local Bartlett spectrum for Example 2 (local mean density being constant over time). 33 / 40

66 Figure: Conditional intensity function of a simulated Hawkes process following Example 2, with T = / 40

67 Figure: Estimation of the local Bartlett spectrum for Example / 40

68 Conclusion Self-exciting point processes ("Hawkes" processes) are somewhat similar to autoregressive processes in time series. 36 / 40

69 Conclusion Self-exciting point processes ("Hawkes" processes) are somewhat similar to autoregressive processes in time series. Unfortunately, unlike linear locally stationary time series, we cannot rely on a simple representation such as the TVMA( ) (or spectral representation). 36 / 40

70 Conclusion Self-exciting point processes ("Hawkes" processes) are somewhat similar to autoregressive processes in time series. Unfortunately, unlike linear locally stationary time series, we cannot rely on a simple representation such as the TVMA( ) (or spectral representation). Local stationary approximations can still be obtained using local Laplace transforms and its derivatives (control of cumulants). 36 / 40

71 Conclusion Self-exciting point processes ("Hawkes" processes) are somewhat similar to autoregressive processes in time series. Unfortunately, unlike linear locally stationary time series, we cannot rely on a simple representation such as the TVMA( ) (or spectral representation). Local stationary approximations can still be obtained using local Laplace transforms and its derivatives (control of cumulants). Denitions of (local) mean intensity and a local Bartlett spectrum follow. 36 / 40

72 Conclusion Self-exciting point processes ("Hawkes" processes) are somewhat similar to autoregressive processes in time series. Unfortunately, unlike linear locally stationary time series, we cannot rely on a simple representation such as the TVMA( ) (or spectral representation). Local stationary approximations can still be obtained using local Laplace transforms and its derivatives (control of cumulants). Denitions of (local) mean intensity and a local Bartlett spectrum follow. Work in progress: Asymptotic estimation theory. 36 / 40

73 Introduction Non-stationary Hawkes processes Locally stationary Hawkes processes Numerical experiments References 37 / 40

74 References I E. Bacry, S. Delattre, M. Homann, and J.F. Muzy. Some limits for Hawkes processes and application to nancial statistics. Stochastic Process. Appl., 123(7): , E. Bacry, I. Mastromatteo, and J-F. Muzy. Hawkes processes in nance. ArXiv e-prints, ( v2), C.G. Bowsher. Modelling security market events in continuous time: Intensity based, multivariate point process models. J. Econometrics, 141(2):876912, Dec F. Chen and P. Hall. Inference for a non-stationary self-exciting point process with an application in ulta-high frequency nancial data modeling. J. Appl. Probab., 50: , R. Dahlhaus. On the Kullback-Leibler information divergence of locally stationary processes. Stochastic Process. Appl., 62: , / 40

75 References II A.G. Hawkes. Spectra of some self-exciting and mutually exciting point processes. Biometrika, 58(1):8390, A.G. Hawkes and D. Oakes. A cluster process representation of a self-exciting process. Journal of Applied Probability, 11(3): , Y. Ogata. On Lewis' simulation method for point processes. Information Theory, IEEE Transactions on, 27(1):2331, Jan Y. Ogata. Statistical Models for Earthquake Occurrences and Residual Analysis for Point Processes. J. Amer. Statist. Assoc., 83 (401):927, Mar P. Reynaud-Bouret and S. Schbath. Adaptive estimation for Hawkes processes; application to genome analysis. Ann. Statist., 38(5): , / 40

76 References III P. Reynaud-Bouret, V. Rivoirard, and C. Tuleau-Malot. Inference of functional connectivity in neurosciences via Hawkes processes. In 1st IEEE Global Conference on Signal and Information Processing, Austin, Texas, USA, François Roue, Rainer von Sachs, and Laure Sansonnet. Locally stationary hawkes processes. Stochastic Processes and their Applications, 126(6): , ISSN doi: URL S / 40