Limit theorems for Hawkes processes

Transcription

1 Limit theorems for nearly unstable and Mathieu Rosenbaum Séminaire des doctorants du CMAP Vendredi 18 Octobre 213 Limit theorems for

2 1 Denition and basic properties Cluster representation of Correlation structure 2 Examples Fitting market data 3 On the scaling of Our main theorem Sketch of the proof Limit theorems for

3 Sommaire Denition and basic properties Cluster representation of Correlation structure 1 Denition and basic properties Cluster representation of Correlation structure 2 3 Limit theorems for

4 Denition and basic properties Cluster representation of Correlation structure Denition (Hawkes 1971) A Hawkes process N is a point process on R + of intensity: t λ t = µ + φ(t s)dn s (1) = µ + Ji <t φ(t J i ) (2) where µ R + is the exogenous intensity and φ is a positive kernel supported in R + which satises φ < 1 and the J i are the points of N. Limit theorems for

5 Basic properties Denition and basic properties Cluster representation of Correlation structure Proposition (Hawkes 1971) The process is well dened and admits a version with stationary increments under the stability condition: φ := φ < 1. Proposition The average intensity of a stationary Hawkes process is E[λ t ] = µ 1 φ. Limit theorems for

6 A population model Denition and basic properties Cluster representation of Correlation structure At time there are no individuals. Migrants arrive as a Poisson process of intensity µ. If a migrant arrives in t, its children are an inhomogenous Poisson process of intensity φ(. t). If a child is born in t, its children are an inhomogenous Poisson process of intensity φ(. t). Etc... Proposition (Hawkes, Oakes 1974) The cumulated number N of individuals who arrived and were born before t is a point process of intensity: λ t = µ + Ji <t φ(t J i ). Limit theorems for

7 Interpretation Denition and basic properties Cluster representation of Correlation structure If we call colony the set of descendants of a migrant, a Hawkes process can be characterized as a Poisson superposition of independant colonies. If there is a migrant in t, the average intensity of its children is φ(. t), the average intensity of its grandchildren is φ φ(. t), etc... Therefore, the average intensity of its descendants is ψ(. t) with ψ = + k=1 φ k. The average cardinal of a colony is thus 1 + ψ = k=1 φ k = 1/(1 φ ). Limit theorems for

8 Denition and basic properties Cluster representation of Correlation structure Proposition (Dayri et al. 212) The correlation of the h-increments of stationary can be computed: C h τ = hλ(g h τ + (g h ψ) τ + (g h ψ) τ + (g h ψ ψ) τ ) where Λ = µ/(1 φ ), ψ(x) = ψ( x) and g h τ = (1 τ /h) +. Given an empirical correlation function, it is possible to numerically nd a φ which ts it. Limit theorems for

9 Sommaire Examples Fitting market data 1 2 Examples Fitting market data 3 Limit theorems for

10 Various applications Examples Fitting market data Ecology. Seismology (Hawkes 1971) reproduces the clusterization of earthquakes. Genomic analysis (cf. Reynaud-Bouret, Schbath 21). Trac network (cf. Brémaud, Massoulié 22). Limit theorems for

11 Examples Fitting market data Financial modeling of market activity Denition The order ow process is the cumulated number of market orders which arrived during the day. are a natural way to reproduce the clusterization of this process (see eg. Bacry, Muzy 213). Most estimation procedures applied to the nancial order ow yield a parameter φ close to one. This is due to the persistence in the order ow. Limit theorems for

12 Scale of the second Examples Fitting market data Figure: Cumulated number of market orders thoughout the day as a function of time (in seconds). Limit theorems for

13 Scale of the hour Examples Fitting market data Figure: Cumulated number of market orders thoughout the day as a function of time (in seconds). Limit theorems for

14 Scale of the day Examples Fitting market data Figure: Cumulated number of market orders thoughout the day as a function of time (in seconds). Limit theorems for

15 A rst result Examples Fitting market data As Poisson processes, at large time scales, behave as deterministic processes. They thus cannot t the data. Theorem (Bacry et al. 213) The sequence of is asymptotically deterministic, in the sense that the following convergence in L 2 holds: sup v [,1] 1 T N Tv E[N Tv ]. Limit theorems for

16 Sommaire On the scaling of Our main theorem Sketch of the proof On the scaling of Our main theorem Sketch of the proof Limit theorems for

17 Formal framework On the scaling of Our main theorem Sketch of the proof We want to study the long term behaviour of close to criticality (whose kernel's norm is close to one). Formally, we consider a sequence of (A T N T Tt ) t indexed by the observation scale T of intensity µ and of kernel with φ = 1 and a T 1. φ T = a T φ Limit theorems for

18 Our asymptotic On the scaling of Our main theorem Sketch of the proof Assumption T (1 a T ) + φ is dierentiable with derivative φ such that Finally, ψ T is bounded. λ. (3) T + sφ(s)ds = m <. (4) φ < + and φ 1 < +. Limit theorems for

19 The theorem On the scaling of Our main theorem Sketch of the proof Let us denote C T t = (1 a T )λ T Tt. Theorem (Jaisson, Rosenbaum 213) The sequence of renormalized Hawkes intensities (Ct T ) converges in law, for the Skorohod topology, towards the law of the unique strong solution of the following Cox Ingersoll Ross stochastic dierential equation on [, 1]: X t = t (µ X s ) λ m ds + λ m t X s db s. Limit theorems for

20 The theorem On the scaling of Our main theorem Sketch of the proof Theorem Furthermore, the sequence of renormalized V T t = 1 a T T N T tt converges in law, for the Skorohod topology, towards the process t X s ds, t [, 1]. Limit theorems for

21 On the scaling of Our main theorem Sketch of the proof We begin by the denition of the intensity t λ T t = µ + φ T (t s)dn T s which can be rewritten λ T t = µ + t φ T (t s)dm T s + t φ T (t s)λ T s ds with dm = dn λdt the martingale compensated point process. This implies that t t λ T t = µ + ψ T (t s)µds + ψ T (t s)dm T s. Limit theorems for

22 Renormalization On the scaling of Our main theorem Sketch of the proof If C T t = (1 a T )λ T Tt : t C T t = (1 a T )µ + µ t + T (1 a T ) λψ T (T (t s))ds λ λψ T (T (t s)) C T s db T s, with B T t = 1 T (1 a T ) tt T λ dm T s λ T s. (5) Limit theorems for

23 On B T On the scaling of Our main theorem Sketch of the proof B T t = 1 T (1 a T ) tt T λ dm T s λ T s. The sequence of processes (B T ) is a sequence of martingales with jumps uniformly bounded by c/ µ. Furthermore, for t [, 1], the quadratic variation of (B T ) at point t is equal to T (1 a T ) λ T Now, remark that E [ ( tt Therefore, B T tt dm T s T λ T s dn T s λ T s = T (1 a T )( tt t + λ ) 2] E [ T dm T s T λ T s ). dn T s (T λ T s ) 2 ds] 4/(T µ). converges towards a Brownian motion. Limit theorems for

24 On x ψ T (Tx) On the scaling of Our main theorem Sketch of the proof Recall that ψ T = a T φ + a 2 T φ φ +... is the average intensity of the descendants of a migrant. We will consider x ρ T (x) = T ψt ψ T 1 (Tx) which is the density of the random variable X T = 1 T I T i=1 X i, where the (X i ) are iid random variables of density φ and I T geometric random variable with parameter 1 a T. is a Limit theorems for

25 Geometric sums On the scaling of Our main theorem Sketch of the proof The asymptotic behaviour of geometric sums can be computed using its Fourier transform: ρ T (z) = E[e izx T ] = (1 a T )(a T ) k 1 E[e i z P k T i=1 X i ] = k=1 (1 a T )(a T ) k 1 ( ˆφ( z ˆφ( z T ))k = ) T 1 a T k=1 1 a T ( ˆφ( z ) 1) T 1 1 using that xφ = m = i ˆφ (). 1 izm T (1 a T ) 1 izm λ Proposition (Rényi's theorem) The sequence of random variable X T converges in law towards an exponential random variable with parameter 1/d = λ/m. Limit theorems for

26 Sketch of the proof On the scaling of Our main theorem Sketch of the proof Heuristically replacing C T by the limiting process C, B T by a Brownian motion B and ψ T (Tx) by 1 m e x λ m in (5), we get C t = µ(1 e t λ m ) + λ m t e (t s) λ m C s db s. This is a characterization of a CIR process: C t = t (µ C s ) λ m ds + λ m t C s db s. Limit theorems for

27 We have essentially shown that if one looks at a Hawkes process of kernel's norm close to one at a time scale of the order 1/(1 φ ) then one sees an integrated CIR process. Limit theorems for

28 Coherence with macroscopic models At macroscopic time scales, the cumulated order ow is empirically proportional to the integrated variance: t V t = κ σ 2 ds. s In many usual frameworks, the macroscopic squared volatility is modelled as a CIR process. This is coherent with the asymptotic behaviour of our order ow. Limit theorems for

29 What if the kernel has no average? Eventhough volatility is often modeled as the square of a CIR process, this does not t empirical data. In fact, estimations yield that the shape of the kernel does not have an average: φ(x) 1/x 1+α with α [,.5]. The correlation scale becomes in 1/(1 φ ) 1/α. In this case, it is much more dicult to obtain strong convergence theorems. We can however asymptotically nd an apparent long memory in the Hawkes process which is coherent with the behaviour of empirical order ows. Limit theorems for

30 Back to geometric sums Let E α C be a random variable whose Fourier transform satises E[e ize α C ] = 1 1 C(iz) α. I T Recall that X T = 1 T i=1 X i and assume that our kernel does not have an average, more precisely ˆφ(z) 1 σ(iz) α for some σ >, < α < 1. Proposition If (1 a T )T α λ >, then X T random variable E α σ. λ converges in law towards the Limit theorems for

31 Thank you for your attention! Limit theorems for