LAN property for ergodic jump-diffusion processes with discrete observations

Transcription

1 LAN property for ergodic jump-diffusion processes with discrete observations Eulalia Nualart (Universitat Pompeu Fabra, Barcelona) joint work with Arturo Kohatsu-Higa (Ritsumeikan University, Japan) & Ngoc Khue Tran (University of Paris 13) Frontier Probability Days, 18th May, 2014 Eulalia Nualart (Universitat Pompeu Fabra) 18th May, / 17

2 Parametric statistical model Consider a real-valued Markov process X θ = (X θ t, t 0) defined on a complete probability space (Ω, F, P) and adapted to the natural filtration {F t} t 0 of a standard Brownian motion B = (B t, t 0). Let P θ be the probability measure induced by X θ. The unkown parameter θ belongs to Θ a closed rectangle of R k. Consider a high frequency observation of X θ denoted X n = (X t0, X t1,..., X tn ), where t k = k n : Distance between observations n 0, as n. Horizon n n, as n. Let P θ n the probability measure on induced by X n under θ. Assume that X n admits a density p n(x; θ). Let p θ (t s, x, y) the transition density of Xt θ Xs θ = x. at y conditionally on Eulalia Nualart (Universitat Pompeu Fabra) 18th May, / 17

3 The LAN property The parametric statistical model {P θ n, θ Θ} is said to have the local asymptotic normality (LAN) property at θ if there exist 1 a k-dimensional vector ϕ n(θ) with strictly positive entries tending to zero as n 2 a k k symmetric positive definite matrix Γ(θ) such that for any u R k, as n, log dpθ+ϕn(θ)u n (X n ) L(Pθ ) u T N ( 0, Γ(θ) ) 1 dp θ n 2 ut Γ(θ)u. N (0, Γ(θ)) : centered R k -valued Gaussian variable with covariance Γ(θ). Γ(θ) :asymptotic Fisher information matrix. References : Le Cam 60, Hájek 70. Extension to Γ(θ) random (mixed normal, LAMN) : Jeganathan Eulalia Nualart (Universitat Pompeu Fabra) 18th May, / 17

4 Remarks Observe that the LAN property is equivalent to log dpθ+ϕn(θ)u n (X n ) = log pn(x n ; θ + ϕ n(θ)u) dp θ n p n(x n ; θ) = u T ϕ n(θ) θ log p n(x n ; θ) 1 2 ut Γ(θ)u + o P θ (1), where ϕ n(θ) θ log p n(x n ; θ) converges in P θ -law to N (0, Γ(θ)). Assume k = 1. By the Markov property log pn(x n ; θ + ϕ n(θ)u) p n(x n ; θ) where θ(l) := θ + lϕ n(θ)u. n 1 = ϕ n(θ)u k=0 1 0 θ log p θ(l) ( n, X tk, X tk+1 )dl Eulalia Nualart (Universitat Pompeu Fabra) 18th May, / 17

5 Expression of the log-likelihood via Malliavin calculus Theorem (Malliavin, Gobet 01) Suppose that Xt θ D 1,2 for all t 0, and there exists a stochastic process u Dom(δ) such that ti+1 t i D tx θ t i u(t)dt = θ X θ t i, i = 0,..., n 1, Then, P θ -a.s., ] θ log p θ ( n, X tk, X tk+1 ) = E θ X tk [δ (u) Xt θ k+1 = X tk+1. Proof If ϕ is a test function, using the chain rule and the duality relation, [ ] [ ] E θ X tk θ ϕ(xt θ k+1 ) = E θ Xtk ϕ (Xt θ k+1 ) θ Xt θ k+1 = E θ X tk ( ϕ(x θ t k+1 )δ(u) ). = E θ X tk ( D(ϕ(X θ t k+1 )), u 2 ) On the other hand, [ ] ( ) E θ X tk θ ϕ(xt θ k+1 ) = E θ Xtk ϕ(xt θ k+1 ) θ log p θ ( n, X tk, Xt θ k+1 ). Eulalia Nualart (Universitat Pompeu Fabra) 18th May, / 17

6 Consequences of the LAN property Assume that {P θ n, θ Θ} satisfies the LAN property. A sequence of estimators (θ n) n 1 is called regular if for all u R k, ϕ n(θ) 1 (θ n (θ + ϕ n(θ)u)) L(Pθ+ϕn(θ)u ) V (θ), as n, for some R k -valued random variable V (θ). Conditional convolution theorem : Let (θ n) n 1 a sequence of regular estimators of θ. Then ( L (V (θ)) = N 0, Γ(θ) 1) G Γ(θ), where G Γ(θ) is independent of N. A sequence of estimators (θ n) n 1 is called asymptotically efficient if ( ϕ 1 n (θ) (θ n θ) L(Pθ ) N 0, Γ(θ) 1), as n. Minimax theorem : Γ(θ) 1 is a lower bound for the asymptotic covariance of any sequence of unbaised estimators. Eulalia Nualart (Universitat Pompeu Fabra) 18th May, / 17

7 LAN property for diffusion processes 1 Gobet 01 derives the LAMN property in the non-ergodic case : t t Xt θ = X 0 + b(θ, s, Xs θ )ds + σ(θ, s, Xs θ )db s, t [0, 1] Gobet 02 shows the LAN property in the ergodic case : t t X θ,β t = X 0 + b(θ, Xs θ,β )ds + σ(β, Xs θ,β )db s, t Delattre and al. 11 have established the LAMN property : t t Xt λ = X 0 + b(s, Xs λ )ds + a(s, Xs λ )db s + c(xt λ k, λ k ), 0 0 k:t k t for t [0, 1], where the jump times T 1, T 2,..., T K are given. 4 Kawai 13, LAN for Ornstein-Uhlenbeck Processes with Jumps Eulalia Nualart (Universitat Pompeu Fabra) 18th May, / 17

8 The density estimates These papers use upper and lower Gaussian type estimates of the transition densities of the diffusion processes. Some computations show that in simpler one dimensional situations (Gaussian type jumps) the upper density estimate is of the type ( ) C y x exp c y x ln, t t and the lower density estimates are of the type ( ) Ce λt y x exp c y x ln, t with a different estimate over the diagonal. This shows that the upper and lower bounds are of different characteristic and Gobet s argument cannot be implemented. Eulalia Nualart (Universitat Pompeu Fabra) 18th May, / 17

9 LAN property for a linear model with jumps Theorem For all z = (u, v, w) R 3, as n, log p (X n ; (θ n, σ n, λ n)) p (X n ; (θ, σ, λ)) X t = x + θt + σb t + N t λt. L(P θ,σ,λ x ) z T N (0, Γ(θ, σ, λ)) 1 2 zt Γ(θ, σ, λ)z, where N (0, Γ(θ, σ, λ)) is a centered R 3 -valued Gaussian vector with covariance matrix Γ(θ, σ, λ) = , σ σ2 λ and we set θ n := θ + u, σ n := σ + v, λ n := λ + w. n n n n n Eulalia Nualart (Universitat Pompeu Fabra) 18th May, / 17

10 Sketch of the proof 1 Markov s property, the integration by parts formula of the Malliavin calculus and Girsanov s theorem, allows to obtain en expansion of the log-likelihood of the form log p (X n ; (θ n, σ n, λ n)) p (X n ; (θ, σ, λ)) n 1 ( ) = ξk,n + H k,n + η k,n + M k,n + β k,n R k,n. k=0 2 Apply a central limit theorem for triangular arrays of random variables to ξ k,n + η k,n + β k,n to show the convergence in law. 3 Negligible terms : the key argument consists in conditioning on the number of jumps J j := {N tk+1 N tk = j} within the conditional expectation which expresses the transition density and outside it. 4 Show a large deviation type inequality. Eulalia Nualart (Universitat Pompeu Fabra) 18th May, / 17

11 Sketch of the proof Large deviation principle : Set M θ, σ, λ 1,p : = M θ, σ, λ 2,p : = j=0 j=0 Lemma Assume that θ θ α (0, 1 ), and n large enough, 2 [ [ ]] j p E θ,σ,λ X tk 1 Jj E θ, σ, λ X θ, σ, λ X tk 1 J c j t k+1 = X tk+1, [ [ E θ,σ,λ X tk 1 Jj E θ, σ, λ ( ) ]] p X θ, σ, λ X tk 1 J c Ntk+1 j N tk t k+1 = X tk+1. C n n and λ λ C n n. Then for all M θ, σ, λ 1,p + M θ, σ, λ 2,p C 1 e 1 C 2 1 2α n. Idea of the proof : Separate the cases 1 { Btk+1 B tk α n } and 1 { Btk+1 B tk > α n }. Eulalia Nualart (Universitat Pompeu Fabra) 18th May, / 17

12 An ergodic diffusion process with jumps dx θ t = b(θ, Xt θ )dt + σ(xt θ )db t + c(xt, θ z) (N(dt, dz) ν(dz)dt) R 0 Usual Lipschitz, ellipticity, ergodicity and regularity assumptions. N(dt, dz) is a Poisson random measure with intensity measure ν(dz)dt, where λ = R 0 ν(dz) < independent of B such that There exist constants q > 1, ρ 1, ρ 2 > 0 and 0 < υ, γ < 1 such that as 2 n, n ( ( n n ν(dz) + { z ρ 2 γ n } { z ρ 1 υ n } ν(dz) )) 1 q 0. There exists n 0 1 such that for all θ Θ, there exists C > 0 such that [ ] sup max e C 1 2γ n Xt 2 k <. E θ x n n 0 k {0,...,n} Eulalia Nualart (Universitat Pompeu Fabra) 18th May, / 17

13 LAN property Theorem For all θ Θ and u R, as n, log p(x n ; θ n) p(x n ; θ) L(P θ x ) un (0, Γ(θ)) u2 2 Γ (θ), where θ n = θ + u n n, and N (0, Γ(θ)) is a centered Gaussian random variable with variance ( ) 2 θ b(θ, x) Γ (θ) = π θ (dx). σ(x) R Eulalia Nualart (Universitat Pompeu Fabra) 18th May, / 17

14 dx θ,β t Another diffusion process with jumps = b(θ, X θ,β t )dt + σ(β, X θ,β t )db t + dz t λ zµ(dz)dt R 0 Z t is a compound Poisson process independent of B rate λ > 0 and jump size distribution µ(dz) = i=1 pa i δa (dz), a i i R 0, 0 p ai 1, i=1 pa i = 1, and for any p > 1, p a i pa i <. For any ω, ω Ω ( ) ( ) Z tk+1 Z tk (ω) Ztk+1 Z tk (ω ) { = 0, or C υ n. Furthermore, for all q > 1 and p {2, 4}, r p P θ,β ( X Ztk+1 tk Z tk = r ) 1 q <, P θ,β ( X Ztk+1 tk Z tk = r ) 1 q <. r r Example : Assume that the jump sizes are positive and for some constant c > 0, e cz µ(dz) 2. Then by the result of Willmot and Lin, 0 for any r > 0, and n large enough P ( Z tk+1 Z tk r ) 2e cr. Eulalia Nualart (Universitat Pompeu Fabra) 18th May, / 17 1

15 LAN property Theorem For all (θ, β) Θ Σ and w = (u, v) R 2, as n, where θ n = θ + log p(x n ; (θ n, β n)) p(x n ; (θ, β)) L(P θ,β x ) w T N (0, Γ(θ, β)) 1 2 w T Γ(θ, β)w, u n n, β n = β + v n, and N (0, Γ(θ, β)) is a centered R 2 -valued Gaussian random variable with covariance matrix ( ) 2 θ b(θ, x) π R θ,β(dx) 0 Γ(θ, β) = σ(β, x) 0 2 ( ) 2 β σ(β, x). π R θ,β(dx) σ(β, x) Eulalia Nualart (Universitat Pompeu Fabra) 18th May, / 17

16 Work in progress 1 The case of infinite jumps requires a convergence argument. 2 The case where the drift is unbounded and we have a general Poisson random measure : dx t = b(θ, X t)dt + σ(β, X t)db t + c(x t, z) (N(dt, dz) ν(dz)dt). R 0 Eulalia Nualart (Universitat Pompeu Fabra) 18th May, / 17

17 References 1 Clément, E., Delattre, S. and Gloter, A. (2013), Asymptotic lower bounds in estimating jumps, to appear in Bernoulli. 2 Corcuera, J.M. and Kohatsu-Higa, A. (2011), Statistical Inference and Malliavin Calculus, Seminar on Stochastic Analysis, Random Fields and Applications VI. 3 Gobet, E. (2001), Local asymptotic mixed normality property for elliptic diffusions : a Malliavin calculus approach, Bernoulli, 7, Gobet, E. (2002), LAN property for ergodic diffusions with discrete observations, Ann. I. H. Poincaré, 38, Kawai, R. (2013), Local Asymptotic Normality Property for Ornstein-Uhlenbeck Processes with Jumps Under Discrete Sampling, J Theor Probab Eulalia Nualart (Universitat Pompeu Fabra) 18th May, / 17