Weierstrass Institute for Applied Analysis and Stochastics Maximum likelihood estimation for jump diffusions

Transcription

1 Weierstrass Institute for Applied Analysis and Stochastics Maximum likelihood estimation for jump diffusions Hilmar Mai Mohrenstrasse Berlin Germany Tel Haindorf Seminar 212

2 Discontinuities in market data Tankov et al. [27] Drift estimation for jump diffusions Haindorf Seminar 212 Page 2 (24)

3 Jump diffusions - an overview dx t = γ(x t, t) dt + σ(x t, t) dl t, t [, T ] X = x Mathematical finance Option pricing: Merton [1976] Stochastic volatility models: Heston [1993], Barndorff-Nielsen and Shephard [21] Interest rate models: Cox, Ingersoll and Ross [1985] Condensed matter physics Atomic diffusion modelling in Hall and Ross [1981] Neuroscience Neuronal membrane potential in Jahn, Berg, Hounsgaard and Ditlevsen [211] Drift estimation for jump diffusions Haindorf Seminar 212 Page 3 (24)

4 Drift estimation for jump diffusions Consider an SDE with driver (L t, t ) on a filtered probability space (Ω, F, (F t) t, P ) dx t = γ(x t, t, θ) dt + σ(x t, t) dl t, t [, T ] X = x L a Lévy process and parameter θ A R. We call the solution X a jump diffusion. Recall that L is a Lévy process if L has independent and stationary increments, L is continuous in probability, i.e. p lim t sl t = L s How to estimate θ efficiently when (X t) t [,T ] observed continuously? given discrete oberservations X t1,..., X tn? Drift estimation for jump diffusions Haindorf Seminar 212 Page 4 (24)

5 Absolute coninuity for jump diffusions Define K t(x, A) = 1 A\{} (γ(t, x)) µ(dy), R d β(θ, s, x) = δ(θ, s, x) + b + y K st(x, dy), x >1 c(s, x) = γ(t, x)γ(t, x) a(θ, θ, s, x) = β(θ, s, X) β(θ, s, x). (1) Under suitable conditions on the coefficients of X the Hellinger process h for the absolute continuity problem induced by X is given by h t(θ, θ ) = t (c(s, X t) 1 a(θ, θ, s, X t)) c(s, X t)(c(s, X t) 1 a(θ, θ, s, X t)). Drift estimation for jump diffusions Haindorf Seminar 212 Page 5 (24)

6 Likelihood theory for jump diffusions Theorem Suppose that the coefficients are locally Lipschitz and c is strictly positive definite. Then (i) P θ t loc Pt θ if and only if P θ (h t(θ, θ ) < ) = P θ (h t(θ, θ ) < ) = 1. (ii) If (ii) holds the likelihood function is given by [ dpt θ t = exp c(s, X s ) 1 a(θ, θ, s, X s ) dx dpt θ s c 1 t ] a(θ, θ, s, X s) c(s, X s ) 1 a(θ, θ, s, X s) ds 2 here X c denotes the continuous martingale part under P θ. Drift estimation for jump diffusions Haindorf Seminar 212 Page 6 (24)

7 Lévy-driven Ornstein-Uhlenbeck processes Let (L t, t ) be a Lévy process on (Ω, F, (F t), P ). For every a R dx t = ax t dt + dl t, t R +, X = x, (2) defines an Ornstein-Uhlenbeck process driven by the Lévy process L. Equivalently, The likelihood function is dp a t dp a t = dp a ( t exp dp a X t = e at X + t e a(t s) dl s. (3) (a a) X σ 2 s dxs c (a a) 2 2σ 2 where X c denotes the continuous martingale part of X under P a. The MLE vor a is â T = T Xs dxc s T X2 s ds. t ) Xs 2 ds, Drift estimation for jump diffusions Haindorf Seminar 212 Page 7 (24)

8 Asymptotic properties of MLE The form of the likelihood function means that we are in curved exponential family setting (cf. Küchler and Sørensen (1997)). Theorem If σ 2 > the MLE â T = T Xs dxc s T X2 s ds. exists and is strongly consistent. If furthermore X is stationary and E a[x 2 ] < then under P a as T. T (ât a) N ( ) σ 2, E a[x 2] weakly Drift estimation for jump diffusions Haindorf Seminar 212 Page 8 (24)

9 Local asymptotic normality Theorem Assume that X is stationary and E a[x 2 ] <, then the following holds: 1. The statistical experiment {P a, a R} is locally asymptotically normal. 2. The maximum likelihood estimator â T is asymptotically efficient in the sense of Hájek-Le Cam. Drift estimation for jump diffusions Haindorf Seminar 212 Page 9 (24)

10 Lévy-driven CIR processes Let (L t, t ) be a Lévy process on (Ω, F, (F t), P ). For every a R dx t = ax tdt + X tdl t; t T defines Cox-Ingersoll-Ross process X driven by the Lévy process L. No explicit solution is known for this SDE. The likelihood function for continuous-time observations is dpt a = exp [ 1σ dpt 2 (XcT X c) aσ T 2 where X c denotes the continuous martingale part of X under P a. The MLE vor a is ] X t dt. â T = Xc T X c T. (4) Xt dt Drift estimation for jump diffusions Haindorf Seminar 212 Page 1 (24)

11 Consistency and estimation error Theorem Let X be stationary, σ 2 > and E[X 2 ] <. Then the MLE â T is consistent and as T. T (ât a) D N(, σ 2 E[ X 1 ]) (5) Moreover, the model is locally asymptotically normal such that the MLE is efficient in the sense of Hájek-Le Cam. Drift estimation for jump diffusions Haindorf Seminar 212 Page 11 (24)

12 Linear stochastic delay equations Let L be a Lévy process with Lévy -Khintchine triplet (γ, σ 2, µ) and consider the following linear stochastic delay differential equation (SDDE) driven by L. dx t = ax t dt + bx t 1 dt + dl t, t > (6) X t = X t, t [ 1, ], (7) where a, b R and X : [ 1, ] Ω R is the initial process with càdlàg trajectories that is assumed to be independent of L. The likelihood function reads where L(θ, X, T ) = dp ( T θ = exp θ V dp (,) T 1 ) 2 θ I T θ, T V T = ( ) T Xt dxc t T Xt 1 dxc t and I T is the observed Fisher information ( T ) T I T = X2 t dt XtXt 1 dt T XtXt 1 dt T. X2 t 1 dt Drift estimation for jump diffusions Haindorf Seminar 212 Page 12 (24)

13 Consistency and asymptotic normality Theorem Assume that X is a stationary solution, then the MLE ˆθ T is strongly consistent, i.e. under P a ˆθ T a.s. θ as t. Theorem Assume that X is a stationary solution, then T 1/2 (ˆθT ) D θ N (, Σ 1) where ( Σ = σ 2 x (s) 2 ds x (s)x (s + 1) ds ) x (s)x (s + 1) ds x (s) 2. ds and x denotes the fundamental solution. Drift estimation for jump diffusions Haindorf Seminar 212 Page 13 (24)

14 Continuous martingale part X c By the Lévy-Itô decomposition of L we can write X as t X t = X a X s ds + σw t + J t, t, where W is a standard Wiener process and J a quadratic pure jump process in the sense of Protter given by J t = { x <1} x(n t(dx) tµ(dx)) + bt + s t X s1 { Xs 1}. Hence, the continuous martingale part of X under P a is t X c = W t a X s ds. Drift estimation for jump diffusions Haindorf Seminar 212 Page 14 (24)

15 Influence of jump noise An interesting question is the influence of jumps on the MLE. Define X j t (ɛ) = x(n t(dx) tµ(dx)), x ɛ then the resulting estimate remains strongly consistent. Theorem Let us assume that X is stationary with E[X 2 ] <, σ 2 > and set X cj (ɛ) = X c + X j (ɛ). If we define ã ɛ T = T Xs dxcj T X2 s ds s (ɛ), then ã ɛ T a with probability 1 as T. Drift estimation for jump diffusions Haindorf Seminar 212 Page 15 (24)

16 Influence of jumps on estimation error Theorem Let X be a stationary Ornstein-Uhlenbeck process with E a[x 2 ] <, then T (ã ɛ T a) N(, Σ(ɛ)) as T where Σ(ɛ) = E a[x 2 ] 1 σ 2 + E a[x 2 ] 1 x <ɛ x 2 µ(dx). Drift estimation for jump diffusions Haindorf Seminar 212 Page 16 (24)

17 Maximum likelihood versus least squares The LSE for the parameter a is a LS T = T T X2 s ds Xs dxs From the theorem it follows for the asymptotic variances AV AR LSE AV AR MLE = E a[x 2 ] x 2 µ(dx) >. This motivates the jump filtering approach that will be discussed in the next section. R Drift estimation for jump diffusions Haindorf Seminar 212 Page 17 (24)

18 Filtering jumps from discrete observations Observation scheme: Observation points = t 1 <... < t n = T n such that T n n and n = max{ t i+1 t i, 1 i n 1} n. Discretized MLE with jump filter: n i=1 ā n := X t n ix1 i { i X v n} n i=1 X2 t n n i i for ix = X ti+1 X ti and some cut-off sequence v n >. Drift estimation for jump diffusions Haindorf Seminar 212 Page 18 (24)

19 Discretized MLE with jump filter Assumption 1. There exists α (, 2) such that as v v x 2 µ(dx) = O(v 2 α ). (8) v 2. There exists δ > such that for all ɛ δ E[ im1 { i M ɛ}] = Theorem If Assumption 1. and 2. hold and v n = β n for β (, 1/2) such that T n 1/2 β n n then Tn 1/2 (ā n a) D N(, σ 2 E[X 2 ] 1 ). Furthermore, the estimator is asymptotically efficient. Hence, the truncated MLE is asymptotically efficient in the sense of Hájek-Le Cam. = o(1) as Drift estimation for jump diffusions Haindorf Seminar 212 Page 19 (24)

20 Main steps of the proof 1. Choose the threshold v n such that the continuous part is approximated in the limit. 2. Show that ā n has the same asymptotic behavior as the following benchmark estimator n i=1 â n = X t n ix(un) i n. i=1 X2 t n n i i 3. Prove a CLT for the benchmark â n. 4. Finally, show that the drift is negligible and Tn 1/2 p (ā n â n). Drift estimation for jump diffusions Haindorf Seminar 212 Page 2 (24)

21 Identifying the jumps Define for n N and i {1,..., n} } A i n = {ω Ω : 1 { i X vn} = 1 { i N(B cun )=} where B un = [ u n, u n], its complement B c u n and N(B un ) counts the jumps of L in B un. Lemma Let v n, u n such that for the Lévy measure µ of L µ(b 2vn \Bun ) µ(b c un ) u 2 nv 2 n = o(t 1 n ). = o(tn 1 ) and Then, it follows that for A n = n i=1 Ai n we have P (A n) 1 as n. Drift estimation for jump diffusions Haindorf Seminar 212 Page 21 (24)

22 The benchmark estimator To proof the CLT for ā n we introduce a benchmark estimator n i=1 â n = Xt ix(un) i n. i=1 X2 t i n i Lemma Under the Assumptions of the previous lemma and if n 1/2 µ(bu c n ) = o(tn 1 ) it follows that n 1 X t n i ( ix1 { i X v n} ix(u n)) = op(1) i= as n. Drift estimation for jump diffusions Haindorf Seminar 212 Page 22 (24)

23 CLT for benchmark estimator The following lemma leads to a CLT for the benchmark estimator. Lemma Let X be stationary with finite second moments. Set X(un) = σw + J(u n) then n 1 Tn 1/2 i= X t n i n i X(u n) D N (, σ 2 E a[x 2 ] ) as n. Drift estimation for jump diffusions Haindorf Seminar 212 Page 23 (24)

24 Summary We have developed a maximum likelihood approach for drift estimation in jump diffusions. The jumps lead to an inefficient LSE in this class of processes. Several examples like Ornstein-Uhlenbeck, Cox-Ingersoll-Ross and linear stochastic delay equations lead to explicit MLEs. Strong optimality properties have been demonstrated for the MLE is these models. For the OU process the discretized MLE attains the efficiency bound from the continuous case when a jump filter is employed. The estimator can be computed directly and performs well for finite sample size. Drift estimation for jump diffusions Haindorf Seminar 212 Page 24 (24)