frequency) rare event ( ) 28 jump ( ) (macro-jumps) (intensity function) Hawkes Hawkes (point process) Hawkes 3 (intensity function) Gran

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1 46, 2, Causality Analysis of Financial Markets by Using the Multivariate Hawkes Type Models Naoto Kunitomo, Ayao Ehara and Daisuke Kurisu Hawkes Granger :,,, Hawkes, G-, 1. (high-frequency financial data) (low

2 frequency) rare event ( ) 28 jump ( ) (macro-jumps) (intensity function) Hawkes Hawkes (point process) Hawkes 3 (intensity function) Granger (G- )

3 139 Hawkes 1) Ait-Sahalia and Jacod (214), Ait-Sahalia et al. (215), Embrechts et al. (211), Grothe et al. (214), Bacry et al. (215) Bacry et al. (215) 2) 3) 2 Hawkes 3 Hawkes 4 Granger (G- ) 5 Hawkes 6 G- 2. Hawkes (intensity function) Hawkes 4) Ogata (1978) Hawkes Grothe et al. (214) Hawkes 1) 2) Ogata (1978) (215) 3) (point processes) Daley and Vere-Jones (23, Vol.1, Vol.2) ( ) ) Hawkes (1971a, b), Ogata (1978)

4 [, T ] n I i = (t n i 1, tn i ] (i = 1,, n) t n = I n i i T n p P j (t) (j = 1,, p ; t n i 1 < t tn i, i = 1,, n) s I i Y n j (s) = P j(s) P j (t n i 1 ) P j (t n i 1 ) (j = 1,, p; i = 1,, n) (2.1) Yj n(s) (tn i 1 < s tn i ) j (threshold) u n j s I i τ n j (i, 1) s tn i 1 < s τ n j (i, 1) Xn j (s) = Y n j (s) s [τ n j (i, 1), tn i ] Xn j (s) = Xj n(τ j n(i, 1)) s I i Xj n (s) 5 ( ) Xj n(s) (s (, T ] N j n(s, un j ) s X n j (s) un j (j = 1,, p) P (N n j (t + t, un j ) N n j (t, un j ) = 1 F t) = λ n j (t, un j ) t + o p( t), (2.2) P (N n j (t + t, un j ) N n j (t, un j ) > 1 F t) = o p ( t) (2.3) F t t σ (σ field) (intensity function) λ n j (t, u j ) = lim t E[Nn j (t + t, un j ) N j n(t, un j ) F t ] (2.4) t (self-exciting) λ n j (t, u n j ) = λ n j + c ji (Xi n (s ))g i (t s)dni n (s, u), (2.5) i=1 g i (t s) = e γ i(t s) (γ i > ), C(X) = (c ji (x)) (impact function) x > u n j P (X n j (s) > x X n j (s) > u n j, F s ) = = [ ] 1 + ξ 1/ξj j σj n [ (s)x ] 1 + ξ 1/ξj (2.6) j σj n(s)u j [ 1 + ξ j σ j (s)(x un j ) ] 1/ξj

5 141 σj (s) = ξ ju n j + σn j (s) s X n j (s) F X s ( ) [ ] f j (x, s) = 1 σj (s) 1 + ξ 1/ξj 1 j σj (s)(x u j) (x > u n j ) (2.7) 5) (conditional intensity function) λ n j (t, u n j ) = λ n j + i=1 [A ji X n i (s ) + B ji ]g i (t s)dn n i (s, u) (2.8) 6) λ j, A ji,b ji, γ i (mark) C ji (x) = A ji x i + B ji C ji = B ji (, i, j = 1,, p) Hawkes Grothe et al. (214) σ j (s) = β j + α j v n j (s, u) (= ξ j u j + σ n j (s)). (2.9) vj n (s, u) Grothe et al. (214) c(x) F j (x) H( ) c(x) = c(f t (x t )) = 1 + H (F t (x t )) (2.1) ( H ( ) ) U = F (X) U(, 1) v = H (u) E[C(F t (X))] = 1 [1 + H (u)]du = 1 + R vdg(v) = 1 + δ (2.11) 5) (216), Coles (21), Resnick (27) 6) u n j n

6 δ G 5 ( α j = (j = 1,, p)) c(x) Hawkes 3. 7) p = 1 Hawkes n σ1 (s) = β 1 (= ξ 1 u 1 + σ1 n (s) > ) (3.1) c(x) = Ax δ + B ( δ 1) (3.2) δ = Hawkes δ = 1 Hawkes Hawkes δ 1 Hawkes Hawkes Hawkes B λ n ji B = β = β 1, ξ = ξ 1 y = (ξ/β)(x u) C = = = c(x)f(x)dx (3.3) [Ax δ + B] 1 [ 1 + ξ ] 1/ξ 1 β β (x u) dx [ A ξ (u + β ] ξ y)δ (1 + y) 1/ξ 1 dy δ 1, a, b (a + b) δ a δ + b δ 7) Hawkes ( Ait-Sahalia and Jacod (214) )

7 143 C = [ Au δ ] + B (1 + y) 1/ξ 1 dy + A ξ ξ [σ ξ ]δ y δ (1 + y) 1/ξ 1 dy (3.4) [ Au δ ] + B 1 z 1 ξ 1 dz + A 1 ξ ξ (β ξ )δ z 1 ξ δ 1 (1 z) δ dz = [ Au δ + B ] + A ξ (β ξ )δ B( 1 δ, δ + 1) ξ B(p, q) 1 : δ 1 C [ Au δ + B ] + A Γ(1/ξ δ)γ(δ + 1) ξ (σ )δ ξ Γ(1/ξ + 1) (3.5) < ξ < 1 C ξ = δ = 1 1 C = A[u + β 1 ξ ] + B. (3.6) Hawkes (p = 1) λ n (t, u n ) = λ n 1 (t, u n ), λ n = λ n 1 (2.5) E[λ n (t, u n )] = λ n + E[c(X n (s ))g(t s)dn n (s)] (3.7) s {Nj n (s)} σ (σ field) F N s Xn (s ) F N s E[c(X n (s ))g(t s)dn n (s)] = E[E(c(X n (s ))g(t s)dn n (s) F N s )] (3.8) = E[E(c(X n (s ) F N s ))g(t s)λ n (t, u n )ds] t v(t) = E[λ n (t, u n )] = v( ) E[dN(s)] = vds v [, T ] T/n T/n = t n i tn i 1 (i = 1,, n) n u n u λ n (t, u n ) λ(t, u)

8 u > ( ) (u )jump 1 : 1 Hawkes X(s ) Fs N n n σ n 1 (s) σ ( ) c(x)g(t s) = [Ax+B]e γ(t s), β = ξ 1 u 1 + σ ( ) u = u 1,ξ = ξ 1 ( ξ 1), γ = γ 1 > [ B + A(u + β ] 1 ξ ) < γ (3.9) ( C = E[c(X)] = B + A(u + β/(1 ξ)) ) Hawkes p λ(t, u) N(t, u) p 1 λ(t, u) = λ 1 (t, u 1 ). λ p (t, u p ), V(t) = E[λ(t, u)] = v 1 (t). v p (t), N(t, u) = N 1 (t, u 1 )., N p (t, u p ) (3.1) p p C(X(s )) = [c ij (X(s ))], G(t s) = [diag(g j (t s))] (3.11) ( diag( ) ) λ(t, u) E[λ(t, u)] = λ + E[E(C(X(s ))]G(t s)dn(s, u Fs N )] (3.12) ( F N s s σ ( ) ) 1 2 : p Hawkes X(s ) F N s C ij (X n (s )) = A ij X j (s ) + B ij, 1 > ξ i, G(t) = (diag(e γ jt )), γ j >, Γ = (diag(γ j )) n n s β j σ j (s) = ξ ju j + σ j (s) β j (j = 1,, p)

9 (i) 145 F = CΓ 1 = ( (B ij + A ij (u j + β ) j )(δ(i, j)γ 1 i ) 1 ξ j (3.13) 1 ( C = E[C(X )] ) (ii) i,j A ij = F F C C Γ t t V(t) v = [I p F ] 1 λ (3.14) A ij = δ(i, j) = 1(i = j); δ = (i j), Γ = (diag(γ j )) G- Granger (1969) Granger-causality(G-causality G- ) G-causality (time series econometrics) G- (Granger-non-causality) causality test causality measure( ) ( (1987), (29) ) G- (martingale) A G-noncausality Granger (1969) G-causality ( ) Florens and Fougere (1996) 4 G-causality Hawkes A (co-jumps)

10 G-noncausality Hawkes p = 2, u n j = u (j = 1, 2) N i n (t, u) (i = 1, 2), t λ n i (t) = λn i (t, u j) Hawkes λ 1 (t) = λ 1 + c 11 (X 1 (s ))e γ1(t s) dn1 n (s) + c 12 (X 2 (s ))e γ2(t s) dn2 n (s) τ i (t) (i = 1, 2) Ni n (t, u) (stopping times, ) t s P (τ 1 > t F N s ) = e R t s λ 1(u)du, (4.1) F N s s N i n (v) (i = 1, 2; v s) σ field F is s Ni n (v) (v s) ( i ) σ field N2 n (v); (v s) N1 n (t) (t s) t s P (τ 1 > t F N s ) = P (τ 1 > t F 1s ) = e R t s λn 1 (v)dv (4.2) G- H : c 12 (X s ) = (4.3) G- G- (variance decomposition) Hawkes (relative intensity contribution, RIC) RIC 11 (t s) = P (τ 1 > t F 1s ) P (τ 1 > t F N s ) (4.4)

11 147 RIC 12 (t s) = P (τ 1 > t F N s ) P (τ 1 > t F 1s ) P (τ 1 > t F N s ). (4.5) RIC ij (t s) = 1, RIC ij (t s) 1 (4.6) j=1 Bartrett Bartlett (1963) Bartrett p N(t) = (N i (t)) λ(t) = λ + γ(t u)dn(u) (4.7) (p p γ(u) = (γ ij (u)) u < γ(u) = (O)( ) ) ξ ξ = E[λ(t)] ξ = [I p Γ ()] 1 λ (4.8) Γ (ω) = e iωτ γ(τ)dτ (4.9) ( i 2 = 1 ) N(t) 2 µ(τ) = E[ dn(t + τ) dt dn (t) ] ξξ (τ > ) (4.1) dt τ < µ( τ) = µ(τ) τ = E[(dN(t)) 2 ] = E[dN(t)] µ (c) (τ) = Dδ(τ) + µ(τ) (4.11) ( δ( ), D = (δ(i, j)d ii ) (i, j = 1,, p) d ii, δ(i, j) ) Bartrett f(ω) = 1 e iωτ µ (c) (τ)dτ (4.12) 2π = 1 [D + M(ω)] 2π

12 M(ω) = 1 e iωτ µ(τ)dτ (4.13) 2π Hawkes (1971b) 8) 3 : (4.7) Hawkes η, a 1, a 2 γ ij (u) < a 1 e ηu, β ij (u) < a 2 e η ( B(u) = (β ij ) (A.28) ) f(ω) = 1 2π [I p Γ (ω)] 1 D[I p Γ ( ω)] 1 (4.14) ( f(ω) = (f ij (ω)), Γ (ω) (4.9) Hawkes (relative power contribution, RPC) R p econometrics 3 Bartrett 9) p = 2, D D = diag(d 11, d 22 ) 2 2 Γ (ω) = (γij (ω)) (i, j = 1, 2) ω f 11 (ω) = [d 11(1 γ 22(ω))(1 γ 22( ω)) + d 22 γ 12(ω)γ 12( ω)] I 2 Γ (ω) 2 RP C 1 1 (ω) = RP C 2 1 (ω) = d 11 (1 γ 22(ω))(1 γ 22( ω)) d 11 (1 γ 22 (ω))(1 γ 22 ( ω)) + d 22γ 12 (ω)γ 12 ( ω) (4.15) d 22 γ 12(ω)γ 12( ω) d 11 (1 γ 22 (ω))(1 γ 22 ( ω)) + d 22γ 12 (ω)γ 12 ( ω) (4.16) ω RP C 1 1 (ω) + RP C 2 1 (ω) = 1 (4.17) 8) Hawkes (1971a) D 9) Anderson (1971) 5

13 149 γ(t) = C(diag(e γjt )) Γ (ω) = C(diag(iω + γ j )) 1 γ j = γ (j = 1, 2) Γ (ω) = (iω + γ) 1 C RPC ω d 11 [(γ c 22 ) 2 + ω 2 ]/(γ 2 + ω 2 ), d 22 [(c 12 ) 2 ]/(γ 2 + ω 2 ) (point processes) T Hawkes (T, T ] P(τ k T F t ) = exp( t λ n k(s, u Ft N )ds), (4.18) P(τ k T F N t ) = E[1 [τk T ] F t ] (4.19) p = 3 (p = 3) σ F t,(2,3) Ft,(1,2,3) N σ N 2, N 3 N 1, N 2, N 3 P(τ k T Ft,(1,2,3) N ) = P(τ k T Ft,(2,3) N ) (4.2) A k1 = G T (T, T ] N k F N t (first arrival time) τ P r(τ T FT N ) = exp( T λ n k(t, u FT N )dt) (4.21) (216)

14 ( ) 5. (Tokyo) (New York) (London) , SP5, FTSE1 Hawkes Tokyo opening, closing jumps (co-jumps) NC : Hawkes jumps (co-jumps)

15 151 -[( )-( )]/( ) c =.2 (u j )(j = 1, 2, 3) (closing time) (opening time ) (VaR) 5% 1) p { i=1 log L = λ n i (s)ds + log(λ n i (s))dn n i (s)} (5.1) { i=1 + { i=1 = L 1 + L 2, λ n i (s)ds + log(λ n i (s))dn n i (s)} (5.2) log f i (X n i (s ))dn n i (s)} L 1 = L 2 = { i=1 { i=1 λ n i (s)ds + log(λ n i (s))dn n i (s)}, log f i (X n i (s ))dn n i (s)} u i f i (x) = 1 σ i x i u i (1 + ξ i σi ) 1 ξ 1 i (i = 1,, p) (5.3) Hawkes L 1 1) 1%

16 Log Likelihood σi ξ i Log Likelihood σi ξ i Log Likelihood σi ξ i L 2 L 2 GPD (generalized Pareto distribution, ) (p = 2) c(x) (1) 1, (2) x, (3) x c ( < c < 1) AIC ( L 1 ) Fisher (1) (intensity) λ n 1 (t) = λ n 1 + λ n 2 (t) = λ n 2 + α 11 e γ 1(t s) dn n 1 (s) + α 21 e γ1(t s) dn n 1 (s) + α 12 e γ 2(t s) dn n 2 (s), (5.4) α 22 e γ2(t s) dn n 2 (s) (5.5) γ i γ (γ > )

17 (1) Log Likelihood AIC α 11 α e e α 21 α 22 γ λ 1 λ e e e e e Log Likelihood AIC α 11 α α 21 α 22 γ λ 1 λ N 1 N 2 N 1 N 2 (2) (Intensity) λ n 1 (t) = λ n 1 + λ n 2 (t) = λ n 2 + α 11 e γ 1(t s) X 1 dn n 1 (s) + α 21 e γ 1(t s) X 1 dn n 1 (s) + α 12 e γ 2(t s) X 2 dn n 2 (s), (5.6) α 22 e γ 2(t s) X 2 dn n 2 (s) (5.7) γ 5-3 (2) Log Likelihood AIC α 11 α e e α 21 α 22 γ λ 1 λ e e e e e Log Likelihood AIC α 11 α α 21 α 22 γ λ 1 λ

18 N 1 N 2 N 1 N 2 (3) (Intensity) λ n 1 (t) = λ n 1 + λ n 2 (t) = λ n 2 + α 11 e γ 1(t s) X 1 c 11 dn n 1 (s) + α 21 e γ 1(t s) X 2 c 21 dn n 1 (s) + α 12 e γ 2(t s) X 2 c 12 dn n 2 (s) (5.8) α 22 e γ 2(t s) X 2 c 22 dn n 2 (s) (5.9) γ i c 11 = c 12, c 21 = c 22 11) 5-4 (3) Log Likelihood AIC α 11 α 12 α 21 α e e e e γ λ 1 λ 2 c 11 = c 12 c 21 = c e e e e e Log Likelihood AIC α 11 α 12 α 21 α γ λ 1 λ 2 c 1,1 =c 1,2 c 2,1 =c 2, (1) (2), (3) AIC (2) (3) 3 Hawkes G- G- 1 Hawkes T (T ) 11) c 11, c 12 AIC

19 155 12) (martingale) Ogata (1978) (Wilks-property) C 4 : Hawkes NC L T (θ), θ L T (θ ), ˆθ ML L T (ˆθ ML ) T 2{L T (ˆθ ML ) L T (θ )} d χ(d) (5.1) Ω θ = (θ k ) Ω Ω χ 2 (d) d 2 Ω Ω d χ 2 (d) 1 d = 1 C(X) = α 11X 1 α 12 X 2 α 21 X 1 α 22 X 2, G(t) = e γt e γt Hawkes α ij = H : α 21 = 2 ( ) ( ). 5-5 (1) Log Likelihood AIC α 11 α α 21 α 22 γ λ 1 λ 2 null null ) compensator ( ) T Kunitomo et al. (217)

20 H : α 12 = 2 ( ) = (2) Log Likelihood AIC α 11 α e-1 null.6887 null α 21 α 22 γ λ 1 λ e e e e e H : α 21 = 2 ( ) = ( ) 5-7 (3) Log Likelihood AIC α 11 α α 21 α 22 γ λ 1 λ 2 null null H : α 12 = 2 ( ) = (4) Log Likelihood AIC α 11 α null.893 null α 21 α 22 γ λ 1 λ

21 157 2 RPC Hawkes ( (2)) RPC 2 Bartrett ( ) Bartrett RPC ( ) 13) Hamao et al. (199) GARCH (genaralized autoregressive conditional heteroscedastic models) 13)

22 Hawkes (SEVT) Hawkes Granger Granger G-causality, G-noncausality G- (i) (ii) causality analysis (G- ) (iii) Hawkes (co-jumps) 14) Hawkes Hawkes ( ) JP ) Solo (27), Kunitomo et al. (217)

23 159 Ait-Sahalia, Y. and Jacod, J. (214). High-Frequency Financial Econometrics, Princeton University Press. Ait-Sahalia, Y., Cacho-Diaz, J. and Laeven, L. (215). Modeling financial contagion using mutually exciting jump processes. Journal of Financial Economics, 117, Anderson, T. W. (1971). Statistical Analysis of Time Series, Wiley. Bacry, E., Mastromatteo, I. and Muzy, J.-F. (215). Hawkes processes in Finance. Market Microstructure and Liquidity, 1, 1555, World Scientific. Bartrett, M. S. (1963). The spectral analysis of point processes, Journal of Royal Statistical Society (B), 25(2), Coles, S. (21). An Introduction to Statistical Modeling of Extreme Value, Springer. Daley, D. J. and Vere-Jones, D. (23). An Introduction to the Theory of Point Processes, Volume I, Volune II, 2nd Edition, Springer. Dellacherie, C. and Meyer, P. A. (196). Probabilities and Potentials, Elsevier Science, Paris. (216).,, ( ). Embrechts, P., Liniger, T. and Lin, L. (211). Multivariate Hawkes processes: an application to financial data, Journal of Applied Probability, Special Volume, 48A, Florens, J.-P. and Fougere, D. (1996). Noncausality in continuous time, Econometrica, 64, Granger, C. W. (1969). Investigating causal relations by econometric models and cross-spectral methods, Econometrica, 37, Grothe, O., Korniichuk, V. and Manner, H. (214). Modeling multivariate extreme events using self-exciting point processes, Journal of Econometrics, 182, Hamao, Y., Masulis, R. W. and Ng, V. (199). Correlations in price changes and volatility across international stock markets, Review of Financial Studies, 3, Hawkes, A. G. (1971a). Point spectra of some mutually exciting point processes, Journal of the Royal Statistical Society. Series B, 33(3), Hawkes, A. G. (1971b). Spectra of some self-exciting and mutually exciting point processes, Biometrika, 58(1), Hirsch, M. W. and Smale, S. (1974). Differential Equations, Dynamical Systems and Linear Algebra, Academic Press. (29). 38(2), Kunitomo, N., Kurisu, D., Amano, Y. and Awaya, N. (217). The simultaneous multivariate Hawkes-type point processes and their application to financial markets, (in preparation). Ogata, Y. (1978). The asymptotic behavior of maximum likelihood estimators of stationary point processes, Annal of Institute of Statistical Mathematics, 3, (215). 63(1), Resnick, S. (27). Heavy-Tail Phenomena, Springer. Solo, V. (27). Likelihood functions for multivariate point processes with coincidences, Proceedings of the 46th IEEE Conference on Decision and Control, (216).. (1987)..

24 Granger (Granger noncausality) A G- Granger (1969) ( G- ) Granger (G- ) G- G Florens and Fougere (1996) (z t, w t, y t ) (Ω, A, F t ) s t σ F s F t F t = F t+ = F s s>t sub-σ L t G t F t L t z t G t (z t, w t ) F t y t ( L t = G t ) G- A-1 : ( G ) s, t E[z t F s ] = E[z t G s ] G t F t (weakly) (globally) z t (cause ) A-2 : ( G ) s, t L t F s G s G t F t (strongly) (globally) z t (cause ) L t F s G s (conditional independence) (Dellacherie and Meyer (196), Florens and Fougere (1996) ) G- A-1 : (i) G t F t (strongly) (globally) z t

25 161 cause L t F t (ii) L t F t F t (strongly) (globally) z t cause G- G- G- (local) z t (special) z t = z + H t + M t (A.1) H t predictable ( ), M t local ( ) G- A-3 : ( G- ) t z t F t G t F t (weakly) (instantaneous) z t cause A-4 : ( G- ) L t G t (special) F t F t (strongly) (instantaneous) z t cause A-2 : (Florens and Fougere (1996)) (i) (strongly) (globally)g- G (ii)l t = G t F t G- F t (strongly) (globally)g- G- Co-jump 15) A-3 : (Florens and Fougere (1996)) N t = (N 1 t, N 2 t ) N 1 t = n> 1 {τ 1 n t}, N 2 t = n> 1 {τ 2 n t} τ 1 n, τ 2 n N 1 t, N 2 t L t = G t N 1 t F t N 2 t NC G- 15) Florence and Fougere (1996) co-jump

26 B Hawkes Daley and Vere-Jones (23) N Borel A B(R d ), N(A), Janossy ( ) Janossy ( ) B-1 Janossy ( ) : Borel A B(R d ), A Janossy J n ( A), n = 1, 2,... N(A) = n n A J n (dx 1 dx n A) = P. (A.2) dx 1,..., dx n Janossy R nd Lebesgue, Radon- Nikodym Janossy j n ( A) B-2 : Borel A B(R d ), R d N A, k Janossy J n (dx 1 dx k A) Janossy j n ( A) Janossy B-3 : Borel A B(R d ) N x 1,, x n L A (x 1,, x n ) = j n (x 1,, x n A). (A.3) Poisson λ > Janossy.. N R + = [, ) Borel A A = [, T ]

27 163 {t 1,, t N(T ) } (, T ) N(T ). {τ i = t i t i 1 }. t =. S k (u t 1,, t k 1 ) = P (τ k > u t 1,, t k 1 ). p 1 (t), p 2 (t t 1 ),... S 1 (u), S 2 (u t 1 ),... S n (t t 1,, t n 1 ) = 1 t n 1 p n (u t 1,, t n 1 )du B-1 : j n (t 1,, t n T ) = j n (t 1,, t n [, T ]) J (T ) = S 1 (T ). j 1 (t 1 T ) = S 2 (T t 1 )p 1 (t 1 ). j 2 (t 1, t 2 T ) = S 3 (T t 1, t 2 )p 2 (t 2 t 1 ).. j n (t 1,, t n T ) = S n+1 (T t 1,, t n )p n (t n t 1,, t n 1 ) p 2 (t 2 t 1 )p 1 (t 1 ). ( ) [, t] {T } {t }, J (t) = J (T ) + k=1 1 T j k (u 1,..., u k T )du 1 du k. k! t t (A.4) p 1 (t) t 1, p 1 (t) = j 1 (t T ) + k=2 1 T j k (t, u 2,..., u k T )du 2 du k. (k 1)! t t (A.5) p 1 (t 1 )S 2 (t t 1 ) t 1 1, t 2, p 1 (t 1 )S 2 (t t 1 ) = j 1 (t 1 T ) + = j 1 (t 1 t). k=2 1 T j k (t 1, u 2,..., u k T )du 2 du k (k 1)! t t 1 T p 1 (t 1 )p 2 (t t 1 ) = j 2 (t 1, t T ) + j k (t 1, t, u 3,..., u k T )du 3 du k. (k 1)! t t k=3 (A.6)

28 ( ),, B-2 : N [,T] t 1,, t n N [, T ] ( ) ( N(T ) ) T L = λ (t i ) exp λ (u)du, i=1 λ (t) h λ 1 (t), < t t 1, (t) = h n (t t 1,, t n 1 ), t n 1 < t t n, 2 n. (A.7), h n (t t 1,, t n 1 ) ( ) h n (t t 1,, t n 1 ) = p n(t t 1,, t n 1 ) S n (t t 1,, t n 1 ). ( ) ( ), ( ) p n (t t 1,, t n 1 ) = h n (t t 1,, t n 1 ) exp h n (u t 1,, t n 1 )du t n 1, (A.8). B-1, N(t) L = p j (t j t 1,..., t j 1 ) S N(t) (t t 1,..., t N(t) ) j=1 ( N(t) = h j (t j t 1,..., t j 1 ) exp j=1 ( ) h j (u t 1,..., t j 1 )du t j 1 ) j exp h N(t) (u t 1,..., t N(t) )du T N(t) ( N(t) ) ( tj = λ (t j ) exp λ (u)du exp j=1 t j 1 ( N(t) ) T = λ (t i ) exp λ (u)du. i=1 T N(t) λ (u)du ) (λ ( ) ) ( )

29 165 Janossy B-4 : E, K N E K N E K Borel A B(E) N g (A) = N(A K) < a.s. N g ( ) N 16) N = (N 1,..., N m ) K = {1,, m} N g ( ) = N( {1,, m}) = m i=1 N( {i}) = m i=1 N i( ) (K, B(K)) l K ( ) 17) R + K N l( ) R + Lebesgue B-5 : R + K N Borel A B(R + ) n 1 Janossy J n ( A K) l l K n Janossy B-6 Janossy : Borel A B(R + ) Janossy j n (x 1,, x n, κ 1,, κ n A K)dx 1 dx n l K (dκ 1 ) l K (dκ n ) N = P g (A) = n n A dx 1,..., dx n, K dκ 1,..., dκ n. 16) ground process 17) reference measure

30 J (T ) = S 1 (T ). j 1 (t 1, κ 1 T ) = p 1 (t 1, κ 1 ) = p 1 (t 1 )f 1 (κ 1 t 1 ). j 2 (t 1, t 2, κ 1, κ 2 T ) = p 1 (t 1 )f 1 (κ 1 t 1 )p 2 (t 2 (t 1, κ 1 ))f 2 (κ 2 (t 1, κ 1 ), t 2 ).. f n (κ (t 1, κ 1 ),, (t n 1, κ n 1 ), t) (t 1, κ 1 ),, (t n 1, κ n 1 ), t h λ 1 (t), < t t 1, (t, κ)= h n (t (t 1, κ 1 ),, (t n 1, κ n 1 ))f n (κ (t 1, κ 1 ),, (t n 1, κ n 1 ), t), t n 1 < t t n, 2 n. (A.9) h n (t (t 1, κ 1 ),, (t n 1, κ n 1 )) h n (t (t 1, κ 1 ),, (t n 1, κ n 1 )) = p n(t (t 1, κ 1 ),, (t n 1, κ n 1 )) S n (t (t 1, κ 1 ),, (t n 1, κ n 1 )) λ (t, κ) h n, f n (A.1) λ (t) = h n (t (t 1, κ 1 ),, (t n 1, κ n 1 )) (t n 1 T t n ), (A.11) f (κ t n ) = f n (κ (t 1, κ 1 ),, (t n 1, κ n 1 ), t n ) (A.12) N g ( (T ) L = λ (t i, κ i ) exp i=1 ) λ (u, κ)dudκ N g (T ) N g ( (T ) = λ g(t i ) f (κ i t i ) exp i=1 i=1 K K ) λ (u, κ)dudκ. (A.13) ( Hawkes ) : p N = (N 1,..., N p ) λ m (t) λ m (t) = λ m + j=1 α m,j e β(t s) dn j (s), (A.14) m = 1, 2,, p p N g Ng (t) = N m (t), λ g(t) = λ m (t). m=1 m=1

31 167 m = 1, 2,, p N m [, T ] {t 1,m,, t nm,m} N g {t 1,m,, t Nm (T ),m} m {t 1,, t Ng (T )} N {(t 1, m 1 ), (t Ng (T ), m Ng (T ) )} m 1 1, 2,, p λ (t i, m i ) = λ mi (t i ), (A.15) p ( log L(T t 1,, t Ng (T )) = m=1 λ m (t)dt + log(λ m (t))dn m (t) ). ( Hawkes ) : Hawkes m X(t) = (X 1 (t),..., X p (t)) X m (t) f m (x t), m = 1,, p p N = (N 1,... N p ) N m λ m (t) λ m (t) = λ m + j=1 α m,j e β(t s) C(X j (s ))dn j (s), (A.16) C( ) p N g N g (t) = N m (t), λ g(t) = m=1 λ m (t). m = 1, 2,, p N m [, T ] {(t 1,m, X m (t 1,m )),, (t nm,m, X m (t nm,m))} N g [, T ] {t 1,m,, t Nm(T ),m} m,{t 1,, t Ng (T )} N {(t 1, m 1, X m1 (t 1)), (t N g (T ), m N g (T ), X mn g (T ) (t N g (T )))} m 1 1, 2,, p {t, m} X m (t) m=1 f (X m (t) t, m) = f m (X m (t) t). (A.17) Hawkes log L(T {(t 1, m 1, X m1 (t 1)), (t Ng (T ), m Ng (T ), X mn g (T ) (t Ng (T )))}) ( ) T N m (T ) = λ m (t)dt + log(λ m (t))dn m (t) + log f m (X m (t k,m ) t k,m ). m=1 m=1 k=1

32 C C-1: 1 2 (i) 1 2 p = 1 λ(t) λ n λ n = λ n + E[E(C(X n (s ))G(t s) Fs N )dn n (s)] (A.18) n E(C(X n (s )) F N s )] C γ > e γ(t s) ds = 1/γ p G(t) = (diag ( e γ jt )) n p p C λ = λ + C G(t s)λds (A.19) ( p = 1 c n (s) = E[C(X n (s ))] λ[1 [ lim n cn (s)]e γ(t s) ds] = λ (A.2) λ ) (ii) γ > p = 1 v(t) = λ + c e γ(t s) v(s)ds dv(t) dt = [c γ]v(t) + γλ (A.21) p > 1 C V(t) = E[λ(t)] dv(t) dt = C Γ e γ 1(t s) v 1 (s)ds. e γp(t s) v p (s)ds + V(t) (A.22) C dv(t) dt = CΓC 1 λ + C [ I p ΓC 1] V(t) (A.23) (C CΓC 1 ) xi p = (A.24)

33 169 C 1, C (C Γ) xi p = (A.25) t ( Hirsch and Smale (1974) ) Q.E.D. C-2: 3 (4.7) (4.1) u = u t (4.11) +τ (t) µ(τ) = E{[λ + γ(t + τ u)dn(u)] dn } ξξ (τ ) (A.26) dt = τ γ(τ u )µ (c) (u )du. µ(τ) = γ(τ)d + γ(τ u)µ(u)du, τ > (A.27) (4.11) µ( τ) = µ (τ) µ c ( τ) = (µ c (τ)), B(τ) = γ(τ)d + γ(τ u)µ(u)du µ(τ), < τ < (A.28) ( B(τ) = O (τ < ) ) B(ω), Γ (ω), M(ω) B(ω) = Γ (ω)d + Γ (ω)m(ω) M(ω) (A.29) M(ω) = [I p Γ (ω)] 1 [Γ (ω)d B(ω)] (A.3) (4.13) M( ω) = M (ω) [I p Γ ( ω)][γ ( ω)d B( ω)] = [Γ (ω)d B (ω)][i p Γ (ω)] (A.31) ( H(ω) 18) ) H(ω) = [I p Γ ( ω)]b (ω) + Γ ( ω)d = B( ω)[i p Γ (ω)] + DΓ (ω) = O (A.32) 18) Hawkes (1971b)

34 B (ω) = [I p Γ ( ω)] 1 [ Γ ( ω)d] (A.33) M(ω) M(ω) = [I p Γ (ω)] 1 [Γ (ω)d(i p Γ ( ω)) + DΓ ( ω)][i p Γ ( ω)] 1 D + M(ω) 3 [I p Γ (ω)] 1 D[I p Γ ( ω)] 1 3 Q.E.D. C-3: 4 Ogata (1978) Ogata (1978) (i) = 1 L T (θ ) L T (θ ) T (ˆθ θ ) T θ T θ θ + T (ˆθ θ ) { α T H(t, ω)dt + β T α α i,j d2 2 β β i,j d2 2 G(t, ω)dn t }(ˆθ θ ) d d Ogata (1978) 3 2 Ogata (1978) martingale ( ) N(, E[ (, 1) (, 1) ]) (, 1) m=1 1 λ m (t) θ 1 2 L T (θ ) T θ θ dt + m=1 1 p E[ 2 L θ θ ] λ m (t) θ A = E[ (, 1) (, 1) ], B = E[ 2 L θ θ ] 1 λ m (t) dn m,t T (ˆθ θ ) d N(, B 1 AB 1 ) (ii) p = 2 L T θ i = ( λ 1 θ i + λ 2 θ i )dt + ( λ 1 θ i dn 1,t λ 1 + λ 2 θ i dn 2,t λ 2 )

35 L T θ i = L T θ j = [ 171 [ ( λ 1 θ i (dt dn 1,t λ 1 (t) ) + ( λ 2 θ i (dt dn 2,t λ 2 (t) )] ( λ 1 (ds dn 1,s θ j λ 1 (s) ) + ( λ 2 (dt dn 2,s θ j λ 2 (s) )] λ 1 λ 1 (dtds dt dn 1,s ds dn 1,t + dn 1,s dn 1,t ) θ i θ j λ 1,s λ 1,t λ 1,s λ 1,t λ 2 θ i λ 2 θ j (dtds dt dn 2,s λ 2,s ds dn 2,t λ 2,t + dn 2,s λ 2,s dn 2,t λ 2,t ) λ 1 θ i λ 2 θ j (dtds dt dn 2,s λ 2,s ds dn 1,t λ 1,t + dn 2,s λ 2,s dn 1,t λ 1,t ) λ 2 θ i λ 1 θ j (dtds dt dn 1,s λ 1,s ds dn 2,t λ 2,t + dn 1,s λ 1,s dn 2,t λ 2,t ) 1 2 Ogata (1978) 2 k=1 λ k λ k 1 θ i θ j (λ k ) dt 3 4 Ogata (1978) (a) s < t (b) t > s(c) s = t 3 4 (a), (b) E[dN k (t) H t ] = λ k (t)dt Hawkes Cov[(N i (t + h) N i (t)), (N j (t + h) N j (t)) F t ] = o(h 2 ) (i j; i, j = 1,, p) (c) 1 T 2 k=1 λ k θ i λ k θ j 1 λ k dt p E[ (, 1) (, 1) ] A = B 2{L T (ˆθ) L T (θ )} = 2 L T (ˆθ) θ (ˆθ θ ) + (ˆθ θ ) 2 L T (θ ) θ θ (ˆθ θ ) + o p (1) 1 A = B T (ˆθ θ ) d N(, A 1 ) 4 Q.E.D.

03.Œk’ì

03.Œk’ì HRS KG NG-HRS NG-KG AIC Fama 1965 Mandelbrot Blattberg Gonedes t t Kariya, et. al. Nagahara ARCH EngleGARCH Bollerslev EGARCH Nelson GARCH Heynen, et. al. r n r n =σ n w n logσ n =α +βlogσ n 1 + v n w