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Asymptotic Expansions under degeneracy Yuji Sakamoto and Nakahiro Yoshida Hiroshima International Univ. and University of Tokyo SAPS4 , Universit e du Maine , Le Mans, France December 19-20, 2002 1 Stationary Ergodic Diffusion Model

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Asymptotic Expansions under degeneracy

Yuji Sakamoto and Nakahiro Yoshida Hiroshima International Univ. and University of Tokyo SAPS4, Universit´ e du Maine, Le Mans, France December 19-20, 2002

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Stationary Ergodic Diffusion Model ✷ Observations : X = (Xt)t∈[0,T ], dim(Xt) = d, Continuous ✷ Parameter : θ ∈ Θ ⊂ Rp, dim(θ) = p, Θ : open bounded convex ✷ Model : dXt = V0(Xt, θ)dt + V (Xt)dwt, X0 ∼ νθ V0 : Rd × Θ → Rd, V : Rd → Rd ⊗ Rr w = (wt) : r-dimensional standard Wiener process νθ : stationary distribution with a positive density dνθ dx ✷ Log-Likelihood : ℓ(θ) = log dνθ dx +

T

0 V ′ 0(V V ′)−1(Xt, θ)dXt − 1

2

T

0 V ′ 0(V V ′)−1V0(Xt, θ)dt

✷ Estimator : ˆ θT : (Xt)t∈[0,T ] → ˆ θT ∈ Θ ✷ Asymptotics : T → ∞

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Estimation for unknown parameter θ in drift function V0 ✷ Parameter of interest : θ = (θ1, . . . , θp) ∈ Θ ⊂ Rp, dXt = V0(Xt, θ)dt + V (Xt)dwt, X0 ∼ νθ, t ∈ [0, T]. ✷ True value : θ0 ∈ Θ. ✷ Conditional MLE ˆ θ(c)

T

: δaℓ(c)(ˆ θ(c)

T ) = 0, a = 1, . . . , p, δa = ∂/∂θa.

ℓ(c)(θ) =

T

0 V ′ 0(V V ′)−1(Xt, θ)dXt − 1

2

T

0 V ′ 0(V V ′)−1V0(Xt, θ)dt.

✷ (Exact) MLE ˆ θT : δaℓ(ˆ θT) = 0, a = 1, . . . , p, δa = ∂/∂θa. ℓ(θ) = log dνθ dx (X0) + ℓ(c)(θ). ✷ M-estimator ˆ θψ

T :

ψa;(ˆ θψ

T) = 0,

a = 1, . . . , p, for an estimating fuction ψ = (ψ1;, . . . , ψp;).

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Literature on distributional expansion for ergodic diffusion ✷ Second order

Yoshida(1997) : martingale expansion, (global approach)

(conditional) MLE, d = 1, p = 1

Sakamoto-Yoshida(1998) : M-estimator, d = 1, p = 1,

Numerical studies on (conditional) MLE for OU, etc

Uchida-Yoshida(2001),

(Mykland(1992, 1993)) ✷ Third or higher order

Kusuoka-Yoshida(2000) : ǫ-Markov mixing, (local approach)

Diffusion functional having stoch. exp. , arbitrary d, p

S-Y(1998) : third order MLE,

arbitrary d, p

S-Y(1999) : representation of expansion,

third order M-estimator, arbitrary d, p

S-Y(2000) : under degeneracy Sakamoto(2000), Kutoyants-Yoshida(2001)

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Second order expansion of MLE for univariate ergodic diffusion Theorem 1 (Yoshida(P.T.R.F.,1997)). For any θ ∈ Θ ⊂ R, Let X = (Xt)t∈[0,T ] be a one-dimensional stationary ergodic diffusion pro- cess satisfying dXt = V0(Xt, θ)dt + dwt with a stationary distribution νθ given by νθ(dx) = nθ(x)

∞

−∞ nθ(u)dudx,

nθ(x) = exp

x

0 2V0(u, θ)du

  • .

Suppose that supx∈R ∂xV0(x, θ0) < 0 and that |δl∂jV0(x, θ)| ≤ Cj,l(1 + |x|Cj,l), ∀x, θ. Then it holds that P( √ IT(ˆ θ(c)

T

− θ0) ≤ x) = Φ(x) + 1 √ T (A − Bx2)φ(x) + ¯

  • (T −1/2),

where I = νθ0((δV0)2), A = −νθ0(δV0 · k)/(2I3/2), B = −{νθ0(δV0 · δ2V0) − νθ0(δV0 · k)}/(2I3/2), k = −nθ0(x)−1

∞

x

2nθ0(u)((δV0(u, θ0))2 − I(θ0))du.

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Numerical study on expansion of ˆ θ(c)

T

for OU in S-Y(1998)

1.0 0.8 0.6 0.4 0.2 0.0

  • 4
  • 2

2 4

Normal Expansion MonteCarlo

T=3, θ=2 1.0 0.8 0.6 0.4 0.2 0.0

  • 4
  • 2

2 4

Normal Expansion MonteCarlo

T=3, θ=1 1.0 0.8 0.6 0.4 0.2 0.0

  • 4
  • 2

2 4

Normal Expansion MonteCarlo

T=6, θ=2 1.0 0.8 0.6 0.4 0.2 0.0

  • 4
  • 2

2 4

Normal Expansion MonteCarlo

T=6, θ=1

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Third order expansion of MLE for multidimensional diffusion Theorem 2 (S-Y(1999)). Let M, γ > 0, and ˆ ρ > (ρab). Assume that [L], [DM1], [DM2], [DM3] hold true. For any β ∈ C2

B(Θ), let

ˆ θ∗

T = ˆ

θT − β(ˆ θT)/T. Moreover assume that the diffusion process X has the geometrically strong mixing property. Then there exist positive constants c, ˜ C,˜ ǫ such that for any f ∈ E(M, γ)

  • E[f(

√ T (ˆ θ∗

T − θ))] −

  • dy(0)f(y(0))qT,2(y(0))
  • ≤ cω(f, ˜

CT −(˜

ǫ+2)/2, ˆ

ρab) + o(T −1), (1) where qT,2(y(0)) =φ(y(0); ρab)

  • 1 +

1 6 √ T c∗

abchabc(y(0); ρab) +

1 √ T ρaa′(µa′ − ˜ βa′)ha(y(0); ρab) + 1 2T A∗

abhab(y(0); ρab) +

1 24T c∗

abcdhabcd(y(0); ρab)

+ 1 72T c∗

abcc∗ defhabcdef(y(0); ρab)

  • .

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Coefficients in asymptotic expansion(1)

ρab =Fa,b, (ρab) = (ρab)−1, c∗

abc = −3Γ(−1/3) ab,c

, ˜ βa = βa − ∆a, µa = −1 2ρaa′ρbcΓ(−1)

bc,a′ ,

A∗

ab = − τab − ρcd

  • Fbcd,a + Fab,cd − Fac,bd − F[a,c],[b,d] + 2F[ab,c],d + 2F[ac,b],d + 4F[b,d],ac

+ F[cd,b],a + 2F[[b,c],a],d + 2F[[b,c],d],a

  • + ρcdρef

1

2Γ(−1)

ce,b Γ(−1) d f,a − Γ(1) ac,eΓ(1) bd,f + Γ(−1) cd,e (Γ(1) ab,f + Γ(−1) fb,a ) + Γ(−1) ce,a (Γ(1) bd,f + Γ(−1) bd,f )

  • + ρaa′ρbb′(µa′ − ˜

βa′)(µb′ − ˜ βb′) + 2ρaa′(∆cη∗a′

c,b − δbβa′),

c∗

abcd = −12(F[[a,b],c],d + F[a,b],cd + F[ab,c],d) + 3F[a,b],[c,d] − 4Fabc,d

+ 12Γ(−1/3)

ab,c

ρdd′(˜ βd′ − µd′) + 12ρef(Γ(−1)

ab,e + Γ(1) ae,b)Γ(−1) cf,d ,

Γ(α)

ab,c = Fab,c − F[a,b],c + 1 − α

2

[3]

  • (ab,c)

F[a,b],c, η∗a

b,c = −ρaa′(Γ(1) a′c,b + Γ(−1) bc,a′ ).

The coefficients c∗

abc, A∗ ab, c∗ abcd, and ρaa′(µa′ − ˜

βa′) are functions of F, τ, and ∆.

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Coefficients in asymptotic expansion(2) ∆a = ρaa′νθ0(δa′ dνθ0 dx ), τab = Cov[δa dνθ0 dx , δb dνθ0 dx ], FA1,A2 = νθ0(BA1 · BA2), FA1,[A2,A3] = νθ0(BA1 · [BA2 · BA3]), F[A1,A2],[A3,A4] = νθ0([BA1 · BA2] · [BA3 · BA4]), F[[A1,A2],A3],A4] = νθ0([[BA1 · BA2] · BA3] · BA4]), where B(x, θ) = V ′

0(V V ′)−1V0(x, θ), BA(x, θ) = δa1 · · · δakB(x, θ),

δaj = ∂/∂θaj, A = a1 · · · ak, [f] = − V ′∇Gf−ν(f), AGf−ν(f) = f − ν(f), A =

d

  • i=1

V i

0(x, θ0) ∂

∂xi + 1 2

d

  • i,j=1

r

  • k=1

V i

k(x)V j k (x)

∂2 ∂xi∂xj

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Third order expansion of MLE for CIR model dXt = (p−qXt)dt+r

  • Xtdwt, θ = (p, q), Θ = {(p, q) : 2p > r > 0, q > 0}

F1,1 = 2q r2(2p − r2), F1,2 = F2,1 = − 1 r2, F2,2 = p qr2, F[1,1],1 = − 4q r2(2p − r2)2, F[1,1],2 = 2 (2p − r2), F[1,2],1 = F[1,2],2 = 0 F[2,2],1 = 1 r2q, F[2,2],2 = − p r2q2 F[1,1],[1,1] = 8q r2(2p − r2)3, F[1,1],[1,2] = 0 F[1,1],[2,2] = − 2 r2(2p − r2)q, F[1,2],[1,2] = F[1,2],[2,2] = 0 F[2,2],[2,2] = p r2q3, F[[1,1],1],1 = 8q r2(2p − r2)3, F[[1,1],1],2 = − 4 r2(2p − r2)2 F[[1,1],2],1 =F[[1,1],2],2 = F[[1,2],1],1 = F[[1,2],1],2 = 0 F[[1,2],2],1 =F[[1,2],2],2 = F[[2,2],1],1 = F[[2,2],1],2 = 0 F[[2,2],2],1 = − 1 r2q2, F[[2,2],2],2 = p r2q3, ∆1 =∆2 = 0, τ11 = 4PolyGamma[1, 1], τ12 = τ21 = 2 r2q, τ22 = 2p q2r2.

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Stochastic Expansion of MLE ˆ θT (p = 1)

  • MLE ˆ

θT δℓT(ˆ θT) = 0, δ = ∂/∂θ, : Likelihood eq.

  • Third order stochastic expansion

√ T (ˆ θT − θ0) =ST(Z1, Z2, Z3) + 1 T √ T R3, ST(z1, z2, z3) = z1 + 1 √ T

  • z1z2 + 1

2¯ υ−1

2 ¯

ν3z2

1

  • + 1

T

1

6(¯ ν4 + 3¯ ν2

3)z3 1 + 3

2¯ ν3z2

1z2 + z1z2 2 + 1

2z2

1z3

  • ,

where ¯

υ2 = − 1

T Eθ0[δ2ℓT(θ0)],

¯ ν3 = 1

T ¯

υ−1

2 Eθ0[δ3ℓT(θ0)],

¯ ν4 = 1

T ¯

υ−1

2 Eθ0[δ4ℓT(θ0)],

Z1 = 1 √ T ¯ υ−1

2 δℓT(θ0),

Z2 = 1 √ T ¯ υ−1

2

  • δ2ℓT(θ0) − Eθ0[δ2ℓT(θ0)]
  • ,

Z3 = 1 √ T ¯ υ−1

2

  • δ3ℓT(θ0) − Eθ0[δ3ℓT(θ0)]
  • .

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Derivation of valid distributional asymptotic expansion

E[f( √ T(ˆ θT − θ0))] ≈E[f(ST(Z1, Z2, Z3))], ST : stoch. exp., ⇐ (Delta method) ≈

  • f(ST(z1, z2, z3))pT(z1, z2, z3)dz1dz2dz3,

pT : valid asymptotic exp. of (Z1, Z2, Z3) =

  • f(z1 + QT(z1, z2, z3))pT(z1, z2, z3)dz1dz2dz3

ST = z1 + QT ≈

  • (f(z1) + ∂f(z1)QT + 1

2∂2f(z1)Q2

T)pT(z)dz

⇐ (formal) Taylor’s expansion =

  • f(z1)
  • (−∂)
  • QTpTdz2dz3 + 1

2(−∂)2

  • QTpTdz2dz3
  • dz1

⇐ IBP over R

For the validity of pT, the regularity of the distribution of (Z1, Z2, Z3) ✷ non-degeneracy of covariance matrix ✷ Cram´ er type condition — non-degeneracy of the Malliavin covariance

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Linearly parametrized diffusion model dXt = −θ m(XT )dt + dwt, θ > 0 For this model,

ℓT(θ) = log dνθ dx z(X0) − θ

T

m(Xt)dXt − 1 2θ2

T

m2(Xt)dt Z1 = 1 √ T g−1

  • δ log dνθ

dx (X0) − E[δ log dνθ dx (X0)] −

T

m(Xt)dwt

  • = Op(1),

Z2 = 1 √ T g−1

  • δ2 log dνθ

dx (X0) − E[δ2 log dνθ dx (X0)] −

T

¯ m2(Xt)dt

  • = Op(1),

Z3 = 1 √ T g−1

  • δ3 log dνθ

dx (X0) − E[δ3 log dνθ dx (X0)]

  • = Op( 1

√ T ), √ T(ˆ θT − θ0) =ST(Z1, Z2) + 1 T √ T R′

3,

ST(z1, z2) =z1 + 1 √ T

  • z1z2 + 1

2¯ υ−1

2 ¯

ν3z2

1

  • + 1

T

1

6(¯ ν4 + 3¯ ν2

3)z3 1 + 3

2¯ ν3z2

1z2 + z1z2 2

  • .

Therefore, the previous asymptotic expansion for the general case can be applied if (Z1, Z2) admit a third order asymptotic expansion.

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Linear and Linearly parametrized diffusion: OU-process ✷ Ornstein-Uhlenbeck process : dXt = −θXtdt + dwt, θ > 0.

ℓT(θ) =1 2 log(θ) − θX2

0 − θ

T

XtdXt − 1 2θ2

T

X2

t dt,

Z1 = 1 √ T g−1

  • −X2

0 + 1

2θ0 −

T

Xtdwt

  • ,

Z2 = − 1 √ T g−1

T

(X2

t −

1 2θ0 )dt, Z3 =0, g = 1 2θ0 + 1 T 1 2θ2 .

Third order stochastic expansion becomes

√ T(ˆ θT − θ0) =ST(Z1, Z2) + 1 T √ T R3, ST(z1, z2) =z1 + 1 √ T

  • z1z2 + 1

2¯ υ−1

2 ¯

ν3z2

1

  • + 1

T

1

6(¯ ν4 + 3¯ ν2

3)z3 1 + 3

2¯ ν3z2

1z2 + z1z2 2

  • ,

Can we obtain the valid expansion of (Z1, Z2) ?

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Degeneracy of stochastic expansion Itˆ

  • ’s formula says that

Z1 − θZ2 = 1 √ T g−1

  • −X2

0 + 1

2θ0 − 1 2X2

T + 1

2X2

  • = op(1) → 0

as T → ∞

Z1 and Z2 are asymptotically linearly dependent : Cov[Z1, Z2] → 0 as T → ∞ ✷ Decomposition

ST(Z1, Z2) =S(0)

T (Z1) + 1

T S(1)

T

+ 1 T √ T ˜ R3, where S(0)

T (z1) =z1 +

1 √ T

1

θ0 + 1 2¯ ν3

  • z2

1 + 1

T

1

6(¯ ν4 + 3¯ ν2

3) + 3

2θ0 ¯ ν3 + 1 θ2

  • z3

1,

S(1)

T

= − 1 θ0 g−1

  • −X2

0 + 1

2θ0 − 1 2X2

T + 1

2X2

  • Z1.

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Asymptotic expansion having degenerate component ST = S(0)

T

+ uTS(1)

T

,E[f(S(0)

T

)] = ΨT[f] + ¯

  • (uT), uT → 0 as T → ∞

Theorem 3. Let m ∈ N, M > 0, γ > 0, ℓ1 > 0 and ℓ2 > 0 s.t. m > γ/2 + 1, ℓ1 ≥ ℓ2 − 1, ℓ2 ≥ (2m + 1) ∨ (d + 3). Assume (1) ∀p > 1, supT ||S(1)

T

||p,ℓ2 + supT ||S(0)

T

||p,ℓ2 < ∞, (2) ∃S(0)

∞ , S(1) ∞ , (S(0) T

, S(1)

T

) d → (S(0)

∞ , S(1) ∞ ).

there exits a functional ξT s.t. (3) ∀p > 1, supT ||ξT||p,ℓ1 < ∞, (4) ∃α > 0, P[|ξT| > 1/2] = O(uα

T),

(5) ∀p > 1, supT E[1|ξT |<1σ−p

S(0)

T

] < ∞. Then for any f satisfying |f(x)| ≤ M(1 + |x|γ), E[f(ST)] = ΨT[f] + uT

  • Rd f(x)g∞(x)dx + ¯
  • (uT),

where g∞(x) = −∂x

  • E[S(1)

∞

| S(0)

∞

= x]pS(0)

∞ (x)

  • .

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Rough sketck of the proof

E[f(ST)] =E[f(S(0)

T )] + uTE[∂f(S(0) T )S(1) T ] + u2 T

1

dv(1 − v)E[∂2f(S(0)

T

+ vuTS(1)

T )[(S(1) T )⊗2]]

=

  • f(x)ΨT(dx) + uTE[f(S(0)

T )ΦS(0)

T

1 (S(1) T )] + o(uT)

=

  • f(x)ΨT(dx) + uT
  • f(x)E[ΦS(0)

T

1 (S(1) T )|S(0) T

= x]pS

(0) T (x)dx + o(uT)

gT(x) :=E[ΦS(0)

T

1 (S(1) T )|S(0) T

= x]pS(0)

T (x)

= 1 (2π)d

  • e−iux
  • eiu˜

xE[ΦS(0)

T

1 (S(1) T )]|S(0) T

= ˜ x]pS

(0) T (˜

x)d˜ xdu = 1 (2π)d

  • e−iuxE[eiuS(0)

T XTΦS (0) T

1 (S(1) T )]du =

1 (2π)d

  • e−iuxE[iueiuS(0)

T S(1)

T ]du

− → 1 (2π)d

  • e−iuxE[iueiuS(0)

∞ S(1)

∞ ]du

= 1 (2π)d

  • e−iux
  • ∂˜

xeiu˜ xE[S(1) ∞ |S(0) ∞ = ˜

x]pS

(0) ∞ (˜

x)d˜ xdu = 1 (2π)d

  • e−iux
  • eiu˜

x

  • − ∂˜

xE[S(1) ∞ |S(0) ∞ = ˜

x]pS

(0) ∞ (˜

x)

  • d˜

xdu = g∞(x).

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SLIDE 18

Validity of the formal expansion for OU-process Mixing property of OU implies that X0, XT, Z1 are mutually asymp- totically independent, and therefore,

S(1)

T

= − 1 θ0 g−1

  • −X2

0 + 1

2θ0 − 1 2X2

T + 1

2X2

  • Z1

d

⇒ − 1 θ0 g−1

  • −ξ2

1 + 1

2θ0 − 1 2ξ2

2 + 1

2ξ2

1

  • S(0)

∞ =: S(1) ∞ ,

where ξ1, ξ2, S(0)

∞

are independent and ξ1, ξ2 ∼ ν(stationary dist.). This yields

E[S(1)

∞ |S(0) ∞ ] = − 1

θ0 g−1S(0)

∞ E

  • −ξ2

1 + 1

2θ0 − 1 2ξ2

2 + 1

2ξ2

1

  • = 0.

In this way, the asymptotic expansion of MLE coincides with that of S(0)

T

. Consequently, the formal expansion for general diffusion is valid in the exceptional case(OU-process).

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