5 Asymptotic behaviour of MLE

1− 1

E(Y∞)E _Y¹_∞

>0 since E(Y∞)E _Y¹_∞

= _2θκ−σ^2θκ2 >1, hence the matrix S is invertible, and we conclude (4.4) (D_TG_T)⁻¹ −→^a.s. (1−%²)σ²S⁻¹ as T → ∞.

The aim of the following discussion is to show convergence

(4.5) D_Th_T −→^a.s. 0 as T → ∞.

The strong law of large numbers for continuous local martingales (see, e.g., Theorem B.1) implies h⁽¹⁾_T −→^a.s. 0 as T → ∞, since, by part (ii) of Theorem 2.2,

1 T

Z T 0

ds Y_s

−→a.s. E 1

Y∞

= 2κ

2θκ−σ² ∈(0,∞) as T → ∞, implying

Z T 0

Y_s =T · 1 T

Z T 0

ds Y_s

−→ ∞a.s. as T → ∞.

Convergences h⁽²⁾_T −→^a.s. 0 as T → ∞, and h⁽³⁾_T −→^a.s. 0 as T → ∞ can be proved in the same way, since, by part (ii) of Theorem 2.2,

1 T

Z T 0

Y_sds−→^a.s. E(Y∞) = θ∈(0,∞) as T → ∞, implying

Z T 0

Ysds=T · 1 T

Z T 0

Ysds −→ ∞^a.s. as T → ∞.

Consequently, we conclude (4.5). By (4.1), (4.4) and (4.5), we obtain G⁻¹_T f_T −→^a.s. H(ψ) as T → ∞, hence we conclude the statement. 2 Remark 4.2 For the case θκ = ^σ₂², Theorem 5.3 implies weak consistency of the MLE of

(θ, κ, µ). 2

Theorem 5.1 If θ, κ ∈(0,∞) with θκ ∈ ^σ₂²,∞

, µ ∈R, σ ∈ (0,∞), % ∈ (−1,1), and (Y₀, S₀) = (y₀, s₀)∈(0,∞)², then the MLE of ψ= (θ, κ, µ) is asymptotically normal, namely,

T^1/2(ψb_T −ψ)−→ N^D ₃(0,V) as T → ∞, (5.1)

where the matrix V is given by

With a random scaling, we have

R_TQ_T(ψb_T −ψ)−→ N^D ₃(0,I₃) as T → ∞, (5.3)

where I₃ denotes the 3×3 identity matrix, and R_T, T ∈(0,∞), and Q_T, T ∈(0,∞), are 3×3 (not uniquely determined) random matrices with properties T^−1/2R_T −→^P C as T → ∞ with some C ∈ R^3×3, R^>_TR_T = G_T, T ∈ (0,∞), and Q_T −→^P Q as T → ∞,

Remark 5.2 Note that the limiting covariance matrix V in (5.1) depends only on the un-known parameters θ and κ, but not on (the unknown) µ. The advantage of the random scaling is that the limiting covariance matrix in (5.3) is the 3×3 identity matrix I₃ which does not depend on any of the unknown parameters. Note also that for R_T and Q_T one can choose, for instance,

R_T = 1

as T → ∞, and Q_T −→^a.s. Q as T → ∞, since as T → ∞ by part (i) of Theorem 2.2. Hence then the random scaling factor has the form

R_TQ_T = 1

Proof of Theorem 5.1. For (5.1), it is enough to prove

(5.4) T^1/2(G⁻¹_T f_T −H(ψ))−→ N^D ₃(0,V₀) as T → ∞,

as T → ∞, where Q⁻¹V₀(Q⁻¹)^> =V, hence we obtain (5.1).

By the first equality in (4.1), we have

(5.5) T^1/2(G⁻¹_T f_T −H(ψ)) =T^1/2G⁻¹_T h_T = (T⁻¹G_T)⁻¹(T^−1/2h_T),

provided that G_T is invertible, which holds almost surely, see Section 3. By part (i) of Theorem 2.2, E(Y∞) = θ ∈ (0,∞) and E _Y¹_∞

= _2θκ−σ^2κ 2 ∈ (0,∞), and hence, part (ii) of Theorem 2.2 and (3.4) imply

T⁻¹G_T −→^a.s. E(G∞) as T → ∞, (5.6)

with

G∞:= 1 (1−%²)σ²Y∞







1 −Y_∞ −%σ

−Y∞ Y_∞² %σY∞

−%σ %σY∞ σ²





 ,

where Y_∞ has Gamma distribution with parameters 2θκ/σ² and 2κ/σ². The matrix E(G_∞) is invertible, namely,

[E(G∞)]⁻¹ = 1 E(Y_∞)E _Y¹_∞

−1

E _Y¹_∞







σ²E(Y_∞)E _Y¹_∞

−%²σ² (1−%²)σ²E _Y¹_∞

%σE(Y_∞)E _Y¹_∞

−%σ (1−%²)σ²E _Y¹_∞

(1−%²)σ² E _Y¹_∞2

%σE(Y∞)E _Y¹_∞

−%σ 0 E(Y∞)E _Y¹_∞

−1





 ,

since E(Y_∞)E _Y¹_∞

= _2θκ−σ^2θκ2 >1, which yields [E(G_∞)]⁻¹ =V₀. Whence we conclude (5.7) (T⁻¹G_T)⁻¹ −→^a.s. V₀ as T → ∞.

By (4.2), the process (h_t)t∈[0,∞) is a 3-dimensional continuous local martingale with (pre-dictable) quadratic variation process hhit = Gt, t ∈ [0,∞). Using (5.6), the central limit theorem for multidimensional continuous local martingales, see Theorem B.2, yields T^−1/2h_T −→ N^D ₃(0,E(G∞)) =N₃(0,V⁻¹₀ ) as T → ∞. Hence, by (5.5) and (5.7),

T^1/2(G⁻¹_T f_T −H(ψ))−→ N^D ₃(0,V₀V⁻¹₀ V₀) = N₃(0,V₀) as T → ∞, thus we obtain (5.4).

With random scaling, by (5.1) and Slutsky’s lemma, we obtain R_TQ_T(ψb_T −ψ) = (T^−1/2R_T)Q_T

T^1/2(ψb_T −ψ) D

−→ N₃(0,(CQ)V(CQ)^>) as T → ∞. Moreover, by the assumptions on R_T, T ∈(0,∞),

T⁻¹G_T = (T^−1/2R_T)^>(T^−1/2R_T)−→^P C^>C as T → ∞.

Thus, comparing with (5.6), we obtain C^>C =E(G∞) =V⁻¹₀ . Using Q⁻¹V₀(Q⁻¹)^> =V, we obtain

(CQ)V(CQ)^>= (CQ)Q⁻¹(C^>C)⁻¹(Q⁻¹)^>(CQ)^>=I3,

and we conclude (5.3). 2

Theorem 5.3 If θ, κ ∈ (0,∞) with θκ = ^σ₂², µ ∈ R, σ ∈ (0,∞), % ∈ (−1,1), and and Z2 are independent standard normally distributed random variables, independent from T. With a random scaling, we have



Note that the limit distribution in Theorem 5.3 (which can be considered as the asymptotic error of the estimator (bθT,bκT,µbT)) is a mixed normal distribution. Moreover, the first and second coordinates of the limit distributions in (5.8) and (5.9) are linearly dependent. In spite of this fact, one can give asymptotic confidence sets for (θ, κ), namely, ellipses together with their interiors and with center (bθ_T,bκ_T). Indeed, the sum of the squares of the first two coordinates of the left-hand side of (5.9), which one can call a normalized squared error of (θ, κ), converges weakly to 2Z₁², being a chi-squared distribution of degree 1 (multiplied by 2). Surprisingly, the mixed normal limit distributions of the third coordinate in (5.8) and (5.9) are not centered.

In Appendix D we derive an explicit formula for the density function of _κT^%σ +^σ

√

Due to Ben Alaya and Kebaier [8, Proposition 4], we have 1

where T^∗ := inf{t∈[0,∞) :W_t^∗ = ^κ_σ} with a standard Wiener process (W_t^∗)t∈[0,∞). Applying the scaling property of a standard Wiener process, we obtain

T^∗ = infn

is monotone decreasing, we obtain 1

For (5.8), it is enough to prove

(5.13) CT(G⁻¹_T f_T −H(ψ))−→^D

with ξ_T = C_T(G⁻¹_T f_T − H(ψ)), T ∈ (0,∞), and with functions F : R³ → R³ and F_T :R³ →R³, T ∈(0,∞), given by

F(x) :=Bx, F_T(x) :=

















T^−1/2x1−θx2

T^−1/2x2+κ

x₂ x₃







if x₂ 6=−T^1/2κ, 0 if x2 =−T^1/2κ, for x= (x1, x2, x3)∈R³ and T ∈(0,∞), where

B :=







0 −_2κ^σ²2 0

0 1 0

0 0 1





 .

We have

F_T(C_T(G⁻¹_T f_T −H(ψ))) =







T^−1/2T((G⁻¹_T f_T)1−θκ)−θT^1/2((G⁻¹_T f_T)2−κ) (G⁻¹_T f_T)2

T^1/2((G⁻¹_T f_T)₂−κ) T((G⁻¹_T f_T)3−µ)







=Ce_T(ψb_T −ψ),

provided that (G⁻¹_T f_T)2 6= 0, which holds almost surely, where

Ce_T :=







T^1/2 0 0 0 T^1/2 0

0 0 T





 .

Moreover, F_T(x_T)→F(x) as T → ∞ if x_T →x as T → ∞, since then, for sufficiently large T ∈(0,∞), we have (x_T)₂ 6=−T^1/2κ. Consequently, (5.13) and Lemma C.1 imply

Ce_T(ψb_T −ψ) = F_T(C_T(G⁻¹_T f_T −H(ψ))) −→^D F(ξ) =Bξ as T → ∞, hence we obtain (5.8).

Now we turn to prove (5.13). By the first equality in (4.1), we have (5.14)

C_T(G⁻¹_T f_T −H(ψ)) =C_TG⁻¹_T h_T = (C_TG⁻¹_T C_T)(C⁻¹_T h_T)

= (C⁻¹_T G_TC⁻¹_T )⁻¹(C⁻¹_T h_T),

provided that G_T is invertible, which holds almost surely, see Section 3. We have

C⁻¹_T G_TC⁻¹_T = 1 (1−%²)σ²







T⁻²RT 0

Yu −T^−1/2 −%σT⁻²RT

0 du Yu

−T^−1/2 T⁻¹RT

0 Y_udu %σT^−1/2

−%σT⁻²RT 0

Yu %σT^−1/2 σ²T⁻²RT 0

du Yu





 ,

and, by (4.2),

Consequently, (5.13) will follow from

(5.17) (C⁻¹_T G_TC⁻¹_T ,C⁻¹_T h_T)−→^D (Ge∞,he∞) as T → ∞

where Z₃ and Z₄ are independent standard normally distributed random variables, inde-pendent from Z₂ and T^∗. Indeed, provided that (5.17) holds, by the continuous mapping theorem,

(CTG⁻¹_T CT,C⁻¹_T hT)−→^D (Ge⁻¹_∞,he∞) as T → ∞, since Ge∞ is invertible almost surely with inverse

Ge⁻¹_∞ = 1

and hence, by (5.14) and the continuous mapping theorem,

C_T(G⁻¹_T f_T −H(ψ))−→^D Ge⁻¹_∞he∞ as T → ∞,

where, with Z₁ :=−p

Now we turn to prove (5.17). It will follow from Slutsky’s lemma, continuous mapping theorem and from

which will be a consequence of (5.10), (5.11), (5.16), Slutsky’s lemma (or part (v) of Theorem 2.7 in van der Vaart [38]), and

Using the SDE (1.3),

(5.19) σ consequently, RT

where we used the independence of Y and B yielding that the conditional distribution of RT

which is the same as the previous expectation except the factor exp{−^u^√²^u³

T }. Ben Alaya and Kebaier [9, proof of Theorem 7] proved

Y_T −θκT +κRT hence, by (5.19),

By Slutsky’s lemma, we obtain

as T → ∞. Using the continuity theorem, we obtain E exp

With a random scaling, we have



Applying (5.10), (5.11), (5.16), (5.18), Slutsky’s lemma (or part (v) of Theorem 2.7 in van der Vaart [38]) and the continuous mapping theorem, we obtain

C_TCe⁻¹_T ,Ce_T(ψb_T−ψ) D

−→













√ 2κ³ σ²√

1−%² 0 0

0 √ ¹

2κ(1−%²) 0

0 0 ^κ_σ





 ,







−^σ^√²^Z¹

2κ³

p1−%² Z₁p

2(1−%²)κ

%σ κT + ^Z²^σ

√

1−%² κ√













as T → ∞.

Using again the continuous mapping theorem, we obtain (5.9). 2 Remark 5.4 Putting formally θκ= ^σ₂² into the formula of V given in (5.2) of Theorem 5.1, one can observe that the joint limit distribution of the first two coordinates in (5.1) of Theorem

5.1 and in (5.8) of Theorem 5.3 coincide. 2

Remark 5.5 According to Theorem 7 in Ben Alaya and Kebaier [9], if a= ^σ₂²¹ and b ∈(0,∞), then, based on continuous time observations (Y_t)t∈[0,T], T ∈(0,∞), for the MLE (ba_T,bb_T) of (a, b) for the first coordinate process of the SDE (1.1), we have

T(ba_T −a) T^1/2(bbT −b)

−→D

" _σ2 1

√bT

2bZ1

as T → ∞, (5.21)

where Z₁ is a standard normally distributed random variable independent of T introduced in Theorem 5.3. Hence, using Slutsky’s lemma and that bb_T converges in probability to b as T → ∞ (following from (5.21)), we get

T^1/2

ba_T bbT

−a b

=T^1/2bba_T −abb_T bbbT

=T^1/2b(ba_T −a)−a(bb_T −b) bbbT

= T^−1/2bT(ba_T −a)−aT^1/2(bb_T −b) bbb_T

−→ −D a b²

√

2bZ₁ =−

√2a

√b³Z₁

as T → ∞. Let us observe that in the special case of %= 0, we have ^b^a^T

bbT

=θb_T and bb_T =κ_T, T >0 (for the explicit formulae for baT and bbT, see Ben Alaya and Kebaier [9, Section 3.1]).

Moreover, in case of a =θκ = ^σ₂² and b =κ, we have −

√√2a

b³ = −^√^σ²

2κ³. Hence, under the conditions of Theorem 5.3 together with %= 0, the joint (weak) convergence of the first two coordinates of (5.8) follows from Theorem 7 in Ben Alaya and Kebaier [9]. 2

In document , Mohamed Ben Alaya (Pldal 25-36)