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Conditional Least Squares Estimation for Fractional Super Levy Processes in Nonlinear

SPDEs

Jaya P. N. Bishwal
Department of Mathematics and Statistics, University of North Carolina at Charlotte, 376 Fretwell Bldg,9201 University City Blvd. Charlotte, NC 28223-0001, USAJ.Bishwal@uncc.edu

Abstract. We consider infinite dimensional extension of affine models as super Levy processes sat-isfying a nonlinear SPDE. We obtain the asymptotics of the conditional least squares estimators.Finally we obtain the Berry-Esseen inequality.
1. Introduction and Preliminaries

Parameter estimation in finite dimensional diffusions is now classical. Bishwal [7] studied a newestimating function for discretely sampled diffusions. Bishwal [8] studied asymptotic theory of like-lihood method and Bayesian method for drift estimation of finite dimensional stochastic differentialequations. Bishwal [12] studied applications of Levy processes in stochastic volatility models infinance. Bishwal [13] studied parameter estimation for SPDEs driven by cylindrical stable pro-cesses. Bishwal [6] studied the Bernstein-von Mises theorem and spectral asymptotics of Bayesestimators for parabolic SPDEs when the number of Fourier coefficients becomes large. In thiscase, the measures generated by the process for different parameters are singular. Bishwal [11]studied Bernstein-von Mises theorem and small noise Bayesian asymptotics for parabolic stochas-tic partial differential equations. Bishwal [10] studied hypothesis testing for fractional stochasticpartial differential equations with applications to neurophysiology and finance.Consider the nonlinear SPDE
dX(t, x) =

1

2
∆X(t, x)dt +

√
X(t, x)dW (t, x) (1.1)

where W (t, x) a cylindrical Brownian motion. Konno and Shiga [26] studied the existence andweak uniqueness of the above equation as a martingale problem for the associated super-Brownianmotion. The pathwise uniqueness of nonnegative solution still remains open. The main difficultycomes from the unbounded drift coefficient and non-Lipschitz diffusion coefficient. Wang et al. [39]studied a comparison theorem and showed that the solution of the nonlinear SPDE is distribution
Received: 4 Dec 2023.
Key words and phrases. Nonlinear stochastic partial differential equations, super processes, fractional Cox-IngersollRoss model, conditional least squares estimator, branching interacting particle system, Berry-Esseen inequality.1

https://adac.ee
https://doi.org/10.28924/ada/ma.4.12


Eur. J. Math. Anal. 10.28924/ada/ma.4.12 2function valued. They also established pathwise uniqueness. As application they obtained well-posedness of martingale problems for two classes of measure-valued diffusions: interacting super-Brownian motions and interacting Fleming-Viot processes. He et al. [21] obtained pathwise uniquesolution to nonlinear SPDE with super Levy process, which is a combination of space-time Gaussianwhite noises and Poisson random measures which is a generalization of work of Xiong [40] wherethe result for a super-Brownian motion with binary branching mechanism was obtained. Usingan extended Yamada-Watanabe argument, Xiong [40] established strong existence and uniquenessof the solution to the SPDE. Super-Brownian motion (SBM), also called the Dawson-Watanabeprocess introduced by Dawson and Watanabe is a measure valued process arising as the limit ofempirical measure process of a branching particle system. SBM satisfies a martingale problem.When the state space is R, SBM has a density w.r.t. Lebesgue measure and this density valuedprocess X(t, x) satisfies the above SPDE. When the space R is s single point, the SPDE becomesan SDE which is CIR diffusion dXt =
√
XtdWt whose uniqueness is established using the Yamada-Watanabe argument. Xiong and Yang (2019) studied existence and pathwise uniqueness to anSPDE with Hölder continuous coefficient driven by α-stable colored noise. The existence of thesolution is shown by considering the weak limit of a sequence of SDE system which is obtained byreplacing the Laplacian operator in the SPDE by its discrete version. The pathwise uniqueness isshown by using a backward doubly stochastic differential equation to take care of the Laplacian.In the case of d = 1, the pathwise uniqueness of a nonnegative solution to the correspondingequation was established by Yang and Zhou [42] for 1 < α <

√
5− 1 and pathwise uniqueness for

√
5− 1 < α < 2 is still open.The existence and pathwise uniqueness of solutions to the SDEs with non-Lipschitz coefficientdriven by spectrally positive Levy processes were studied in Fu and Li [20].Consider the SPDE with multiplicative noise:

duθ(t, x) = (A0 + θA1)uθ(t, x)dt +Muθ(t, x)dZ(t, x), t ≥ 0, x ∈ [0, 1] (1.2)

where M is a known nonlinear operator.Priola et al. [32] obtained exponential convergence to the invariant measure, in the total variationnorm, for solutions to SDEs driven by α-stable noises in finite and infinite dimensions using twoapproaches: Lyapounov’s function approach by Harris and Doeblin’s coupling argument. In bothapproaches irreducibility and uniform strong Feller property play crucial role.Equation (1.2) is called diagonalizable if A0, A1 and M have point spectrum and a commonsystem of eigenfunction {hj , j ≥ 1}. Denote by ρk , νk and µk , the eigenvalues of the operators
A0, A1 and M respectively. Then

uθ(t, x) =

∞∑
j=1

uj,thj . (1.3)

We consider fractional stable CIR model as example.Using fractional Levy process as the driving term, maximum quasi-likelihood estimation in frac-tional Levy stochastic volatility model was studied in Bishwal [9].

https://doi.org/10.28924/ada/ma.4.12


Eur. J. Math. Anal. 10.28924/ada/ma.4.12 3Fractional Levy Process (FLP) is defined as
MH,t =

1

Γ(H + 1
2 )

∫
R

[(t − s)
H−1/2
+ − (−s)

H−1/2
+ ]dLs , t ∈ R (1.4)

where {Lt , t ∈ R} is a Levy process on R with E(L1) = 0, E(L2
1) <∞.Here are some properties of the fractional Levy process:1) the covariance of the process is given by

cov(MH,t ,MH,s) =
E(L2

1)

2Γ(2H + 1) sin(πH)
[|t|2H + |s|2H − |t − s|2H]. (1.5)

2) MH is not a martingale. For a large class of Levy processes, MH is neither a semimartingale.3)MH is Hölder continuous of any order β less than H − 1
2 . 4) MH has stationary increments. 5)

MH is symmetric. 6) L is self-similar, but MH is not self-similar. 7) MH has infinite total variationon compacts.Thus FLP is a generalization and a natural counterpart of FBM. Fractional stable motion is aspecial case of FLP.
2. Conditional Least Squares Estimation

Let H be a real separable Hilbert space with inner product 〈·〉 and norm | · |. By L(H) we denotethe Banach space of bounded linear operators from H into H endowed with the operator norm
‖ · ‖L(H). We fix an orthonormal basis (en) in H. Through the basis (en) we will often identify Hin l2. More generally, for a given sequence ρ = (ρn) of real numbers we set

l2ρ = {(xn) ∈ R∞ :
∑
n≥1

x2
nρ

2
n <∞}.

where R∞ = RN. The space l2ρ becomes a separable Hilbert space with the inner product: 〈x, y〉 =∑
n≥1 xnynρ

2
n for x = (xn), y = (yn) ∈ l2ρ . Let us fix θ0, the unknown true value of the parameter θ.Let (Ω,F , P ) be a complete probability space and Z(t, x) be a process on this space with valuesin the Schwarz space of distributions D′(G) such that for φ,ψ ∈ C∞0 (G), ‖φ‖−1

L2(G)
〈W (t, ·), φ(·)〉is a one dimensional stable process.This process is usually referred to as the cylindrical α-stable process (C.S.P.), α ∈ (0, 2). Weassume that there exists a complete orthonormal system {hi}∞i=1 in L2(G)) such that for every

i = 1, 2, . . . , hi ∈ Zm,20 (G) ∩ C∞(G) and
Λθhi = βi(θ)hi , and Lθhi = µi(θ)hi for all θ ∈ Θ

where Lθ is a closed self adjoint extension of Aθ, Λθ := (k(θ)I − Lθ)1/2m, k(θ) is a constantand the spectrum of the operator Λθ consists of eigenvalues {βi(θ)}∞i=1 of finite multiplicities and
µi = −β2m

i + k(θ).

https://doi.org/10.28924/ada/ma.4.12


Eur. J. Math. Anal. 10.28924/ada/ma.4.12 4A Levy process (Zt) with values in H is an H-valued process defined on some stochastic basis
(Ω,F , (Ft)t≥0, P ) having stationary independent increments, cadlag trajectories such that Z0 = 0,P-a.s. One has that

E[e i〈Zt ,s〉] = exp(−tψ(s)), s ∈ Hwhere ψ : H → C is Sazonov continuous, negative definite function such that ψ(0) = 0. Thefunction ψ is called the exponent of (Zt).The exponent ψ can be expressed by the infinite dimensional Levy-Khintchine formula

ψ(s) =
1

2
〈Qs, s〉 − i〈a, s〉 −

∫
H

(
e i〈s,y〉 − 1−

i〈s, y〉
1 + |y |2

)
ν(dy), s ∈ H

where Q is the non-negative trace class operator on H, a ∈ H and ν is the Levy measure or thejump intensity measure associated to (Zt).Cylindrical α-stable process (C.S.P.) is a Levy process taking values in the Hilbert space H = l2ρ ,with a properly chosen weight ρ.Consider the linear SPDE
dXt = θAXtdt + dZt , x ∈ H

C.S.P. Z(t) is a cylindrical α-stable process, α ∈ (0, 2) which can be expanded in the series
Z(t) =

∞∑
i=1

γiZi(t)hi , t ≥ 0

where {Zi(t)}∞i=1 are independent, real valued, one dimensional, normalized, symmetric, α-stableprocesses and (γi)
∞
i=1 is a given sequence of, possibly unbounded, positive numbers, and hi is afixed orthonormal basis in H. The latter series converges P -a.s. in H−α for α > d/2. Indeed

‖Z(t)‖2
−α =

∞∑
i=1

γ2
i Z

2
i (t)‖hi‖2

−α =

∞∑
i=1

Z2
i (t)β−2α

i

and the later series converges P -a.s.For any j ∈ N, t ≥ 0,
E[e iZj (t)h] = e−t|h|

α

.

Stable one-dimensional density: A one-dimensional, normalized, symmetric α-stable distribution
µα, α ∈ (0, 2] has characteristic function

µ̂α(s) = e−|s|
α

, s ∈ R.

The density of µα with respect to Lebesgue measure will be denoted by pα. This even functionis known in closed form only if α = 1 or 2. The precise asymptotic behavior of the density
pα, α ∈ (0, 2) is as follows:For any α ∈ (0, 2), there exists Cα such that

pα(x) ∼
Cα
xα+1

as x →∞.

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Eur. J. Math. Anal. 10.28924/ada/ma.4.12 5

Stable measures on Hilbert space: A random variable ξ on H is called α-stable (α ∈ (0, 2]) if forany n there exists a vector an ∈ H such that for any independent copies ξ1, ξ2, . . . , ξn of ξ, therandom variable n−1/α(ξ1 + ξ2, . . .+ ξn)− an has the same distribution as ξ. A Borel probabilitymeasure µ on H is said to be α-stable if it is the distribution of a stable random variable withvales in H.Consider the SPDE with multiplicative noise:
duθ(t, x) = (A0 + θA1)uθ(t, x)dt +Muθ(t, x)dZ(t, x), t ≥ 0, x ∈ [0, 1] (2.1)

where M is a known nonlinear operator and Z(t, x) is a cylindrical subfractional Levy process.Equation (2.1) is called diagonalizable if A0, A1 and M have point spectrum and a commonsystem of eigenfunction {hj , j ≥ 1}. Denote by ρk , νk and µk , the eigenvalues of the operators
A0, A1 and M respectively. Then uθ(t, x) =

∑∞
j=1 uj,thj .Consider SPDE model with multiplicative noise and mean reversion, where the j-th Fouriercoefficient is the stable Cox-Ingersoll-Ross (SCIR) model:

duj,t = (a − θuj,t)dt + σu
1/α
j,t−dZj,t , j ≥ 1 (2.2)

where a is the mean reverting level and θ is mean reverting speed. Recall that for α = 2, for every
j ≥ 1, the process Zj,t is a standard Brownian motion, this is the famous Cox-Ingersoll-Ross (CIR)model used for modeling interest rate, which is also used a stochastic volatility process in Hestonmodel. Note that there are Brownian CIR models with additive compound Poisson type jumps.When 1 < α < 2, Zj,t is stable process with Levy measure

να(dz) =
1{z>0}dz

αΓ(−α)zα+1
. (2.3)

The discontinuous SCIR model captures the heavy tailed property in the sense of infinite variance.There is empirical evidence from high frequency data available in support of application of purejump models in financial modeling.The SCIR model has the unique stationary distribution µ with Laplace transform given by
Lµ(λ) =

∫ ∞
0

e−λxµ(dx) = exp

{
−
∫ λ

0

αa

αθ + σαzα−1
dz

}
, λ ≥ 0. (2.4)

Now we focus on the fundamental semimartingale behind the CIR model. Define
κH := 2HΓ(3/2−H)Γ(H + 1/2), kH(t, s) := κ−1

H (s(t − s))
1
2
−H,

ηH :=
2HΓ(3− 2H)Γ(H + 1

2 )

Γ(3/2−H)
, vt ≡ vHt := η−1

H t2−2H,MH
t :=

∫ t

0

kH(t, s)dMH
s .For using Girsanov theorem for Brownian motion, since a Radon-Nikodym derivative process is al-ways a martingale, a central problem is how to construct an appropriate martingale which generatesthe same filtration, up to sets of measure zero, as the non-semimartingale called the fundamental

martingale.Extending Norros et al. (1999) it can be shown that MH
t is a martingale, called the funda-mental martingale whose quadratic variation 〈MH〉t is vHt . Moreover, the natural filtration of the

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Eur. J. Math. Anal. 10.28924/ada/ma.4.12 6

martingale MH coincides with the natural filtration of the FLP MH since MH
t :=

∫ t
0 K(t, s)dMH

sholds for H ∈ (1/2, 1) where KH(t, s) := H(2H − 1)
∫ t
s r

H− 1
2 (r − s)H−

3
2 dr, 0 ≤ s ≤ t and for

H = 1/2, the convention K1/2 ≡ 1 is used.Define Qi(t) := d
dvt

∫ t
0 kH(t, s)ui(s)ds. It is easy to see that

Qi(t) = ηH
2(2−2H)

{
t2H−1Zi(t) +

∫ t
0 r

2H−1dZi(s)
}
. Define the process Zi = (Zi(t), t ∈ [0, T ]) by

Zi(t) :=
∫ t

0 kH(t, s)dui(s).Extending Kleptsyna and Le Breton (2002), we have:(i) Zi is the fundamental semimartingale associated with the process ui .(ii) Zi is a (Ft) -semimartingale with the decomposition Zi(t) = µi(θ)
∫ t

0 Qi(s)dvs + β−νi M
H
t .(iii) ui admits the representation ui(t) =

∫ t
0 KH(t, s)dZi(s). (iv) The natural filtration (Zi(t)) of

Zi and (Ui(t)) of ui coincide.We describe our observations now. Note that for equally spaced data (homoscedastic case)
vtk − vtk−1

= η−1
H

(
T
n

)2−2H
[k2−2H − (k − 1)2−2H], k = 1, 2, · · · , n.For H = 0.5, vtk − vtk−1

= η−1
H

(
T
n

)2−2H
[k2−2H − (k − 1)2−2H] = T

n , k = 1, 2, . . . , n. We have
Qi(t) =

d

dvt

∫ t

0

kH(t, s)ui(s)ds = κ−1
H

d

dvt

∫ t

0

s1/2−H(t − s)1/2−Hui(s)ds

= κ−1
H ηHt

2H−1 d

dt

∫ t

0

s1/2−H(t − s)1/2−Hui(s)ds

= κ−1
H ηHt

2H−1

∫ t

0

d

dt
s1/2−H(t − s)1/2−Hui(s)ds

= κ−1
H ηHt

2H−1

∫ t

0

s1/2−H(t − s)−1/2−Hui(s)ds.

The process Qi depends continuously on ui and therefore, the discrete observations of ui does notallow one to obtain the discrete observations of Qi . The process Qi can be approximated by
qi(n) = κ−1

H ηHn
2H−1

n−1∑
j=0

j1/2−H(n − j)−1/2−Hui(j).

It is easy to show that qi(n)→ Qi(t) almost surely as n →∞, see Tudor and Viens (2007).Define a new partition 0 ≤ r1 < r2 < r3 < · · · < rmk = tk , k = 1, 2, · · · , n. Define
qi(tk) = κ−1

H ηHt
2H−1
k

mk∑
j=1

r
1/2−H
j (rmk − rj)

−1/2−Hui(rj)(rj − rj−1), k = 1, 2, · · · , n.

It is easy to show that qi(tk)→ Qi(t) almost surely as mk →∞ for each k = 1, 2, · · · , n. We usethis approximate observation in the calculation of our estimators.Applying Itô’s formula, for t ≥ r ≥ 0, we obtain
qj,t = e−θ(t−r)qj,r + a

∫ t

r

e−θ(t−s)ds + σ

∫ t

r

e−θ(t−s)q
1/α
j,s−dZj,s , j ≥ 1. (2.5)

Let the process be observed at {kh, k = 0, 1, . . . , n} from a single realization {qj,t , t ≥ 0} for fixed
h. For simplicity, we take h = 1. This equation can be considered as a first order autoregressive

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Eur. J. Math. Anal. 10.28924/ada/ma.4.12 7(AR(1)) equation
qj,k = ρ+ γqj,k−1 + εj,k , j ≥ 1 (2.6)where γ = e−θ, ρ = aθ−1(1− γ) and

εj,k = σ

∫ k

k−1

e−θ(k−s)q
1/α
j,s−dZj,s , k ≥ 1, j ≥ 1. (2.7)

For B ∈ B(R+), let
S2,j,n(B) =

n∑
k=1

qj,k−1εk IB(|qj,k−1εj,k |), S1,j,n(B) =

n∑
k=1

q2
j,k−1IB(qj,k−1), j ≥ 1. (2.8)

It is easy to see that
εj,k = qj,k − E(qj,k |Fk−1), k ≥ 1, j ≥ 1. (2.9)is a sequence of martingale differences for every fixed j .Let S1,j,n := S1,j,n(0,∞), S2,j,n := S2,j,n(0,∞) and recall that γ = e−θ .Then

θ̂j,n − θ =
S2,j,n

S1,j,n
(2.10)

where θ̂n is the conditional least squares estimator (CLSE) which minimizes
n∑
k=1

ε2
j,k =

n∑
k=1

[qj,k − E(qj,k |Fk−1)]2 =

n∑
k=1

[qj,k − ρ− γqj,k−1]2 (2.11)

and are given by
γ̂j,n =

∑n
k=1 qj,k−1

∑n
k=1 qj,k − n

∑n
k=1 qj,k−1qj,k

(
∑n
k=1 qj,k−1)2 − n

∑n
k=1 q

2
j,k−1

,

ρ̂j,n =
1

n

n∑
k=1

qj,k − γ̂n
1

n

n∑
k=1

qj,k−1,

θ̂j,n = − log γ̂j,n, âj,n =
ρ̂nθ̂n

1− γ̂n
.Let (S1, S2) have the characteristic function given by

E[exp{iλ1S1 + iλ2S2}] := exp

{
−

σα

θ2Γ(−α)

∫ ∞
0

E
(

1− exp{iλ1y
2 + iλ2y

(α+1)/αVj,1}
)

× E

(
exp

{
ie−2θλ1y

2

1− e−2θ
+
ie−θ(α+1)/αλ2y

(α+1)/αVj,2

(1− eθ(α+1))1/α

})
dy

yα+1

}
(2.12)

and
Vj,k := σ

∫ k

k−1

e−θ(k−s)e−θ(s−k+1)/αdZj,s , k = 1, 2, j ≥ 1 (2.13)

which are i.i.d. with the same distribution as
σ

(
e−θ − 1

(α− 1)θ

)1/α

Zj,1

which is regularly varying with index α. The limit distribution is normal only in the Gaussian case
α = 2.

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Eur. J. Math. Anal. 10.28924/ada/ma.4.12 8Following Li and Ma [28] it can be shown that for every fixed j , if we have 1 < α < (1 +
√

5)/2,then we have as n →∞
(d−2
n S1,j,n, c

−1
n S2,j,n)

D→(S1, S2) on R2

where dn = n1/α and cn = n(α+1)/α2
= d

(α+1)/α
n .For the stable SPDE model, we have the following result on the consistency and the limit dis-tribution of the CLSE:

Theorem 2.1 If we have 1 < α < (1 +
√

5)/2, then for every fixed j ≥ 1a)
θ̂j,n →P θ as n →∞.b)

n(α−1)/α2

(θ̂j,n − θ)→D
(
σ2

ν2
j

)1/α
S2

S1
as n →∞.

c) If in addition, limj→∞
∣∣νj ∣∣ =∞, then for every fixed n ≥ 1,

θ̂j,n →P θ as j →∞

and ∣∣νj ∣∣ (θ̂j,n − θ)→D σ
(
n−(α−1)/α2

)1/α S2

S1
as j →∞.where S2 and S1 are defined in (2.12).

Remarks1) The limit distribution in the case (1 +
√

5)/2 < α < 2 is still open.2) The process (Xj) is exponentially ergodic and hence strongly mixing.3) For the Gaussian case (α = 2), the limit results are based on ergodic theory and martingaleconvergence theorem. For the non-Gaussian case (1 < α < 2), limit results are obtained by thetheory of regular variation and convergence of point processes.4) Let 0 < α < 2 and let Zt be a one dimensional α-stable process with Levy measure ν(dz).Then as n →∞, nP (n−1/αZt ∈ ·)→v tν(·).

We consider the Stable Cox-Ingersoll-Ross Model as an example. Xiong and Yang [41] studiedexistence and strong uniqueness of the following SPDE:
duk(t) = (θνk + ρk)uk(t)dt + σk(uk(t))1/αdZk(t), k ≥ 1.

The existence of the solution in the case of space-time white noise is shown by considering theweak limit of a sequence of SDE systems which is obtained by replacing the Laplacian operatorin the SPDE by its discrete version. The weak uniqueness follows from the uniqueness of solutionto the martingale problem for the associated super-Brownian motion. In the case of α-stable noisethe existence and pathwise uniqueness of the solution is studied in Xiong and Yang [41].

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Eur. J. Math. Anal. 10.28924/ada/ma.4.12 9

3. Interacting Particle Systems

First consider the fractional Cox-Ingersoll-Ross (FCIR) model
d Yt = a(b − Yt)dt + σ

√
YtdW

H
t (3.1)

where WH
t is a fractional Brownian motion with Hurst parameter H > 1/2.Then by Proposition 5.7 of Buchmann and Kluppelberg [15], we have

Yt = f (Xt) (3.2)

where
dXt = a(b −Xt)dt + dWH

t , X0 = f −1(Y0), t ∈ [0, T ] (3.3)

and f (x) = sgn(x)σ2x2/4.Let b = 0, σ = 1 and a > 0. Then Xt is described by the Ornstein-Uhlenbeck SDE
dXt = −aXtdt + dWH

t , X0 = f −1(Y0). (3.4)

For H = 0.5, let us consider maximum likelihood estimator (MLE) for the simple mean-field model
dXj(t) = αXj(t)dt − β(Xj(t)− X̄n(t))dt + dWj(t), Xj(0) = xj(0), j = 1, 2, · · · , n (3.5)

where X̄n(t)) = n−1
∑n
j=1Xj(t), β 6= α, and α 6= 0. The middle term on the right side of (3.5)can be viewed as an interaction among the subsystems which create a tendency for the subsystemsto relax towards the center of gravity of the ensemble. Thus the system provides a simple exampleof a cooperative interaction. Mean-field type models have applications in physics, biology andeconomics, see Dawson [19]. The case β = 0 corresponds to sampling independent replications ofOrnstein-Uhlenbeck processes on [0, T ]. Our parameter here is θ = (α, β).Suppose 1

n

∑n
j=1 xj(0)→ ν0 almost surely and 1

n

∑n
j=1 x

2
j (0)→ γ2

0 +ν2
0 almost surely as n →∞.Then the estimator θ̂n →P θ as n →∞ and √n(θ̂n − θ)→D N (0, I−1(T )) as n →∞

where
I(T ) =

(
A(T ) −B(T )

−B(T ) B(T )

)
with
A(T ) :=

ν2
0

2α
(e2αT − 1) + B(T ), B(T ) :=

e2(α−β)T − 1

4(α− β)2
−

T

2(α− β)
+
γ2

0 (e2(α−β)T − 1)

2(α− β)
.

The case β = 0 corresponds to sampling independent replications of the same process given below:
dXj(t) = αXj(t)dt + dWj(t), j = 1, 2, · · · , n (3.6)

https://doi.org/10.28924/ada/ma.4.12


Eur. J. Math. Anal. 10.28924/ada/ma.4.12 10In the classical case when β = 0, the MLE is given by
α̂n =

∑n
j=1

∫ T
0 Xj(t)dXj(t)∑n

j=1

∫ T
0 (Xj(t))2dt

.

Sampling n independent Ornstein-Uhlenbeck processes on [0, T ] and letting n → ∞ give weakconsistency and asymptotic normality of the MLE: α̂n →P α and √n(α̂n−α)→D N (0, 2α
ν2

0 (e2αT−1)
)as n →∞. See also Bishwal (2010) for independent sampling case.For H ≥ 0.5, let us consider maximum likelihood estimator (MLE) for the fractional mean-fieldmodel

dXj(t) = αXj(t)dt − β(Xj(t)− X̄n(t))dt + dWH
j (t), Xj(0) = xj(0), j = 1, 2, · · · , n (3.7)

where X̄n(t)) = n−1
∑n
j=1Xj(t), β 6= α, and α 6= 0.The case β = 0 corresponds to sampling independent replications of the same process givenbelow:
dXj(t) = αXj(t)dt + dWH

j (t), j = 1, 2, · · · , n (3.8)First consider the FCIR model
d Yj(t) = a(b − Yj(t))dt + σ

√
Yj(t)dW

H
j (t), j = 1, 2, · · · , n (3.9)

where WH
j (t) is a fractional Brownian motion with Hurst parameter H > 1/2.Then by Proposition 5.7 of Buchmann and Kluppelberg [15], we have

Yj(t) = S(Xj(t)) (3.10)

where
dXj(t) = a(b −Xj(t))dt + dWH

j (t), Xj(0) = S−1(Yj(0)), t ∈ [0, T ], j = 1, 2, · · · , n (3.11)

and S(x) = sgn(x)σ2x2/4. Here S is the state space transform.Let b = 0, σ = 1 and a > 0. Then Xj(t) is described by the Ornstein-Uhlenbeck SDEs
dXj(t) = −aXj(t)dt + dWH

j (t), Xj(0) = S−1(Yj(0)), j = 1, 2, · · · , n (3.12)

Consider the model of n interacting particles of fractional diffusions satisfying the Itô stochasticdifferential equations
dXj(t) =

p∑
l=1

θlµj l(X(t)) + σj(X(t))dWH
j (t), j = 1, 2, · · · , n (3.13)

where X(t) = (X1(t), X2(t), · · · , Xn(t))′ and (WH
j (t); t ≥ 0), j = 1, 2, · · · , n are independentfractional Wiener processes. Here θl(·) ∈ L2([0, T ], dt), l = 1, . . . , p are unknown functions to beestimated based on observation of the process X in the time interval [0, T ]. Let θ = (θ1, θ2, . . . , θp)and µj(x) = (µj1(x), µj2(x), . . . , µjp(x))′. The processes Xj(t), j = 1, 2, · · · , n are observed on

[0, T ].The functions µj , σj ; j = 1, 2, · · · , n are assumed to be known such that the system has a uniquesolution.

https://doi.org/10.28924/ada/ma.4.12


Eur. J. Math. Anal. 10.28924/ada/ma.4.12 11We need the following assumption and results to prove the main results.
(A0) Suppose that bj l := µj l(s)σ−1

j (s); j = 1, 2, · · · , n; l = 1, 2 . . . , p are measurable and adaptedprocesses satisfying
1

n

n∑
j=1

∫ t

0

bj l(s)bjm(s)ds → clm(t) a.s. as n →∞
l , m = 1, 2 . . . , p where clm(t) are finite and continuous nonrandom functions of t ∈ [0, T ]. Thelimiting matrix I(t) = (clm(t))l ,m=1,2...,p is positive definite, δ′I(t)δ is increasing for all δ ∈ Rpand I(0) = 0.In the exchangeable case, (A0) follows from McKean-Vlasov Law of Large Numbers. In particular,(A0) will be satisfied when µj l(X) = µlXj and σj(X) = σ(Xj) which corresponds to the independentreplicated sampling on [0, T ]. See Oelschlager [29].We also need the following version of Rebolledo’s Central Limit Theorem for Martingales, seeRebolledo [34]:
Theorem 3.1 Let Mn, n ∈ Z+ be a sequence of locally square integrable martingales with Mn(0) =

0. Suppose the following condition holds:
∑
s≤t E{|∆Mn(s)|2I(|∆Mn(s)| > ε)} → 0 for all

t ∈ [0, T ], ε > 0; and 〈Mn〉(t)→ c(t) a.s. for all t ∈ [0, T ], where c(t) is a continuous increasing
function with c(0) = 0. Then Mn →D M , a continuous Gaussian martingale with zero mean and
covariance function K(s, t) = c(s ∧ t), s, t ∈ [0, T ] where ∆Ms = Ms −Ms− denotes the jump of
M at the point s.The model is given by

dXj(t) =

p∑
l=1

θlµj l(X(t)) + σj(X(t))dWH
j (t), j = 1, 2, · · · , n (3.14)

where X(t) = (X1(t), X2(t), · · · , Xn(t))′ and (WH
j (t); t ≥ 0), j = 1, 2, · · · , n are independentfractional Wiener processes. Here θ = (θ1, θ2, . . . , θp) is the unknown parameter. The functions

µj l , σj , j = 1, . . . , n; l = 1, . . . , p are assumed to be known such that there exists a unique solution
X(t) to the above SDE.Our aim is to estimate the parameter θ based on n particles q1(·), q2(·), · · · , qn(·) of q(t) on
[0, T ]. We denote this data by qn,T .The Radon-Nikodym derivative (likelihood) is given by

Λθn(qn,T ) := dPθ
dP0

(qn,T ) = exp
{∑p

l=1 θl
∑n
j=1

∫ T
0 µj l(q(t))σ−2

j (q(t))dqj(t)

−1
2

∑p
l=1

∑p
m=1 θlθm

∑n
j=1

∫ T
0 µj l(q(t))σ−2

j (q(t))µjm(q(t))dt
}
.

(3.15)

4. Approximate Maximum Likelihood Estimation

https://doi.org/10.28924/ada/ma.4.12


Eur. J. Math. Anal. 10.28924/ada/ma.4.12 12The approximate maximum likelihood estimator is defined as
θ̂n = arg max

θ
Λθn(qn,T ).Extending Kasonga [24], using McKean-Vlasov Law of Large Numbers and Rebolledo’s CentralLimit Theorem for Martingales, we obtain the consistency and asymptotic normality of the approx-imate maximum likelihood estimator θ̂n which is given below:

Theorem 4.1 Under (A0), we have a) θ̂n →P θ as n →∞.
b)
√
n(θ̂n − θ)→D N (0, I−1(T )) as n →∞ where I(T ) is the Fisher information.

5. Berry-Esseen Inequality

In this section we consider the case H = 0.5 and α = 2, i.e., the standard Brownian motion case.
dXj(t) = fj(θ,X(t)) + σj(X(t))dWj(t), Xj(0) = X0

j , j = 1, 2, · · · , nWe assume the following conditions for j = 1, 2, · · · , n:
(A1) |fj(θ, x)| ≤ aj(θ)(1 + |x |), |fj(θ, x)− fj(θ, y)| ≤ aj(θ)|x − y |.(A2) |fj(θ, x)− fj(φ, y)| ≤ bj(x)|θ − φ| for all θ, φ ∈ Θ, x, y ∈ Rwhere supθ∈Θ |aj(θ)| = a <∞, E|bj(X0

j )|r <∞ for any integer r.(A3) The diffusion process X is stationary and ergodic with invariant measure ν, i.e., for any gjwith E[gj(·)] <∞, 1
n

∑n
j=1

∑m
i=1 gj(Xti )→ Eν [g(X0)] a.s. as n →∞ and h → 0.(A4) supt≥0 E|Xj(t)|r <∞ for all r ≥ 0.(A5) E|fj(θ,X0

j )− fj(θ0, X
0
j )|2 = 0 iff θ = θ0.(A6) fj is twice continuously differentiable function in x for all θ.(A7) fj(·, x) and all its derivatives are three times continuously differentiable with respect to θfor all x ∈ R. Moreover, these derivatives upto third order with respect to θ are of polynomialgrowth in x uniformly in θ.The Fisher information is given by 0 < I(θ) :=

∫∞
−∞(f ′j (θ, x))2dν(x) <∞ and for any δ > 0, orany compact Θ̄ ⊂ Θ,

inf
θ0∈Θ̄

sup
|θ−θ0|>δ

Eθ0
|f ′j (θ,X0)− f ′j (θ0, Xj(0))|2 > 0.

(A8) The Malliavin covariance of the process is nondegenerate.
Let fj = µj/σj , j = 1, 2, · · · , n. The model is given by

dXj(t) =

p∑
l=1

θlµj l(X(t)) + σj(X(t))dWj(t), j = 1, 2, · · · , n.

https://doi.org/10.28924/ada/ma.4.12


Eur. J. Math. Anal. 10.28924/ada/ma.4.12 13The Radon-Nikodym derivative (likelihood) is given by
Lθn(Xn,T ) := dPθ

dP0
(Xn,T ) = exp

{∑p
l=1 θl

∑n
j=1

∫ T
0 µj l(X(t))σ−2

j (X(t))dXj(t)

−1
2

∑p
l=1

∑p
m=1 θlθm

∑n
j=1

∫ T
0 µj l(X(t))σ−2

j (X(t))µjm(X(t))dt
}
.

We observe the process {Xt} at times 0 = t0 < t1 < · · · tm = T with ti − ti−1 = T
m = h, i =

1, 2 · · · , n. We assume equispaced sampling for simplicity with T being fixed, m →∞ and n →∞.The dataset is n particles X1(·), X2(·), · · · , Xn(·) of X(t) on [0, T ]. The approximate log-likelihood based on observations Xj(t1), Xj(t2), . . . , Xj(tn), j = 1, 2, · · · , n with ti = iT/m = ihis defined as
Kn,m(θ) =

∑p
l=1 θl

∑n
j=1

∑m
i=1 µj l(X(ti−1))σ−2

j (X(ti−1))(Xj(ti)−Xj(ti−1)

− 1
2

∑p
l=1

∑p
k=1 θlθk

∑n
j=1

∑m
i=1 µj l(X(ti−1))σ−2

j (X(ti−1)))µjk(X(ti−1))(ti − ti−1).

We start with some preliminary lemmas. The first lemma is from Michel and Pfanzagl (1971)which will be needed to prove our main results.
Lemma 5.1 Let ξ, ζ and η be any three random variables on a probability space (Ω,F , P ) with
P (η > 0) = 1. Then, for any ε > 0, we have

(a) sup
x∈R
|P{ξ + ζ ≤ x} −Φ(x)| ≤ sup

x∈R
|P{ξ ≤ x} −Φ(x)|+ P (|ζ| > ε) + ε,

(b) sup
x∈R
|P{

ξ

η
≤ x} −Φ(x)| ≤ sup

x∈R
|P{ξ ≤ x} −Φ(x)|+ P{|η − 1| > ε}+ ε.

The strong rate of convergence of particle approximations of McKean-Vlasov SDEs with Lipschitzcoefficients is O(n−1/2) where n is the number of particles. This rate is driven by the statisticalerror. The bias is of the order O(n−1). Talay and Tubaro [37] showed that for smooth coefficientsthe the weak error is O(h). Bencheikh and Jourdain [2] showed that weak error between a SDEwith nonlinear in the sense of McKean given by moments and its approximation by the Eulerdiscretization with time step h of a system of n interacting particles is O(n−1 + h).From Talay and Tubaro [37] and Bencheikh and Jourdain [2], we have
Lemma 5.2 Let fj = µj/σj . Then

sup
t∈Π
|E[fj(X

n
t )]− E[fj(X

n,m
t )]| ≤ C

T

m
, j ≥ 1.

The following lemma follows from Yoshida [43,44].
Lemma 5.3 Let In(θ) := 1

nI(θ0)

∑n
j=1

∫ T
0 µ2

j (θ,Xt)dt. Then under the conditions (A1)-(A8),
sup
θ∈Θ

E[In(θ)− 1]2 ≤ Cn−1.

https://doi.org/10.28924/ada/ma.4.12


Eur. J. Math. Anal. 10.28924/ada/ma.4.12 14The following lemma follows from Theorem 1 in Yoshida [44].
Lemma 5.4 Let Mn := 1√

nI(θ0)

∑n
j=1

∫ T
0 µj(θ0, Xt)dWt . Then under the conditions (A1)-(A8),

sup
x∈R
|Pθ0
{Mn ≤ x} −Φ(x)| ≤ Cn−1/2.

In this section, our main result is the following theorem.
Theorem 5.5 Under the conditions (A1)–(A8), we have

sup
x∈R

∣∣∣Pθ0

{√
nI(θ0)(θm,n − θ0) ≤ x

}
−Φ(x)

∣∣∣ = O

(
n−1/2

∨ T

m

)
.

Proof By Taylor expansion, we have
K′m,n(θm,n) = K′m,n(θ0) + (θm,n − θ0)K′′m,n(θ̄m,n)

where ∣∣θ̄m,n − θ∣∣ ≤ |θm,n − θ0|. Since K′m,n(θm,n) = 0, hence we have
√
nI(θ0)(θm,n − θ0) = −

1√
nI(θ0)

K′m,n(θ0)

1
nI(θ0)K

′′
m,n(θ̄m,n)

= −
1√
nI(θ0)

∑n
j=1

∑m
i=1 µ

′
j(θ0, Xti−1

)∆Wi

1
nI(θ0)

∑n
j=1

∑m
i=1 µ

′′
j (θ̄m,n, Xti−1

)∆ti
=:

Um,n
Vm,n

Note that
Vm,n =

1

nI(θ0)

n∑
j=1

m∑
i=1

µ′′j (θ̄m,n, Xti−1
)∆ti =

1

nI(θ0)

n∑
j=1

m∑
i=1

µ′j(θ̄m,n, Xti−1
)2∆ti .

Let lim Vm,n = Vn in L2 as T
m → 0. Similar to Lemma 5.3, it can be shown that E(Vn − 1)2 ≤

Cn−1 (see also Pardoux and Veretennikov (2001) and Yoshida (2011)). It can be shown that
E(Vm,n − Vn)2 ≤ C T

m (see Altmeyer and Chorowski (2018)). Hence
E(Vm,n − 1)2 = E[(Vm,n − Vn) + (Vn − 1)]2 ≤ C(n−1

∨ T

m
).

Further by Lemma 5.1 (b), we have
sup
x∈R

∣∣∣Pθ {√nI(θ)(θm,n − θ) ≤ x
}
−Φ(x)

∣∣∣
= sup

x∈R

∣∣∣∣Pθ {Um,nVm,n
≤ x

}
−Φ(x)

∣∣∣∣
= sup

x∈R
|Pθ {Um,n ≤ x} −Φ(x)|+ Pθ {|Vm,n − 1| ≥ ε}+ ε

≤ C(n−1/2
∨ T 2

m
) + ε−2C(n−1

∨ T

m
) + ε.

https://doi.org/10.28924/ada/ma.4.12


Eur. J. Math. Anal. 10.28924/ada/ma.4.12 15since by Lemma 5.1 (a), Lemma 5.2 and Lemma 5.4, we have
sup
x∈R
|Pθ {Um,n ≤ x} −Φ(x)|

≤ sup
x∈R
|Pθ {Mn ≤ x} −Φ(x)|+ Pθ {|Um,n −Mn| ≥ ε}+ ε

≤ Cn−1/2 + ε−2E |Um,n −Mn|2 + ε

≤ Cn−1/2 + ε−2C
T

m
+ ε.

Choosing ε = n−1/2, we have the result.
Remarks We considered fractional Levy process driving term in this paper whose incrementsare stationary. Using fractional Levy process as the driving term which include jumps, maximumquasi-likelihood estimation in fractional Levy stochastic volatility model was studied in Bishwal [9].Recently, sub-fractional Brownian (sub-FBM) motion which is a centered Gaussian process withcovariance function

CH(s, t) = s2H + t2H −
1

2

[
(s + t)2H + |s − t|2H

]
, s, t > 0

for 0 < H < 1 introduced by Bojdecki, Gorostiza and Talarczyk [14] has received some attentionrecently in finite dimensional models. The interesting feature of this process is that this processhas some of the main properties of FBM, but the increments of the process are nonstationary,more weakly correlated on non-overlapping time intervals than that of FBM, and its covariancedecays polynomially at a higher rate as the distance between the intervals tends to infinity. Itwould be interesting to see extension of this paper to sub-FBM case. We generalize sub-fBM toSub-fractional Levy process (sub-FLP).Sub-fractional Levy process (SFLP) is defined as
SH,t =

1

Γ(H + 1
2 )

∫
R

[(t − s)
H−1/2
+ − (−s)

H−1/2
+ ]dMs , t ∈ R

where Mt , t ∈ R is a Levy process on R with E(M1) = 0, E(M2
1 ) < ∞ and without Browniancomponent. SFLP has the following properties:1) The covariance of the process is given by

Cov(SH,t , SH,s) = s2H + t2H +
E[L(1)2]

2Γ(2H + 1) sin(πH)
[|t|2H + |s|2H − |t − s|2H].

2) SH is not a martingale. For a large class of Levy processes, SH is neither a semimartingalenor a Markov process. 3) SH is Hölder continuous of any order β less than H − 1
2 . 4) SH hasnonstationary increments. 5) SH is symmetric. 6) SH is self similar. 7) SH has infinite totalvariation on compacts.

https://doi.org/10.28924/ada/ma.4.12


Eur. J. Math. Anal. 10.28924/ada/ma.4.12 16It would be interesting to investigate estimation in SPDE driven by subfractional Levy processeswhich incorporate both jumps and long memory apart from nonstationarity.
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