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Parameter Estimation for SPDEs Driven by Cylindrical Stable Processes

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, USACorrespondence: J.Bishwal@uncc.edu

Abstract. We consider infinite dimensional extension of affine models with heavy tails in finance. Westudy several estimators of the drift parameter in the stochastic partial differential equation drivenby cylindrical stable processes. We consider several sampling schemes. We also consider randomsampling scheme, e.g, when the solution process is observed at the arrival times of a Poisson process.We obtain the consistency and the asymptotic normality of the estimators.
1. Introduction

Parameter estimation in stochastic partial differential equations is a very young area of researchin view of its applications in finance, physics, biology and oceanography. Loges [32] initiated thestudy of parameter estimation in infinite dimensional stochastic differential equations. When thelength of the observation time becomes large, he obtained consistency and asymptotic normality ofthe maximum likelihood estimator (MLE) of a real valued drift parameter in a Hilbert space valuedSDE. Koski and Loges [28] extended the work of Loges [32] to minimum contrast estimators. Koskiand Loges [27] applied the work to a stochastic heat flow problem. Martingale estimation functionfor discretely observed diffusions was studied in Bibby and Srensen [2]. Bishwal [6] studied a newestimating function for discretely sampled diffusions by removing the stochastic integral in Girsanovlikelihood. Bishwal [7] contains asymptotic theory on likelihood method and Bayesian method fordrift estimation of finite and infinite dimensional stochastic differential equations. Bishwal [12]studied applications of Levy processes in stochastic volatility models in finance.Huebner, Khasminskii and Rozovskii [23] started statistical investigation in SPDEs. They gavetwo contrast examples of parabolic SPDEs in one of which they obtained consistency, asymptoticnormality and asymptotic efficiency of the MLE as noise intensity decreases to zero under thecondition of absolute continuity of measures generated by the process for different parameters (the
Received: 12 Apr 2022.
Key words and phrases. stochastic partial differential equations; space-time colored noise; cylindrical stable process;stable random field; super levy process; poisson sampling; martingale estimating function; quasi likelihood estimator;stable Ornstein-Uhlenbeck process; stable Black-Scholes model; stable Cox-Ingersoll-Ross model; consistency; asymp-totic normality. 1

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Eur. J. Math. Anal. 10.28924/ada/ma.3.4 2
situation is similar to the classical finite dimensional case) and in the other they obtained theseproperties as the finite dimensional projection becomes large under the condition of singularity ofthe measures generated by the process for different parameters. The second example was extendedby Huebner and Rozovskii [24] and the first example was extended by Huebner [22] to MLE forgeneral parabolic SPDEs where the partial differential operators commute and satisfy differentorder conditions in the two cases.Huebner [21] extended the problem to the ML estimation of multidimensional parameter. Lototskyand Rozovskii [33] studied the same problem without the commutativity condition. Small noiseasymptotics of the nonparmetric estimation of the drift coefficient was studies by Ibragimov andKhasminskii [29].Based on continuous observations, usually there can be two asymptotic settings in SPDE: 1)
T → ∞ 2) n → ∞ where T is the length of the observations and n is the number of Fouriercoefficients of the SPDE solution.In a Bayesian approach, using the first setting, Bishwal [3] proved the Bernstein-von Misestheorem and asymptotic properties of regular Bayes estimator of the drift parameter in a Hilbertspace valued SDE when the corresponding ergodic diffusion process is observed continuously overa time interval [0,T ]. The asymptotics are studied as T → ∞ under the condition of absolutecontinuity of measures generated by the process. Results are illustrated for the example of anSPDE.Using the second setting, Bishwal [5] proved the Bernstein-von Mises theorem and spectralasymptotics of Bayes estimators for parabolic SPDEs when the number of Fourier coefficientsbecomes large. In this case, the measures generated by the process for different parameters aresingular.Bishwal [10] studied Bernstein-von Mises theorem and small noise Bayesian asymptotics for par-abolic stochastic partial differential equations. Bishwal [9] studied hypothesis testing for fractionalstochastic partial differential equations with applications to neurophysiology and finance.In this paper we study the asymptotic properties of the quasi maximum likelihood estimatorwhen we have observations of finite-dimensional projections at Poisson arrival time points. Theasymptotic setting is only the large number of observations at random time points which are thearrivals of a Poisson process.The rest of the paper is organized as follows: Section 2 contains model, assumptions andpreliminaries. In Section 3 we prove estimation results with additive noise. Section 4 and 5, weprovide estimation results with multiplicative noise. In section 6, we give several examples.
2. Model and Preliminaries

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 endowded with the operator norm

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Eur. J. Math. Anal. 10.28924/ada/ma.3.4 3
‖ · ‖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

x2nρ
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‖−1L2(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).We assume that there exists a complete orthonormal system {hi}∞i=1 in L2(G)) such that forevery 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 constant andand the spectrum of the operator Λθ consists of eigenvalues {βi (θ)}∞i=1 of finite multiplicities and
µi = −β2mi + k(θ).A 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[ei〈Zt,s〉] = exp(−tψ(s)), s ∈ H

where 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

(
ei〈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 U = l2ρ ,with a properly chosen weight ρ.Consider the linear SPDE
dXt = θAXtdt + dZt,x ∈ HC.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

where {Zi (t)}∞i=1 are independent, real valued, one dimensional, normalized, symmetric, α-stableprocesses and (γi ) is a given sequence of, possibly unbounded, positive numbers, and hi is a fixed

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Eur. J. Math. Anal. 10.28924/ada/ma.3.4 4
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

γ2i Z
2
i (t)‖hi‖

2
−α =

∞∑
i=1

Z2i (t)β
−2α
i

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

α

.

Stable one-dimensional density :A one-dimensional, normalized, symmetric α-stable distribution µα, α ∈ (0, 2] has characteristicfunction
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 →∞.
Stable measures on Hilbert space :A random variable ξ on H is called α-stable (α ∈ (0, 2]) if for any n there exists avector an ∈ H such that for any independent copies ξ1,ξ2, . . . ,ξn of ξ, the random variable
n−1/α(ξ1 + ξ2, . . . + ξn) −an has the same distribution as ξ. A Borel probability measure µ on His said to be α-stable if it is the distribution of a stable random variable with vales in H.
Stable OU Process:

dXt = −θXtdt + σdZt, X0 = x0The solution is
Xt = e

−θtx0 +

∫ t
0

e−θ(t−s)σdZs.The stochastic integral can be defined as the limit in probability of Riemann sums.Let
Yt =

∫ t
0

e−θ(t−s)σdZs.Then
E[eihYt ] = exp

[
−σα|h|α

∫ t
0

e−αθsds

]
= e−|h|

αcα(t)

where
cα(t) = σ

(
1 −e−αθt

αθ

)
.

We show that the process X is stochastically continuous.First we show that Y is stochastically continuous, i.e.,
lim
h→0+

sup
t≥0

P (|Yt+h −Yt| > �) = 0

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Eur. J. Math. Anal. 10.28924/ada/ma.3.4 5
Note that for any t ≥ 0, h ≥ 0,

Yt+h −Yt =
∫ t+h
t

e(t+h−s)AdZs + e
hA

∫ t
0

e(t−s)AdZs −
∫ t

0

e(t−s)AdZs

= ehAYt −Yt +
∫ t+h
t

e(t+h−s)AdZs

Let us choose p ∈ (0,α). We have
P (|Yt+h −Yt| > �) ≤ P

(
|ehAYt −Yt| >

�

2

)
+ P

(∣∣∣∣∫ t+h
t

e(t+h−s)AdZs

∣∣∣∣ > �2
)

≤ 2p
E|ehAYt −Yt|p

�p
+ 2p

E|
∫h

0
esAdZs|p

�p
= I1(t,h) + I2(h).But

E|Yt|p ≤ cp

(
∞∑
n=1

1 −e−αθt

αθ

)p/α
and so [I2(h)]α/p → 0 as h → 0. Concerning I1, by Khintchine inequality

|ehAYt −Yt| =

∑
n≥1

∣∣(e−θh − 1)Y nt ∣∣2
1/2 ≤ Cp

Ẽ
∣∣∣∣∣∣
∑
n≥1

rn(e
−θh − 1)Y nt

∣∣∣∣∣∣
p1/p .

where Ẽ denotes expectation w.r.t. to the measure P̃ P̃ (rn = 1) = P̃ (rn = 1) = 1/2 where aRademacher sequence (rn) with rn : Ω̃ → {−1, 1} is defined on the probability space (Ω̃,F̃, P̃ ).Hence
E|ehAYt −Yt|p ≤ CppẼE

∑
n≥1

∣∣(e−θh − 1)Y nt ∣∣2
1/2

≤ Cp

∑
n≥1

∣∣(1 −e−θht)βn∣∣α (1 −e−θht)
αθ

p/α ≤ Cp
αp/α

∑
n≥1

∣∣(1 −e−θht)βn∣∣α
θ

p/α
Since

lim
h→0+

∑
n≥1

∣∣(1 −e−θht)βn∣∣α
θ

p/α = 0,
we get

lim
h→0+

sup
t≥0

2p
E|ehAYt −Yt|p

�p
= 0.

Since
E|Yt|p ≤ Cp

∑
n≥1
|βn|α

(1 −e−θht)
αθ

p/α ,
hence

lim
h→0

∑
n≥1
|βn|α

(1 −e−θht)
αθ

p/α = 0

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Eur. J. Math. Anal. 10.28924/ada/ma.3.4 6
hence

lim
h→0+

2p
E|
∫h

0
esAdZs|p

�p
→ 0.

Thus
lim
h→0+

sup
t≥0

I1(t,h) = 0

This proves stochastic continuity of Yt.
Using the stochastic continuity and Ft-adaptedness of X, we conclude that the process X hasa predictable version.

Time Change:

Let L be a one dimensional α-stable process, α ∈ (0, 2). Then there exists an α-stable process,
α ∈ (0, 2) Z = (Zt) such that∫ t

0

e−θsdLs = Z(u(t)) where u(t) = 1 −e−αθt
αθ

.

Recall that u ∈ C∞([0,∞]) with u′(t) 6= 0, t ≥ 0.In the limiting Gaussian case of α = 2, it becomes time change for Brownian motion.
Infinite Dimensional Stable OU Process

dXnt = −θX
n
t dt + σdZ

n
t , X

n
0 = xn,n ∈Nwith x = (xn) ∈ L2 = H. The solution is a stochastic process X = Xxt with values in R∞ withcomponents

Xxt = e
−θtxn +

∫ t
0

e−θ(t−s)σdZns .(The stochastic integral can be defined as the limit in probability of Riemann sums.)
Xxt =

∞∑
n=1

Xnt en = e
tAx + ZA(t)

where
ZA(t) =

∫ t
0

e(t−s)AdZs =

∞∑
n=1

(∫ t
0

e−θ(t−s)σdZns

)
en.

The process Xxt is an Ft-adapted irreducible Markov process and its transition semigroup isstrong Feller.Let
Y nt = Z

n
A(t) =

∫ t
0

e−θ(t−s)σdZns ,n ∈N, t ≥ 0.Then
E[eihY

n
t ] = exp

[
−σα|h|α

∫ t
0

e−αθsds

]
= e−|h|

αcαn (t)

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Eur. J. Math. Anal. 10.28924/ada/ma.3.4 7
where

cn(t) = σ

(
1 −e−αθt

αθ

)1/α
.

It follows that
E[eihY

n
t ] = E[eihcn(t)Ln ], h ∈R

where (Ln) are independent α-stable random variables having the same law µα. Thus Xxt is
Ft-adapted.The Markov property easily follows from the identity

ZA(t + h) −ehAZA(t) =
∫ t+h
t

e(t+h−s)AdZs, t,h ≥ 0.

If the cylindrical Levy process Z takes values in Hilbert space H, the by the Kotelenez regularityresults trajectories of the process X are cadlag with values in H.
Moments of the process

The OU process is stochastically continuous and trajectories in Lp([0,T ]; H) for any 0 < p < αa.s. Set Yt := ZA(t). Then we have
E|Yt|p ≤ c̃pσp

(
∞∑
n=1

1 −e−αθt

αθ

)p/α
where c̃p depends on p.
Moments of the stochastic integral

Suppose (Zt) is an α-stable Levy process with 0 ≤ α ≤ 2 and y(t) is a predictable processsatisfying ∫T
0
|y(t)|αdt < ∞. Then for any 0 < r < α, there exists a constant C such that

E

[
sup
t≤T

∣∣∣∣∫ t
0

y(s)dZs

∣∣∣∣r] ≤ E
[(∫ t

0

|y(t)|αdt
)r/α]

.

Equivalence of Transition Probabilities

Assume
sup
n≥1

e−γntγ
1/α
n

βn
= Ct < ∞, E

∫ T
0

∑
n≥1
|Y nt |

2

p/2 dt < ∞.
Let pα be the density of the one dimensional stable measure. Then the laws µxt and µyt of Xxt and
X
y
t respectively are equivalent for any t > 0, x,y ∈ H,α ∈ (0, 2). Moreover, the density dµxtdµxt of

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Eur. J. Math. Anal. 10.28924/ada/ma.3.4 8
µxt with respect to µyt is given by

dµxt
dµ

y
t

= lim
n→∞

n∏
k=1

pα

(
Zk−e−θtxk

c(t)

)
pα

(
Zk−e−θtyk

c(t)

).
The corresponding MLE is denoted as θ̂n.

Priola et al. [38] 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.First we consider the method of moments estimation in modified tempered stable-Ornstein-Uhlenbeck model. Masuda and Uehara [36] studied two-step estimation in ergodic Levy drivenSDE
dXt = a(θ,Xt)dt + b(β,Xt−)dZt, X0 = x0.Masuda [35] studied multi-step estimation in stable OU Model:

dXt = −θXtdt + σdZt, X0 = x0.

For the least squares estimator (LSE) θ̃n of θ, Hu and Long [20] obtained(
T

log n

)1/α
(θ̃n −θ0) →D

S′α
S′′+
α/2where Sα is stable distribution of order β.While in Gaussian OU case, for different parts θ > 0, θ < 0 and θ = 0, LAN, LAMN and LABFhold respectively (see Bishwal [11]), in stable case entirely different phenomena occur.The solution of the SDE is given by

Xt = e
−θ(t−s)Xs + σ

∫ t
s

e−θ(t−s)dZu,t ≥ 0.

Due to the stable integral property,
L
(∫ t

s

e−θ(t−s)dZu

)
= Sα(κ∆(θ))

where
κ∆(θ) =

{
1 −e−θ∆

θα

}1/α
∼ ∆1/α.

For each j ≤ n, the transition probability is given by
L(Xtj |Xtj−1 = x) = δx exp(−θ∆) ? Sα(κ∆(θ)).

LAMN holds for θ ∈R when T is fixed.
n1/α−1/2(θ̂n −θ) →D MN(0, Iθ(T )−1).

where Iθ(T ) is the Fisher information of the process.

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Eur. J. Math. Anal. 10.28924/ada/ma.3.4 9
We study estimation in MTS-OU SV model. The Inverse Gaussian-OU and Gamma-OU modelsare special cases.An infinitely divisible distribution is said to be α-modified tampered stable distribution (α-MTS)distribution if its Levy triplet is given by

σ2 = 0,

ν(dx) = C

λα+
1
2

+ Kα+ 1
2
λ+x

xα+
1
2

Ix>0 +
λ
α+ 1

2
+ Kα+ 1

2
λ−x

xα+
1
2

Ix<0

dx,
γ = µ + C

(
Γ( 1

2
−α)

2α+
1
2

(λ2α−1+ −λ
2α−1
− ) −λ

α−1
2

+ Kα−1
2

(λ+) + λ
α−1

2
− Kα−1

2
(λ−)

)
where C > 0,λ+,λ− > 0, µ ∈ R, α ∈ (−∞, 1)\{12} and Kp(x) is the modified Bessel functionof second kind. We denote the MTS random variable by X ∼ MTS(α,C,λ+,λ−,µ). The Levymeasure ν(dx) is called the MTS Levy measure with parameter (α,C,λ+,λ−).The MTS distribution is obtained by taking a symmetric α-stable distribution with α ∈ (0, 1)and multiplying by a Levy measure with √|x|λα+ 12 K

α+ 1
2

(λ|x|) on each half of the real axis. Themeasure can be extended to the case α ≤ 0. If α = 1
2
, then γ may not be defined, so it is removed.The MTS distribution was introduced by Kim, Rachev and Chung [25].The tails of the α-MTS distribution are thinner than those of the 2α-stable and fatter (heavier)than those of the 2α-TS distribution. At the zero neighborhood, all three have the same asymptoticbehavior.If λ+ > λ−, then the distribution is skewed to the left. If λ+ < λ−, then the distribution is

skewed to the right. If λ+ = λ−, then the distribution is symmetric.
C controls the kurtosis of the distribution. If C increases, the peakedness of the distributionincreases.As α decreases, the distribution has fatter tails and increased peakedness. The Levy processcorresponding to the MTS distribution has finite activity if α < 0 and infinite activity if α > 0. Ithas finite variation if α < 1

2
and infinite variation if α > 1

2
.With proper choice of C and µ, MTS distribution has zero mean and unit variance, and thedistribution is called standard MTS distribution and denoted X ∼ stdMTS(α,λ+,λ−).CGMY process proposed in Carr et al. [14] is a tempered stable process. In order to obtain aclosed form solution of the European option price, CGMY used the generalised Fourier transformof the distribution of the stock price under the assumption of Markov property.The stochastic volatility model is given by

dYt = (µ + βXt) dt +
√
Xt dWt + ρ dZt

dXt = −θ Xt dt + dZt

where µ is the drift parameter, β is the risk premium, θ > 0 is the drift of the volatility and Zt isa MTS process.

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Eur. J. Math. Anal. 10.28924/ada/ma.3.4 10
We estimate θ from the observations of {Yt} at the time points tk = k∆, k = 0, 1, 2, . . . ,n,

∆ > 0.
cm(Z) :=

dm

dum
log φTS(u)|u=0For the tempered stable distribution TS(b,δ,γ) where 0 < b < 1,δ > 0,γ ≥ 0, the m-th cumulantis given by

cm(Z) = −δ(−2)mγ(b−m)/bb(b− 1) . . . (b− (m− 1))for γ > 0. When γ = 0, it is positive b-stable distribution for which the moments of only order
k < b exist. For b = 1/2, TS distribution reduces to Inverse Gaussian (IG) distribution.The infinite divisibility of this distribution allows one to construct the corresponding Levy process.A Levy process Z = (Zt)t≥0 is said to be a tempered stable process if Z1 follows a temperedstable distribution. The tempered stable process is of finite activity if α < 0 and infinite activity if
0 < α < 2. The tempered stable process is of finite variation if 0 < α < 1 and infinite variation if
1 < α < 2.The MTS-GARCH model is given by

log
St
St−1

= rt −dt + λtσt −g(σt; α,λ+,λ−) + σt�t

σ2t = (α0 + α1σ
2
t−1�

2
t−1 + β1σ

2
t−1) ∧ρ, �0 = 0

α0,α1,β1 ≥ 0, α1 + β1 < 1, 0 < ρ < λ2+, �t ∼ stdMTS(α,λ+,λ−), rt is the risk-free rate, dtis the dividend rate, λt is the market price of risk, g is the characteristic exponent of the Laplacetransform for the distribution stdMTS(α,λ+,λ−), i.e., g(x; α,λ+,λ−) = log(E(exp(x�t)).The characteristic function of Z is given by
φZ(u) = exp(iuµ + GR(u; α,C,λ+,λ−) + GI(u; α,C,λ+,λ−))

where for u ∈R,
GR(u; α,C,λ+,λ−) = 2

−α+3
2
√
πCΓ

(
1 −

α

2

)[
(λ2+ + u

2)
α
2 −λα+ + (λ

2
− + u

2)
α
2 −λα−

]
,

GI(u; α,C,λ+,λ−)

= iuC2−
α+1

2 Γ

(
1 −α

2

)[
λα−1+ F

(
1,

1 −α
2

;
3

2
, ;−

u2

λ2+

)
−λα−1− F

(
1,

1 −α
2

;
3

2
, ;−

u2

λ2−

)]
where F is the hyper-geometric function. The value of GI for symmetric MTS distribution is alwayszero.The m-th cumulant is given by

cm(Z) = µ if m = 1,

cm(Z) = 2
m−α+3

2

(
m− 1

2

)
!CΓ

(
m−α

2

)
(λα−m+ −λ

α−m
− ) if m = 3, 5, 7, . . .

cm(Z) = 2
−α+3

2
√
π

(
m!
m
2

!

)
CΓ

(
m−α

2

)
(λα−m+ + λ

α−m
− ) if m = 2, 4, 6, . . .

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Eur. J. Math. Anal. 10.28924/ada/ma.3.4 11
The mean, variance, skewness and excess kurtosis are given by

E(Z) = c1(Z) = µ + 2
−α+1

2 CΓ

(
1 −α

2

)
(λα−1+ −λ

α−1
− ),

V (Z) = c2(Z) = 2
−α+1

2
√
πCΓ

(
1 −

α

2

)
(λα−2+ + λ

α−2
− ),

s(Z) =
c3(Z)

c2(Z)
3/2

=
2
α+9

4 Γ
(

3−α
2

)
(λα−3+ −λ

α−3
− )

π3/4C1/2(Γ( 1−α
2

)(λα−2+ + λ
α−2
− ))

3/2
,

κ(Z) =
c4(Z)

c2(Z)
2

=
3 · 2

α+3
2 CΓ

(
2 − α

2

)
(λα−4+ + λ

α−4
− )√

πC(Γ( 1−α
2

)(λα−2+ + λ
α−2
− ))

2
.

If α ∈ (0, 2)\{1}, the Levy measure of α-stable, α-TS and α-MTS have the same asymptotic behav-ior at the zero neighborhood. However, the tails of the Levy measures for the α-MTS distributionare thinner than those of α-stable and heavier than those of α-TS distribution.When Z is a IG process, the moment estimators of ρ and θ are given by
θ̂n :=

γȳ

∆δρ̂n
, ρ̂n :=

γ(γs2y − ∆δ)
2ȳwhere

ȳ :=
1

n

n∑
j=1

yj, yj := Yj∆ −Y(j−1)∆,

s2y :=
1

n

n∑
j=1

(yj − ȳ)2 =
1

n

n∑
j=1

y2j − (ȳ)
2.

When Z is a Gamma process, the moment estimators are given by
θ̂n :=

1
n2

[∑n
i=1(Yi∆ −Y(i−1)∆)

]2
1
n2

∑n
i=1(Yi∆ −Y(i−1)∆)2 −

∆
n

[∑n
i=1(Yi∆ −Y(i−1)∆)

] 2a3(a + 1)
b4∆

,

ρ̂n :=
1
n2

∑n
i=1(Yi∆ −Y(i−1)∆)

2 − ∆
n

[∑n
i=1(Yi∆ −Y(i−1)∆)

]
1
n2

[∑n
i=1(Yi∆ −Y(i−1)∆)

] b3∆
2a2(a + 1)

.

For the MTS-OU model, the estimating functions are given by
c1(y1) = λρ∆c1(Z),

c2(y1) = ∆c1(Z) + 2λρ
2∆c1(Z),

c3(y1) = ∆c1(Z) + 2λρ
2∆c2(Z),

c4(y1) = ∆c1(Z) + 2λρ
2∆c3(Z)which give

E(y1) = c1(y1) = µ + 2
−α+1

2 CΓ

(
1 −α

2

)
(λα−1+ −λ

α−1
− ),

V (y1) = c2(y1) = 2
−α+1

2
√
πCΓ

(
1 −

α

2

)
(λα−2+ + λ

α−2
− ).This gives the moment estimators for the SOU model

θ̂n :=
1
n2

[∑n
i=1(Yi∆ −Y(i−1)∆)

]2
1
n2

∑n
i=1(Yi∆ −Y(i−1)∆)2 −

∆
n

[∑n
i=1(Yi∆ −Y(i−1)∆)

]

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Eur. J. Math. Anal. 10.28924/ada/ma.3.4 12
× [2−

α+1
2 CΓ

(
1 −α

2

)
(λα−1+ −λ

α−1
− )]

2[2−
α+1

2
√
πCΓ

(
1 −

α

2

)
(λα−2+ + λ

α−2
− )]2∆

−1.

ρ̂n :=
1
n2

∑n
i=1(Yi∆ −Y(i−1)∆)

2 − ∆
n

[∑n
i=1(Yi∆ −Y(i−1)∆)

]
1
n2

[∑n
i=1(Yi∆ −Y(i−1)∆)

]
× [2−

α+1
2 CΓ

(
1 −α

2

)
(λα−1+ −λ

α−1
− )2

−α+1
2
√
πCΓ

(
1 −

α

2

)
(λα−2+ + λ

α−2
− )]

−12−1∆.

Let ϑ = (ρ,θ) and ϑ̂n = (ρ̂n, θ̂n). By using Theorem 2.2 in Masuda [34] (see also Theorem 4.1 Vander Vaart [41]), we obtain the strong consistency and asymptotic normality of the MM estimators:
Proposition 2.1 For fixed ∆ > 0 as n →∞,

(a) ϑ̂n → ϑ0 a.s. as n →∞.
(b)

√
n(ϑ̂n −ϑ0) →D N2(0, (J−1(ϑ0)) as n →∞.

where J(ϑ0) is the Fisher information.
3. SPDEs with Additive Noise

Consider the parabolic SPDE
duθ(t,x) = θuθ(t,x) +

∂2

∂x2
uθ(t,x)dt + dZ(t,x), t ≥ 0, x ∈ [0, 1] (3.1)

u(0,x) = u0(x) ∈ L2([0, 1]), (3.2)

uθ(t, 0) = uθ(t, 1), t ∈ [0,T ]. (3.3)

Here θ ∈ Θ ⊆ R is the unknown parameter to be estimated on the basis of the observations ofthe field uθ(t,x),t ≥ 0, x ∈ [0, 1].Let S3 and S4 be independent stable random variables, S3 is positive α/2-stable with distri-bution Sα/2(σ1, 1, 0) and S4 is symmetric α-stable random variable with distribution Sα(σ2, 0, 0),
σ1 = C

−2/α
α/2

, σ2 = C
−1/α
α , Cα = (

∫∞
0
x−α sin xdx)−1 = [Γ(1 −α) cos(πα/2)]−1.

In this case, in the limiting distribution, S3 and S4 are independent stable random variables witha rate faster than the cylindrical Brownian motion case.For x ∈ [0, 1], we observe the process {ut,t ≥ 0} at times {t0,t1,t2, . . .}. We assume thatthe sampling instants {ti, i = 0, 1, 2, . . .} are generated by a Poisson process on [0,∞), i.e.,
t0 = 0,ti = ti−1 + αi, i = 1, 2, ... where αi are i.i.d. positive random variables with a commonexponential distribution F (x) = 1 − exp(−λx). Note that intensity parameter λ > 0 is theaverage sampling rate which is assumed to be known. It is also assumed that the sampling process

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Eur. J. Math. Anal. 10.28924/ada/ma.3.4 13
ti, i = 0, 1, 2, ... is independent of the observation process {Xt,t ≥ 0}. We note that the probabilitydensity function of tk+i − tk is independent of k and is given by the gamma density

fi (t) = λ(λt)
i−1 exp(−λt)It/(i − 1)!, i = 0, 1, 2, .... (3.4)

where It = 1 if t ≥ 0 and It = 0 if t < 0.Consider the Fourier expansion of the process
uθ(t,x) =

∞∑
t=1

uθi (t)φi (x) (3.5)

corresponding to some orthogonal basis {φi (x)}∞i=1. Note that uθi (t), i ≥ 1 are independent onedimensional stable Ornstein-Uhlenbeck processes
duθi (t) = µ

θ
i u
θ
i (t)dt + β

−α
i
dZi (t) (3.6)

uθi (0) = u
θ
0i,Recall that µi (θ) = k(θ) −β2mi . Thus

duθi (t) = (k(θ) −β
2m
i )u

θ
i (t)dt + β

−α
i
dZi (t) (3.7)

The random field u(t,x) is observed at discrete times t and discrete positions x. Equivalently, theFourier coefficients uθi (t) are observed at discrete time points.Define
ρ := ρ(λ,θ) =

λ

λ−κ(θ) + β2m
i

.

The quasi-likelihood estimator is the solution of the estimating equation:
G∗n(θ) = 0 (3.8)

where
G∗n(θ) =

β2αi λ(ρ(λ,θ))
2

ρ(λ, 2θ)

n∑
i=1

uti−1
(

(uti−1θρ(λ,θ))
2 + λ

)−1
(uti −ρ(λ,θ)uti−1 ). (3.9)

We call the solution of the estimating equation the quasi-likelihood estimator. There is no explicitsolution for this equation.The optimal estimating function for estimation of the unknown parameter θ is
Gn(θ) = β

2α
i

n∑
i=1

uti−1 [uti −ρ(λ,θ)uti−1 ]. (3.10)

The martingale estimation function (MEF) estimator of ρ is the solution of
Gn(θ) = 0 (3.11)and is given by

ρ̂n :=

∑n
i=1 uti−1uti∑n
i=1 u

2
ti−1

.

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Eur. J. Math. Anal. 10.28924/ada/ma.3.4 14

We do the parameter estimation in two steps: The rate λ of the Poisson process can be estimatedgiven the arrival times ti , therefore it is done at a first step. Since we observe total number ofarrivals n of the Poisson process over the T intervals of length one, the MLE of λ is given by
λ̂n :=

n

T
.

Theorem 3.1 We have
λ̂n → λ a.s. as n →∞.

√
n(λ̂n −λ) →D N(0, eλ(1 −e−λ)) as n →∞.

Proof. Let Vi be the number of arrivals in the interval (i − 1, i]. Then Vi, i = 1, 2, . . . ,n are i.i.d.Poisson distributed with parameter λ. Since Φ is continuous, we have I{0}(Vi ) = I{0}(u(ti )) a.s. i =
1, 2, . . . ,n. Note that

1

n

n∑
i=1

I{0}(uti ) →
a.s. E(I{0}V1) = P (V1 = 0) = e

−λ as n →∞.
LLN and CLT and delta method applied to the sequence I{0}(uti ), i = 1, 2, . . . ,n give the results.

The CLT result above allows us to construct confidence interval for the jump rate λ.
Corollary 3.1 A 100(1 −α)% confidence interval for λ is given by[

n

T
−Z1−α

2

√
1

n
−

1

T
,
n

T
+ Z1−α

2

√
1

n
−

1

T

]
where Z1−α

2
is the (1 − α

2
)-quantile of the standard normal distribution.

We obtain the strong consistency and asymptotic normality of the MEF estimator.
Theorem 3.2 When α = 2, we have

ρ̂n →a.s. ρ as n →∞
√
n(ρ̂n −ρ) →D N(0, λ−i (1 −e−ρ)) as n →∞.

Proof: By using the fact that every stationary mixing process is ergodic, it is easy to show that if utis a stationary ergodic O-U Markov process and ti is a process with nonnegative i.i.d. incrementswhich is independent of ut, then {uti, i ≥ 1} is a stationary ergodic Markov process. Hence
{uti, i ≥ 1} is a stationary ergodic Markov process.

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Eur. J. Math. Anal. 10.28924/ada/ma.3.4 15
Observe that uθi (t) := vi is a stationary ergodic Markov chain and vi ∼ N(0,σ2) where σ2 isthe variance of u0. Thus by SLLN for zero mean square integrable martingales, we have

1

n

n∑
i=1

uti−1uti →
a.s. E(ut0ut1 ) = ρE(u

2
t0

)

1

n

n∑
i=1

u2ti−1 →
a.s. E(u2t0 )

Thus ∑n
i=1 uti−1uti∑n
i=1 u

2
ti−1

→a.s. ρ.

Further,
√
n(ρ̂n −ρ) =

n−1/2
∑n
i=1 uti−1 (uti −θuti−1 )
n−1

∑n
i=1 u

2
ti−1

.

Since
E(ut1ut2|ut1 ) = θu

2
t1it follows by Lemma 3.1 in Bibby and Srensen [2]

n−1/2
n∑
i=1

uti−1 (uti −θuti−1 )

converges in distribution to normal distribution with mean zero and variance equal to
E[(ut1ut2 ) −E(ut1ut2|ut1 )]

2 = 1 −e2(θ−βiδ){2(βi −θ)(βi + 1)}−1.

Applying delta method the result follows.
In the next step, we use the estimator of λ to estimate θ.Note that

1

ρ̂n
=

∑n
i=1 u

2
ti−1∑n

i=1 uti−1uti
.

Thus
1 +

β2m1 −κ(θ)
λ

=

∑n
i=1 u

2
ti−1∑n

i=1 uti−1utiwhich gives
β2m1 −κ(θ)

λ
=

∑n
i=1 u

2
ti−1∑n

i=1 uti−1uti
− 1 = −

∑n
i=1 uti−1 [uti −uti−1 ]∑n

i=1 uti−1utiNow replace λ by its estimator MLE λ̂n.
β2m1 −κ(θ) = −

∑n
i=1 uti−1 [uti −uti−1 ]
T
n

∑n
i=1 uti−1utiThus

θ̂n = κ
−1

(
β2mi +

∑n
i=1 uti−1 [uti −uti−1 ]
T
n

∑n
i=1 uti−1uti

)
.

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Eur. J. Math. Anal. 10.28924/ada/ma.3.4 16
Since the function κ−1(·) is a continuous function, by application of delta method, the followingresult is a consequence of Theorem 3.2.
Theorem 3.3 When α = 2,

a) θ̂n → θ a.s. as n →∞
b)
√
n(θ̂n −θ) →D N(0, (κ′(θ))−2λ2(1 −e−2λ

−1(κ(θ)−β2m1 ))) as n →∞.
In the second stage, we substitute λ by its estimator λ̂n.

Theorem 3.4 When 0 < α < 2,
a) θ̂n →a.s. θ as n →∞

b) n(α−1)/α
2

(θ̂n −θ) →D
(
κ′(θ))−2λ2(1 −e−2λ

−1(κ(θ)−β2m1 )
)1/α S4

S3
as n →∞.

where S4 and S3 are independent stable random variables.
In the second stage, we substitute λ by its estimator λ̂n. The limit distribution is normal onlyin the Gaussian case α = 2.

4. SPDEs with Linear Multiplicative Noise

Consider the SPDE with multiplicative noise:
duθ(t,x) = (A0 + θA1)u

θ(t,x)dt + Muθ(t,x)dZ(t,x), t ≥ 0, x ∈ [0, 1] (4.1)

where M is a known linear operator.Equation (4.1) 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.

The Fourier coefficients have the dynamics
duk(t) = (θνk + ρk)uk(t)dt + σkuk(t)dZk(t), k ≥ 1

which is the Stable Black-Scholes Model whose solution is geometric stable process.Let
θνk + ρk =: µk(θ), ṽk,T := ln(uk,T/uk,0).Conditional characteristic function (CCF) estimator is given by
µ̂k(θ) =

ṽk,T

T 2(α−1)/α
2 +

σ2k
b1T

−((α−1)2+1)/α2 .

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Eur. J. Math. Anal. 10.28924/ada/ma.3.4 17
Since µk(θ) is strictly monotone function of θ, by invariance principle of CCFE, under invertibletransformations, we can find the CCFE of the parameter θ

θ̂k,T =
ṽk,T

νkT
2(α−1)/α2 +

σ2k
b1νkT

−((α−1)2+1)/α2 −
ρk
νkwhich can be represented as

θ̂k,T = θ0 +
σkMT

νkT
2(α−1)/α2where MT is a square-integrable martingale. Due to the LLN for martingales, we have strongconsistency.Note that in the standard Black-Scholes case where α = 2, σk = σ, the MLE of the driftcoefficient of the geometric Brownian motion is given by

θ̂T =
ln(uT/u0)

T
+
σ2

2
= θ0 + σ

Wt
T
.

Due to the law of iterated logarithm for Brownian motion, the MLE is strongly consistent as T →∞.
Theorem 4.1 When 0 < α < 2,

a) θ̂k,T is an unbiased estimator of θ.b)
θ̂k,T → θ a.s. as T →∞.c)

T (α−1)/α
2

(θ̂k,T −θ) →D
(
σ2k
ν2
k

)1/α
S4
S3

as T →∞
where S4 and S3 are independent stable random variables.d) If in addition,

lim
k→∞

∣∣∣∣σkνk
∣∣∣∣ = 0,then for every fixed T > 0,

θ̂k,T → θ a.s. as k →∞and ∣∣∣∣νkσk
∣∣∣∣ (θ̂k,T −θ) →D (T (α−1)/α2)1/α S4S3 as k →∞.

Remark: The parabolicity condition and the MLE consistency condition in general are notconnected. In terms of operator’s order, parabolicity states that the order of operator M from thediffusion term is smaller than half of the order of operators A0 and A1 from deterministic part.The consistency condition assumes that the order of the operator M from the diffusion part doesnot exceed the order of the operator A1 from the deterministic part that contains the parameter ofinterest θ.

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Eur. J. Math. Anal. 10.28924/ada/ma.3.4 18

5. SPDEs with Nonlinear Multiplicative Noise

Consider the SPDE with multiplicative noise:
duθ(t,x) = (A0 + θA1)u

θ(t,x)dt + Muθ(t,x)dZ(t,x), t ≥ 0, x ∈ [0, 1] (5.1)

where M is a known nonlinear operator.Equation (5.1) 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.

We consider stable CIR model as example. Here S1 and S2 are dependent stable randomvariables unlike the linear case where S3 and S4 are independent stable random variables.
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 [17].Consider the nonlinear SPDE

dX(t,x) =
θ

2
Xxx (t,x)dt +

√
X(t,x)dW (t,x)

where W (t,x) is a space-time white noise. Konno and Shiga [26] studied the existence and weakuniqueness of the above equation as a martingale problem for the associated super-Brownian mo-tion. 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. [42]studied proved a comparison theorem and showed that the solution of the nonlinear SPDE is distri-bution function valued. They also established pathwise uniqueness. As application they obtainedwell-posedness of martingale problems for two classes of measure-valued diffusions: interactingsuper-Brownian motions and interacting Fleming-Viot processes. He et al. [18] obtained pathwiseunique solution to nonlinear SPDE with super Levy process, which is a combination of space-time Gaussian white noises and Poisson random measures which is a generalization of work ofXiong [43] where the result for a super-Brownian motion with binary branching mechanism wasobtained. Using an extended Yamada-Watanabe argument, Xiong [43] established strong exis-tence and uniqueness of the solution to the SPDE. Super-Brownian motion (SMB), also calledthe Dawson-Watanabe process introduced by Sawson and Watanabe is a measure valued processarising as the limit of empirical measure process of a branching particle system. SBM satisfiesa martingale problem. When the state space is R, SBM has a density w.r.t. Lebesgue measureand this density valued process X(t,x) satisfies the above SPDE. When the space R is s single

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Eur. J. Math. Anal. 10.28924/ada/ma.3.4 19
point, the SPDE becomes an SDE which is CIR diffusion dXt = √XtdWt whose uniqueness isestablished using the Yamada-Watanabe argument. Xiong and Yang (2019) studied existence andpathwise uniqueness to an SPDE with Hölder continuous coefficient driven by α-stable colorednoise. The existence of the solution is shown by considering the weak limit of a sequence of SDEsystem which is obtained by replacing the laplacian operator in the SPDE by its discrete version.The pathwise uniqueness is shown by using a backward doubly stochastic differential equation totake care of the Laplacian. In the case of d = 1, the pathwise uniqueness of a nonnegative solutionto the corresponding equation was established by Yang and Zhou [45] for 1 < α < √5 − 1 andpathwise uniqueness for √5 − 1 < α < 2 is still open.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 (5.2)

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
. (5.3)

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. (5.4)

Applying Itô’s formula, for t ≥ r ≥ 0, we obtain
uj,t = e

−θ(t−r)uj,r + a

∫ t
r

e−θ(t−s)ds + σ

∫ t
r

e−θ(t−s)u
1/α
j,s−dZj,s, j ≥ 1. (5.5)

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

uj,k = ρ + γuj,k−1 + �j,k, j ≥ 1 (5.6)

where γ = e−θ, ρ = aθ−1(1 −γ) and
�j,k = σ

∫ k
k−1

e−θ(k−s)u
1/α
j,s−dZj,s, k ≥ 1, j ≥ 1. (5.7)

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Eur. J. Math. Anal. 10.28924/ada/ma.3.4 20
For B ∈B(R+), let

S2,j,n(B) =

n∑
k=1

uj,k−1�kIB(|uj,k−1�j,k|), S1,j,n(B) =
n∑
k=1

u2j,k−1IB(uj,k−1), j ≥ 1. (5.8)

It is easy to see that
�j,k = uj,k −E(uj,k|Fk−1), k ≥ 1, j ≥ 1. (5.9)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

(5.10)

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

�2j,k =

n∑
k=1

[uj,k −E(uj,k|Fk−1)]2 =
n∑
k=1

[uj,k −ρ−γuj,k−1]2 (5.11)

and are given by
γ̂j,n =

∑n
k=1 uj,k−1

∑n
k=1 uj,k −n

∑n
k=1 uj,k−1uj,k

(
∑n
k=1 uj,k−1)

2 −n
∑n
k=1 u

2
j,k−1

,

ρ̂j,n =
1

n

n∑
k=1

uj,k − γ̂n
1

n

n∑
k=1

uj,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{iλ1y2 + 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

}
(5.12)

and
Vj,k := σ

∫ k
k−1

e−θ(k−s)e−θ(s−k+1)/αdZj,s, k = 1, 2, j ≥ 1 (5.13)

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.

Following Li and Ma [31] it can be shown that for every fixed j, if we have 1 < α < (1 +√5)/2,then we have as n →∞
(d−2n S1,j,n,c

−1
n S2,j,n)

D→(S1,S2) on R2

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Eur. J. Math. Anal. 10.28924/ada/ma.3.4 21
where dn = n1/α and cn = n(α+1)/α2 = d(α+1)/αn .For the stable SPDE model, we have the following result on the consistency and the limitdistribution of the CLSE:
Theorem 5.1 If we have 1 < α < (1 + √5)/2, then for every fixed j ≥ 1a)

θ̂j,n →P θ as n →∞.b)
n(α−1)/α

2

(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,T →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 (5.12).
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(·).
6. Examples

(a) Consider the linear stochastic heat equation with additive noise
du(t,x) = θuxx (t,x)dt + dZ(t,x)

for 0 ≤ t ≤ T and x ∈ (0, 1) and θ > 0 with periodic boundary conditions.Here 2m = m1 = 2 and µj = −θπ2j2,γ > 1/2. The eigenfunctions are hj(x1, . . . ,xn) =
(
√

2/π)d(sin(n1x1), . . . , sin(ndxd)), x = (x1, . . . ,xn) ∈ Rd, j = (n1, . . . ,nd) ∈ Nd. The corre-sponding eigenvalues are −νj where νj = (n21 + . . . + n2d).As n →∞,h → 0, nh1+α/ log n → 0, nh2α−1 log n →∞, nh2−α/2+ρ →∞ for some ρ > 0,(
n

log n

)1/α
h1/α(θ̂n −θ0) →D 2θ0(αθ0)−1/α

S4
S3

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Eur. J. Math. Anal. 10.28924/ada/ma.3.4 22
where S3 and S4 are independent stable random variables, S3 is positive α/2-stable with distri-bution Sα/2(σ1, 1, 0) and S4 is symmetric α-stable random variable with distribution Sα(σ2, 0, 0),
σ1 = C

−2/α
α/2

, σ2 = C
−1/α
α , Cα = (

∫∞
0
x−α sin xdx)−1 = [Γ(1 −α) cos(πα/2)]−1.

Observe the rate of convergence (nh)1/α(log n)−1/α = (T )1/α(log n)−1/α. For α = 2, this rate is
T 1/2(log n)−1/2.
(b)Consider the linear stochastic heat equation with multiplicative noise

du(t,x) = θuxx (t,x)dt + u(t,x)dZ(t,x)

for 0 ≤ t ≤ T and x ∈ (0, 1) and θ > 0 with zero boundary conditions and nonzero initial value
u(0) ∈ L2(0, 1). Here A1 is the Laplace operator on (0, 1) with zero boundary conditions that hasthe eigenfunctions hk(x) = √2/π sin(kx), k > 0 and the eigenvalues νk = −k2,ρk = 0, σk =
1,k > 0.

uk(t) =

∫ 1
0

hk(x)u(t,x)dx,

duk(t) = (θνk + ρk)uk(t)dt + σkuk(t)dZk(t).Recall that
ṽk,T := ln(uk,T/uk,0).The CCFE has the form
θ̂k,T =

ṽk,T − 1
k2

.

(c) Consider the following SPDE
du(t,x) = [∆u(t,x) + θu(t,x)]dt + (1 − ∆)ru(t,x)dZ(t,x).

In this case A0 = ∆,A1 = I,M = (1 − ∆)r with the eigenvalues νk = 1,ρk = σk,µk = (1 + σk)r .It has a unique solution for any r ≤ 1/2.The CCFE has the form̂
θk,T =

ṽk,T

k2T 2(α−1)/α
2 −

(1 −σk)2r

k2T−((α−1)
2+1)/α2

−
1

σk(d) Stable Cox-Ingersoll-Ross Model
Xiong and Yang [44] studied existence 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 consideringthe weak limit of a sequence of SDE systems which is obtained by replacing the Laplacianoperator in the SPDE by its discrete version. The weak uniqueness follows from the uniqueness

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Eur. J. Math. Anal. 10.28924/ada/ma.3.4 23
of solution to the martingale problem for the associated super-Brownian motion. In the case of α-stable noise the existence and pathwise uniqueness of the solution is studied in Xiong and Yang [44].
Concluding Remark We considered Levy process driving term in this paper. Using fractionalLevy process as the driving term, maximum quasi-likelihood estimation in fractional Levy stochasticvolatility model was studied in Bishwal [8]. Recently, sub-fractional Brownian (sub-FBM) motionwhich is a centered Gaussian process with covariance function

CH(s,t) = s
2H + t2H −

1

2

[
(s + t)2H + |s − t|2H

]
, s,t > 0

for 0 < H < 1 introduced by Bojdecki, Gorostiza and Talarczyk [13] 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(M21 ) < ∞ and without Browniancomponent. SFLP has the following properties:1) The covariance of the process is given by
Cov(SH,t,SH,s) = s

2H + 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 − 12 . 4) SH hasnonstationary increments. 5) SH is symmetric. 6) SH is self similar. 7) SH has infinite totalvariation on compacts.It would be interesting to investigate QML estimation in SPDE driven by subfractional Levyprocesses which incorporate both jumps and long memory apart from nonstationarity.

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