Reweighted Nadaraya-Watson estimation of jump-diffusion models

SCIENCE CHINAMathematics

. ARTICLES . May 2012 Vol. 55 No. 5: 1005–1016

doi: 10.1007/s11425-011-4340-4

c© Science China Press and Springer-Verlag Berlin Heidelberg 2011 math.scichina.com www.springerlink.com

Reweighted Nadaraya-Watson estimationof jump-diffusion models

HANIF Muhammad, WANG HanChao & LIN ZhengYan∗

Department of Mathematics, Zhejiang University, Hangzhou 310027, China

Email: [email protected], [email protected], [email protected]

Received June 23, 2010; accepted July 12, 2011; published online December 30, 2011

Abstract In this paper, we study the nonparametric estimation of the second infinitesimal moment by using

the reweighted Nadaraya-Watson (RNW) approach of the underlying jump diffusion model. We establish strong

consistency and asymptotic normality for the estimate of the second infinitesimal moment of continuous time

models using the reweighted Nadaraya-Watson estimator to the true function.

Keywords continuous time model, Harris recurrence, jump-diffusion model, local time, nonparametric esti-

mation, RNW estimator

MSC(2010) 60J60, 62G15

Citation: Hanif M, Wang H C, Lin Z Y. Reweighted Nadaraya-Watson estimation of jump-diffusion models. Sci

China Math, 2012, 55(5): 1005–1016, doi: 10.1007/s11425-011-4340-4

1 Introduction

Recently, the nonparametric approach has become popular in estimating the continuous-time models in

finance by the availability of large data sets. Nonparametric estimation based on continuous sampling

observations has been considered in the literature for many years. However, continuous-time sampling

is difficult to achieve in practice. It is therefore natural that estimations concerned with discrete time

observations have been considered, and some progress has been made in both parametric and nonpara-

metric estimations. In nonparametric estimation based on discrete-time observations, [11] first proposed

an estimator for the diffusion coefficient as a regression problem. Since then, some authors have discussed

this problem under weaker conditions.

Consider a continuous-time Ito diffusion model defined by the following stochastic differential equation:

dXt = μ(Xt)dt+ σ(Xt)dWt, (1.1)

where the functions μ(x) and σ2(x) are the drift function and diffusion function, respectively, and {Wt, 0 �t � T } is a standard Brownian motion. This time homogeneous diffusion model is widely used to

describe properties of underlying economic variables. [17] and [24] considered the drift and diffusion

functions as the conditional moments, and employed a kernel estimator to estimate these functions

under a stationarity assumption. An extension has been provided by [5], who used Stanton’s method to

establish the asymptotic theory of the estimator without assuming stationarity. [20] studied nonparametric

estimation for both drift and diffusion functions under a recurrence assumption. [10] studied the local

∗Corresponding author

1006 Hanif M et al. Sci China Math May 2012 Vol. 55 No. 5

linear fitting under a strict stationarity assumption. The diffusion process with jumps is more complex,

since the types of jumps in the models vary. [4] established the asymptotic normality by considering the

functional estimation of the jump diffusion models.

Recently, some effective statistical inference methods are applied to study the diffusion model and jump-

diffusion models. [25] and [26] applied the empirical likelihood method and the reweighted kernel method

to the recurrent diffusion processes. The empirical likelihood method can be used to construct more

effective confidence intervals. [18] considered the empirical likelihood inference for jump-diffusion models.

The reweighted kernel method is an effective method to construct a better point estimator. Although local

linear fitting possesses simple bias representation and corrects the boundary bias automatically, it may

produce a negative result to estimate the nonnegative quantity. The reweighted nonparametric estimator

based on Nadaraya-Watson estimation can improve local linear fitting. It possesses the bias properties of

the local linear estimator and is guaranteed to be nonnegative in finite samples. [26] considered the RNW

estimator of the second infinitesimal moment for diffusion processes. In this paper, we study the RNW

estimator of the second infinitesimal moment of jump-diffusion models. We obtain strong consistency

and asymptotic normality of the estimator.

The paper is structured as follows. In Section 2, we introduce the jump diffusion model and RNW

estimation. In Section 3, we concentrate on assumptions and some useful preliminary results about the

cadlag local time of semimartingales. In this section we also explore the strong consistency and asymptotic

normality of the estimator under study. Technical proofs are confined to Section 4.

2 Jump-diffusion model and RNW estimator

The stochastic processes with jumps are becoming an important issue (see [2, 6, 8]). Many statisticians

have studied the statistical inference for models with jumps (see [1,4]). Some of them discussed diffusion

processes with jumps. We consider a time-homogeneous Markov process Xt defined by the equation

dXt =

[μ(Xt−)− λ(Xt−)

∫Y

c(Xt−, y)Π(dy)]dt+ σ(Xt−)dWt + dJt, (2.1)

where {Wt, t � 0} is a standard Brownian motion and {Jt, t � 0} is a jump process independent of

{Wt, t � 0}; μ(x) and σ(x) are smooth functions, λ(x) is the conditional intensity of the jumps, ξ is a

random variable with range Y and Π(y) is the distribution function of ξ. Denoting

ΔXt := Xt −Xt−, dJt = ΔXt =

∫Y

c(Xt−, y)N(dt, dy), (2.2)

and

ν(ds, dy) = N(dt, dy)− E(N(dt, dy)) = N(dt, dy)− λ(Xt−)Π(dy)dt, (2.3)

where N is a Poisson counting measure with independent increments. The integral form is as follows,

Xt+Δ = Xt +

∫ t+Δ

t

μ(Xs−)ds+∫ t+Δ

t

σ(Xs−)dWs +

∫ t+Δ

t

∫Y

c(Xt−, y)ν(ds, dy). (2.4)

The following relation is from (2.2) and (2.3),

∫ t+Δ

t

∫Y

c(Xs−, y)ν(ds, dy) =∫ t+Δ

t

dJs −∫ t+Δ

t

λ(Xs−)EY [c(Xs−, ξ)]ds. (2.5)

For further details see [21]. Such adjustment ensures that we can construct some local martingales which

are based on the compensated measure in (2.3), and the nature of local martingales will be heavily used

in the proof of our results.

Since the emergence of jumps, the statistical inference for the jump-diffusion model has become more

complex. [4] studied the Nadaraya-Watson estimator of infinitesimal moments for the jump-diffusion

Hanif M et al. Sci China Math May 2012 Vol. 55 No. 5 1007

model. [18] considered the empirical likelihood inference of infinitesimal moments. The infinitesimal

conditional moments can be expressed in terms of the functions μ(·), σ(·), c(·, y) and λ(·). Note discrete

observations of Xt : XiΔ, i = 1, . . . , n, where Δ = T/n is the frequency of observations and T is the time

span. In this paper, we discuss the high frequency case, Δ → 0. We have

M1(x) := limΔ→0

1

ΔE[Xt+Δ −Xt|Xt = x] = μ(x), (2.6)

M2(x) := limΔ→0

1

ΔE[(Xt+Δ −Xt)

2|Xt = x] = σ2(x) + λ(x)EY [c2(x, ξ)] (2.7)

(see [4]). These functions are infinitesimal moments of the jump-diffusion model. Based on these relations,

we can deal with the statistical inference of jump-diffusion models as a regression problem. We shall

discuss (2.7).

[14] introduced the reweighting idea in bootstrap techniques. [7] used this method to estimate the mean

function and the variance function, respectively, for discrete-time stationary mixing time series. [16]

employed this method in the distribution function estimation. [13] used a similar reweighting idea to

improve general kernel-type estimators.

The RNW estimator of the diffusion function M2(·) at a spatial point x is computed subject to a

discrete bias-reducing moment condition satisfied by the local linear estimator. The RNW estimator

is asymptotically equivalent to the local linear estimator and ensures the nonnegativity condition of the

diffusion function (see [26]). [15] pointed out that local linear fitting may assign negative weights to certain

sample points, and the resulting estimator of a positive quantity (e.g., diffusion function) may produce a

negative result in finite samples. Because M2(x) reflects the infinitesimal variation of the process around

a spatial point x, the local linear estimator for M2(x) may have some shortcomings which are mentioned

above. One might suggest truncating negative estimates as needed, but it does not seem credible. In this

sense, the local constant estimator is better, and the weights based on observations are nonnegative.

The local polynomial estimator of M2(x) is defined by M2(x, h) = β0, where β0, . . . , βp minimizes the

weighted sum of squares

n∑i=1

((X(i+1)Δ −XiΔ)

2/Δ−p∑

j=0

βj(XiΔ − x)j)2

K

(XiΔ − x

h

).

The kernel K(·) is a continuously differentiable symmetric nonnegative function whose derivative K ′ isabsolutely integrable and h is the bandwidth. Let Kh(·) = K(·/h)/h.

A local linear estimator has the following explicit form:

M2LL(x) =

∑ni=1 w

LLi (x, h)Kh(XiΔ − x)

(X(i+1)Δ−XiΔ)2

Δ∑ni=1 w

LLi (x, h)Kh(XiΔ − x)

,

where wLLi = Sn,2 − (XiΔ − x)Sn,1 with Sn,j =

∑ni=1(XiΔ − x)jKh(XiΔ − x), j = 1, 2. The local linear

weights wLLi (x, h) are not guaranteed to be nonnegative and thus the resulting estimates may be negative.

The RNW estimator for the diffusion function is defined by

M2RNW (x) =

∑ni=1 w

RNWi (x, h)Kh(XiΔ − x)

(X(i+1)Δ−XiΔ)2

Δ∑ni=1 w

RNWi (x, h)Kh(XiΔ − x)

, (2.8)

where the weights wRNWi (x, h), i = 1, . . . , n, solve the following constrained optimization problem,

wRNWi (x, h) = argmaxwi

n∑i=1

lognwi (2.9)

subject to

n∑i=1

wi = 1, (2.10)


wi � 0, and

n∑i=1

wi(XiΔ − x)Kh(XiΔ − x) = 0. (2.11)

The weights wRNWi (x, h) in the RNW diffusion function estimator (2.8) are easy to compute. Let the

order observations be X(Δ) � X(2Δ) � · · · � X(nΔ) and assume that the spatial point x satisfies X(Δ) <

x < X(nΔ). The Lagrangian function for the constrained optimization problem is

Ln(wi, γ, λ) =1

n

n∑i=1

lognwi − γ

( n∑i=1

wi − 1

)− λ

n∑i=1

wi(XiΔ − x)Kh(XiΔ − x), (2.12)

where γ, λ are Lagrange multipliers. The first-order condition of Ln(wi, γ, λ) with respect to wi, γ and

λ gives

1

n− γwi − λwi(XiΔ − x)Kh(XiΔ − x) = 0,

n∑i=1

wi = 1,

n∑i=1

wi(XiΔ − x)Kh(XiΔ − x) = 0,

from which γ = 1 and

wRNWi (x, h) =

1

n(1 + λ(XiΔ − x)Kh(XiΔ − x)), (2.13)

where λ satisfies

n∑i=1

(XiΔ − x)Kh(XiΔ − x)

n(1 + λ(XiΔ − x)Kh(XiΔ − x))= 0. (2.14)

The discrete moment condition (2.11) describes the properties of the local linear estimator such as bias

reduction and automatic boundary correction (see [9]). Without constraint (2.11), the maximization

problem (2.9) is solved, subject to (2.10) only. In this paper we study the strong consistency and

asymptotic normality for the RNW estimate of the second infinitesimal moment of jump-diffusion models.

3 Some assumptions and useful preliminaries

The basic assumptions are summarized here for easy reference. Let D = (l, u) with −∞ � l < u � ∞the range of the process Xt.

Assumption 1. (i) The functions μ(·), σ(·), c(·, y) and λ(·) are doubly continuously differentiable. Thereexists a constant C1 such that for all x and z in a compact subset of range D,

|μ(x) − μ(z)|+ |σ(x) − σ(z)|+ λ(x)

∫Y

|c(x, y)− c(z, y)|Π(dy) � C1|x− z|. (3.1)

Furthermore, there exists a constant C2 such that for any x ∈ D,

|μ(x)|+ |σ(x)| + λ(x)

∫Y

|c(x, y))|Π(dy) � C2{1 + |x|}. (3.2)

(ii) There exist constants α > 2 and C3 such that for any x ∈ D,

λ(x)

∫Y

|c(x, y))|αΠ(dy) � C3{1 + |x|α}. (3.3)


(iii) λ(·) � 0 and σ2(·) > 0 on D.

Remark 1. Assumption 1 is a very common assumption that guarantees the existence and uniqueness

of a Cadlag strong solution to (2.1) (see [23]).

Assumption 2. The solution to (2.1) is positive Harris recurrent.

Remark 2. Harris recurrence guarantees the existence and uniqueness of invariant measure φ. This

assumption is much weaker than the usual assumption. Researchers usually assume that the process has

stationary density. This assumption is stronger than ours. We do not assume the existence of stationary

density. We use the local time to replace the density function as the normalized factor and obtain the

asymptotic results by means of an ergodic theorem for the diffusion process.

Assumption 3. (i) The kernel K(·) is a continuous differentiable, bounded and symmetric function.

It has bounded compact support (−1, 1), satisfying∫K(s)ds = 1 and

∫ ∣∣∣∣dK(s)

ds

∣∣∣∣ds < ∞. (3.4)

(ii) The kernels K±(·) : R± → R+ are continuously differentiable and bounded functions. They have

bounded compact support and satisfy

∫R±

K±(s)ds = 1 (3.5)

and∫R±

∣∣∣∣dK±(s)ds

∣∣∣∣ds < ∞. (3.6)

In this paper, we take the local times of Xt as the normalized factor. Let Xt be a semimartingale

satisfying∑

0<s�t |ΔXs| < ∞ a.s., ∀t > 0. Then, for any x and t, we have

LX(t, x+) = LX(t, x) = limε→0

1

ε

∫ t

0

1(x�Xs�x+ε)d[X ]cs a.s.,

and

LX(t, x−) = limε→0

1

ε

∫ t

0

1(x−ε�Xs�x)d[X ]cs a.s.,

where [X ]c is the quadratic variation process of continuous parts of X . Also,

L∗X(t, x) :=

LX(t, x+) + LX(t, x−)

2= lim

ε→0

1

2ε

∫ t

0

1(|Xt−x|�ε)d[X ]cs a.s.,

is a symmetrized version of the local time (see [22]).

Furthermore, let

LX(t, x) =1

σ2(x)limε→0

1

ε

∫ t

0

1(x�Xs�x+ε)σ2(Xs)ds a.s.,

LX(t, x−) =1

σ2(x)limε→0

1

ε

∫ t

0

1(x−ε�Xs�x)σ2(Xs)ds a.s.,

L∗X(t, x) =

1

σ2(x)limε→0

1

2ε

∫ t

0

1(|Xs−x|�ε)σ2(Xs)ds a.s.

Assumption 4. T → ∞, Δ = T/n → 0 and h = hn → 0 as n → ∞ such that

(L∗X(T, x)

h

)(Δ log

(1

Δ

)) 12

= oa.s.(1).


Remark 3. Assumption 4 stipulates that the divergence or convergence rates of L∗X(T, x), Δ and h,

are necessary for a consistent estimation of nonparametric drift and diffusion estimators (see [5]).

We now state our main theorem.

Theorem. Under Assumptions 1–4, for each x ∈ D,

M2RNW (x)

a.s.−−→ M2(x). (3.7)

Furthermore, if h5L∗X(T, x) = Oa.s.(1), then

√hL∗

X(T, x)

(M2

RNW (x) −M2(x)− 1

2h2K1(M

2(x))′′)

⇒ N(0, 4K2(M2(x))2), (3.8)

where K1 =∫s2K(s)ds and K2 =

∫K2(s)ds.

The theorem shows strong consistency and asymptotic normality for the RNW estimate of the second

infinitesimal moment.

4 Proof

The proof of the theorem relies on the following lemma which characterizes the order of the Langrange

multiplier λ.

Lemma 1. Under Assumptions 1–4, there exists the σ-finite invariant measure φ of the underlying

discontinuous semimartingale. The Langrange multiplier can be written as

λ =hK1φ

′(x)υ2φ(x)

+Oa.s.(h3), (4.1)

where υ2 =∫s2K2(s)ds.

The result (4.1) can be compared with the corresponding results obtained under the stationarity in

the discrete-time series context (see [7]).

Proof of Lemma 1. For simplicity we write the weights wRNWi (x, h) in the RNW estimator as wi. We

also write Kjk(u) = ujKk(u), where K(·) is the kernel function satisfying Assumption 3 and j, k are

nonnegative integers.

First we show that

n∑i=1

Δ

hKjj

(XiΔ − x

h

)=

∫ T

0

1

hKjj

(Xs− − x

h

)ds+ oa.s.(1). (4.2)

We have

∣∣∣∣n∑

i=1

Δ

hKjj

((XiΔ − x)

h

)−

∫ T

0

1

hKjj

(Xs− − x

h

)ds

∣∣∣∣

� 1

h

∣∣∣∣n−1∑i=0

∫ (i+1)Δ

iΔ

Kjj

(XiΔ − x

h

)ds−

n−1∑i=0

∫ (i+1)Δ

iΔ

Kjj

(Xs− − x

h

)ds

∣∣∣∣+

Δ

h

∣∣∣∣Kjj

(X0 − x

h

)∣∣∣∣+ Δ

h

∣∣∣∣Kjj

(XT − x

h

)∣∣∣∣� 1

h

n−1∑i=1

∫ (i+1)Δ

iΔ

∣∣∣∣Kjj

(XıΔ − x

h

)−Kjj

(Xs− − x

h

)∣∣∣∣ds+O

(Δ

h

)

=1

h

n−1∑i=0

∫ (i+1)Δ

iΔ

∣∣∣∣K ′jj

(Xis− − x

h

)∣∣∣∣ ·∣∣∣∣XiΔ −Xs−

h

∣∣∣∣ds+ o(1)


� kn,Th2

n−1∑i=0

∫ (i+1)Δ

iΔ

|K ′jj

(Xis− − x

h

)|ds+ o(1), (4.3)

where Xis− is some value between and XiΔ and Xs− and

kn,T = maxi�n

supiΔ�s�(i+1)Δ

|XiΔ −Xs−|.

It follows from the Levy continuity of modulus for diffusion processes (see [4]) that

kn,T = Oa.s.

(√Δ log

1

Δ

). (4.4)

Using (4.4) we obtain

K ′jj

(Xis− − x

h

)= K ′

jj

(Xs− − x

h+Oa.s.

(1

h

√Δ log

1

Δ

)).

Then the first term of right-hand side of (4.3) is bounded by

kn,Th2

∫ T

0

∣∣∣∣K ′jj

(Xs− − x

h+Oa.s.

(1

h

√Δ log

1

Δ

))∣∣∣∣ds.Using the well-known occupation time formula (see [4]), we have

kn,Th2

∫ T

0

∣∣∣∣K ′jj

(Xs− − x

h+Oa.s.

(1

h

√Δ log

1

Δ

))∣∣∣∣ds=

kn,Th2

∫ ∞

−∞

∣∣∣∣K ′jj

(a− x

h+ oa.s.(1)

)∣∣∣∣L∗X(T, a)da

=kn,Th

∫ ∞

−∞|K ′

jj(q + oa.s.(1))|L∗X(T, hq + x)dq

� Ckn,Th

Oa.s.(L∗X(T, x))

= oa.s.(1),

where we have used the absolute integrability ofK ′jj(·), which follows from the Cauchy-Schwarz inequality.

Let Aj = 1n

∑ni=1 Kjj((XiΔ − x)/h) for any integer j � 1. From (4.2) and the similar argument in [5],

we have

Aj

A2=

∑ni=1

ΔhKjj(

XiΔ−xh )∑n

i=1Δh K22(

XiΔ−xh )

=

∫ T

01hKjj(

Xs−−xh )ds+ oa.s.(1)∫ T

01hK22(

Xs−−xh )ds+ oa.s.(1)

=

∫1hKjj(

z−xh )φ(z)dz + oa.s.(1)∫

1hK22(

z−xh )φ(z)dz + oa.s.(1)

+ oa.s.(1)

=

∫ujKj(u)φ(x+ uh)du+ oa.s.(1)∫u2K2(u)φ(x + uh)du+ oa.s.(1)

+ oa.s.(1)

=

∫ujKj(u)[φ(x) + uhφ′(x) +O(h2)]du + oa.s.(1)∫u2K2(u)[φ(x) + uhφ′(x) +O(h2)]du+ oa.s.(1)

+ oa.s.(1). (4.5)

By symmetry of K(·), for odd j = 1, 3, . . . ,

Aj

A2=

h(∫uj+1Kj(u)du)φ′(x) +O(h2) + oa.s.(1)

υ2φ(x) +O(h2) + oa.s.(1)+ oa.s.(1); (4.6)


for even j = 4, 6, . . . ,

Aj

A2=

∫ujKj(u)duφ(x) +O(h2) +Oa.s.(1)

ν2φ(x) +O(h2) + oa.s.(1)+ oa.s.(1). (4.7)

Let qi = K11(XiΔ−x

h ); then Aj =1n

∑ni=1 q

ji . By Assumption 3,

max1�i�n

|qi| � C < ∞. (4.8)

By (2.10) and (2.12), we have∑n

i=1[n(1 + λqi)]−1 = 1 or

n∑i=1

−qi1 + λqi

= 0, (4.9)

and therefore noting that

1 + λqi = (nwi)−1 � 0,

we have

0 =

∣∣∣∣ 1nn∑

i=1

(λq2i

1 + λqi− qi

)∣∣∣∣ � 1

n

n∑i=1

|λ|q2i1 + λqi

−∣∣∣∣ 1n

n∑i=1

qi

∣∣∣∣ � |λ|A2

1 + C|λ| − |A1| = (A2 − C|A1|)|λ| − |A1|1 + C|λ| ,

which implies (A2 − C|A1|)|λ| � |A1| or (1− C|A1/A2|)|λ| � |A1/A2|. So in the view of (4.6) and (4.7),

we have

|λ| = Oa.s.(h). (4.10)

By (4.9) and the Taylor expansion

0 =1

n

n∑i=1

qi1 + λqi

= A1 − λA2 + λ2A3 − λ3A4 +Oa.s.(h4).

Then we have

λ =A1

A2+ λ2A3

A2− λ3A4

A2+Oa.s.(h

4) =hK1φ

′(x)υ2φ(x)

+Oa.s.(h3)

in the view of (4.6) and (4.7) again. This completes the proof of Lemma 1.

Proof of Theorem. We start by considering the expression

∑n−1i=1 nwiK(XiΔ−x

h )((X(i+1)Δ−XiΔ)2

Δ )∑ni=1 K(XiΔ−x

h )

=1∑n

i=1 K(XiΔ−xh )Δ

( n−1∑i=1

nwiK

(XiΔ − x

h

)∫ (i+1)Δ

iΔ

(σ2(Xs−)

+

∫Y

c2(Xs−, y)Π(dy)λ(Xs−))ds

)

+1∑n

i=1 K(XiΔ−xh )

( n−1∑i=1

nwiK

(XiΔ − x

h

)[(X(i+1)Δ −XiΔ)

2

Δ− 1

Δ

∫ (i+1)Δ

iΔ

(σ2(Xs−)

+

∫Y


])

=1∑n

i=1 K(XiΔ−xh )Δ

( n−1∑i=1

1

1 + λ(XiΔ − x)Kh(XiΔ − x)K

(XiΔ − x

h

)∫ (i+1)Δ

iΔ

(σ2(Xs−)

+

∫Y

c2(Xs−, y)Π(dy)λ(XS−))ds

)


+1∑n

i=1 K(XiΔ−xh )

( n−1∑i=1

1


(XiΔ − x

h

)((X(i+1)Δ −XiΔ)

2

Δ

− 1

Δ

∫ (i+1)Δ

iΔ

(σ2(Xs−) +

∫Y


))

=: α(x) + β(x) (4.11)

by (2.12).

Firstly, we examine α(x), by the Taylor expansion, and following [26], we have

α(x) =

∑n−1i=1 K(XiΔ−x

h )∫ (i+1)Δ

iΔ (σ2(Xs−) +∫Y c2(Xs−, y)Π(dy)λ(Xs−))ds∑n

i=1 K(XiΔ−xh )Δ

+Oa.s.(h)

=1h

∫ T

0K(Xs−−x

h )(σ2(Xs−) +∫Yc2(Xs−, y)Π(dy)λ(Xs−))ds+ oa.s.(

L∗X(T,x)

h (Δ log( 1Δ))1/2)

1h

∫ T

0 K(Xs−−xh )ds+ oa.s.(

L∗X(T,x)

h (Δ log( 1Δ ))1/2)

+Oa.s.(h).

Using the quotient limit theorem for Harris recurrent Markov processes (see [3]), we can write

limn→∞α(x)

(∫RK(q)(σ2(x + qh) +

∫Yc2(x+ qh, y)Π(dy)λ(x + qh))φ(x + qh)dq∫

RK(q)φ(x+ qh)dq

)−1

= 1 a.s.,

where φ(dx) is the σ-finite invariant measure of the underlying discontinuous semimartingale. Such a

measure is known to be absolutely continuous with respect to the Lebesegue measure (see [19]) φ(dx) =

φ(x)dx. Provided that h converges to zero slowly enough then we have that

(L∗X(T, x)

h

)(Δ log

(1

Δ

))1/2

= oa.s.(1),

∀ x ∈ D. As n and T diverge jointly, we can show that

α(x)a.s.−−→ σ2(x) +

(∫Y

c2(x, y)Π(dy)

)λ(x) = σ2(x) + EY [c

2(x, ξ)]λ(x) = M2(x). (4.12)

Now we turn to the term β(x),

β(x) =


h )((X(i+1)Δ−XiΔ)2

Δ − 1Δ

∫ (i+1)Δ

iΔ (σ2(Xs−) +∫Y c2(Xs−, y)Π(dy)λ(Xs−))ds)∑n

i=1 K(XiΔ−xh )

+Oa.s.(h).

We have

β(x) =


h )((X(i+1)Δ−XiΔ)2

Δ − 1Δ

∫ (i+1)Δ

iΔ(σ2(Xs−) +

∫Yc2(Xs−, y)Π(dy)λ(Xs−))ds)∑n

i=1 K(XiΔ−xh )

.

Following [4],

(X(i+1)Δ −XiΔ)2 = X2

(i+1)Δ −X2iΔ − 2XiΔ[X(i+1)Δ −XiΔ]

=

∫ (i+1)Δ

iΔ

(σ2(Xs−) +

(∫Y

c2(Xs−, y)Π(dy))λ(Xs−)

)ds

+ 2

∫ (i+1)Δ

iΔ

(Xs− −XiΔ)μ(Xs−)ds

+ 2

∫ (i+1)Δ

iΔ

(Xs− −XiΔ)σ(Xs−)dWs


+

∫ (i+1)Δ

iΔ

∫Y

((Xs− + c)2 −X2s− − 2XiΔc(Xs−, y))ν(ds, dy).

Then

β(x) =1∑n

i=1 K(XiΔ−xh )

·{ n−1∑

i=1

K

(XiΔ − x

h

)(1

Δ(X(i+1)Δ −XiΔ)

2 − 1

Δ

∫ (i+1)Δ

iΔ

(σ2(Xs−)

+

∫Y


)}

=1∑n

i=1 K(XiΔ−xh )

·{ n−1∑

i=1

K

(XiΔ − x

h

)(1

Δ2

∫ (i+1)Δ

iΔ

(Xs− −XiΔ)μ(Xs−)ds)}

+1∑n

i=1 K(XiΔ−xh )

·{ n−1∑

i=1

K

(XiΔ − x

h

)(1

Δ2

∫ (i+1)Δ

iΔ

(Xs− −XiΔ)σ(Xs−)dWs

)}

+1∑n

i=1 K(XiΔ−xh )

·{ n−1∑

i=1

K

(XiΔ − x

h

)(1

Δ

∫ (i+1)Δ

iΔ

∫Y

((Xs− + c)2 −X2s−

− 2XiΔc(Xs−, y))ν(ds, dy))}

=: β1(x) + β2(x) + β3(x).

The quantities β1(x), β2(x) and β3(x) are the sample averages of MDGSs converging to zero (see [4]).

These imply that

β(x)a.s.−−→ 0. (4.13)

Then, we obtain that

∑ni=1 nwiK(XiΔ−x

h )(X(i+1)Δ−XiΔ)2

Δ∑ni=1 K(XiΔ−x

h )

a.s.−−→ M2(x). (4.14)

Similarly, we have


h )∑ni=1 K(XiΔ−x

h )

a.s.−−→ 1. (4.15)

By combining (4.14) and (4.15), we obtain (3.12)

M2RNW(x)

a.s.−−→ M2(x).

This shows the strong consistency for the RNW estimator of the second infinitesimal moment of jump-

diffusion models.

To prove asymptotic normality, we proceed with

M2RNW(x) −M2(x)

=


h )((X(i+1)Δ−XiΔ)2

Δ −M2(x))∑ni=1 K(XiΔ−x

h )

=


h )( 1Δ

∫ (i+1)Δ

iΔ (σ2(Xs−) +∫Y c(Xs−, y)Π(dy)λ(Xs−))ds−M2(x))∑n

i=1 K(XiΔ−xh )

+


h )((X(i+1)Δ−XiΔ)2

Δ − 1Δ

∫ (i+1)Δ

iΔ(σ2(Xs−) +

∫Yc2(Xs−, y)Π(dy)λ(Xs−))ds)∑n

i=1 K(XiΔ−xh )

=: Bn + Vn.


By (4.4), we have

1

Δ

∫ (i+1)Δ

iΔ

(σ2(Xs−) +

∫Y

c2(Xs−, y)Π(dy)λ(Xs−))ds−M2(x)

= M2(XiΔ)−M2(x) +Oa.s.

(√Δ log

1

Δ

).

By the Taylor expansion and Assumption 4, we have

Bn =


h )((XiΔ − x)(M2(x))′ + (M2(x))′′ (XiΔ−x)2

2 ) + oa.s.(h2)∑n

i=1 K(XiΔ−xh )

=


h )(M2(x))′′ (XiΔ−x)2

2 + oa.s.(h2)∑n

i=1 K(XiΔ−xh )

by (2.11). Similar to (4.11), we have

Bn =1

2h2K1(M

2(x))′′ + oa.s.(1). (4.16)

Next we turn to the term Vn.

Vn =1∑n

i=1 K(XiΔ−xh )

( n∑i=1

1


(XiΔ − x

h

)

×((X(i+1)Δ −XiΔ)

2

Δ− 1

Δ

∫ (i+1)Δ

iΔ

(σ2(Xs−) +

∫Y


)),

√L∗X(T, x)Vn =

√L∗X(T, x) · 1∑n

i=1 K(XiΔ−xh )

·{ n∑

i=1

K

(XiΔ − x

h

)((X(i+1)Δ −XiΔ)

2

Δ

− 1

Δ

∫ (i+1)Δ

iΔ

(σ2(Xs−) +

∫Y


)}

− λ√

L∗X(T, x) · 1∑n

i=1 K(XiΔ−xh )

·{ n∑

i=1

(XiΔ − x)Kh(XiΔ − x)K

(XiΔ − x

h

)

×((X(i+1)Δ −XiΔ)

2

Δ− 1

Δ

∫ (i+1)Δ

iΔ

(σ2(Xs−) +

∫Y


)}

+ λ2√

L∗X(T, x) · 1∑n

i=1 K(XiΔ−xh )

·{ n∑

i=1

((XiΔ − x)Kh(XiΔ − x))2K

(XiΔ − x

h

)

×((X(i+1)Δ −XiΔ)

2

Δ− 1

Δ

∫ (i+1)Δ

iΔ

(σ2(Xs−) +

∫Y


)}

+Oa.s.(h3)

=√

L∗X(T, x) · 1∑n

i=1 K(XiΔ−xh )

·{ n∑

i=1

K

(XiΔ − x

h

)((X(i+1)Δ −XiΔ)

2

Δ

−∫ (i+1)Δ

iΔ

(σ2(Xs−) +

∫Y


)}

− λOP

(1√

L∗X(T, x)

)+ λ2OP

(1√

L∗X(T, x)

)

=√

L∗X(T, x) · 1∑n

i=1 K(XiΔ−xh )

·{ n∑

i=1

K

(XiΔ − x

h

)((X(i+1)Δ −XiΔ)

2

Δ

−∫ (i+1)Δ

iΔ

(σ2(Xs−) +

∫Y


)}


− λOP (1) + λ2OP (1).

Following from [5],

√L∗X(T, x)Vn ⇒ N(0, 4K2(M

2(x))2). (4.17)

Thus by combining (4.16) and (4.17), we get (3.13),

√L∗X(T, x)(M2

RNW(x)−M2(x)−Bn) ⇒ N(0, 4K2(M2(x))2).

This proves asymptotic normality for the RNW estimate of the second infinitesimal moment of jump-

diffusion models.

Acknowledgements This work was supported by National Natural Science Foundation of China (Grant Nos.

10871177, 11071213), and Research Fund for the Doctor Program of Higher Education of China (Grant No.

20090101110020).

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Reweighted Nadaraya-Watson estimation of jump-diffusion models

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