presentation - Eurandom

Recent results on volatility change point estimation 

for stochastic differential equations 

Stefano M. Iacus (University of Milan & Core Team) 

Workshop on Parameter Estimation for Dynamical Systems, Eurandom, 4-6 June 2012 

S.M. Iacus 2012 PEDS2 Eurandom – 1 / 95

The change point 

problem 

Historical remarks 

Diffusion processes 

General Itô processes 

YUIMA R Package 

The change point problem 



problem 





Change point problem 

Discover from the data a structural change in the generating model. 



problem 







Next example shows historical change points for the Dow-Jones index. 



problem 







Next example shows historical change points for the Dow-Jones index. 

750 850 950 1050 

-0.06 0.00 0.04 

Dow-Jones closings 

1972 1973 1974 

Dow-Jones returns 

1972 1973 1974 

break of gold-US$ linkage (left) Watergate scandal (right) 



problem 


The i.i.d. case and time 

series 







problem 



series 




The i.i.d. case and time series 

Originally, the problem was considered for i.i.d samples; see Hinkley 

(1971), Csörgő and Horváth (1997), Inclan and Tiao (1994) 



problem 



series 







It moved quickly to time series analysis: Bai (1994, 1997), Kim et al. 

(2000), Lee et al. (2003), Chen et al. (2005) and the papers cited therein. 



problem 



series 







It moved quickly to time series analysis: Bai (1994, 1997), Kim et al. 

(2000), Lee et al. (2003), Chen et al. (2005) and the papers cited therein. 

The list is vastly incomplete. 

Main approaches: least squares, MLE, Bayesian and CUSUM 



problem 



Continuous time obs. 

Discrete sampling 

Least Squares 

CUSUM 






problem 





Least Squares 

CUSUM 



Change point for diffusion processes 

For continuous time observations, Kutoyants (1994, 2004) considered 

change point (in space) problem for the drift of an ergodic diffusion 

process solution to 

dXt = S(θ, Xt)dt + σ(Xt)dWt, X0, t ≥ 0 (1) 

with the trend function discontinuous along the two points of the state 

space of Xt, say x (1) 

∗ (θ) and x (2) 

∗ (θ), θ ∈ [α, β] ⊂ R and the interest is in 

the estimation of θ. 

Lee, Nishiyama and Yoshida (2003) considered CUSUM tests statistic for 

the time change point for the model in (1) where θ takes different values 

before and after a time instant τ ∗ . 

Habibi R. (2012) considered a recursive-MLE approach for the estimation 

of the change point in the parameter of the drift from 1-d continuous time 

observations in the very special case of gBm with stochastic volatility: 

dXt = θXtdt + σtXtdWt. 



problem 





Least Squares 

CUSUM 



Estimation of volatility under discrete time sampling 

The problem of parametric estimation of the volatility (and drift) for 

discretely observed diffusion processes under different sampling schemes, 

dates back to the works of Genon-Catalot and Jacod (1993, 1994), 

Yoshida (1992) and Kessler (1997). 

Change point analysis for the volatility appears in the few works reviewed 

in the following. 

The main peculiarity is the non regularity of the statistical model with 

respect to the structural change, i.e. the (quasi-)likelihood is not smooth 

with respect to the parameter τ, the change point instant. This fact add 

some technicalities in the proofs and different limiting distributions are 

obtained and rates of convergence are faster than usual √ n. 



problem 





Least Squares 

CUSUM 



Least squares approach 

De Gregorio and I. (2008) considered the change point problem for the 

ergodic model 

dXt = b(Xt)dt + √ θσ(Xt)dWt, 0 ≤ t ≤ T,X0 = x0, 

observed at discrete time instants ti = i∆n, i =0,...,n, ∆n = ti+1 − ti 

under the sampling scheme ∆n → 0, n →∞, n∆n = T , T fixed. The 

coefficients b and σ are supposed to be known. The change point problem 

is formulated as follows 

Xt = 

 

X0 + t 

0 b(Xs)ds + √ θ1 

S.M. Iacus 2012 PEDS2 Eurandom – 9 / 95 

t 

Xτ ∗ + t 

τ ∗ b(Xs)ds + √ θ2 

0 σ(Xs)dWs, 0 ≤ t ≤ τ ∗ 

t 

τ ∗ σ(Xs)dWs, τ ∗


problem 





Least Squares 

CUSUM 



i=1 


Let Zi = Xi+1−Xi−b(Xi)∆n 

√ be the standardized residuals. Let 

∆nσ(Xi) 

k0 =[n · τ ∗ ], then the LS estimator is obtained via 

ˆk0 = 

 

k 

argmin min (Z 

k θ1,θ2 

i=1 

2 i − θ1) 2 n 

+ (Z 

i=k+1 

2 i − θ2) 2 

= 

 

 

k 

argmin (Z 

k 

2 i − ¯ θ1) 2 n 

+ (Z 2 i − ¯ θ2) 2 

 

, (2) 

where 

¯θ1 = 1 

k 

k 

i=1 

Z 2 i =: Sk 

k 

i=k+1 

and ¯ θ2 = 1 

n − k 


n 

i=k+1 

Z 2 i =: S∗ n−k 

n − k .


problem 





Least Squares 

CUSUM 



Solving (2) is equivalent to solve the following 

Dk = k 

n 


Sk 

− . 

Sn 

Therefore, least squares change point estimator ˆ k0 (ˆτn = ˆ k0/n)isgiven 

by 

ˆk0 =argmax|Dk| 

k 

Theorem: Under H0 : θ1 = θ2, i.e. “no change point”, we have that 

 

n 

2 |Dk| d →|B0(τ)|, B0(τ) Brownian bridge on [0, 1] 

The estimator ˆτn is consistent. 



problem 





Least Squares 

CUSUM 



Let ϑn = |θ2 − θ1|. Under the conditions 

ϑn → 0, 

nϑ 2 n(ˆτn − τ ∗ ) 

2 ˆ θ 2 

Under contiguous alternatives 

√ nϑn 

√ log n →∞, 

 

d 

→ arg max W(v) − 

v 

|v| 

 

2 

with ˆ θ some consistent estimator of the common limit θ0 of θ1 and θ2; W 

is a two sided Brownian motion, i.e. 

 

W1(−u), u < 0 

W(u) = 

(3) 

W2(u), u ≥ 0 

where W1,W2 are two independent Brownian motions. 



problem 





Least Squares 

CUSUM 



The estimators ˆ θ1, ˆ θ2 are also well behaved, i.e. 

 

√ ˆθ1 − θ1 

n 

ˆθ2 − θ2 

Under contiguous alternatives 

d→ N (0, Σ) , where Σ= 


 

2 θ2 0 

τ ∗ 0 

0 2 θ2 0 

1−τ ∗ 

The above results hold in the high frequency case ∆n → 0 with n →∞ 

and n∆ =T fixed, but also for the rapidly increasing design case 

n∆n = T →∞. 

 

.


problem 





Least Squares 

CUSUM 



Case of unknown drift 

We assume now to observe a diffusion process that is a solution to the 

reduced stochastic differential equation 

dXt = b(Xt)dt + √ θdWt, 

where b(·) is unknown and estimated using nonparametric methods. Let 

K be a suitable kernel, and consider the estimator 

ˆ b(x) = 

Xi−x 

n i=1 K 

n i=1 K 


hn 

Xi+1−Xi 

Xi−x 

the above is a particular case of the nonparametric estimators of the drift 

considered in Bandi and Phillips (2003), but other approaches exist. 

For fixed T , the drift b(·) cannot be estimated consistently, hence it is 

required that T →∞. 

hn 

∆n


problem 





Least Squares 

CUSUM 



Let 

¯LX(t, x) = lim 

ɛ→0 

Bandwidth hn and local time 


1 

2ɛ 

t 

0 

1 {x,x+ɛ}(Xs)ds 

be the the chronological local time of X. Bandi and Phillips (2003) shown 

that under the following additional assumption 

 

¯LX(T,x) 

 

1 

∆n log = oa.s.(1) 

hn 

and hn ¯ LX(T,x) a.s. 

→∞, ˆb(·) is a consistent estimator of b(·). In the 

stationary case, the above condition may be replaced by 

 

T 

1 

∆n log = oa.s.(1). 

hn 

∆n 

∆n


problem 





Least Squares 

CUSUM 



Pluggin-in the standardized residuals 

ˆZi = Xi+1 − Xi − ˆ b(Xi)∆n 

√ ∆n 

into the statistic Dk, all previous results hold. 

Case of unknown drift 

This approach has been used in Smaldone (2009) to re-analyze the recent 

global financial crisis using data from different markets. It emerged that 

the analysis of some of the assets could have predict by the last week of 

June 2008 the global structural change of late september 2008. 


June 30 

July 4 

Lehman 

Brothers 

July 7 

July 11 

July 14 

July18 

DJ Stoxx America 600 Banks 

DJ DJ Stoxx 600 Banks 

Deutsche Bank 

HBSC 

Barclays 

Deutsche Bank (GER) 

CAC 

DJ Stoxx Global 

1800 Banks 

August 18 

August 22 

S&P MIB 

Nikkei 225 

August 25 

August 29 

FTSE 

DJ Stoxx 600 

Time 

Sept. 1 

Sept. 5 

Lehman 

Brothers 

Sept. 8 

Sept. 12 

DJ Stoxx 

600 Banks 

Sept. 15 

Sept. 19 

Goldman 

Sachs 

Sept. 22 

Sept. 26 

Sept. 29 

Oct. 3 

Oct. 6 

Oct. 10 

Deutsche Bank 

HSBC Nasdaq 

Nyse 

Dow Jones 

S&P 500 

FTSE 

DAX 

S&P MIB 

CAC 

IBEX 

SMI 

Nikkei 225 

DJ Stoxx 600 Banks 

DJ Stoxx Global 1800 

MCSI World 

Morgan Stanley 

Bank of America 

Barclays 

RBS 

Unicredit 

Intesa Sanpaolo 

Deutsche Bank (GER) 

Commerzbank 

Oct. 20 

Oct 24 

Dj Stoxx Asia Pacific 

600 Banks 

JP Morgan Chase

8000 9000 10000 11000 12000 13000 

−0.20 −0.15 −0.10 −0.05 0.00 0.05 0.10 

Financial crisis 2008 - Real time analysis 

Dow Jones Jan08−Jan09 

Jan Mar May Jul Sep Nov Jan 

weekly returns 



8000 9000 10000 11000 12000 13000 

−0.20 −0.15 −0.10 −0.05 0.00 0.05 0.10 







8000 9000 10000 11000 12000 13000 

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8000 9000 10000 11000 12000 13000 

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8000 9000 10000 11000 12000 13000 

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8000 9000 10000 11000 12000 13000 

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8000 9000 10000 11000 12000 13000 

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8000 9000 10000 11000 12000 13000 

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8000 9000 10000 11000 12000 13000 

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8000 9000 10000 11000 12000 13000 

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8000 9000 10000 11000 12000 13000 

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8000 9000 10000 11000 12000 13000 

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8000 9000 10000 11000 12000 13000 

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8000 9000 10000 11000 12000 13000 

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8000 9000 10000 11000 12000 13000 

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8000 9000 10000 11000 12000 13000 

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8000 9000 10000 11000 12000 13000 

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8000 9000 10000 11000 12000 13000 

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8000 9000 10000 11000 12000 13000 

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8000 9000 10000 11000 12000 13000 

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8000 9000 10000 11000 12000 13000 

−0.20 −0.15 −0.10 −0.05 0.00 0.05 0.10 







8000 9000 10000 11000 12000 13000 

−0.20 −0.15 −0.10 −0.05 0.00 0.05 0.10 







8000 9000 10000 11000 12000 13000 

−0.20 −0.15 −0.10 −0.05 0.00 0.05 0.10 







8000 9000 10000 11000 12000 13000 

−0.20 −0.15 −0.10 −0.05 0.00 0.05 0.10 







8000 9000 10000 11000 12000 13000 

−0.20 −0.15 −0.10 −0.05 0.00 0.05 0.10 







8000 9000 10000 11000 12000 13000 

−0.20 −0.15 −0.10 −0.05 0.00 0.05 0.10 







8000 9000 10000 11000 12000 13000 

−0.20 −0.15 −0.10 −0.05 0.00 0.05 0.10 







8000 9000 10000 11000 12000 13000 

−0.20 −0.15 −0.10 −0.05 0.00 0.05 0.10 







8000 9000 10000 11000 12000 13000 

−0.20 −0.15 −0.10 −0.05 0.00 0.05 0.10 







8000 9000 10000 11000 12000 13000 

−0.20 −0.15 −0.10 −0.05 0.00 0.05 0.10 







8000 9000 10000 11000 12000 13000 

−0.20 −0.15 −0.10 −0.05 0.00 0.05 0.10 







8000 9000 10000 11000 12000 13000 

−0.20 −0.15 −0.10 −0.05 0.00 0.05 0.10 







8000 9000 10000 11000 12000 13000 

−0.20 −0.15 −0.10 −0.05 0.00 0.05 0.10 







8000 9000 10000 11000 12000 13000 

−0.20 −0.15 −0.10 −0.05 0.00 0.05 0.10 







8000 9000 10000 11000 12000 13000 

−0.20 −0.15 −0.10 −0.05 0.00 0.05 0.10 







8000 9000 10000 11000 12000 13000 

−0.20 −0.15 −0.10 −0.05 0.00 0.05 0.10 







8000 9000 10000 11000 12000 13000 

−0.20 −0.15 −0.10 −0.05 0.00 0.05 0.10 







8000 9000 10000 11000 12000 13000 

−0.20 −0.15 −0.10 −0.05 0.00 0.05 0.10 







8000 9000 10000 11000 12000 13000 

−0.20 −0.15 −0.10 −0.05 0.00 0.05 0.10 







8000 9000 10000 11000 12000 13000 

−0.20 −0.15 −0.10 −0.05 0.00 0.05 0.10 







8000 9000 10000 11000 12000 13000 

−0.20 −0.15 −0.10 −0.05 0.00 0.05 0.10 







8000 9000 10000 11000 12000 13000 

−0.20 −0.15 −0.10 −0.05 0.00 0.05 0.10 







cpoint function in the sde package 

> library(tseries) 

> S require(sde) 

> cpoint(S) 

$k0 

[1] 123 

$tau0 

[1] "2005-01-11" 

$theta1 

0.1020001 

$theta2 

0.3770111 

> X plot(X, type = "l", main = "log returns") 

> abline(v = cpoint(X)$tau0, lty = 3, col = "red") 


X 

−0.04 −0.02 0.00 0.02 0.04 

log returns 

0 50 100 150 200 

Index 

cpoint function in the sde package 


22 

20 

18 

16 

S [2004−07−23/2005−05−05] 

Lug 23 

2004 

Last 20.45 

Bollinger Bands (20,2) [Upper/Lower]: 21.731/20.100 

Ott 01 

2004 

Dic 01 

2004 

Feb 01 

2005 

Apr 01 

2005


problem 





Least Squares 

CUSUM 



Song and Lee (2009) considered the model 

CUSUM approach 

dXt = b(Xt)dt + σ(θ, Xt)dWt, 0 ≤ t ≤ T,X0 = x0, 

under the sampling scheme ∆n → 0, n∆n = T →∞, with b and σ 

known. 

The object is to study a test statistics to verify the presence of the change 

point rather than estimate the instant of the change point. The results of 

this paper are based also on Kessler (1997). 



problem 





Least Squares 

CUSUM 



The CUSUM test statistics is given the form 

Tn = max 

l≤k≤n 

k 2 

n (ˆ θn,k − ˆ θn,n) 2 ˆ 

Jn 


where ˆ θn,k is the estimator based on the first k observations and ˆ θn,n the 

one based on all the n observations. ˆ Jn is a consistent estimator of 

J =2 

µ0 the invariant law, for θ = θ0. 

2 ∂θσ(θ0,x) 

σ(θ0,x) 

dµ0(x) 



problem 





Least Squares 

CUSUM 




The main regularity conditions are contained in Assumption [A]k: let k 

be a positive integer. 

(i) The function σ(·, ·) is continuously differentiable with respect to x up 

to order k for all θ and its derivatives are twice differentiable in θ for 

all x. All derivatives belong to the sets of functions 

{f(x, θ) :|f(θ, x)| ≤C(1 + |x|)) C for some C not depending on θ}. 

(ii) The function b(·) is continuously differentiable with respect to x up to 

order k and its derivatives are at most of polynomial growth in x. 



problem 





Least Squares 

CUSUM 




Under the additional assumptions n∆ p n → 0, n∆ q n →∞, p>q>4 and 

[A]k with k =2k0, k0 =[p/2], under H0 : θ1 = θ2 = θ0, i.e. “no 

change”, 

Tn d → sup B 

0≤s≤1 

2 0(s) 

with B0 the Brownian bridge and 

J 1 

2 

[ns] 

√n 

ˆθn,[ns] − θ0 

d→ Ws in D[0, 1], 

No properties of the change point estimator but only on the estimator of 

parameters. 



problem 





Least Squares 

CUSUM 




Under the above conditions on p and q, CUSUM test may loose efficiency 

because the time series has to be too long, e.g., for daily data at least 5 

years of observations are needed in order to get good rate of convergence. 

Recently, S. Lee (2011), considered a new type of CUSUM-type test 

statistics based on the residuals which provide better performance than his 

previous result above. 



problem 




Numerical experiment 





problem 






Change point for multidimensional Itô processes 

As in I. and Yoshida (2012), we consider 

dYt = btdt + σ(θ, Xt)dWt, t ∈ [0,T], 

where Wt is an r-dimensional standard Wiener process, on a stochastic 

basis, bt and Xt are vector valued progressively measurable processes, 

and σ(θ, x) is a matrix valued function. 

The coefficient σ(θ, x) is assumed to be known up to the parameter θ, 

while bt is completely unknown and unobservable, therefore possibly 

depending on θ and τ ∗ . 

The parameter space Θ of θ is a bounded domain in Rd0 , d0 ≥ 1, and the 

parameter θ is a nuisance in estimation of τ ∗ . Denote by θ∗ k the true value 

of θk for k =1, 2. 



problem 






dYt = btdt + σ(θ, Xt)dWt 

The sample consists of (Xti ,Yti ), i =0, 1, ..., n, where ti = i∆ for 

∆=∆n = T/n. In this work T is fixed. Pure high-frequency setup. 



problem 









It will be assumed that the process Y generating the data is an Itô process 

and satisfies the stochastic integral equation 

 

Y0 + 

Yt = 

t 

0 bsds + t 

0 σ(θ∗ 1 ,Xs)dWs for t ∈ [0, τ ∗ ) 

Yτ ∗ + t 

τ ∗ bsds + t 

τ ∗ σ(θ ∗ 2 ,Xs)dWs for t ∈ [τ ∗ ,T]. 



problem 











 

Y0 + 

Yt = 

t 

0 bsds + t 

0 σ(θ∗ 1 ,Xs)dWs for t ∈ [0, τ ∗ ) 

Yτ ∗ + t 



We assume that consistent estimators for θ exist and afterward propose a 

way to obtain them. The focus is on the estimation of the change point 

and its properties. 



problem 











 

Y0 + 

Yt = 

t 

0 bsds + t 

0 σ(θ∗ 1 ,Xs)dWs for t ∈ [0, τ ∗ ) 

Yτ ∗ + t 



We assume that consistent estimators for θ exist and afterward propose a 

way to obtain them. The focus is on the estimation of the change point 

and its properties. 

Remark: clearly this model includes diffusion models by taking, e.g., 

Yt = Xt and bt = b(Xt). 



problem 






Write ∆iY = Yti 

where 

− Yti−1 and let 

Φn(t; θ1,θ2) = 

[nt/T ] 

 

Quasi ML approach 


i=1 

Gi(θ1)+ 

n 

i=[nt/T ]+1 

Gi(θ2), 

Gi(θ) =logdetS(Xti−1 ,θ)+∆−1 S(Xti−1 ,θ)−1 [(∆iY ) ⊗2 ] 

with S(x, θ) =σ(θ, x) ⊗2 with A[B ⊗2 ]=B ′ AB. This is a modification 

of the QL introduced in Genon-Catalot and Jacod (1993).


problem 






Change point for multidimensional Itô processes 

Suppose that there exists an estimator ˆ θk for each θk, k =1, 2. In case θ ∗ k 

are known, we define ˆ θk just as ˆ θk = θ ∗ k . 

for the estimation of τ ∗ . 

ˆτn =arg min 

t∈[0,T ] Φn(t; ˆ θ1, ˆ θ2) 

More precisely, ˆτn is any measurable function of (Xti )i=0,1,...,n satisfying 

Φn(ˆτn; ˆ θ1, ˆ θ2) = min 

t∈[0,T ] Φn(t; ˆ θ1, ˆ θ2). 



problem 






Regularity conditions 

[H ]j (i) σ(θ, x) is a measurable function defined on Θ ×X satisfying 

(a) inf (x,θ)∈X ×Θ λ1(S(x, θ)) > 0,(λ1 first eigenvalue of S) 

(b) derivatives ∂ ℓ θ σ (0 ≤ ℓ ≤ j+[d0/2]) exist and those functions are 

continuous on Θ ×X, 

(c) there exists a locally bounded function L : X×X×Θ → R+ such 

that 

|σ(θ, x) − σ(θ, x ′ )|≤L(x, x ′ ,θ)|x − x ′ | α 

for some constant α>0. 

(x, x ′ ∈X,θ∈ Θ) 

(ii) (Xt) t∈[0,T ] is a progressively measurable process taking values in X 

such that 

w [0,T ] 

as n →∞. 

 

1 

n ,X 

 

= op(ϑ 1/α 

n ), (w : modulus of continuity func.) 

(iii) (bt) t∈[0,T ] is a progressively measurable process taking values in R 


d 

such that (bt − b0) t∈[0,T ] is locally bounded.


problem 






Identifiability conditions & consistent estimators 

[A] P [S(Xτ ∗; θ ∗ 1) = S(Xτ ∗; θ ∗ 2)] = 1; 

[B] Ξ(Xτ ∗,θ ∗ ) is positive-definite a.s., where 

Ξ(x, θ) = 

 

Tr((∂θ (i1 )S)S −1 (∂θ (i2 )S)S −1 d0 )(x, θ) 

i1,i2=1 

, θ =(θ (i) ). 

[C] Let ϑn = |θ ∗ 2 − θ∗ 1 | and |ˆ θk − θ ∗ k | = op(ϑn) as n →∞for k =1, 2. 

In case the parameters are known, ˆ θk should read θ ∗ k , and then Condition [C] 

requires nothing. 



problem 






Two kinds of limit theorems 

Let ϑn = |θ∗ 2 − θ∗ 1 |. We will consider the following two different 

situations. 

A : “Non shrinking case”: θ ∗ 1 and θ∗ 2 

are fixed and do not depend on n. 

B : “Shrinking case” (contiguous alternatives): θ∗ 1 and θ∗ 2 depend on n, 

and as n →∞, θ∗ 1 → θ∗ ∈ Θ, ϑn → 0 and nϑ2 n →∞. 

In Case A, ϑn is a constant ϑ0 independent of n. 



problem 






Consistency 

Theorem 1: The family {nϑ 2 n(ˆτn − τ ∗ )}n∈N is tight under any one of the 

following conditions. 

(a) [H]1, [A] and [C] hold in Case A. 

(b) [H]2, [B] and [C] hold in Case B. 

In Case B, this result gives immediately consistency of ˆτn since 

nϑ 2 n →∞by assumption. 



problem 






Let 

Asymptotic distribution: shrinking case 

 

H(v) =−2 

Γ 1 

2 

η W(v) − 1 

2 Γη|v| 

for Γη =(2T ) −1 Ξ(Xτ ∗,θ ∗ )[η ⊗2 ]. Here W is a two-sided standard 

Wiener process independent of Xτ ∗. 

Theorem 2: Suppose that the limit η = limn→∞ ϑ−1 n (θ∗ 2 − θ∗ 1 ) exists. 

Suppose that [H]2, [C] and [B] are fulfilled in Case B. Then 

as n →∞. 

nϑ 2 n(ˆτn − τ ∗ )→ d arg min 

v∈R H(v) 

Remind that in the simple 1-d case we have 

nϑ2 n(ˆτn − τ ∗ ) 

2ˆ θ2 

d 

→ arg max W(v) − 

v 

|v| 

 

2 



problem 






Let 

K(v) = 

i=1 

Asymptotic distribution: non shrinking case 

v 

 

Tr 

tσ(θ∗ 2,Xτ∗)S(Xτ ∗,θ∗ 1) −1 − S(Xτ ∗,θ∗ 2) −1 σ(θ ∗ 2,Xτ∗)ζ⊗2 

 

− log det S(Xτ ∗,θ∗ 1) −1 S(Xτ ∗,θ∗ 

2) , 

where ζi are independent r-dimensional standard normal variables and 

independent of Xτ ∗. 

Theorem 3: Suppose that [H]1, [C] and [A] are fulfilled in Case A. Then 

 

nτ ∗ 

ˆkn − → 

T 

d arg min 

v∈Z K(v) 

as n →∞. 


i


problem 






Two stage procedure 

When θ1 and θ2 are not known, we propose a two stage estimation 

procedure: 

First θ1 is estimated with ˆ θ1 using the first part of the series [0,an] 

and θ2 with ˆ θ2 using the last part [T − an,T]. 

Then a first stage change point estimator ˆτ is obtained plugin in the 

contrast function the first stage estimators ˆ θ1 and ˆ θ2. 

Then, a second stage estimator of the change point ˇτ is obtained, and 

further refinement of the estimators of θ1 and θ2 can be calculated, 

say ˇ θ1 and ˇ θ2. 



problem 






Two stage procedure 

Under a suitable sequence an, Theorems 1-3 hold for (ˇτ, ˇ θ1, ˇ θ2). 

For example, we assume that there exitst a constant β ∈ (0, 1/2) such that 

an ≥ 1/(nϑ 1/β 

n ) and that | ˆ θk − θ ∗ k | = op((nan) −β ) for k =1, 2. 

When lim sup n→∞ ϑn > 0, we also assume nan →∞. 

The first condition implies nϑ 2 n →∞, which is required in our theorems. 

The second condition is natural because the number of data is 

proportional to nan. 



problem 






A modified QML 

For small sample sizes or when the asymptotic is not realized, it may be 

the case to modify the QL in the following way 

with 

and 

Φ ′ n(t; θ1,θ2) = 

[nt/2T ] 

 

i=1 

G ′ i(θ1)+ 


n 

i=[nt/2T ]+1 

G ′ i(θ2), (4) 

G ′ i(θ) =logdetS(Xt2i−1 ,θ)+∆−1 

n S(Xt2i−1 ,θ)−1 [( ˜ ∆iY ) ⊗2 ] 

˜∆Yi = Y2i+1 − 2Y2i + Y2i−1 

√ 2 

With this approach we use less observation in the estimator of τ ∗ but we 

control better for the drift effect. Asymptotically the same results hold. So 

no change in the proofs and in the construction of the estimators.


problem 







To test the finite sample behaviour of all estimators considered in this 

review, we set up a simulation study. 

We consider different combinations of the sample size 

n = 500, 1000, 2000 and time horizon T =1, 2, 5, 10, which correspond 

to different ∆n =0.01, 0.001. Each experiment is replicated 1000 times. 

We consider three different statistical models. 

We compare the estimators: LS (ˆτn), CUSUM (˜τn), QML (ˇτn) and the 

modified QML (ˇτ ′ n). 



problem 






Finite sample properties. Driftless model. 

n ∆n T = n∆n τ ∗ ˆτn ˇτn ˜τn ˇτ ′ n 

L.S. Q.MLE CUSUM Modified Q.MLE. 

1000 0.001 1 0.6 0.606 0.602 0.751 0.604 

(0.061) (0.012) (0.150) (0.031) 

2000 0.001 2 1.2 1.202 1.205 1.784 1.214 

(0.196) (0.095) (0.316) (0.109) 

500 0.01 5 3 3.020 3.024 4.206 3.075 

(0.698) (0.512) (0.801) (0.623) 

1000 0.01 10 6 5.981 6.087 8.804 6.129 

(1.122) (1.617) (1.547) (1.296) 

dXt =(1+X 2 t ) θ∗ 

dWt 



problem 






Finite sample properties. Model with drift. 


L.S. Q.MLE CUSUM Modified Q.MLE 

1000 0.001 1 0.6 0.609 0.600 0.692 0.602 

(0.018) (0.005) (0.159) (0.011) 

2000 0.001 2 1.2 1.230 1.200 1.272 1.202 

(0.049) (0.005) (0.223) (0.010) 

500 0.01 5 3 3.644 2.977 3.005 3.002 

(0.320) (0.067) (0.012) (0.052) 

1000 0.01 10 6 8.179 3.607 5.983 6.001 

(0.221) (0.375) (0.045) (0.011) 

dXt = Xtdt +(1+X 2 t ) θ∗ 

dWt 



problem 






Finite sample properties. Mean reverting. 


L.S. Q.MLE CUSUM Modified Q.MLE 

1000 0.001 1 0.6 0.604 0.601 0.895 0.602 

(0.008) (0.007) (0.136) (0.015) 

2000 0.001 2 1.2 1.204 1.202 1.842 1.204 

(0.009) (0.009) (0.258) (0.017) 

500 0.01 5 3 3.030 3.033 4.299 3.013 

(0.051) (0.084) (0.691) (0.145) 

1000 0.01 10 6 6.036 6.027 8.947 6.022 

(0.076) (0.078) (1.367) (0.146) 

dXt =(2− Xt)dt + θ ∗ dWt 



problem 






Empirical vs True Asymptotic Distribution. Model 1 

Due to mixed-normal limit, we studied the distribution of the studentized 

limiting distribution of nθ 2 n(ˇτn − τ ∗ ) under the true model, i.e. 

Z = nθ 2 n(ˇτn − τ ∗ ) ˆ Γ(Xτ ∗,θ0), 

with ˆ Γ(Xτ ∗,θ0) = (log(1 + X2 τ ∗)) 2 . Then Z converges to W(v) − 1 

2 |v| 

with density 

f(x) = 3 

2 e|x| 

 

1 − Φ 

 

3 

|x| − 

2 

1 

2 

 

1 − Φ 

 

1 

|x| 

2 

and distribution function F (x) = 

 

g(x), 

1 − g(−x), 

x > 0 

, Φ(x) the 

x ≤ 0 

distribution function of N (0, 1), and 

 

x x 

g(x) =1+ e− 8 − 

2π 1 

√ 

x 

(x +5)Φ − + 

2 2 

3 

2 ex 

Φ − 3 

 

√ 

x 

2 

(see e.g. Csörgő and Horváth, 1997). 


Density 

0.0 0.1 0.2 0.3 0.4 0.5 

Empirical vs True Asymptotic Distribution. Driftless model. 

histogram vs limit distribution (second stage) 

−30 −20 −10 0 10 20 30 

Fn(x) 

−30 −20 −10 0 10 20 30 


0.0 0.2 0.4 0.6 0.8 1.0 

EDF vs limit (second stage) 

● ● ●● ●●●● ●●● ●● ●● ●● ● ●●● ●● ● ●● ●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● 

●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ● ● ● ●● ● ● ●●●●● ●● ●●●●●●●● 

●●●


problem 







Example of Volatility Change-Point Estimation 

Consider the 2-dimensional stochastic differential equation 

 

dX1 t 

dX2 t 

 

sin(X1 t ) 

= 

3 − X2 

θ1.k · X 

dt + 

t 

1 t 0 · X1 0 · X 

t 

2 t 

with change point instant at time τ =4 

x1 

x2 

2.0 3.0 4.0 

2 3 4 5 

X 1 0 =1.0, X 2 0 =1.0, 

θ2.k · X 2 t 

dW 1 t 


dW 2 t 

0 2 4 6 8 10 

t

The following model can be specified in the yuima package 

 

dX1 t 

dX2 t 

 

sin(X1 t ) 

= 

3 − X2 

θ1.k · X 

dt + 

t 

1 t 0 · X1 0 · X 

t 

2 t 

> library(yuima) 

θ2.k · X 2 t 

YUIMA Package 

dW 1 t 

> diff.matrix drift.c drift.matrix ymodel yuima


problem 





We can call CPoint on the yuima object 

Example with YUIMA package 

> t1 t2 t.est t.est$tau 

[1] 3.99 

> t.est$param1 

theta1.k theta2.k 

0.4723067 0.2899005 

> t.est$param2 

theta1.k theta2.k 

0.2515379 0.5518635 

http://R-Forge.R-Project.org/projects/yuima 



problem 





The model specification 

We consider here the three main classes of SDE’s which can be easily 

specified. All multidimensional and eventually parametric models. 



problem 








Diffusions dXt = a(t, Xt)dt + b(t, Xt)dWt 



problem 









Fractional Gaussian Noise, with H the Hurst parameter 

dXt = a(t, Xt)dt + b(t, Xt)dW H t 



problem 









Fractional Gaussian Noise, with H the Hurst parameter 

Diffusions with jumps, Lévy 

dXt = a(t, Xt)dt + b(t, Xt)dW H t 

dXt = a(Xt)dt + b(Xt)dWt + 

+ 

 

01 

c(Xt−,z)µ(dt, dz) 

c(Xt−,z){µ(dt, dz) − ν(dz)dt}

egular 

irregular 

tick times 

space disc. 

... 

multigrid 

asynch. 

Sampling 

random 

deterministic 

univariate 

multivariate 

Data 

Yuima 

ts, xts, ... 

diffusion 

Lévy 

CoGarch 

Model 

fractional 

BM 

Markov 

Switching HMM

Exact 

Asymptotic 

expansion 

Monte 

Carlo 

Euler- 

Maruyama 

Simulation 

Option 

pricing 

Covariation 

LASSOtype 

Space 

discr. 

Akaike’s 

Nonparametrics 

Yuima 

Model 

selection 

p-variation 

High freq. 

Low freq. 

Adaptive 

Bayes 

MCMC 

Hypotheses 

Testing 

Parametric 

Inference 

Quasi 

MLE 

Diff, fBM, 

Jumps 

Change 

point


problem 





For more informations and software see 

The YUIMA Project 

http://R-Forge.R-Project.org/projects/yuima 

A. Brouste (Univ. Le Mans, FR) 

M. Fukasawa (Osaka Univ. JP) 

H. Hino (Waseda Univ., Tokyo, JP) 

S.M. Iacus (Milan Univ., IT) 

K. Kengo (Tokyo Univ., JP) 

H. Masuda (Kyushu Univ., JP) 

Y. Shimitzu (Osaka Univ., JP) 

M. Uchida (Osaka Univ., JP) 

N. Yoshida (Tokyo Univ., JP) 

. . . more to come 


References 

Bai, J. (1994) Least squares estimation of a shift in linear processes, Journal of Times Series Analysis, 15, 453-472. 

Bai, J. (1997) Estimation of a change point in multiple regression models, The Review of Economics and Statistics, 79, 551-563. 

Chen, G., Choi, Y.K., Zhou, Y. (2005) Nonparametric estimation of structural change points in volatility models for time series, Journal of Econometrics, 126, 

79-144. 

Csörgő, M., Horváth, L. (1997) Limit Theorems in Change-point Analysis. New York: Wiley. 

De Gregorio, A., Iacus, S.M. (2008) Least squares volatility change point estimation for partially observed diffusion processes, Communications in Statistics, 

Theory and Methods, 37(15), 2342-2357. 

Genon-Catalot, V., Jacod, J. (1993) On the estimation of the diffusion coefficient for multidimensional diffusion processes, Ann. Inst. Henri Poincaré, 29, 

119–151. 

Genon-Catalot, V., Jacod, J. (1994) On the estimation of the diffusion coefficient for diffusion processes: random sampling, Scand. J. Stat., 21, 192–221. 

Habibi, R. (2012) Change point detection in Black-Scholes models using recursive estimation, Int. J. of E-Business Dev., 2(1), 13-15. 

Hinkley, D.V. (1971) Inference about the change-point from cumulative sum tests, Biometrika, 58, 509-523. 

Iacus, S.M., Yoshida, N. (2012) Estimation for the change point of volatility in a stochastic differential equation, Stochastic Processes and Their Applications, 

122(3), 1068-1092. 

Inclan, C., Tiao, G.C. (1994) Use of cumulative sums of squares for retrospective detection of change of variance, Journal of the American Statistical 

Association, 89, 913-923. 

Kessler, M. (1997) Estimation of an ergodic diffusion from discrete observations, Scand. J. Stat., 24, 211–229. 

Kim, S., Cho, S., Lee, S. (2000) On the cusum test for parameter changes in GARCH(1,1) models. Commun. Statist. Theory Methods, 29, 445-462. 

Kutoyants, Y. (1994) Identification of Dynamical Systems with Small Noise, Kluwer, Dordrecht. 

Kutoyants, Y. (2004) Statistical Inference for Ergodic Diffusion Processes, Springer-Verlag, London. 

Lee, S. (2011) Change point test for dispersion parameter based on discretely observed sample from SDE models, Bull. Korean. Math. Soc, 48(4), 839-845. 

Lee, S., Ha, J., Na, O., Na, S. (2003) The Cusum test for parameter change in time series models, Scandinavian Journal of Statistics, 30, 781-796. 

Lee, S., Nishiyama, Y., Yoshida, N. (2006) Test for parameter change in diffusion processes by cusum statistics based on one-step estimators, Ann. Inst. Statist. 

Mat., 58, 211-222. 

Song, J., Lee, S. (2009) Test for parameter change in discretely observed diffusion processes, forthcoming in Statistical Inference for Stochastic Processes. 

Yoshida, N. (1992) Estimation for diffusion processes from discrete observation, J. Multivar. Anal., 41(2), 220–242.

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