In this case
^( X ) = D ^ a n d *{x) = D^
9t(c) = — c1 - Xti{t,u)ttc ; c > 0 7
h{F) = — F7 - \n{t,u)ZTF ; F > 0 7
for all t e [0, T] and u G 0 . We have g't(c) = 0 for
c = c , M = [ ^ ] ^ and by concavity this is the maximum point of gt.
Similarly
F = FX(OJ)
D2
(30) (31)
(32)
(33)
is the maximum point of ht.
We now seek A* such that (35) holds, i.e.
E or
/ fat
Jo
'^fat'
L Di \
X^N = z,
(34)
where
N = E f
Jo
fat fat
Di T „^±T
fir
Jo Dy-
L>2
EUr1 )dt+^E[r,f tl
D7 _ 1
(35)
Hence
A* = (s) z \ 7 - i (36)
Substituted into (32) and (33) yields the optimal consumption rate c\'{t,u) = fay-1
N\DiJ and
*•<ô>-*(^r
logx
If </>(x) — ^ p and tp(x) = - ^ p , then in similar way we have A*
D2
and
c*(*,w;
X*^TVT
D2
^fat It is easy to see that c*(t,u) is Ht adapted.
(37)
(38)
(39)
(40)
(41)
4 . C o n c l u s i o n
In this paper we study the optimal consumption and portfolio problem (9) under t h e constraint (l)-(5). T h e problem is reduced t o Problem III, namely, (25)-(26), which can be solved explicitly for some specific utility functions. Solving Problem III yields ct and F. Here, (ct,£ > 0) is t h e optimal consumption r a t e process. To find t h e optimal portfolio process 9t = (at,Pt), one uses (19)-(20) t o find ftt- One m a y use (16) to find zt a n d t h e n use (15) to find at.
R e f e r e n c e s
1. Beran, J. Statistics for long-memory processes. Monographs on Statistics and Applied Probability, 61. Chapman and Hall, New York, 1994.
2. Biagini, F.; Hu, Y.; 0ksendal, B. and Sulem, A. A stochastic maximum prin- ciple for processes driven by fractional Brownian motion. Stochastic Process.
Appl. 100 (2002), 233-253.
3. Cox J. and Huang C.F. Optimal consumption and portfolio policies when asset prices follow a diffusion process. Journal of Economic Theory 49 (1989), 33-83.
4. Cox J. and Huang C.F. A variational problem arising in financial economics.
J. Mathematical Economics 20 (1991), 465-487.
5. Duncan T.E., Hu, Y. and Pasik-Duncan B. Stochastic calculus for fractional Brownian motion. I. Theory. SIAM J. Control Optim. 38 (2000), 582-612.
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7. Hu Y. Option pricing in a market where the volatility is driven by fractional Brownian motions. Recent developments in mathematical finance (Shanghai, 2001), 49-59, World Sci. Publishing, River Edge, NJ, 2002.
8. Hu, Y. and Oksendal, B. Fractional white noise calculus and applications to finance. Preprint, University of Oslo, 1999.
9. Hu, Y.; Oksendal, B. and Sulem, A. Optimal portfolio in a fractional Black &
Scholes market. Mathematical physics and stochastic analysis (Lisbon, 1998), 267-279, World Sci. Publishing, River Edge, NJ, 2000.
10. B. B. Mandelbrot: Fractals and Scaling in Finance: Discontinuity, Concen- tration, Risk. Springer-Verlag, 1997.
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W I T H A C H A N G I N G D R I F T *
SHIQING LING Department of Mathematics
Hong Kong University of Science and Technology Clear Water Bay
E-mail: maling@ust.hk
T h i s paper investigates the maximum likelihood estimator (MLE) of structure -changed ARMA-GARCH models. The convergent rates of the estimated change- point and other estimated parameters are obtained. After suitably normalized, it is shown that the estimated change-point has the same asymptotic distribution as t h a t in Picard(1985) and Yao (1987). Other estimated parameters are shown to be asymptotically normal. As special cases, we obtain the asymptotic distributions of MLEs for structure-changed GARCH models, structure-changed ARMA models with structure-unchanged GARCH errors, and structure-changed ARMA models with i.i.d. errors, respectively.
1. Introduction
Consider the following autoregressive moving-average (ARMA) model with the generalized autoregressive conditional heteroscedasticity (GARCH) er- rors:
p ô
Vt = '52<Piyt-i + '52'<Pi£t-i+£t, (1)
i=\ i=\
T S
£t = r\t\fhu ht = a0 + ] P o n e \ - i + ^2^ht-u (2) where p, q, r and s are known positive integers and r\t are indepen-
dent and identically distributed (i.i.d.). Equations (1.1)-(1.2) is called the ARMA-GARCH model and denoted by M(A), where A = {m',6')' with m = ((pi,--- ,4>P,^i,--- ,il>q)' and 5 = (a0,ai,--- , ar, / 3 i , - - - ,/?s)'. Denote
" T h i s research is supported by the competitive earmarked research grant
#hkust6113/02p.
174
Yf = (j/i,--- ,2/fc)'- Yi 6 M(A0) means that yit--- ,yk are generated by model (1.1)-(1.2) with the true parameter A = Ao- We say that Y™ follows a structure-changed ARMA-GARCH model if there exist fco S [l,n — 1], Ao £ 0 and Aoi € © with Ao / Aoi so that
Y{° e M(A0) and Ffe"0+1 G M(A01). (3) This structure-changed ARMA-GARCH model is denoted by
M(ko, Ao, Aoi). ko is called the change-point of this structure-changed model. Yj™ 6 M(fco, Ac Aoi) means that (1.3) holds. The focus of this paper is to investigate the maximum likelihood estimator (MLE) of model M(fc0,A0,Aoi).
Structural change has been recognized to be an important issue in econo- metrics, engineering, and statistics for a long time. The literature in this area is extensive. The earliest references can go back to Quandt (1960) and Chow (1960). Many approaches have been developed to detect whether or not structural change exists in a statistical model. Examples are the weighted likelihood ratio test in Picard (1985), and Andrew and Ploberger (1994), Wald and Lagrange multiplier tests in Hansen (1993), Andrews (1993), and Bai and Perron (1998), the exact likelihood ratio test in Davis, Huang and Yao (1993), the empirical methods in Bai (1996), and the se- quential test in Lai (1995). A general theory for exact testing change-points in time series models was established by Ling (2002a). Empirically, we want to know not only that structural change exists, but also the location of change-point.
Hinkley (1970) and Hinkley and Hinkley (1970) investigated the MLE of change-points in a sequence of i.i.d. Gaussian random variables and the binomial model, respectively. Their changed parameters are fixed and the limiting distributions of the estimated change-points seem not to be useful in practice. Picard (1985) allowed the difference between parameters before and after the change-point in AR models to tend to zero but not too fast when the sample size tends to infinity, and obtained a nice limiting distribution for the estimated change-point. This distribution can be used to construct the confidence intervals of the change-point, and hence it is very useful in applications, as these confidence intervals indicate the degree of estimation accuracy. Yao (1987) used a similar idea for independent data and obtained the same limiting distribution as Picard's. Picard's method has been developed for the regression models by Bai (1994, 1995, 1997).
In particular, Bai, Lumsdaine and Stock (1998) used Picard's method for the structure-changed multivariate AR model and cointegrating time series
model, and derived the asymptotic distributions of the change-points in these models. Chong (2001) developed a comprehensive theorem for the structure-changed AR(1) model. A general theory for estimating change- points in time series models with a fixed drift was established by Ling
(2002b).
In this paper, we use Picard's method to model M(kQ, Ao, Aoi). The con- vergent rates of the estimated change-point and other estimated parame- ters are obtained. After suitably normalized, it is shown that the estimated
change-point has the same asymptotic distribution as that in Picard(1985) and Yao (1987). Other estimated parameters are shown to be asymptoti- cally normal. As special cases of model M(fco, Ao,Aoi), this paper obtains the asymptotic distributions of MLEs for structure-changed GARCH mod- els, structure-changed ARM A models with structure-unchanged GARCH errors, and structure-changed ARMA models with i.i.d. errors, respectively.
This paper proceeds as follows. Section 2 presents main results. In Section 3 we report some Monte Carlo results. We give the proofs of main results in Apendix A.