Revisiting Parameter Estimation for Diffusion Processes

See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/255569485 REVISITING PARAMETER ESTIMATION FOR DIFFUSION PROCESSES Article CITATIONS READS 0 14 2 authors, including: Stergios Fotopoulos Washington State University 76 PUBLICATIONS 614 CITATIONS SEE PROFILE All content following this page was uploaded by Stergios Fotopoulos on 26 December 2014. The user has requested enhancement of the downloaded file. All in-text references underlined in blue are added to the original document and are linked to publications on ResearchGate, letting you access and read them immediately. REVISITING PARAMETER ESTIMATION FOR DIFFUSION PROCESSES STERGIOS B. FOTOPOULOS & VENKATA K. JANDHYALA Washington State University The problem of estimating the parameter of both continuous and discrete time diffusion processes is revisited. Various limiting results are obtained including strong consistency, uniform consistency, asymptotic normality and other related distributional results. The derived results are illustrated through the Ornstein-Uhlenbeck process. Key words: Diffusion processes, Girsanov's theorem, Ornstein-Uhlenbeck process, time change for martingales. 1. INTRODUCTION The problem of estimating parameters of a continuous time process has been well studied in the literature. Some of the relevant studies in the area include Feigin (1976, 1978, 1979, 1981, 1985), Heyde (1978, 1987), Hudson (1982) and Küchler and Sørenson (1994, 1997, 1998), among others. Considering relatively simpler forms of the diffusion equation, authors thus far derived estimators of parameters and established important properties such as strong consistency and asymptotic normality. For example, it has been customary to assume that the innovation process follows the standard Brownian motion. There has been greater interest in recent years in applying diffusion processes for modeling physical phenomena in wide ranging areas including finance, environmental modeling and biological systems (e.g., see Chapter 7, Kloeden and Platen, 1995). Due to the presence of high volatility in the markets, most financial models require that the innovation process be nonnormal. They also demand greater scope and flexibility in the model itself. Keeping such issues in mind, we revisit the problem of parameter estimation for continuous time diffusion processes. We first begin by formulating the problem in a fairly general form with the standard Wiener process representing the innovations in the stochastic differential equation of the diffusion. From the viewpoint of deriving parameter estimates and their optimality properties, we let the family of probability measures of the process to belong to the conditional exponential family of Hunt processes. Thus, we express the non-anticipative functional of the stochastic differential equation as a product of a differentiable function of the parameter alone and a continuously differentiable function of the observations alone. Under this set-up, we first derive the explicit form of the maximum likelihood estimator (mle) of the above parametric function. PARAMETER ESTIMATION FOR DIFFUSION PROCESSES Subsequently, applying the random time change and optional sampling properties, we establish asymptotic normality of the mle of the parameter itself. In the next section, we move away from the standard Brownian motion set-up and formulate the problem under a physical white noise. Such a formulation is quite appealing for modeling volatile processes arising from financial applications. It turns out that under this set-up, the mle estimator of the process parameter is inconsistent. We then suggest a modified estimator and establish its strong consistency as well as its asymptotic normality. In observing a continuous process, often times, the observations are obtained at discrete time points (perhaps equidistant) of the sampling interval. Thus, parameter estimation for the discrete analog of the continuous time stochastic differential equation turns out to be quite important. In the last section, we undertake this problem and again deduce estimators of the parameter that satisfy the properties of strong consistency and asymptotic normality. Finally, the stochastic differential equation representing the familiar Ornstein-Uhllenbeck process along with its discrete analog are considered for illustrating the results derived in the paper. This example has been considered previously by many authors. Here, we reestablish some of their results and also identify further properties that have been hitherto not observed. Let (Ω, F , P ) be a complete probability space with a non-decreasing family of right continuous 2. DEVELOPMENT AND DIFFUSION PROCESSES sub- σ -algebras Ft , 0 ≤ t ≤ T . This section is concerned with the pair (λ , X t ) , 0 ≤ t ≤ T , where λ ∈ Θ ⊆ R is a parameter needed to be estimated from a single realization of the process X t , 0 ≤ t ≤ T . We set B = (Bt , Ft ) , 0 ≤ t ≤ T , to be a standard Wiener process and let X 0 be an F0 - measurable random variable independent of λ . Further, let the observable process X permit the stochastic differential dX t = µ ( X t ; λ ) dt + dBt , t ≤ T , X 0 = x , (2.1) where for convenience, the non-anticipative functional µ ( x; λ ) in (2.1) above is assumed to becontinuously differentiable and of bounded slope. The bounded slope condition guarantees that the diffusion process does not reach infinity in finite time. Under these conditions, the solution of the process X exists, and is unique. To this end, one may also generalize the solution to the Ito Processes. In particular, we may define Yt , 0 ≤ t ≤ T , to be an Ito process Y expressed as dYt = [γ (t , ω ; λ ) + µ (Yt ; λ )] dt + dBt , t ≤ T , Y0 = x , 2 (2.2) PARAMETER ESTIMATION FOR DIFFUSION PROCESSES where γ (s, ω ; λ ) : [0, ∞ ) × Ω × Θ → R satisfies the following conditions: i.) (s, ω ; λ ) → γ (s, ω ; λ ) on [0, ∞ ) . is B × F - measurable, where B denotes the Borel σ -algebra ii.) γ (s, ω ; λ ) is Ft -adapted, and [ ] iii.) E ∫ 0 γ 2 (s, ω ; λ )ds < ∞ ∀λ ∈ Θ . T Then, according to Girsanov’s Theorem (see e.g., Øksendal p.156, 1998), γ (s; λ ) = − µ (x; λ ) , a.s., Y0 is an F0 -measurable random variable independent of λ with P ( Y0 < ∞ ) = 1 . Moreover, if (CT , BT ) denotes the measurable space of the continuous functions x = (x s ), s ≤ T , then, the measures µ X and µ Y , corresponding to the processes X and Y, are equivalent and { dPX (ω ) = exp ∫ 0T µ (Bs , ω ; λ ) dBs − dPY and 1 2 T 2 ∫ 0 µ (Bs , ω ; λ ) ds t Bˆ t := − ∫ 0 µ (B s ; λ ) ds + Bt , t ≤ T (dYt }= M T (λ ; µ ) on FT (2.3) = dBt ) , (2.4) where B̂t is a Brownian motion with respect to PY . This, in turn, implies (from the weak uniqueness theorem) that the PY -law of Yt = Bt coincides with the PX -law of X tx ( X 0 = x ), so [ ( ) that ( )] [ ( )] , ( ) E f 1 X tx1 L f k X txk = E M T f 1 Bt1 L f k Btk (2.5) for any real continuous functions, f 1 , L , f k , with compact support. From (2.4) and (2.5), we may further conclude that the measures derived from different λ ’s are also equivalent. Specifically, for any λ1 , λ 2 ∈ Θ and fixed T dPλT1 dPλ2 T { = exp ∫ 0 [µ ( X s ; λ1 ) − µ ( X s ; λ 2 )] dX s − T = M T (λ1 , λ 2 ; µ ) . 1 2 T 2 2 ∫ 0 [µ ( X s ; λ1 ) − µ ( X s ; λ 2 )] ds } (2.6) clear that M T (λ1 , λ 2 ; µ ) is an FT -measurable function. This, obviously, defines the likelihood ratio as a Radon-Nikodym derivative. From (2.6), it is formulations presented in Feigin (1976), we let l t (λ ; µ ) = log M T (λ , λ0 ; µ ) , where λ0 is the true value of the parameter. Adapting some aspects of the Thus, since µ ( x; λ ) is a non-anticipative differentiable functional, 3 PARAMETER ESTIMATION FOR DIFFUSION PROCESSES U t (λ ; µ ) = ∂ l t (λ ; µ ) exists with finite expectation (because of iii.) and is FT -measurable for ∂λ all T ≥ 0 and λ ∈ Θ . Using (2.1) and some of the arguments presented in Feigin (1976), it can be now seen that for any finite t ≥ 0 U t (λ ; µ ) = ∫ 0 t = ∫0 t ∂ µ ( X s ; λ ) dB s ∂λ ∂ ∂ t µ ( X s ; λ ) dX s − ∫ 0 µ ( X s ; λ ) µ ( X s ; λ ) ds , a.s. [Pλ ] . ∂λ ∂λ (2.7). 2  T ∂   µ ( X t ; λ ) dt  < ∞ , ∀T ≥ 0 and using (2.7), the stochastic process Assuming that E λ  ∫ 0     ∂λ  {U t (λ ; µ ), Ft , t ≥ 0} paths of {U t (λ ; µ ), t ≥ 0} being right continuous with left limits that exist. Formally, we obtain that ∀t ≥ 0 , is a square integrable zero mean local martingale with almost all sample [ ] E [U t (λ ; µ )] = 0 , E U t2 (λ ; µ ) < ∞ U t (λ ; µ ) is Ft -measurable and E [U t (λ ; µ ) Fs ] = U s (λ ; µ ) a.s., ∀t ≥ s , λ ∈ Θ . (2.8) In light of (2.8), U is a continuous local martingale, where the quadratic variation of the process t ∂  µ ( X s ; λ ) ds , is [Pλ ] integrable. If we further assume U given by [U (λ ; µ ), U (λ ; µ )]t = ∫ 0  λ ∂   2 that U 0 ∈ L2 -bounded martingales, then it can be shown (see e.g., Revuz and Yor, p.129, 1999) that U 2 (λ ; µ ) − [U (λ ; µ ),U (λ ; µ )] is uniformly integrable and that for any pair s, t ∈ [0, ∞ ), s ≤ t , of stopping times, we have [ E U t2 (λ ; µ ) − U s2 (λ ; µ ) Fs ] ( ) = E  U t (λ ; µ ) − U s (λ ; µ )  2 [ ] Fs  = E [U (λ ; µ ), U (λ ; µ )]s Fs .  t to express the bounded continuous non-anticipative functional µ (x ; λ ) in terms of the Lipschitz Another way of looking for the unique solution of the stochastic differential equation (2.1) is condition given by d (µ ( x; λ ), µ ( y; λ )) ≤ ϕ (λ ) d (x, y ) , x, y ∈ R and λ ∈ Θ , 4 (2.9) PARAMETER ESTIMATION FOR DIFFUSION PROCESSES where ϕ : Θ → R is continuous and differentiable and d : R × R → R + denotes the usual metric (Euclidean metric). It should be noted that µ (x ; λ ) depends on x and λ only, not on t. Then, according to Øksendal (Ch. 7, 1998), the Ito diffusion process {X t , t ≥ 0} is time homogeneous. Further, for any bounded Borel function f , one may show that a. b. E x [ f ( X t + h ) Ft ] = E X t [ f ( X t + h )] , t , h ≥ 0 , X 0 = x , and E x [ f ( X τ + h ) Fτ ] = E Xτ [ f ( X τ + h )] , h ≥ 0 , X 0 = x and τ is a stopping time w.r.t. Fτ , τ < ∞ , a.s. Property (a.) yields the Ito diffusion process {X t , t ≥ 0} to be Markov and (b.) makes the diffusion process (2.1) to be strong Markov. To this end, we recommend that µ ( x ; λ ) = ϕ (λ )h(x ) (this is related to the exponential conditionally additive functionals), where h is a non-anticipative Borel functional of x only and ϕ : Θ → R is continuous and differentiable. This alteration on the non-anticipative functional µ (x ; λ ) leads one to Proposition 5.4 in Küchler and Sørensen (1998), which suggests that the process {X t , t ≥ 0} is a Brownian motion and the family of measures {Pλ , λ ∈ Θ} belongs to a conditional exponential family of Hunt processes. Thus, the Radon { Nikodym derivative in (2.3) may now be modified to M t (ω ; λ ; ϕ , h ) = exp ϕ (λ )∫ 0 h( X s , ω ) dB s − 12 ϕ (λ ) t 2 t 2 ∫ 0 h ( X s , ω ) ds }, t ≥ 0 and λ ∈ Θ . (2.10) Now, in place of h( X t ) , t ≥ 0 , if we have a continuous functional of only t, say γ (t ) , adapted to Ft , t ∈ [0, T ) with ∫ 0 γ (s, ω ) [ { t 2 ds < ∞ , ω ∈ Ω , such that E exp ϕ (λ )∫ 0 γ (s, ω ) dB s − 12 ϕ (λ ) t 2 t 2 ∫ 0 γ (s, ω ) ds }] = 1, for all t>0 and λ ∈ Θ , where Θ is some closed interval of the real numbers, then ∃ {Pλ , λ ∈ Θ} under which {X t , t ≥ 0} would be a solution of the stochastic differential equation ~ dX t = ϕ (λ )γ (t , ω )dt + dBt , t ≤ T , X 0 = x , ~ for some standard Brownian motion B on {Ω, F , Pλ , λ ∈ Θ} . For the process X to be Markov, it is, moreover, necessary that γ (t , ω ) = γ ( X t (ω )) , i.e., the process X should be a diffusion process In conjunction with (2.7) and letting µ ( x ; λ ) = ϕ (λ )h(x ) , where ϕ is differentiable, we obtain (see e.g., Küchler and Sørensen, 1998). 5 PARAMETER ESTIMATION FOR DIFFUSION PROCESSES [ ] U t (λ ; ϕ , h ) = ϕ& (λ ) ∫ 0 h( X s ) dX s − ϕ (λ )∫ 0 h 2 ( X s ) ds , a.s. [Pλ ] . t t (2.11). Without loss of generality, we let the random variable X 0 = ξ be independent of Wt such that E [ξ ] = 0 . Then, the solution of the stochastic differential equation (2.1) is given by X t = ξ + ϕ (λ )∫ 0 h( X s ) ds + Bt t ≥ 0 and λ ∈ Θ . { t } (2.12) Note that the process {G (λ ), λ ∈ Θ} = Gt (λ ) := X t − ϕ (λ )∫ 0 h( X s ) ds − ξ , Ft , t ≥ 0 , is also a t square integrable zero mean local martingale with quadratic variation [G (λ ), G (λ )]t = t , t ≥ 0 . Some simple consequences of defining the process G may be formulated as follows: for any t ≥ 0 , λ ∈Θ , Gt (λ ) = dGt (λ ) t = − ∫ 0 h( X s ) ds , dϕ (λ ) [ ] (2.13) U t (λ ; ϕ , h ) = ϕ& (λ ) ∫ 0 h( X s ) {dX s − ϕ (λ )h( X s ) ds} = ϕ& (λ )∫ 0 t and ϕ& (λ ) Gt (λ ) = −[G (λ ), U (λ ; ϕ , h )]t . t dG s (λ ) d [G (λ ), G (λ )]t dG s (λ ) , (2.14) (2.15) Applying Kunita-Watanabe’s inequality into equation (2.15), one may obtain that [U (λ ; ϕ , h ), U (λ ; ϕ , h )]t ≥ ϕ& 2 (λ )(Gt (λ )) [G(λ ), G(λ )]t 2 , t ≥ 0 , λ ∈Θ . (2.16) Clearly, the expected value with respect to [Pλ ] of the left hand side in (2.16) is just the Fisher information at time t, for λ ∈ Θ . Thus, inequality (2.16) provides a lower bound of the estimators of the functional ϕ (λ ) , λ ∈ Θ . Fisher information. {G (λ ), λ ∈ Θ} Clearly, the estimator ϕ (λ )t for the family Equation (2.15) also plays an important role in constructing efficient is constructed as a solution of the equation U t (λ ; ϕ , h ) = 0 (assume that ϕ& (λ ) ≠ 0 and bounded). Specifically, the mle estimator of ϕ (λ ) is given by ϕ (λ )t = [∫ h (X ) ds] {∫ h(X ) dX }, t ≥ 0 . t 0 −1 2 s t 0 s s In view of (2.17), the process {U (λ ; ϕ , h ), λ ∈ Θ} in (2.14) can then be written as ( ) U t (λ ; ϕ , h ) = ϕ& (λ )∫ 0 h 2 ( X s ) ds ϕ (λ )t − ϕ (λ ) t 6 (2.17) PARAMETER ESTIMATION FOR DIFFUSION PROCESSES ( ) = ϕ& −1 (λ ) [U (λ ; ϕ , h ), U (λ ; ϕ , h )]t ϕ (λ )t − ϕ (λ ) , t ≥ 0 . (2.18) As already pointed out via (2.16), the quadratic variation of the process {U (λ ; ϕ , h ), λ ∈ Θ} has tremendous implications to parameter estimation of statistical random processes. As U t (λ ; ϕ , h ) , t ≥ 0 , is cadlag, that is, it has right continuous paths with left hand limits, the indicated below, there are further important ways that this process may be utilized. Since predictable quadratic variation {[U (λ;ϕ , h),U (λ;ϕ , h)] , t ≥ 0} t always exists. It is unique, U t2 (λ ; µ ) − [U (λ ; µ ), U (λ ; µ )]t , t ≥ 0 , is a local martingale vanishing almost surely at zero. To predictable, and increasing in t. Consequently, one may easily show that the random process this end, we may further assume that lim t →∞ [U (λ ; ϕ , h ), U (λ ; ϕ , h )]t = ϕ& 2 (λ ) lim t →∞ t 2 ∫ 0 h ( X s ) ds = ∞ a.s. In view of this, one may continue this analytical approach and bridge diffusion processes and statistical inference. Specifically, referring to the “Time-Change for Martingales” result, we obtain various statistical properties for estimators of the parameter λ . More precisely, in Gt = Fτ t , the stopping rule τ s = inf {t > 0 : [U (λ ; ϕ , h ), U (λ ; ϕ , h )]t > s} and Bt = U τ t (λ ; ϕ , h ) , t ≥ 0 , connection with the Lévy’s characterization of Brownian motion, we can define subspaces λ ∈Θ . {Bt , Gt }, t ≥ 0 , is a standard Brownian motion, {[U (λ ; ϕ , h ), U (λ ; ϕ , h )]t : t ≥ 0} are stopping times for It can be shown that Gt and U t (λ ; ϕ , h ) = B[U (λ ;ϕ ,h ),U (λ ;ϕ , h )]t a.s., for all t ≥ 0 and at any λ ∈ Θ . (2.19) In light of (2.19), it clearly follows that if we divide (2.18) by the square root of the quadratic variation of U, the resulting quantity is just a standard normal random variable. We have thus established that ( [U (λ ; ϕ , h ), U (λ ; ϕ , h )]t )−1 2 U t (λ ; ϕ , h ) = ϕ& −1 (λ )( [U (λ ; ϕ , h ), U (λ ; ϕ , h )]t ) 12 (ϕ (λ ) t ) − ϕ (λ ) = B1 a.s. (2.20) Thus, in (2.20) we establish normality property of the mle of the parametric function ϕ . Next, we consider the problem of estimating the process parameter λ . To this end, we begin by providing few restrictions on the parametric function ϕ . Since ϕ is a continuous function on Θ ⊂ R , we have that ∀λ ∈ Θ (interior) ∃δ > 0 and c > 0 such that 7 PARAMETER ESTIMATION FOR DIFFUSION PROCESSES ( ) ( ( )) ( ) d λ , λˆ < δ ⇒ d ϕ (λ ), ϕ λˆ < cd λ , λˆ , ∀λ̂ ∈ Θ . (2.21) Setting ϕ (λ ) = λ and U t (λ ;1, h ) = U t (λ ; h ) , (2.18) may be rewritten as ( ) U t (λ ; h ) = [U (λ ; h ), U (λ ; h )]t λˆt − λ , t ≥ 0 , where λ̂t = [∫ h (X ) ds] t −1 2 ∫ 0 h( X s ) dX s t variation [U (λ ; h ), U (λ ; h )]t → ∞ a.s. s 0 (2.22) is the mle estimator of λ . Further, the quadratic Now, since U t (λ ; h ) is a zero mean local martingale, according to Birkhoff’s ergodic theorem for continuous times, ([U (λ ; h ), U (λ ; h )]t ) U t (λ ; h ) → 0 a.s. in L1 . Consequently, the ( −1 ) Euclidean distance d λ̂t , λ → 0 a.s. in L1 . (( ) ) d ϕ λˆt , ϕ (λ ) → 0 a.s. in L1 . Since (2.21) is satisfied, it easily follows that continuous times to the original parametric function ϕ (λ ) , we may also conclude that the Repeating, once again, the Birkhoff’s ergodic theorem for ( ) distance d (ϕ (λ ) , ϕ (λˆ )) → 0 a.s. in L . In light of this, (2.21) may now be tailored by replacing ϕ (λ ) with ϕ (λ̂ ) . In addition, letting ϕ ( y ) := inf {s : ϕ (s ) ≥ y} and noting that ϕ is a distance d ϕ (λ )t , ϕ (λ ) → 0 a.s. in L1 . Thus, using the triangle inequality we can deduce that the 1 t t ← t t continuous function, we may obtain that the mle estimator for λ in the general case of (2.18) is [  given by λ̂t = ϕ ←  ∫ 0 h 2 ( X s ) ds t ] −1 ∫ 0 h( X s ) dX s  , a.s.,  t ( ) {( ) From the mean value theorem, it is clear that ( t ≥ 0. }ϕ& (1λ ) a.s., λˆt − λ = ϕ ← o ϕ λˆt − ϕ ← o ϕ (λ ) = ϕ λˆt − ϕ (λ ) ) ( ) (2.23) t ≥ 0, ∗ (2.24) where λ∗ ∈ λˆt , λ ∨ λ , λˆt . Secondly, assuming that ϕ& is a bounded function, it also follows that ( ) ( ) for the general case, d λ̂t , λ → 0 a.s. in L1 , which, in turn, implies d λ∗ , λ → 0 a.s. in L1 . (( ) ) Consequently, ϕ& d λ∗ , λ + λ → ϕ& (λ ) a.s. as t → ∞ . ( [U (λ ; ϕ , h ), U (λ ; ϕ , h )]t )1 2 (λˆt − λ ) → D B1 as t → ∞ . In view of (2.24) and (2.20), we may conclude that 8 (2.26) PARAMETER ESTIMATION FOR DIFFUSION PROCESSES In the preceding paragraphs, we have seen how random time change and optional sampling lead to establishing the property of normal approximations for the mle estimators of the parameter λ. situations, it is customary to replace the stopping times {[U (λ ; ϕ , h ), U (λ ; ϕ , h )]t , t ≥ 0} in (2.26) by Here, it is of special interest to adapt some related work in sequential analysis. In this type of deterministic functionals. Such adaptations are of particular interest in sequential analysis and in There exists a non-decreasing deterministic function C t (λ ; ϕ , h ) such that the theory of scale mixtures of normal distributions. To this end, we impose the following C t (λ ; ϕ , h ) → ∞ as t → ∞ and assumption. [U (λ ; ϕ , h ), U (λ ; ϕ , h )]t C t (λ ; ϕ , h ) → W (λ ; ϕ , h ) [Pλ ] as t → ∞ , (2.27) for some random function W (λ ; ϕ , h ) , W (λ ; ϕ , h ) > 0 a.s. [Pλ ] . Then, arguing as in Feigin (1976, 1981) or Heyde (1978), it can be shown that as t → ∞ (( [U (λ;ϕ , h),U (λ;ϕ , h)] ) (λˆ 12 t t ) ) − λ , [U (λ ; ϕ , h ), U (λ ; ϕ , h )]t C t (λ ; ϕ , h ) → D (B1 , W (λ ; ϕ , h )) , (2.28) where the positive random function W (λ ; ϕ , h ) is independent of the standard normal variable B1 . Consequently, as t → ∞ ( C t (λ; ϕ , h ))1 2 (λˆt ) − λ → D D(λ ; ϕ , h ) , where D(λ ; ϕ , h ) is a scale mixture of a normal distribution. In many statistical problems, model (2.1) is acceptable only with the proviso that B = (Bt , Ft ) , 3. PARAMETER ESTIMATION FOR SYSTEMS WITH PHYSICAL WHITE NOISE 0 ≤ t ≤ T is not a standard Brownian motion, but for a fixed time its distribution is close in some ( sense to a Brownian motion. Formally, one considers the innovations process B n = Btn , Ft n ), n ∈ N (natural numbers), to be a stationary process with asymptotically independent increments. ( ) [ ] Moreover, one lets B n to be uniformly square integrable for each t, tight, and that E Btn 2 E  Btn  → σ 2 t as n → ∞ .   ( B n = Btn , Ft n ), =0, It is clear that under these conditions the random process n ∈ N , converges weakly to a standard Brownian motion, i.e., Btn → D Bt as n → ∞ . An example of the process B n , n ∈ N , is the sequence of processes 9 PARAMETER ESTIMATION FOR DIFFUSION PROCESSES 0 ≤ t , n ∈ N and σ 2 = 2∫ 0 E [ξ 0ξ s ] ds , ξ ns ds , σ ∫0 Btn = n ∞ t [ ] where {ξ t : t ∈ R} is a strictly stationary ergodic process with E [ξ 0 ] = 0 , E ξ 02 < ∞ , and satisfies ∞ 2 ∫ 0 E [ξ 0 the condition of weak dependency F0 ξ ] 12 ds < ∞ and F0 ξ = σ (ξ s : s ≤ 0 ) . Under the above set-up, model (2.1) may now be reformulated as ( ) X tn = ξ + ∫ 0 µ X sn ; λ ds + Btn , t ≥ 0 , n ∈ N , X 0n = ξ . t (3.1) The focus of the present section then is to provide strongly consistent and asymptotically normal estimators of the parameter λ under the condition that the random process ( ) , n ∈ N , replaces the process B = (B , F ) . The key to the analysis is to show that the process {X B n = Btn , Ft } n t t ≥ 0, n ∈ N converges weakly to the [ ] n t ,t process {X t , t ≥ 0} . We assume, for convenience, that E Bt2 = σ 2 t and that both processes B and B n start from the same point, i.e., X 0 = X 0n = x . Note that for 0 ≤ t ≤ T ( E x  X tn − X t  )  = E 2 x { (( ) ) } 2 t  n  n  Bt − Bt − ∫ 0 µ X s ; λ − µ ( X s ; λ ) ds  . Using the Ito’s isometry property and the Lipschitz condition (2.9), it yields that ( E x  X tn − X t  )  ≤ 2E (B 2 x ( − Bt n t )  + 2tϕ 2 In view of (3.2), the function v(t ) = E x  X tn − X t  ( where a = 2 E x  Btn − Bt  conclude that )  2 2 (λ )∫ 0t E x (X sn − X s )2  ds .   (3.2) )  satisfies the inequality v(t ) ≤ a + bt ∫ 2 t 0 v(s ) ds , and b = 2ϕ 2 (λ ) . Therefore, applying the Gronwall’s inequality we v(t ) ≤ a exp(bt ) , t ≥ 0 . ( Since Btn → D Bt , n → ∞ , and B n = Btn , Ft ( easily shown that lim sup n→∞ E x  Btn − Bt  ) is uniformly square integrable and tight, it can be )  = 0 . n 2 follows that 10 Hence, by the continuity of t a X tn − X t , it PARAMETER ESTIMATION FOR DIFFUSION PROCESSES ( ) P lim sup n →∞ X tn − X t = 0, ∀t ∈ [0, T ] =1. (3.3) Thus, X tn converges weakly to the process X t , i.e., X tn → D X t , n → ∞ . From the inference point of view, it is natural to estimate λ adapting the same form as in λ̂t given in (2.23). That is, one may estimate λ by [  ( ) λtn = ϕ ←  ∫ 0 h 2 X tn ds ~ t ] −1 ∫ 0 h (X t t ) dX n n t   , a.s., t ≥ 0 .  (3.4) The interest then is to investigate whether the above estimator admits the same asymptotic ~ properties as the estimator λ̂t . Specifically, it will be shown below that the estimator λtn does not converge (always) weakly to the estimator λ̂t , t ≥ 0 , as n → ∞ and thus does not inherit the ~ same optimal asymptotic properties. Subsequently, we modify λtn in a certain way so as to We start our analysis by examining estimators of the form similar to ϕ (λ )t . Subsequently, obtain desired asymptotic properties. applying similar arguments as in the preceding section, we are able to establish properties of the ~ estimator λtn . Note that the process X in (2.12) is continuous with finite quadratic variation on [0, t ] given by [X , X ]t = t . Let the continuous functional h be differentiable. We then define f : [0, ∞ ) ∋ t a ∫ 0 t h(u ) du = f t (s ) ∈ R s to be a functional on C ([0, ∞ )) of some continuous functions s t , t ≥ 0 . In light of (3.3), it is not ( f (X ), X ) ⇒ ( f ( X ), X n hard to show that t n t t t ), ( ) n → ∞ . Clearly, the function f t X n can be ( ) ( ) expressed as the Lebesgue integral of the form f t X n = ∫ 0 h X tn dX tn . Applying the onedimensional Ito formula for the process f t ( X ) , we obtain that f t ( X ) = ∫ 0 h( X s ) dX s − t (∫ h(X ) dX ) (∫ h(X ) dX 1 2 t ∫ 0 h ′( X s ) ds , t ≥ 0 . t ), t ≥ 0 , n → ∞ . (3.5) In view of (3.5), the following weak convergence result may now be formulated as: t 0 n s ( n s , X tn → D ) t 0 s s − 1 2 ∫ 0 h ′( X s ) ds, X t t (3.6) Assuming that P ∫ 0 h 2 ( X s ) ds > 0 = 1 , and adapting a similar approach as in the limit result (3.6), t it may further be shown that the followings weak convergence results also hold: 11 PARAMETER ESTIMATION FOR DIFFUSION PROCESSES ( (X ) ds ) ∫ 0 h ′(X s ) ds → ∫ 0 h ′( X s ) ds , and ∫ 0 h t [U n t D n t 2 −1 n s (λ ; µ ), U n (λ ; µ )]t = ∫ 0t  ∂ µ (X tn ; λ )   ∂λ 2 →D (∫ h (X ) ds ) t 0 −1 2 s , t ≥ 0 , n → ∞ , (3.7) ds → D [U (λ ; µ ), U (λ ; µ )]t , t ≥ 0 , n → ∞ . (3.8) The above limit results suggest that, given a path of the process X n , the right form of an estimator ϕ (λ )t , t ≥ 0 and n ∈ N , for the function ϕ (λ ) may be expressed as n ∫ h(X s ) dX s − 12 ∫ 0 h ′(X s ) ds , ϕ (λ )t = 0 t 2 n ∫ 0 h (X s ) ds t n t n n n t ≥ 0 and n ∈ N . (3.10) In view of Section 2 and the results established in (3.6)-(3.10), the following theorem is in order. THEOREM 1. Let the function ϕ : Θ → R be continuous and differentiable with ϕ& (λ ) = dϕ (λ ) dλ being bounded. Let the function h : R → R be continuously differentiable satisfying the Lipschitz conditions: d (h(x ), h( y )) ≤ Ld (x, y ) d (h ′(x ), h ′( y )) ≤ Ld (x, y ) , and where d ( x, y ) denotes the usual Euclidean distance and L is some positive constant. Let ( ) X tn = ξ + ∫ 0 µ X sn ; λ ds + Btn , t ≥ 0 , n ∈ N , X 0n = ξ , t { } where µ ( x ; λ ) = ϕ (λ )h(x ) and the random variable ξ is independent of the process B n , n ∈ N . { } = {B , t ≥ 0} are differentiable with respect to t at each n ∈ N , Further, assume that the sequence B n , n ∈ N satisfies the following conditions: { } i.) the trajectories of B n ii.) (B ) , t ≥ 0 , is uniformly integrable for each t B = {B , t ≥ 0} is tight, and E [B ] = 0 , E (B )  → t as n → ∞ .   iii.) iv.) v.) n t B n = Btn , t ≥ 0 has asymptotically independent increments n 2 t n n t n t n 2 t Let, in addition, the following condition hold for any t>0 12 PARAMETER ESTIMATION FOR DIFFUSION PROCESSES ( ) P ∫ 0 h 2 ( X s ) ds > 0 = 1 . t Then, A.) For all λ ∈ Θ , the estimator (3.10) satisfies the following consistency results: a. b. c. n P lim sup n →∞ ϕ (λ )t − ϕ (λ )t = 0, ∀t ∈ [0, T ] = 1   ( P (lim ) P lim sup n →∞ λˆtn − λˆt = 0, ∀t ∈ [0, T ] = 1 n →∞ ) lim sup t →∞ λˆtn − λ = 0 = 1 . ( ) B.) The following joint convergence in distribution results hold a. b.  X n , ϕ (λ )n  → d X , ϕ (λ ) , and   (X n ( [U ( [U ) ( ) , λˆn → d X , λˆ as n → ∞ . (λ ; ϕ , h ), U n (λ ; ϕ , h )]t )−1 2 U tn (λ; ϕ , h ) → B1 as C.) The following normal approximation limits are true: a. b. n n (λ ; ϕ , h ), U n (λ ; ϕ , h )]t )1 2 (λˆtn − λ ) → D n → ∞ and t → ∞ B1 as n → ∞ and t → ∞ . D.) Suppose ∃ a non-decreasing deterministic sequence of functions C tn (λ ; ϕ , h ) , n ∈ N , such that C tn (λ ; ϕ , h ) → ∞ as both n → ∞ and t → ∞ and [U n (λ ; ϕ , h ), U n (λ ; ϕ , h )]t C tn (λ ; ϕ , h ) → W (λ ; ϕ , h ) [Pλ ] as n → ∞ and t → ∞ , for some random function W (λ ; ϕ , h ) , W (λ ; ϕ , h ) > 0 a.s. [Pλ ] . Then (C n t (λ ; ϕ ,))1 2 (λˆtn − λ ) → D D(λ ; ϕ , h ) , where D(λ ; ϕ , h ) is a scale mixture of an independent standard normal variable. The process X = {X t : t ≥ 0} satisfies equation (2.12), B = (Bt , Ft ) , 0 ≤ t ≤ T , is a standard Wiener process and λ̂t is given by (2.23). Consider the univariate diffusion process X = {X t , t ≥ 0} , obtained as a solution of the stochastic 4. ESTIMATING PARAMETERS UNDER DISCRETE DIFFUSION PROCESSES differential equation (2.1). Here, we are interested in estimating λ from discrete observations 13 {X } : i = 1, 2, L , n ti PARAMETER ESTIMATION FOR DIFFUSION PROCESSES of the process X. Let the step size between two consecutive observations be fixed and given by δ = t i +1 − t i , i = 1, 2, L , n (we consider time equidistant observations). If the transition densities p(s, x, t , y; λ ) of X are known, then it is clear that the log likelihood function ( l n (λ ) = ∑i =1 log p t i −1 , X ti −1 , t i , X ti ; λ n ) (4.1) may be used to obtain estimators of λ that satisfy consistency, asymptotic normality, efficiency, etc (see, e.g., Billingsley, 1968; Dacunha-Castelle & Florens-Zmirnou, 1986). However, difficulties prevail in evaluating the transition densities exactly. Therefore, in resolving this problem, we adapt some ideas proposed by Pedersen (1995). In the process, we provide strongly consistent and asymptotically normal estimators for λ . Mainly, this is attained by considering various approximations of the transition probabilities p(s, x, t , y; λ ) . Let p ( N ) (s, x, t , y; λ ) , N ∈ N , denote the approximate transition density for all fixed 0 ≤ s < t , x, y ∈ R and λ ∈ Θ . The approximation of the process X, used here, is due to Euler- Maruyama (see, e.g., Kloeden and Platen, Ch. 9, 1995). Define for k = 0, 1, L , N , τ k = s + kδ (δ = t−s ). The Euler method utilized here approximates the stochastic differential equation N ( (2.1) by the stochastic difference equation ) Yτ(kN ) = Yτ(kN−1) + δµ Yτ(kN−1) ; λ + ∆Bτ k , Ys( N ) = x and N ∈ N . (4.2) Assuming that (2.9) is satisfied, it can be seen that Yτ(NN ) = Yt ( N ) → X t , as N → ∞ in L1 (P(s, x,⋅,⋅; λ )) . It may also be seen (see also Pedersen, 1995) that for N=1 ( ) p (1) (s, x, t , y; λ ) = exp − ( y − x − (t − s )µ (x; λ )) 2(t − s ) 2 2π (t − s ) , and p ( N ) (s, x, t , y; λ ) = ∫ R N −1 ∏ kN=1 p (1) (τ k −1 , x k −1 , τ k , x k ; λ ) dx1 L dx N −1 , N ≥ 2 , (4.3) (4.4) with x 0 = x and x N = y . Note that the log-likelihood function, l n(1) (λ ) , n ∈ N , can be expressed as ( ) ( ) l n(1) (λ ) = K n + log M s(,Nt ) = K n + ∑i =1 µ X τ i −1 ; λ ∆X τ i −1 − 12 δ ∑i =1 µ 2 X τ i −1 ; λ , n n where K n is some random variable independent of the parameter λ . 14 (4.5) PARAMETER ESTIMATION FOR DIFFUSION PROCESSES If µ = 0 , then for i = 1, L , N , X τ k = X τ k −1 + ∆Bτ k , X 0 = x , is the one-dimensional Brownian motion after time s. If, on the other hand, µ ≠ 0 then from expression (4.2), the difference ( ) { } is a ) ) , k ≥ 1 . By stochastic equation Yτ(kN ) = Yτ(kN−1) + δµ Yτ(kN−1) ; λ + ∆Bτ k , Ys( N ) = x , is in place. Clearly, Yτ(kN ) ( Brownian Motion (Markov chain) with drift after time s. Let Fk ( N ) = σ Y0( N ) , L , Yτ(kN direct calculations it is easy to see that dPλs ,t Y ( N ) s ,t dQ X ( {∑ µ (x ) ( N ) x1 , L , x N = exp τ k −1 ; λ N k =1 ) ∆xτ )} ( − 12 δ ∑k =1 µ 2 xτ k −1 ; λ = M s(,Nt ) . N k N k =0 (4.6) Since µ is a continuous function, the right hand side of (4.6) converges in probability under Q as N → ∞ to the continuous version as expressed in (2.9) (see for details, Pedersen, 1995). obtain that p ( N ) (s, x, t , y; λ ) → p(s, x, t , y; λ ) as N → ∞ . Note that (4.6) and (4.5) share exactly This can be considered as the discrete version of Girsanov’s Theorem. Thus, one may easily the same information as far as the parameter λ is concerned. Hence, letting µ ( x ; λ ) = ϕ (λ )h(x ) and s=0, where ϕ is the same differentiable function as defined in (2.11) and h satisfies the usual [∑ h(Y ( ) ){∆Y ( conditions as in Sections 2 and 3, we obtain U t( N ) (λ ; ϕ , h ) = ϕ& (λ ) N k =1 τ k −1 N [ ( N) τk ( − δϕ (λ ) h Yτ(kN−1) ){ )} ] , a.s. [P ( ) ] . λ ( As in Section 2, we have E ϕ& (λ )h Yτ(kN−1) ∆Yτ(kN ) − δϕ (λ )h Yτ(kN−1) { ( E  ∆Yτ(kN ) − δϕ (λ )h Yτ(kN−1)  )} 2 N) (λ ; ϕ , h ), U ( N ) (λ ; ϕ , h )] t ( N Let ϕ (λ )t = arg max λ∈Θ ( ) = ϕ& 2 (λ ) δ ∑k =1 h 2 Yτ(kN−1) a.s. ] The quantity U ( N ) (λ ; ϕ , h ), U ( N ) (λ ; ϕ , h ) (N ) ) { ( = ϕ& 2 (λ ) ∑ k =1 h 2 Yτ(kN−1) E  ∆Yτ(kN ) − δϕ (λ )h Yτ(kN−1)  N [ )} F ( ) ] N k −1 (4.7) = 0 a.s. and } Fk −(1N )  = δ a.s. It then follows that U t( N ) (λ ; ϕ , h ), FN( N ) , N ≥ 1 is a  square integrable martingale. It can further be seen that [U ( { N dPλt ⋅ Y ( N ) dQ t X ( N ) )} 2 Fk −(1N )   (4.8) is obviously a form of conditional information. (x ,L, x ) denote the mle estimator of the function ϕ (λ ) . 1 t N Under the same conditions as in Section 2, the estimator of ϕ (λ ) is constructed as a solution of 15 PARAMETER ESTIMATION FOR DIFFUSION PROCESSES the equation U t( N ) (λ ; ϕ , h ) = 0 (assume that ϕ& (λ ) ≠ 0 and bounded). Specifically, it can be shown that ϕ (λ )t (N ) is given by ϕ (λ )t( N ) = [∑ N h2 k =1 (Y ( ) )] τ k −1 N −1 ∑k =1 h(Yτ(kN−1) )∆Yτ(kN ) , N ){ We thus may continue as in (2.20) to obtain that ( } (λ ; ϕ , h )] {ϕ (λ ) U t( N ) (λ ; ϕ , h ) = ϕ& (λ ) δ ∑ k =1 h 2 Yτ(kN−1) ϕ (λ )t  N [ (N ) = ϕ& −1 (λ ) U ( N ) (λ ; ϕ , h ), U ( N ) t ≥ 0. − ϕ (λ )   t (N ) t (4.9) } − ϕ (λ ) , t ≥ 0 . (4.10) Applying similar arguments as for the case of approximate densities, it can be shown that [ ] lim sup N →∞ U ( N ) (λ ; ϕ , h ), U ( N ) (λ ; ϕ , h ) This, in turn, shows that ϕ (λ )t −1 U t( N ) (λ ; ϕ , h ) = [U (λ ; ϕ , h ), U (λ ; ϕ , h )] t (N ) t −1 U t (λ ; ϕ , h ) a.s. (4.11) → ϕ (λ ) , a.s. as N → ∞ , i.e., the estimator is strongly consistent. [ ] The last statement follows from the fact that U ( N ) (λ ; ϕ , h ), U ( N ) (λ ; ϕ , h ) and t → ∞ . Arguing exactly as in Section 2, we may then estimate λ by ( ) ( ) −1 t → ∞ a.s. as N → ∞  N h Yτ( N ) ∆Yτ( N )  ←  ∑i =1 (N ) i −1 i  , N = 1,2, L , t > 0 and fixed. ˆ λt = ϕ  N (N ) 2  h Y τ i −1   ∑i =1 (4.12) In conjunction with Sections 2 and 3, the conclusions of this section may be summarized in the following theorem THEOREM 2. Let the function ϕ : Θ → R be continuous and differentiable and ϕ& (λ ) = dϕ (λ ) dλ be bounded. Let the function h : R → R be continuously differentiable satisfying the Lipschitz conditions: d (h(x ), h( y )) ≤ Ld (x, y ) , where d ( x, y ) denotes the usual Euclidean distance and L is some positive constant. Define for k = 0, 1, L , N , τ k = kδ ( δ = t ). We, thus define the stochastic difference equation, namely N 16 PARAMETER ESTIMATION FOR DIFFUSION PROCESSES ( ) Yτ(kN ) = Yτ(kN−1) + δµ Yτ(kN−1) ; λ + ∆Bτ k , Y0( N ) = ξ , N = 1, 2, L , where µ ( x ; λ ) = ϕ (λ )h(x ) and the random variable ξ {∆Bτ k } : k ∈ N , where B = (Bt , Ft ) , 0 ≤ t ≤ T , is a standard Wiener process. ( is independent of the process ) Let, in addition, the following condition hold for any t>0 P ∫ 0 h 2 ( X s ) ds > 0 = 1 . t Then, A.) For all λ ∈ Θ , the estimator (4.9) satisfies the following consistency results: P lim sup N →∞ ϕ (λ )t  ( P (lim a. (N ) − ϕ (λ )t = 0, ∀t ∈ [0, T ] = 1  ) P lim sup N →∞ λˆ(t N ) − λˆt = 0, ∀t ∈ [0, T ] = 1 b. c. N →∞ ) lim sup t →∞ λˆ(t N ) − λ = 0 = 1 . ( ) B.) The following joint convergence in distribution results hold:  X ( N ) , ϕ (λ )( N )  → d X , ϕ (λ ) , and   (X ( a. b. N) ) ( ) , λˆ( N ) → d X , λˆ as N → ∞ . C.) The following normal approximation limits are true: [ ]  U ( N ) (λ ; ϕ , h ), U ( N ) (λ ; ϕ , h )  t   a. [ ]  U ( N ) (λ ; ϕ , h ), U ( N ) (λ ; ϕ , h )  t   b. −1 2 12 U t( N ) (λ ; ϕ , h ) → B1 as N → ∞ and t → ∞ (λˆ( t N) ) − λ → D B1 as N → ∞ and t → ∞ . D.) Suppose ∃ a non-decreasing deterministic sequence of functions C t( N ) (λ ; ϕ , h ) , N ∈ N , such that C t( N ) (λ ; ϕ , h ) → ∞ as both N → ∞ and t → ∞ and [U ( N) (λ ; ϕ , h ), U ( N ) (λ ; ϕ , h )] t C t( N ) (λ ; ϕ , h ) → W (λ ; ϕ , h ) [Pλ ] as N → ∞ and t → ∞ , for some random function W (λ ; ϕ , h ) , W (λ ; ϕ , h ) > 0 a.s. [Pλ ] . Then (C( t N) (λ ; ϕ ,))1 2 (λˆ(t N ) − λ ) → D D(λ ; ϕ , h ) , 17 PARAMETER ESTIMATION FOR DIFFUSION PROCESSES where D(λ ; ϕ , h ) is a scale mixture of an independent standard normal variable. The process X = {X t : t ≥ 0} satisfies equation (2.12), B = (Bt , Ft ) , 0 ≤ t ≤ T , is a standard Wiener process and λ̂t is given by (2.23). 5. THE ORNSTEIN-UHLENBECK PROCESS The results derived in the previous sections are illustrated here through the continuous time version of model (2.1) where we let ϕ (λ ) = λ and h(x ) = x . Letting t=1 and partitioning the Ornstein-Uhlenbeck process as well as a discretized version of it. We start with the discrete interval [0,1] with δ as the step size between two consecutive observations such that τ k = kδ , for k = 0, 1, L , N , and δ = 1 (we again consider time equidistant observations), the stochastic N difference equation in (4.2) may be expressed as: Yτ(kN ) = (1 + δλ )Yτ(kN−1) + ∆Bτ k , Y0( N ) = ξ , and N ≥ 1 . { } (5.1) The initial random variable ξ above is independent of the process ∆Bτ k , k ∈ N , and is also independent of the parameter λ . Furthermore, the process B = (Bt , Ft ) , 0 ≤ t ≤ T , is assumed to be the standard Wiener process. For reasons of simplicity we may rewrite equation (5.1) in the form Yn = (1 + δλ )Yn −1 + ξ n , Y0 = ξ , N ≥ 1 , and n ≤ N , where ξ k = ∆Bτ k and Yk = Yτ(kN ) , k = 0, 1, L , N . (5.2) This is an AR(1)-type model, where the coefficient, (1 + δλ ) , depends upon the number of divisions of the sampling interval. It is easy to see that Yn = (1 + δλ ) ξ + ∑i =1 (1 + δλ ) ξ i , N ≥ 1 , and n ≤ N . n −i n n (5.3) Setting Y N (t ) = Yn , for [Nt ] ≤ n < [Nt ] + 1 , t > 0 , and 0 otherwise, we may express equation (5.3) in the following Lebesgue-Stieltjes integral form: Y N (t ) = (1 + δλ ) [ Nt ] ξ + ∫ 0 (1 + δλ )[Nt ]−1−[ Ns − ] dB N (s ) , t > 0 , t 18 (5.4) PARAMETER ESTIMATION FOR DIFFUSION PROCESSES where B N (s ) = Bn , if [Ns ] ≤ n < [Ns ] + 1 , and 0, otherwise. Arguing as in Section 4, it is not hard to see that Y N (t ) ⇒ X t as N → ∞ , i.e., Y N (t ) converges weakly to the process X t , t > 0 , (see also e.g., Chan and Wei, 1987), where the limit solution is given by X t = e λt ξ + ∫0 e λ (t − s ) dB s , t > 0 . t (5.5) Note that the process, X = {X t , t ≥ 0} , is the Ornstein-Uhlenbeck process satisfying the diffusion equation dX t = λX t dt + dB s , which is of the form (2.1) and {B s , s > 0} is the standard Brownian motion. Also, one may note that ( P (X t ∈ A X 0 = ξ ) = ∫A dt exp − λ x − e λt ξ  ) (e 2 U t( N ) (λ ;1,1) = U ( N ) (λ ;1,1), U ( N ) (λ ;1,1) ] where U ( N ) (λ ;1,1), U ( N ) (λ ;1,1) ) ( − 1   ] {λˆ( [ Employing (4.10), we may then obtain [ 2 λt t ) π e 2 λt − 1 λ . N) } −λ , t ≥ 0, (5.6) (5.7) = δ ∑ k =1 Yk2−1 . Reiterating some of the arguments presented in N t the previous section it is not hard to see that λ̂( N ) is a strongly consistent estimator of λ . It follows that the expression for λ̂( N ) is given by λ̂( N ) = ∑iN=1 Yi −1 ∆Yi ∑i =1 Yi −21 , N N = 1,2, L , t > 0 . (5.8) In conjunction with Theorems 2 and 3, the following corollary is now in order. COROLLARY. Consider the Ornstein-Uhlenbeck stochastic differential equation dX t = λX t dt + dBt , t ≤ T , X 0 = ξ , where the random variable ξ is independent of the process B = (Bt , Ft ) , 0 ≤ t ≤ T . Define for k = 0, 1, L , N , τ k = kδ ( δ = t ). We, thus, define the stochastic difference equation (5.2). N Then, A.) For all λ ∈ Θ , the estimator (5.8) satisfies the following consistency results: ( P (lim ) P lim sup N →∞ λˆ( N ) − λˆt = 0, ∀t ∈ [0, T ] = 1 and N →∞ ) lim sup t →∞ λˆ( N ) − λ = 0 = 1 . B.) The following joint convergence in distribution result holds: 19 PARAMETER ESTIMATION FOR DIFFUSION PROCESSES (X ( N) ) ( ) , λˆ( N ) → d X , λˆ a.s. as N → ∞ . C.) The following normal approximation limit is true for all λ ≠ 0 : [ ]  U ( N ) (λ ;1,1), U ( N ) (λ ;1,1)  t   12 (λˆ( N) ) − λ → D B1 as N → ∞ and t → ∞ . Note that for λ = 0 , Feigen (1979) showed that the normalized form in the above Corollary ){∫ } (part C) is not normally distributed. In particular, Feigen showed that for each t [ ] (  U ( N ) (λ ;1,1), U ( N ) (λ ;1,1)  λˆ( N ) → 1 B 2 − t t t 2   12 t B 2 ds 0 s under [P0 ] . Clearly, under [P0 ] , it can be seen that −1 2 a.s. as N → ∞ ∫ 0 Bs ds → ∞ , t 2 (5.9) a.s. The study of the distribution of the right hand side of (5.9) has been considered by many authors in the past. It is not our intention to get into the details of such results. Finally, we shall conclude our observations by providing exact forms of the mixing scale random variable appearing in part D. of Theorem 2, for the Ornstein-Uhlenbeck process. As in [ ] t → − 1 2λ , a.s. in [Pλ ] as N → ∞ . Note that, previous studies (see for more details, e.g., Küchler and Sørensen, p.52, 1997), it can also be shown that for λ < 0 , U ( N ) (λ ,1,1), U ( N ) (λ ,1,1) ( ) t in this case, the appropriate function is not random, thus t (− 2λ ) λˆ( N ) − λ → D B1 , as N → ∞ . [ ] When λ > 0 , on the other hand, we obtain that U ( N ) (λ ,1,1), U ( N ) (λ ,1,1) (5.10) t e 2 λt (H ∞ + ξ ) 2 → 1 2λ , where H ∞ = ∫ 0 e − λs dB s . Thus, the conditional distribution of 2λ (H ∞ + ξ ) on ξ is ∞ 2 χ 2 (1) -distribution with non-centrality ξ 2λ . Therefore, (C( t N) (λ ;1,1)) 12 (λˆ( N) ) − λ → D D(λ ;1,1) , as N → ∞ , (5.11) where C t( N ) (λ ;1,1) = e 2 λt A(λ , ξ ) , A(λ , ξ ) is a χ 2 (1) -distribution with non-centrality ξ 2λ and D(λ ;1,1) = A1 2 (λ , ξ )B1 , i.e., the expression is of a scale mixture of the positive random variable A(λ , ξ ) and the standard normal variable. 20 PARAMETER ESTIMATION FOR DIFFUSION PROCESSES REFERENCES Billingsley, P. (1968). Convergence of Probability Measures. John Wiley & Sons, New York. Chan, N.H. and Wei, C.Z. (1987). Asymptotic inference for nearly nonstationary AR(1) processes. Ann. Stat. 15, 1050-1063. Dacunha-Castelle, D. and Florens-Zmirnou, D. (1986). Estimation of the coefficients of a diffusion from discrete observations. Stochastics, 19, 263-184. Feigin, P.D. (1976). Maximum likelihood estimation for continuous time stochastic processes. Adv. Appl. Prob. 8, 712-736. Feigin, P.D. (1978). The efficiency criteria problem for stochastic processes. Stoch. Proc. Appl. 6, 115-128. Feigin, P.D. (1979). Some comments concerning a curious singularity. J. Appl. Prob. 16, 440444. Feigin, P.D. (1981). Conditional exponential families and representation theorem for asymptotic inference. Ann. Stat. 9, 597-603. Feigin, P.D. (1985). Stable convergence of semimartingales. Stoch. Proc. Appl. 19, 125-134. Heyde, C.C. (1978). On an optimal asymptotic property of the maximum likelihood estimator of a parameter from a stochastic process. Stoch. Proc. Appl. 8, 1-9. Heyde, C.C. (1987). On combining quasi-likelihood estimating functions. Stoch. Proc. Appl. 25, 281-287. Hudson, I.L. (1982). Large sample inference for Markovian exponential families with applications to branching processes with immigration. Aus. J. Statist. 24, 98-112. Kloeden, P.E. and Platen, E. (1995). Numerical Solution of Stochastic Differential Equations. Springer-Verlag, Berlin. Küchler, U. and Sørenson, M. (1994). Exponential families of stochastic processes and Lévy processes. J. Stat. Plan. Inf. 39, 211-237. Küchler, U. and Sørenson, M. (1994). Exponential families of Markov processes. J. Stat. Plan. Inf. 66, 3-19. Küchler, U. and Sørenson, M. (1997). Exponential Families of Stochastic Processes. SpringerVerlag, New York. Øksendal, B. (1998). Stochastic Differential Equations: An Introduction with Applications. Springer-Verlag, Berlin. Pederson, A.R. (1995). A new approach to maximum likelihood estimation for stochastic differential equations based on discrete observations. Scand. J. Statist. 22, 55-71. 21 PARAMETER ESTIMATION FOR DIFFUSION PROCESSES Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion. Springer-Verlag, Berlin. Department of Management & Decision Sciences and Program in Statistics Washington State University Pullman, WA 99164-4736 [email protected] Department of Mathematics and Program in Statistics Washington State University Pullman, WA 99164-3113 [email protected] 22 View publication stats