Asymptotic Properties of an Approximate Maximum Likelihood Estimator for Stochastic PDEs

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1 Asymptotic Properties of an Approximate Maximum Likelihood Estimator for Stochastic PDEs M. Huebner S. Lototsky B.L. Rozovskii In: Yu. M. Kabanov, B. L. Rozovskii, and A. N. Shiryaev editors, Statistics and Control of Stochastic Processes the Liptser festschrift, pp World Scientific, Singapore, Abstract A maximum likelihood estimate of a scalar parameter is constructed for a stochastic evolution system using the Galerkin approximation of the original equation. Conditions are established to guarantee the consistency and asymptotic normality of the estimator as the dimension of the approximation tends to infinity. Examples are given to illustrate the results. 1 Introduction Many parameter estimation problems for stochastic partial differential equations can be reduced to the following model: dut, ω = A tut, ω + θa 1 tut, ω + ft dt + BdW t, 1.1 where A, A 1, and B are linear operators, W is a cylindrical Brownian motion, all defined in some Hilbert space H, and θ is an unknown scalar parameter. A computable estimate of θ based on the observations of u can usually involve only a finite-dimensional approximation of the random field u. If the operators A, A 1, and B have a common system of eigenfunctions, then equation 1.1 is diagonalizable so that the maximum likelihood estimator MLE of θ can be easily constructed and its properties studied. In [1, 2] model 1.1 is considered under the assumption that A, A 1, and B are elliptic, selfadjoint operators with a common system of eigenfunctions in a bounded domain of IR d. It was demonstrated in these works that under certain conditions the MLE is consistent, asymptotically normal, and asymptotically efficient as the dimension of Department of Statistics and Probability, Michigan State University, East Lansing, MI his work was partially supported by NSF Grant #DMS 959 Center for Applied Mathematical Sciences, University of Southern California, Los Angeles, CA his work was partially supported by ONR Grant #N Center for Applied Mathematical Sciences, University of Southern California, Los Angeles, CA his work was partially supported by ONR Grant #N and ARO Grant DAAH

2 the approximation tends to infinity. It was mentioned in [2] that some of the results remain valid if the operators in 1.1 do not commute. In this paper, equation 1.1 is studied in the general Hilbert space setting without any assumptions on the eigenfunctions of the operators A and A 1, and the MLE of θ is based on the Galerkin approximation of 1.1. he objective is to establish conditions under which the MLE is consistent and asymptotically normal as the dimension of the approximation tends to infinity. hese conditions are related to the separation of the corresponding sequences of the probability measures and can be verified in a number of examples Section 4. Unless the system is diagonalizable, the Galerkin approximation is usually not observable and therefore the results presented in this paper are mainly of theoretical interest. However the conditions for asymptotic optimality for maximum likelihood estimators obtained with this approach can be applied to approximate maximum likelihood estimators based on other finite-dimensional approximations of Stochastic Evolution Systems in Hilbert Scales Let H be a real separable Hilbert space with the inner product, and the corresponding norm. Consider a linear operator Λ on H such that Λf c f for all f from the domain of Λ. hen the powers Λ s of Λ are defined for all s IR and generate the spaces H s as follows [3]: for s, H s is the domain of Λ s ; for s <, H s is the completion of H with respect to the norm s := Λ s. he collection of spaces {H s } s IR is a Hilbert scale with the following properties [3]: for s 2 > s 1, the space H s 2 is continuously embedded in H s 2 ; for every s IR and r >, the spaces H s+r, H s, H s r form a normal triple with the canonical bilinear functional y 1, y 2 s,r = Λ s r y 1, Λ s+r y 2, where y 1 H s r, y 2 H s+r ; Λ s H r = H s r. Let Ω, F, F = F t, P be a filtered probability space with the usual assumtions, and let W = W t t be a cylindrical Brownian motion on this space. his means that a family of continuous martingales W t f, f H, is defined on Ω, F, F, P so that the quadratic covariation W f, W g t = tf, g for every f, g H. he stochastic integral ξ t, dw t is defined [4, 5] for predictable processes ξ = ξ t t L 2 Ω [, ]; H, that is ξ t, y is a predictable stochastic process for every y H. Consider the following stochastic evolution system: du θ t, ω = A θ tut, ω + ftdt + B dw t, < t, u θ t= = u, 2.1 where θ belongs to some parameter space Θ IR, and A θ = A + θa 1. he linear operators A, A 1, and B are deterministic. Equation 2.1 will be studied under the following assumptions: 2

3 A1 here exists α such that Λ α B is a Hilbert-Schmidt operator in H; A2 here exists γ such that for every t [, ] the operators A and A 1 are bounded and linear from H γ α to H γ α ; A3 For every θ Θ there exist positive numbers δ = δθ and K = Kθ such that for all t [, ] and y H γ α, A θ y, y α,γ δ y 2 γ α + K y 2 α, and A θ y γ α K y γ α ; A4 u is an F -measurable random element with values in H α and E u 2 α < ; A5 f = ft is an H γ α -valued deterministic function and ft 2 γ α <. Assumption A1 implies that BW t is an H α -valued Wiener process, therefore by heorem in [6] for every θ Θ equation 2.1 has a unique solution u θ in the normal triple H γ α, H α, H γ α and he solution u satisfies u θ L 2 Ω; C[, ]; H α L 2 Ω [, ]; H γ α. 2.2 E sup ut 2 α + E t ut 2 γ αdt <. 2.3 he approximate maximum likelihood estimator for the parameter θ will be based on a finite-dimensional approximation of 2.1. o study the properties of this estimator, it is further assumed that A6 Operators Λ and B have a common system of eigenfunctions {h k } k with the following properties: {h k } k is a complete orthonormal system in H; h k s H s for all k ; A7 he eigenvalues of B are nonzero, and the operator A 1 is not identically zero. 3 Consistency and Asymptotic Normality of the Approximate Maximum Likelihood Estimator For a positive integer N the operator Π N acting from s H s to s H s is defined by Π N : u N k= u, h k h k, where {h k } k is the orthonormal basis of H defined in A6. hen Π N is an orthogonal projection in the space H, and Π n u s u s for all s IR. he finite-dimensional Galerkin approximation of 2.1 is defined by u θ,n t = Π n u + Π N A θ su θ,n s + fs ds + Π N BW N t, 3.1 where W N = W N t is a standard Wiener process with the components W t h k, k N. It follows from assumption A6 that Π N B is a diagonal matrix whose diagonal elements are the first N eigenvalues of the operator B. 3

4 Lemma 3.1. Under the assumptions A1 A6, lim E u θ,n t u θ t 2 γ αdt =. 3.2 N Proof. Property 2.2 implies that lim E Π N u θ t u θ t 2 γ αdt =. N On the other hand, for N, assumption A3 and the Gronwall inequality imply that E u θ,n t Π N u θ t 2 γ αdt CθE Π N A θ ut Π N ut 2 αdt CθE u θ t Π N u θ t 2 γ αdt, and 3.2 follows. o simplify the future presentation, the solutions of 2.1 and 3.1 corresponding to the unknown but fixed value of the parameter θ Θ will be denoted by u and u N, respectively. he objective is to estimate θ given the trajectory of u N t, t. Denote by P θ,n t the measure on C[, t]; H α generated by the solution u θ,n s of 3.1 for s t. Since for each θ Θ the process u θ,n is finite dimensional, assumption A7 and the results from [7] Chapter 7 imply that for each θ Θ the measure P θ,n t is absolutely continuous with respect to the measure P θ,n t with the likelihood ratio Radon-Nikodym derivative given by dp θ,n { t dp θ u N = exp θ θ,n t θ θ t θ2 θ 2 2 Π N B 2 A 1 su N s, du N s Π N B 1 A 1 su N s 2 ds Π N B 1 A 1 su N s, Π N B 1 A su N s + fs ds }. 3.3 he maximum likelihood estimate MLE ˆθ N of θ is obtained by maximizing the likelihood ratio 3.3, where t is replaced by, with respect to θ Θ. ˆθ N = Π N B 2 A 1 tu N t, du N t Π N A tu N t + ft dt ΠN B 1 A 1 su N t 2 dt. 3.4 By convention, ˆθ N = if ΠN B 1 A 1 su N t 2 dt =. If ΠN B 1 A 1 su N t 2 dt >, then 3.4 implies that ˆθ N satisfies ˆθ N Π N B 1 A 1 tu N t, dw N t = θ + ΠN B 1 A 1 tu N t 2 dt

5 It follows from assumption A7 that the operator Π N B 1 A 1 t is not identical zero for all N that are greater or equal to some N. hus, by Lemma 7.1 in [7], P Π N B 1 A 1 tu N t 2 dt > = 1, N N, so that 3.5 holds with probability 1 for all sufficiently large N. Direct computations show that Fisher s information I N θ corresponding to the likelihood ratio 3.3 with t = is given by heorem 3.1. I N θ = E Π N B 1 A 1 tu θ,n t 2 dt. 3.6 i If then If, in addition, P lim N Π N B 1 A 1 tu N t 2 dt =, 3.7 P lim ˆθ N = θ. N ii If P lim Nθ 1 N Π N B 1 A 1 tu N t 2 dt = 1, 3.8 then ˆθ N is asymptotically normal with normalizing factor I N θ, i.e., I N θ θ θ converges in distribution, as N, to a Gaussian random variable with zero mean and unit variance. for all g L 2 [, ]; H γ α, then B 1 A 1 tgt 2 dt C P lim ˆθ N = θ + N gt 2 γ αdt 3.9 B 1 A 1 tut, dw t B 1 A 1 tut 2 dt. 3.1 Proof. Part i. Consistency of ˆθ N follows from 3.5 and the following result: Lemma 3.2. If {f n t} n 1 is a sequence of predictable random functions in L 2 Ω [, ]; H such that P lim N f n t 2 dt =, then f n t, dw t P lim N =. f n t 2 dt 5

6 his result is not new, but, for the sake of completeness, we give a short proof in the Appendix. o prove the asymptotic normality, consider M N t := ΠN B 1 A 1 su N s, dw N s ds E. ΠN B 1 A 1 su N s 2 ds hen Mt N, F t t is a continuous square integrable martingale with quadratic characteristic M N t = ΠN B 1 A 1 su N s 2 ds E ΠN B 1 A 1 su N s 2 ds. By assumption 3.8, P lim N M N = 1. On the other hand, if w 1 t, F t t is a one dimensional Wiener process and M t := w 1 t/, then M t, F t t is a continuous square integrable martingale and M = 1. herefore, by heorem 5.5.4II in [8] lim N M N = M in distribution. Since M is a Gaussian random variable with zero mean and unit variance, the asymptotic normality follows. Part ii. he right-hand side of 3.1 is well defined because the process B 1 A 1 tut is in the space L 2 Ω [, ]; H, by 3.9 and 2.2, and, for every y H, the process B 1 A 1 tut, y is continuous and thus predictable. It suffices to show that lim E Π N B 1 A 1 tu N t B 1 A 1 tu 2 dt = N hen the assertion 3.1 follows from 3.5 and the properties of the stochastic integral [4, 6]. By the triangle inequality, E Π N B 1 A 1 tu N t B 1 A 1 tut 2 dt 2E + 2E Π N B 1 A 1 tu N t Π N B 1 A 1 tu 2 dt Π N B 1 A 1 tut B 1 A 1 tu 2 dt he first term on the right of 3.12 tends to zero as N since B 1 A 1 u L 2 Ω [, ]; H, and the second term tends to zero by 3.9 and 3.2. his proves he conditions 3.7 and 3.9 of heorem 2.1 hold for a large class of SPDE s. In [2] they were verified for elliptic operators A and A 1 which have the same eigenfunctions as the operators Λ and B. Other examples to which heorem 2.1 applies are given in Section 4 below. 6

7 Despite their technical nature, conditions 3.7 and 3.9 can be related to some standard statistical notions. Definition. Let {P n } n 1 and { P n } n 1 be sequences of probability measures on measurable spaces Ω n, F n. he sequences are called completely separated if there exist sets A n F n such that lim n P n A n = 1 and lim n Pn A n =. he sequences are called entirely separated if there is a sequence n k, k such that the sequences {P nk } k 1 and { P nk } k 1 are completely separated. heorem If the sequences of measures {P θ,n } N 1 are completely separated, then 3.7 holds. {P θ,n } N 1 and 2 If 3.7 holds, then {P θ,n } N 1 and {P θ,n } N 1 are entirely separated. Proof. It follows from Corollary IV.1.37 in [9] that the Hellinger process h N t 1/2 corresponding to the likelihood ratio 3.3 and the reference measure P θ,n is given by h N t 1/2 = θ θ 2 8 Π N B 1 A 1 tu N t 2 dt. he first statement of the theorem is proved in the same way as property i from heorem V.2.4a in [9] and the second statement follows from [9] heorem V.2.4b. Denote by P θ the measure on C[, ]; H α generated by the solution of 2.1. heorem 3.3. he measures P θ are equivalent for all θ Θ if 3.9 holds. Proof. It follows from Corollary 1 in [4] that the measures P θ 1 and P θ 2 are equivalent if and only if E B 1 A 1 tu θ t 2 dt < 3.13 for θ = θ 1 and θ = θ 2. his follows from the conditions 3.9 and 2.2. For the model with commuting operators discussed in [2] the conditions 3.7 and 3.9 of heorem 3.1 are equivalent to the absolute continuity or singularity of the measures P θ for different θ Θ, respectively. 4 Examples In this section, heorem 3.1 is applied to three parameter estimation models of the type 2.1. o simplify the notation, the superscript θ is omitted: 1. du = θ + axudt + 1 1/2 dw, x [, 1], with zero boundary conditions; 2. du = + θax / xudt + dw, x [, 1], with periodic boundary conditions; 3. du = + θaxudt + dw, x [, 1], with either zero or periodic boundary conditions. For the first two examples, conditions 3.7 and 3.8 of heorem 3.1 are fulfilled so that the maximum likelihood estimator 3.4 for θ is consistent and asymptotically 7

8 normal, whereas for the third example, condition 3.9 holds so that the MLE 3.4 converges to 3.1. In the following, C denotes a real number, independent of N and independent of the solutions of 2.1 and 3.1. he constant C may depend on other parameters, as indicated, and the value of C may be different at different places. For sequences {a n } n 1 and {b n } n 1 of positive real numbers the notation a n b n means that < lim inf n a n /b n lim sup a n /b n <. n Example 1 his system is governed by the following equation with zero boundary conditions. dut, x = θ + axut, xdt + 1 1/2 dw t, x, <t, x, 1, u, x =, ut, = ut, 1 =. 4.1 It is assumed that the parameter θ is positive, the function a = ax belongs to C [, 1], and a n + = a n 1 = for n 1, where a n x is the n-th derivative of a. Let H = L 2 [, 1], with orthonormal basis h k x = 2 sinπkx for k 1. he operator Λ : L 2 [, 1] L 2 [, 1] defined by Λh k = πk h k satisfies the assumptions from Section 2. herefore Λ generates a Hilbert scale {H s } s IR. he operators corresponding to the evolution system 2.1 in Section 1 are as follows: A 1 = satisfies A 1 h k = kπ 2 h k. A = ax satisfies A h k = l=1 a l k a l+k h l. B = 1 1/2 satisfies Bh k = 1 + πk 2 1/2 h k, and B is a Hilbert-Schmidt operator in H = L 2 [, 1]. he assumptions on ax imply that a k Cr/k r for every r > so that A f s Cs f s 4.2 for all s IR. herefore conditions A1 A7 are fulfilled for all θ > with α =, γ = 1. Consequently, for every θ >, there is a unique solution u of 4.1 satisfying E sup ut 2 + E ut 2 1dt <. t he objective now is to show that 3.7 and 3.8 hold for all θ >. o simplify the presentation it is assumed that θ = 1. he variation of parameters formula yields: u N t, h k = e πk2 t s A u N s, h k ds πk 2 1/2 e πk2 t s dw k s. he first step is to show that Fisher s information 3.6 diverges as N tends to infinity: I N θ = E Π N B 1 A 1 u N t 2 dt N

9 he following notation is used for the sake of brevity: ξ k t = With this notation, πk 2 e πk2 t s dw k s, X N t = ξkt, 2 ηk N t = πk πk 2 e πk2 t s A u N s, h k ds. Π N B 1 A 1 u N t 2 = X N t + 2 ξ k tηk N t + ηk N t By the inequality 1/2x 2 y 2 x+y 2 2x 2 +y 2, the norm 4.4 can be bracketed as follows. 1 2 XN t ηk N t 2 Π N B 1 A 1 u N t 2 2X N t + 2 ηk N t Since for each t the random variables {ξ k t} k 1 are independent and Gaussian, it follows that E X N tdt k 2 N he asymptotics in 4.3 holds if the expectation E ηn k t2 dt is bounded. For this the following lemma is necessary. Lemma 4.1. If a > and ft then 2dt e at s fsds f 2 tdt a 2. he above inequality can be easily verified by direct computation; for the sake of completeness, the main steps are given in the Appendix. By Lemma 4.1, 2 E ηk N t 2 dt E 1 + πk 2 A u N t, h k dt, so that by 3.2, 4.2, and 2.3, lim E N ηk N t 2 dt CE ut 2 1dt <. 4.7 he last inequality together with 4.5 and 4.6 implies the asymptotics for Fisher s information in 4.3. Now, 3.7 can be rewritten as follows. ΠN B 1 A 1 u N t 2 dt E ΠN B 1 A 1 u N t 2 dt = + XN t dt N ξ k tηk N + 2 I N I N N ηk Nt2 dt. I N t dt 9

10 Both the second and third terms on the right-hand side converge to zero by 4.3 and 4.7. For the second term it is necessary to first apply the inequality 2xy δx 2 + δ 1 y which holds for every δ > and every real x, y. he precise arguments are given below. It will be shown next that the first term converges to 1.By direct computation, Define varx N t = varξkt 2 CN 5, t [, ]. Y N := XN t EX N tdt E. XN tdt hen XN tdt E XN tdt = 1 + Y N and varx N tdt EYN 2 E 2 C as N, XN tdt N so that by Chebychev s inequality, P lim N Y N = and P lim XN tdt N E = XN tdt he δ-inequality 4.8 can be written in the following form: 1 δx δ 1 y 2 x + y δx δ 1 y 2. his and 4.5 imply 1 δe X N tdt δ E E 1 + δe Π N B 1 A 1 ut 2 dt ηk N t 2 dt X N tdt δ E Since δ is arbitrary, 3.8 follows from ηk N t 2 dt. Conclusion. For the model 4.1, the MLE 3.4 is consistent and asymptotically normal with the normalizing factor I N θ N 3/2. Example 2 Here the system s equation is, with periodic boundary conditions, dut, x = + θax ut, xdt + dw t, x, < t, x [, 1], x u, x =, ut, = ut,

11 he parameter θ is real, and the function a = ax is an infinitely differentiable and periodic with period 1. Let H = L 2 [, 1], with orthonormal basis h x = 1, h 2n x = 2 cos2πnx, h2n 1 x = 2 sin2πnx for n 1. For k define the numbers µ k = 2π [k 1/2] + 1, where [x] is the largest integer less than or equal to x. he Hilbert scale {H s } s IR is generated by the operator Λ with Λh k = µ k h k, k. he operator A = satisfies A h k = µ 2 k h k, k, and / x satisfies x h k = { µk h k+1, if k = 2n 1 µ k h k 1, if k = 2n. Multiplication by ax is defined according to the formulas for the Fourier coefficients of the product of two functions, and then the operator A 1 = ax / x is defined in an obvious way. he assumptions on ax imply that A 1 f s Cs f s+1 for all s IR. he operator Λ 1 is Hilbert-Schmidt in H = L 2 [, 1] and B = 1, therefore conditions A1 A7 are fulfilled for all θ IR with α = γ = 1. As a result, for every θ R, there is a unique solution u of 4.1 satisfying E sup ut E ut 2 dt <. t he objective now is to show that 3.7 and 3.8 hold for all θ IR. Again, it is assumed that θ = 1. For technical reasons, equation 4.1 is rewritten as follows: du = 1 + ax + 1udt + dw. x As in Example 1, the following notation is introduced: ξ k t = e 1+µ2 k t s dw k s, X N t = where a lk = N n= A 1 h l, h n A 1 h k, h n, and η N k t = l,k= e 1+µ2 k t s à 1 u N s, h k ds, a lk ξ l tξ k t, where Ã1 = A his notation, equation 3.1, and the variation of parameters formula yield: Π N B 1 A 1 u N t 2 = X N t + 2 l,k= a lk ξ l tη N k t + l, a lk η N l tη N k t. For every t > the random variables {ξ k t} k are independent and Gaussian with zero mean and variance 1 R k t = 21 + µ 2 k 1 e 1+µ2t k. 11

12 It follows that and E X N tdt = varx N t = 2 a kk R k tdt k= k,l= a 2 klr k tr l t. Direct computations similar to those in Example 1 yield and 1 a kk R k tdt k= 1 + µ 2 A 1 h k, h n 2 a 2 N N, k,n= k varx N t CN, lim E N l, a lk ηl N tηk N tdt CE ut 2 dt <. Now, conditions 3.7 and 3.8 of heorem 3.1 follow in the same way as in Example 1. Conclusion. For the model 4.1, the MLE 3.4 is consistent and asymptotically normal with normalizing factor I N θ N 1/2. Example 3 Let a = ax be either as in Example 1 or in Example 2 and consider the equation dut, x = ut, x + θaxut, xdt + dw t, x, < t 4.11 with zero initial conditions and zero or periodic boundary conditions. hen it follows from the results of Examples 1 and 2 that the solution u of 4.11 exists and is unique in the corresponding Hilbert scale, and E sup ut E t ut 2 dt <. In this case A 1 = ax and B = 1, so that 3.9 holds with α = γ = 1. his means that for the model 4.11 the MLE 3.4 is asymptotically biased, and the MLE converges to θ + a ut, dw t a ut 2 dt, as N tends to infinity. Appendix Proof of Lemma 3.2 For an arbitrary ε, 1 define x ε n := ε f n s, dw s and xε n := ε 2 f n s 2 ds. hen f n s, dw s P f ns 2 ds > ε = P exp x ε n x ε n /2 > exp x ε n /2, 12

13 and to complete the proof of the lemma it remains to use the Chebuchev inequality and note that sup n E exp x ε n x ε n /2 1 heorem 5.2 and Lemma 6.1 of [7]. Proof of Lemma 4.1. Note that 2 t s e as fsds = 2 e as e au fufsduds. If U := t 2dt, e at s fsds then direct computations yield U = 2 s and the result follows. e a2t s u fufsdudsdt a 1 1/2U f sds 2 1/2, References [1] M. Huebner, R. Khasminskii, and B. L. Rozovskii, wo Examples of Parameter Estimation, in Stochastic Processes, ed. Cambanis et al. Springer, New York, [2] M. Huebner and B. Rozovskii, On Asymptotic Properties of Maximum Likelihood Estimators for Parabolic Stochastic PDE s, Probability heory and Related Fields, 13, [3] S. G. Krein, Ju. I. Petunin, and E. M. Semenov. Interpolation of Linear Operators American Mathematical Society, Providence, RI, [4] R. Mikulevicius and B.L. Rozovskii, Uniqueness and Absolute Continuity of Weak Solutions for Parabolic SPDE s, Acta Applicandae Mathematicae, 35, [5] J. B. Walsh. An Introduction to Stochastic Partial Differential Equations, In Ecole d été de Probabilités de Saint-Flour, XIV, ed. P. L. Hennequin Lecture Notes in Mathematics, volume 118, Springer, Berlin, [6] B. L. Rozovskii, Stochastic Evolution Systems Kluwer Academic Publishers, 199. [7] R. Sh. Liptser and A. N. Shiryayev, Statistics of Random Processes Springer, New York, [8] R. Sh. Liptser and A. N. Shiryayev, heory of Martingales Kluwer Academic Publishers, Boston, [9] J. Jacod and A. N. Shiryayev, Limit heorems for Stochastic Processes Springer, Berlin,