Bayesian Estimations for Diagonalizable Bilinear SPDEs

Transcription

1 ayesian Estimations for Diagonalizable ilinear SPDEs Ziteng Cheng Igor Cialenco uoting Gong Department of Applied Mathematics, Illinois Institute of Technology W 32nd Str, John T. ettaliata Engineering Center, oom 208, Chicago, IL 60616, USA First Circulated: May 29, 2018 Abstract: The main goal of this paper is to study the parameter estimation problem, using the ayesian methodology, for the drift coefficient of some linear parabolic SPDEs driven by a multiplicative noise of special structure. We tae the spectral approach by assuming that one path of the first Fourier modes of the solution are continuously observed over a finite time interval. We derive ayesian type estimators for the drift coefficient, and as custom for ayesian statistics, we prove a ernstein-von Mises theorem for the posterior density. Consequently, we obtain some asymptotic properties of the proposed estimators, as. Finally, we present some numerical examples that illustrate the obtained theoretical results. Keywords: statistical inference for SPDEs, ayesian statistics, ernstein-von Mises, parabolic SPDE, multiplicative noise, stochastic evolution equations, identification problems for SPDEs MSC2010: 60H15, 65L09, 62M99 1 Introduction The analytical theory for Stochastic Partial Differential Equations SPDEs have been extensively studied over the past few decades. It is well recognized that SPDEs can be used as important modeling tool in various applied disciplines such as fluid mechanics, oceanography, temperature anomalies, finance, economics, biological and ecological systems; cf. [Cho07, Hai09, L17]. On the other hand, the literature on statistical inference for SPDEs is relatively speaing limited. We refer to the recent survey [Cia18] for an overview of the literature and existing methodologies on statistical inference for parabolic SPDEs. Most of the obtained results are done within the socalled spectral approach, when it is assumed that one path of Fourier modes of the solution are observed continuously over a finite interval of time, in which case usually the statistical problems are addressed via maximum lielihood estimators MLEs. Asymptotic properties of the estimators are studied in the large number of Fourier modes regime,, while time horizon is fixed. In particular, there are only few wors related to ayesian statistics for infinite dimensional evolution equations [is02, is99, P00]. As usual, studying SPDEs driven by multiplicative noise is more involved, and parameter estimation problems for such equations are not an exception; the literature on this topic is also limited [CL09, Cia10, PT07, CH17, T17]. The main goal of this paper is twofold: to study the parameter estimation problem for the drift coefficient of linear SPDEs driven by a multiplicative noise of special structure and by applying the ayesian estimation procedure. Similar to the existing literature, we are assuming the 1

2 2 Cheng, Cialenco, and Gong spectral approach, and the main objective is to derive ayesian type estimators and to study their asymptotic properties as. esides contributing to these two important and undeveloped topics, the obtained results will prepare the foundation for studying similar problems for more complex nonlinear equations. We consider a multiplicative noise of special structure, which is customary considered in the SPDE applied literature cf. [GHZ09, GHZ08, GHKVZ14] and references therein. This wor is first attempt to study parameter estimation problems for SPDEs driven by such noise. The case of large time asymptotics is omitted here, since it is easily reduced to study the corresponding statistical problem for finite dimensional stochastic differential equations, which is a well developed field. It is worth mentioning that the asymptotic properties for MLEs for these equations are trivially obtained, and we mention them here only briefly since they are used in derivation of convergence of proposed ayesian estimators. The paper is organized as follows. In Section 2 we setup the problem and show that the considered SPDEs are well defined. Section 3 is devoted to MLE and its asymptotic properties. The main results are presented in Section 4. We start in Section 4.1 with derivation of the ayesian estimators, and prove a ernstein-von Mises type result see Theorem 4.2. The consistency and asymptotic normality of the ayesian estimator are proved in Section 4.2. Using simulation technique we apply the obtained theoretical results to stochastic heat equation; see Section 5. Some auxiliary results are deferred to Appendix A. Finally, in Appendix we provide some reasoning on the form of the posterior density within this framewor. 2 Setup of the Problem Let Ω, F, {F t } t 0, P be a stochastic basis satisfying the usual assumptions, on which we consider a sequence of independent standard rownian motions {w, }. Assume that H is a separable Hilbert space, with the corresponding inner product, H. Let A be a positive definite self-adjoint operator in H that has only point spectrum, denoted by {µ }, with the corresponding eigenvectors {h }. We mae the standing assumption that {h } forms a complete orthonormal system in H, and µ. We will denote by {H γ, γ } the scale of Hilbert spaces generated by the operator A, i.e. H γ is equal to the closure of the collection of all finite linear combinations of {h : } with respect to the norm γ := A γ H, with A γ v = 1 µγ v h, and v := v, h H. We consider the following stochastic evolution equation { dut + Aut dt = σ u th q dw t, 2.1 u0 = u 0 H, where u t := ut, h H, t 0, are the Fourier modes of the solution u with respect to {h },, σ := 0,, and {q } is a sequence in. In what follows we will assume that lim µ = +, q 2 Cµ1 ε, 2.2 for some ε > 0, C > 0. Although the equation 2.1 is driven by a multiplicative noise, due to the special structure of the noise, it is a diagonalizable SPDE, namely the Fourier modes satisfy an infinite dimensional system of decoupled equations du t + µ u t dt = σq u t dw t, t [0, T ],, 2.3

3 ayesian Estimations for SPDEs 3 with initial condition u 0 = u 0, h H. Hence, we have that u t = u 0 exp µ σ2 q 2 t + σq w t, t [0, T ],. 2.4 Without loss of generality we will assume that u 0 0,. ext, in view of 2.2, we deduce that 2 E u th = E µ 2γ γ u2 t = = µ 2γ u2 2 0 exp µ σ2 q 2 t E exp2σq w t µ 2γ u2 0 exp 2tµ + σ 2 tq 2 <. Hence the equation 2.1 admits a uniqe solution in L 2 Ω; C[0, T ], H γ, for any γ. We will tae the continuous-time observation framewor by assuming that the solution u, as an object in H, or in a finite dimensional projection of H, is observed continuously in time for all t [0, T ], and for some fixed horizon T. We assume that σ, and q,, are nown constants, since generally speaing under the continuous-time sampling scheme, using quadratic variation arguments, these parameters can be found exactly. We will be interested in estimating the unnown parameter, with 0 being the true value of this parameter of interest. In what follows we will denote by u the solution to 2.1 that corresponds to the parameter, and correspondingly, we put u := u, h H,. For notational simplicity, we will continue to write u and u instead of u 0 and u 0. We will also assume that q 0 for all. If q = 0 for some, then can be found exactly, and the considered statistical problem becomes trivial. In this study, we will assume that one path of the first Fourier modes u 1 t,..., u t is observed continuously over a fixed time interval [0, T ], for some T > 0. We will focus on the asymptotic properties in large number of Fourier modes,, while T is fixed. The large time asymptotics T, with fixed, reduces to existing results for finite dimensional systems of stochastic differential equations, which is well understood. The mixed case with both, T is left for further studies. 3 Maximum Lielihood Estimator In this section, we will investigate the asymptotic properties of the Maximum Lielihood Estimator MLE for the unnown parameter. The obtained results, albeit simple, are important on their own, and we will also use them later to study the ayesian estimators. Let C[0, T ]; denote the space of all -valued continuous functions on [0, T ]. For every, let P be the probability measure on C[0, T ]; induced by the projected solution U := {u 1 t,..., u t, t [0, T ]}. As before, we simple write U and P instead of U 0 and P 0. The measures P and P are equivalent, for any, and the Lielihood atio, or the adon-yodim derivative, is given by cf. [LS00, Section 7.6.4] dp 0 T dp U 0 = exp σ 2 µ q 2 du t 2 0 u t T 2σ 2 µ 2 q

4 4 Cheng, Cialenco, and Gong y maximizing the lielihood ratio with respect to, we obtain the following MLE for := µ q 2 T 0 T µ2 q 2 du t u t Together with 2.3, the MLE ˆ can be conveniently written as = 0 σ T where ξ n := µ q 1 w nt, and b n := n Lemma A.2, we have that. 3.2 µ q 1 w T = 0 σ ξ n, 3.3 µ2 q 2 T b µ2 q 2. Clearly, b n +. Moreover, in view of n=1 Varξ n b 2 n <. y Law of Large umbers cf. [Shi96, Theorem IV.3.2], n=1 ξ n/b 0, and thus by 3.3, we get lim = 0, P a.s., 3.4 namely is a strongly consistent estimator of 0. Finally, we note that 1 µ q 1 w T = L 0, T µ 2 q 2,, which combined with 3.3 gives the following trivial asymptotic normality result I 0 L 0, 1,, 3.5 where 4 ayesian Estimator I := T σ 2 µ 2 q In this section, we propose a ayesian type estimator for, and we will investigate its asymptotic behavior. As custom for ayesian statistics cf. [KP71] and [is02], we first prove a ernstein- Von Mises theorem for the posterior density, which will then allow us to estimate the asymptotic difference between the MLE and the ayesian estimator. Throughout this section, let Θ : Ω, F, P, be the unnown random parameter which admits a prior density function with respect to the Lebesgue measure ρ on. The realizations of Θ will be denoted by. law. 1 We use the symbol L = to denote equality in law between random variables, while L is used for convergence in

5 ayesian Estimations for SPDEs The ernstein-von Mises Theorem We recall that, generally speaing, the ernstein-von Mises type theorem states that the posterior distribution of the normalized distance between Θ and the MLE is asymptotically normal. This type of result implies that the posterior distribution measure approaches the Dirac measure as the number of observations increases; see emar 4.3 below for more details. Moreover, it also serves as an essential tool in derivation of some asymptotic properties of the ayesian estimator. We start by defining the posterior density of Θ as p U := dp U ρ dp η,, 4.1 dp U ρη dη dp where we recall that U = {u 1 t,..., u t, t [0, T ]} is the projected solution associated to the true parameter value 0. Some detailed discussions of the rational behind 4.1 are deferred to the Appendix. Together with 3.1 and 3.3, we deduce that p U = exp exp 0 σ 2 0 η σ 2 exp σ 2 = exp η σ 2 exp I 2 = exp I 2 T µ q 2 0 T µ q 2 0 µ q 2 T 0 T µ q ρ emar 4.1. In view of 4.2, we note that η 2 ρη dη du t 2 u t T 2σ η 2 T du t u t + du t u t 2 T 2σ 2 du t u t η2 T 2σ 2 2σ 2 µ 2 q 2 µ 2 q 2 µ 2 q 2 µ 2 q 2 ρ ρη dη ρ ρη dη. 4.2 a one possible choice of the conjugate prior is the truncated normal µ, σ 2 ; supported on I, in which case the posterior is also a truncated normal 1 µ+σ2 σ, 2 I 1 ;. σ 2 +I 1 σ 2 +I 1 b one can conveniently tae the non-informative prior ρ 1,. Although such prior admits no probabilistic meaning, the corresponding posterior is well defined and preserves the convergence results presented later in Section 4. ext, let Λ := I Θ be the normalized difference between Θ and. The posterior density of Λ is then given by p U = p + d U I d = C 1 e 2 /2ρ +, I,, 4.3 I

6 6 Cheng, Cialenco, and Gong where C := I exp I 2 η ρη dη = + 2 e 2 /2 ρ + d. 4.4 I I For notational convenience, we will extend the domains of ρ and p to, with ρ = 0 for, 0] and p U = 0 for, I ]. We are now in the position of presenting the ernstein-von Mises theorem. Theorem 4.2. Assume that ρ is a density function on which is strictly positive and continuous in a neighborhood of 0. Let f be a non-negative, orel-measurable function on such that C1 there exists α 0, 1 so that fx e αx2 /2 dx < ; C2 for any ε > 0 and δ > 0, lim e εi x >δ f I x ρ + x dx = 0, P a.s.. Then, lim f p U e 2/2 d = 0, P a.s emar 4.3. The above theorem implies that, for P a. s. ω, the posterior distribution measure of Θ converges wealy to the Dirac measure centered at 0, as the number of Fourier modes increases. Indeed, let g be any continuous and bounded function on. Without loss of generality, assume that g is non-negative otherwise, consider g + and g separately. y 4.3, for any ω Ω, g p U ω d = g + ω p U ω d. 4.6 I For each given ω Ω, we put f ω := g/ I + ω,. In light of the boundedness of g, f ω satisfies condition C1 and condition C2 above, for any given ω. From the proof of Theorem 4.2 see 4.11 below, equality 4.5 holds for f ω at any given ω. Together with the strong consistency of recalling 3.4, for P a.s. ω Ω, we deduce lim g p U ω d = lim = lim = lim = lim g + ω p U ω d I f ω p U ω d 1 1 where we used the dominated convergence theorem in the last equality. f ω e 2 /2 d g + ω e 2 /2 d = g 0, I

7 ayesian Estimations for SPDEs 7 The proof of Theorem 4.2 is based on the following technical lemma. Lemma 4.4. Under the conditions of Theorem 4.2, a there exist δ 0 > 0 such that lim fe 2 /2 ρ + ρ 0 δ 0 I I d = 0, P b for any δ > 0, lim >δ f e 2 /2 ρ + ρ 0 I I d = 0, P a.s.; a.s.. Proof. Pic δ 0 0, 0 such that ρ is continuous on [ 0 δ 0, 0 + δ 0 ]. ecall that the Fisher information I is unbounded, and thus, for any C > 0, there exists so that δ 0 I > C. We decompose the integral from part a as follows = 1 { 0 >δ 0 } +1 { 0 δ 0 +1 } { 0 δ 0 }. 4.7 δ 0 I δ 0 I C C< δ 0 I y the strong consistency of, ω 0 δ 0, P a.s., for sufficiently large that may depend on ω. Hence lim 1 { 0 >δ 0 } f e 2 /2 ρ + ρ 0 δ 0 I I d = 0, P a.s Moreover, by the strong consistency of and the continuity of ρ on [ 0 δ 0, 0 + δ 0 ], for any κ [ C, C], lim ρ + ρ 0 I = 0, P a.s.. Hence, by condition C1 and dominated convergence theorem, lim { 0 δ 0 f e 2 /2 } ρ + ρ 0 I d = 0, P a.s C Finally, by C2 and the boundedness of ρ on [ 0 δ 0, 0 + δ 0 ], 1 { 0 δ 0 } f e 2 /2 ρ + ρ 0 C< δ 0 I I d 2M f e α2 /2 e 1 α2 /2 d 2M e 1 αc2 /2 f e α2 /2 d, 4.10 κ >C where M := sup [0 δ 0, 0 +δ 0 ] ρ. Combining , and since C > 0 is arbitrarily chosen, we complete the proof of part a. As for part b, note that for any δ > 0, >δ f e 2 /2 ρ + ρ 0 I I d I f I e I 2 /2 ρ + d + ρ 0 >δ >δ f e 2 /2 d I I e I δ 2 /2 f I ρ + d + ρ 0 >δ f e 2 /2 d. I >δ

8 8 Cheng, Cialenco, and Gong and in view of the conditions C1 and C2, b follows at once. The proof is complete. Proof of Theorem 4.2. Since the constant function f 1 satisfies C1 and C2, by Lemma 4.4, e κ2 /2 κ ρ + ρ 0 I dκ = 0, P a.s., lim which, together with 4.4, implies that κ lim C := lim e κ2 /2 ρ + dκ = ρ 0 I e κ2 /2 dκ = ρ 0, P a.s.. Therefore, as, fκ p κ U,T e κ2/2 dκ C 1 fκ e κ2 /2 κ ρ + s ρ 0 I dκ + ρ 0 1 fκ e κ2 /2 dκ 0, P a.s., which completes the proof of the theorem. C Corollary 4.5. If the prior density ρ is positive and continuous in the neighborhood of 0, and has a finite r-th moment, for some r 1, then lim p r U,T e 2 /2 d = 0, P a.s.. Proof. Clearly, the power function f = r satisfies the condition C1, so it remains to chec that f satisfies the condition C2. Indeed, for any ε > 0 and δ > 0, e εi f I ρ + d = I r/2 e εi r ρ d >δ >δ 2 r 1 I r/2 e εi r r ρ d + 0, P a.s., as. The corollary then follows immediately from Theorem Asymptotic Properties of the ayesian Estimator Throughout this subsection, let l : [0, be a orel measurable loss function. That is, l is locally bounded on such that lx 1 lx 2, for any x 1, x 2 with x 1 x A ayesian estimator for the unnown parameter, with respect to a loss function l, is defined as β := arg min l β p U d, 4.13 β given that the minimum is strict and attainable. When β is well defined for all, the following theorem provides a sufficient condition for the consistency and asymptotic normality of β.

9 ayesian Estimations for SPDEs 9 Theorem 4.6. Assume that the prior density ρ is positive and continuous in a neighborhood of 0, and that β is well-defined with respect to a loss function l, for every. Moreover, assume that there exists {a }, a test function f as in Theorem 4.2 satisfying C1 and C2, and another loss function l, such that i a l f, for any and ; I ii lim sup a l l I = 0, for any compact set A ; A iii r l + re 2 /2 d has strict minimum at r = 0. Then, a b lim I β = 0, P a.s.; lim a l β p U d = l e 2 /2 d, P a.s.. In particular, β is strongly consistent and asymptotically normal, as, β 0, P a.s. and I β 0 L 0, 1. Proof. The strong consistency and asymptotic normality of β are immediate consequences of part a together with the strong consistency and asymptotic normality of recalling 3.4 and 3.5. The proof of a and b is split in four steps. Step 1. We will first show that y the definition of β, l lim sup a l β p U d l e 2 /2 d, P a.s β p U d l p U d = l p U d. I Hence, to prove 4.14, it suffices to show that lim a l p U d = l e 2 /2 d, P a.s I Using conditions i and ii, we obtain that l f, for any. Therefore, by conditions

10 10 Cheng, Cialenco, and Gong i and ii, Theorem 4.2, and dominated convergence theorem, we deduce lim a l p U d l e 2 /2 d I lim a l l I p U e 2/2 d + lim l p U e 2/2 d + lim a l l e 2 /2 I d lim 3 f p U e 2/2 d + lim a l l e 2 /2 I d K 0 + lim lim K a l l e 2 /2 I d lim lim sup K a l l K e 2 /2 I d = 0, [,K] which completes the proof of 4.15, and therefore, of Step 2. ext, we will show that the sequence of random variables Y := I β,, are uniformly bounded, P a.s.. That is, for P a.s. ω, there exists Kω 0,, such that Y ω Kω for all. For any ω A := {ω Ω : lim sup Y ω = } and any K, there exists an increasing sequence of integers j = j ω, K, j, such that j, as j, and Y j ω K, for any j. Consequently, l β j ω p + Yj ω U j ω d = l p U j ω d Ij K + K l p U j ω d, 4.16 Ij where we used 4.12 in the last inequality. On the other hand, since l is locally bounded, the function 1 [,K] l + K is bounded on and thus satisfies conditions C1 and C2. Hence, by Theorem 4.2 and condition ii, as j, a j K K + K l p K U j d l + K e 2 /2 d Ij a + K j l l + K Ij p K U j d + l + K p e 2/2 U j d sup a K j l l Ij + l+k p e 2/2 U j d 0, P a.s [ 2K,2K] Let denote the exceptional subset of Ω in which the limit in 4.17 does not hold, then P = 0.

11 ayesian Estimations for SPDEs 11 Without loss of generality, assume that A \. y 4.16 and 4.17, for any ω A \, lim sup a j l j β j ω p K + K U j ω d lim l p U j ω d + j Ij = K l + K e 2 /2 Since K is arbitrary, by condition iii and monotone convergence theorem, we have lim sup a j l j β j ω p K U j ω d lim l+k e 2 /2 d > l e 2 /2 d. + K In view of 4.14, we must have PA \ = 0 so that PA = 0, completing the proof of Step 2. Step 3. We now prove a and b. y Step 2, for P a.s. ω, there exists j = j ω, j, such that the sequence Y j ω j is convergent, as j, and we denote its limit by Y ω. For any K > 0, a j l β j ω p U j ω d = aj l + Moreover, for j large enough, K a j l K a j l + K K + + K K + Yj ω p K U j d Ij a j K d. + Yj ω Ij p U j d l + Yj ω Ij p U j d l + Y ω e 2 /2 + Yj ω Ij l + Y j ω p U j ω d l +Y j ω p e 2 /2 K U j ω d + a j l + Yj ω Ij l + Y j ω p U j ω d d l +Y j ω l +Y ω d e 2 /2 l +Y ω+11 [0, + l +Y ω 11,0 p U j ω e 2 /2 d l + Y j ω l + Y ω d e 2 /2 y the same argument as in 4.17, the first two integrals in 4.19 vanish as j, for P a.s. ω. Moreover, the monotone property 4.12 for l implies that it is almost surely with respect to the Lebesgue measure continuous on. Hence, conditions i and ii which implies that l is bounded by f, together with dominate convergence, implies that the last integral in 4.19 vanishes as j, for P a.s. ω. Therefore, for P a.s. ω, we have K + lim a Yj ω j l p K U j d = l + Y ω e 2 /2 d j Ij

12 12 Cheng, Cialenco, and Gong Combining 4.18 and 4.20, and noting that K > 0 is arbitrary, we obtain that lim inf j a j l β j p U j d l + Y ω e 2 /2 d l e 2 /2 d, P a.s., 4.21 where we have used condition iii in the second inequality. ow assume that PY 0 > 0. Then inequality on 4.21 would be strict in {Y 0} by assumption 3, which is clearly a contradiction to Therefore, PY 0 = 0. Combining 4.14 in Step 1 with 4.21 completes the proof of b. To prove a, it remains to show that every subsequence of Y converges to 0 P a.s.. Indeed, assume the contrary. Applying similar arguments as in Step 3 to this exceptional subsequence leads to a contradiction to The proof is now complete. 5 umerical Example In this section, we provide an illustrative numerical example of the asymptotics of the MLE and the ayesian estimator derived in the previous sections. Specifically, we consider the following equation { dut, x + 0 u xx t, x dt = σ u th x α dw t, u0, x = π2 4 x π , where h x := 2/π sinx, x [0, π], and α < 1 so that the solution to 5.1 exists. Throughout this section, we fix the parameters σ = 1, and T = 1, and tae the true value of the parameter of interest to be 0 = 1. ote that, in view of 3.6, in this case the Fisher information I for parameter taes the form I = T σ 2 4 2α. When α 1, the rate of divergence of I as, and thus the rates of convergence of both the MLE and the ayesian estimator, become smaller, as. We will illustrate this below by considering two different regimes α = 0 and α = We simulate = 20 independent rownian sample paths {w t, t [0, 1]}, = 1,..., 20, with time-step of , and compute the corresponding Fourier modes {u t, t [0, 1]}, = 1,..., 20, using 2.4 and 0 = 1. Then, we compute the MLE according to 3.2, where the Itô integral is approximated by the finite sum with time-step of Moreover, we simulate the posterior density p U using 4.2, where the integral in the denominator is approximated using the integral Matlab built-in function on the time interval [0, 5] with error tolerance In Figures 1 and 2, we present and compare the posterior density p U under two different prior distributions, the uniform distribution U0, 1.5 and truncated normal 0, 1;, for α = 0 and α = 0.999, respectively. For both choices of α, the posterior densities under both priors converge to the Dirac measure concentrated at the MLEs, which is consistent with the discussion in emar 4.3. Moreover, under both priors, the posterior densities with α = 0 exhibit faster convergence rates than those with α = as expected. ext, we tae the quadratic loss function lx = x 2,

13 ayesian Estimations for SPDEs 13 Figure 1: Posterior densities for = 2, = 4, two different priors, and α = 0 Figure 2: Posterior densities for = 2, = 4, two different priors, and α = in which case the ayesian estimator is given by β = p U d. In Figures 3 and 4, we compare the MLE with the ayesian estimator β under both priors U0, 1.5 and truncated normal 0, 1; for different numbers of Fourier modes = 1,..., 20, with α = 0 and α = 0.999, respectively. Again, while both the MLEs and the ayesian estimators converge to the true parameter 0 = 1, as the number of Fourier modes increases, for both choices of α, the case of α = 0 tends to have a better convergence rate. A Auxiliary esults For the sae of completeness, we recall a version of strong law of large number, which will be used in the proof of the strong consistency of MLE. We refer the reader to [LS78, Theorem IV.3.2] for its detail proof.

14 14 Cheng, Cialenco, and Gong Figure 3: MLE vs. ayesian Estimator with α = 0 Figure 4: MLE vs. ayesian Estimator with α = Theorem A.1 Strong Law of Large umbers. Let {ξ n } n be a sequence of independent random variables with finite second moments. Let {b n } n be a sequence of non-decreasing real numbers such that lim n b n =, and n=1 Varξ n/b 2 n <. Then, n ξ Eξ b n 0, P a.s., n. Also here we present a simple technical lemma used in Section 3. Lemma A.2. If the sequence {a n } n satisfies a 1 > 0 and a 0, 2, then n=1 a n n a 2 < +.

15 ayesian Estimations for SPDEs 15 Proof. ote the following which finishes the proof. n=1 a n n a 2 1 a = 1 a n=2 n=2 = 1 a a 1 a n n 1 a n a 1 n 1 a 1 a, 1 n a Discussions on Derivation of the Posterior Density In this appendix, we will present a formal derivation of the posterior density 4.1. We mae the following standing assumptions i The random variable Θ is independent of the rownian motions {w, }; ii σθ F 0. ecall that U denotes the first Fourier modes of the solution u to 2.1 with parameter. Let U Θ be the -valued process obtained by substituting with Θ in U. y condition i, for any [0,T ] -measurable functional f on [0,T ] with EfU Θ, Θ <, we have that E f U Θ, Θ = f U, ρ d..1 E ext, we will show that, for any, P Θ U Θ = where, r η U := exp η σ 2 r η U Θ ρη dη r η T µ q 2 0 U Θ ρη dη,.2 du t u t η2 T 2σ 2 µ 2 q 2 for any, η For the sae of argument, we assume that r η is [0,T ] -measurable. Clearly,.2 follows immediately from the following equality E 1 {Θ } U Θ ρη dη U Θ = r η U Θ ρη dη..3 r η To prove.3, for any A [0,T ], we first deduce from.1 that E 1 {U Θ A} 1 {Θ } r η U Θ ρη dη = E 1 {U A} r η =, U ρη dη ρ d E 1 {U A} rη U ρ d ρη dη.

16 16 Cheng, Cialenco, and Gong In view of 3.1, we have dp η dp U = rη r U U..4 Hence, by Girsanov theorem, E 1 {U Θ A} 1 {Θ } r η U Θ ρηdη = = dp η E E dp U 1{U A}r U ρd ρηdη 1 {U η A} r U η ρ d ρη dη..5 On the other hand, E 1 {U Θ A} r η U Θ ρη dη = = E 1 {U A} r η U ρη dη ρ d E 1 {U A} rη U ρη dη ρ d..6 Combining.5 and.6 leads to.3. eferences [is99] [is02] [KP71] J. P.. ishwal. ayes and sequential estimation in Hilbert space valued stochastic differential equations. J. Korean Statist. Soc., 281:93 106, J. P.. ishwal. The ernstein-von Mises theorem and spectral asymptotics of ayes estimators for parabolic SPDEs. J. Aust. Math. Soc., 722: , J. orwaner, G. Kallianpur, and. L. S. Praasa ao. The ernstein-von Mises theorem for Marov processes. Ann. Math. Statist., 42: , [T17] M. ibinger and M. Trabs. Volatility estimation for stochastic PDEs using highfrequency observations. Preprint, arxiv: , [CH17] [Cho07] I. Cialenco and Y. Huang. A note on parameter estimation for discretely sampled SPDEs. Preprint, arxiv: , P. Chow. Stochastic partial differential equations. Chapman & Hall/CC Applied Mathematics and onlinear Science Series. Chapman & Hall/CC, oca aton, FL, [Cia10] I. Cialenco. Parameter estimation for SPDEs with multiplicative fractional noise. Stoch. Dyn., 104: , [Cia18] I. Cialenco. Statistical inference for SPDEs: an overview. Forthcoming in Statistical Inference for Stochastic Processes, Feb [CL09] I. Cialenco and S. V. Lototsy. Parameter estimation in diagonalizable bilinear stochastic parabolic equations. Stat. Inference Stoch. Process., 123: , 2009.

17 ayesian Estimations for SPDEs 17 [GHKVZ14]. Glatt-Holtz, I. Kuavica, V. Vicol, and M. Ziane. Existence and regularity of invariant measures for the three dimensional stochastic primitive equations. J. Math. Phys., 555:051504, 34, [GHZ08] [GHZ09]. Glatt-Holtz and M. Ziane. The stochastic primitive equations in two space dimensions with multiplicative noise. Discrete Contin. Dyn. Syst. Ser., 104: , Glatt-Holtz and M Ziane. Strong pathwise solutions of the stochastic avier-stoes system. Adv. Differential Equations, 145-6: , [Hai09] M. Hairer. Introduction to Stochastic PDEs. Unpublished lecture notes, [L17] [LS78] [LS00] S. V. Lototsy and. L. ozovsy. Stochastic partial differential equations. Universitext. Springer, Cham, S. Liptser and A.. Shiryayev. Statistics of random processes. II. Springer-Verlag, S. Liptser and A.. Shiryayev. Statistics of random processes I. General theory. Springer-Verlag, ew Yor, 2nd edition, [P00]. L. S. Praasa ao. ayes estimation for some stochastic partial differential equations. J. Statist. Plann. Inference, 912: , Prague Worshop on Perspectives in Modern Statistical Inference: Parametrics, Semi-parametrics, onparametrics [PT07] [Shi96] J. Pospíšil and. Tribe. Parameter estimates and exact variations for stochastic heat equations driven by space-time white noise. Stoch. Anal. Appl., 253: , A.. Shiryaev. Probability, volume 95 of Graduate Texts in Mathematics. Springer- Verlag, ew Yor, second edition, 1996.