Parameter estimation for the stochastic heat equation with multiplicative noise from local measurements

For the stochastic heat equation with multiplicative noise we consider the problem of estimating the diﬀusivity parameter in front of the Laplace operator. Based on local observations in space, we ﬁrst study an estimator, derived in [3] for additive noise. A stable central limit theorem shows that this estimator is consistent and asymptotically mixed normal. By taking into account the quadratic variation, we propose two new estimators. Their limiting distributions exhibit a smaller (conditional) variance and the last estimator also works for vanishing noise levels. The proofs are based on local approximation results to overcome the intricate nonlinearities and on a stable central limit theorem for stochastic integrals with respect to cylindrical Brownian motion. Simulation results illustrate the theoretical ﬁndings.


Introduction
We consider estimation of the diffusivity parameter ϑ > 0 in the stochastic heat equation with multiplicative noise = ϑ∆X(t) dt + σ(X(t)) dW (t), 0 < t ≤ T, X(0) = X 0 , X(t)| ∂Λ = 0, 0 < t ≤ T. (1.1) Here, W is a cylindrical Brownian motion with values in L 2 (Λ), where Λ is an open bounded interval in R, dW (t)/dt is also referred to as space-time white noise.The function σ : R → R + generates a multiplicative noise, see Section 2 below for precise assumptions.Multiplicative noise appears naturally in stochastic partial differential equations (SPDEs) as a scaling limit or to ensure positivity of the solution, see e.g. the examples given in [17], [18] or [4].
Diffusivity estimation has emerged as a benchmark inference problem for SPDEs.The spectral estimation approach, initiated by [11], has been shown to give reliable estimation results even for more general semi-linear equations like the stochastic Navier-Stokes equation [8], yet always assuming additive noise.In [9] a specific case of multiplicative noise has been treated which leads to geometric Brownian motions in the spectral decomposition of the Laplacian.In [6] also Bayesian estimators have been developed and analysed in this setting.
Similarly, discrete observations of the solution X in time and space give rise to realised p-variation estimators for quite general classes of SPDEs.Most notably, in [18] a precise convergence analysis of p-variation of X(t, x) in space x with p = 2 and in time t with p = 4 is given, which leads to a consistent diffusivity estimator in the multiplicative noise case, while convergence rates or asymptotic normality are not considered.Estimation of the multiplicative noise function σ(•) from discrete observations is treated in [7] with intriguing phenomena arising in central limit theorems for p-variations.
Recently, methods for local observations in space have provided a new methodology for linear and semi-linear SPDEs with additive noise [3,2].This has enabled the estimation of diffusivity in a stochastic cell motility model from experimental data [1].The underlying SPDE with additive noise describes chemical concentrations, for which, however, a multiplicative noise structure might be more natural as well as more in line with the empirical data than additive noise.
Starting point of our work is the question whether the additive noise estimator (ANE) derived in [3] for local observations of a stochastic heat equation with additive noise is robust against a multiplicative noise misspecification the same way, as it is against nonlinear reaction terms [2].Technically, we cannot use a splitting technique to separate the nonlinear from the linear part and we must derive new tools to analyse the estimation error.This is achieved by a stepwise disentanglement and localisation of the statistics, carried out in Proposition 5.7 below.The result is that the estimator has the same rate as for additive noise, but it is asymptotically mixed normal under stable convergence with a suboptimal conditional variance for varying σ(•).
Therefore we improve the ANE by taking into account the varying quadratic variation of the martingale term in the ANE.The multiplicative noise estimator (MNE) obtained this way satisfies a central limit theorem with smaller variance provided the multiplicative noise σ(•) is bounded away from zero.Since in many cases it is natural that σ(•) vanishes at some boundary values, we improve the MNE to the stabilised multiplicative noise estimator (SMNE), satisfying a stable central limit theorem with small conditional variance even when σ(•) vanishes sometimes.
The exact setting is introduced in Section 2. The construction of the estimators, the main asymptotic results and an application to confidence intervals are presented in Section 3. In Section 4 we discuss the implementation of the estimators and their behaviour for three fundamentally different noise specifications.The detailed proofs are delegated to Section 5.The stable convergence results require a martingale representation theorem in terms of cylindrical Brownian motion and rely on asymptotic orthogonality of martingales by spatial localisation, which might be of independent interest.This material is therefore gathered in Section 6.

The model 2.1 Notations
We write R + := [0, ∞), a∧b := min(a, b) and a∨b := max(a, b).By A δ ≲ B δ we mean that there exists some constant C > 0 such that A δ ≤ CB δ for all values δ under consideration.Here, we work with δ ∈ (0, 1) or with the convergence δ → 0. Convergence in probability and convergence in distribution are denoted by P → and d →, respectively.The symbol stably − −−− → denotes stable convergence, see e.g.[13], Chapter VIII.Section 5. We say that X δ stably − −−− → X holds on an event Sobolev space of order s on Λ and H 1 0 (Λ) the space of all f ∈ H 1 (Λ) with f (x) = 0 for x ∈ ∂Λ.We use the standard Laplace operator notation ∆z = z ′′ for z ∈ H 2 (R), even in the simple one-dimensional case.

The stochastic heat equation
Let (Ω, F , (F t ) 0≤t≤T , P) be a stochastic basis equipped with the cylindrical Brownian motion W taking values in L 2 (Λ).The filtration (F t ) 0≤t≤T is assumed to be generated by the cylindrical Brownian motion and augmented by P-null sets.We study the stochastic heat equation (1.1) with multiplicative noise.The initial value X 0 ∈ L 2 (Λ) is supposed to be deterministic and continuous on Λ.We require throughout the following two assumptions.

The observation scheme
As motivated in [3,1], we observe the solution process (X(t, x), t ∈ [0, T ], x ∈ Λ) only locally in space around some point x 0 ∈ Λ.That point x 0 as well as the terminal time T ∈ (0, ∞) remain fixed.More precisely, the observations are given by a spatial convolution of the solution process with a kernel K δ,x0 , localising at x 0 as the resolution δ tends to zero.This kernel might for instance model the point spread function in microscopy.

The additive noise estimator
We study first the augmented maximum likelihood estimator θδ from [3], derived for the stochastic heat equation with additive space-time white noise.
Definition 3.1.The additive noise estimator (ANE) θδ of the parameter ϑ > 0 is defined as According to (2.5), the numerator and the fundamental error decomposition is given by where The term I δ is incorporated because it gives the quadratic variation of the martingale M δ in time.It turns out that δ 2 I δ converges in probability to the limit (2ϑ T 0 σ 2 (X(t, x 0 )) dt, compare Proposition 5.10 below for bounded σ(•).Since the quadratic variation I δ does not become asymptotically deterministic, we cannot rely on a standard martingale central limit theorem (e.g.Theorem 1.19 in [15]) to prove asymptotic normality of M δ /I 1/2 δ .Therefore we employ the concept of stable convergence, which is stronger than convergence in distribution and allows to formulate mixed normal limits and to derive feasible confidence intervals, see e.g.[13] for a general introduction.In Section 6 we prove a general martingale representation theorem and a stable limit theorem for martingales with respect to cylindrical Brownian motion filtrations.As a consequence, we obtain the following result, when specialising Corollary 6.3 to our setting involving the kernels K δ,x0 : Proposition 3.2.Let (Y δ (t), 0 ≤ t ≤ T ) for δ ∈ (0, 1) be L 2 (Λ)-valued processes, progressively measurable with respect to the cylindrical Brownian filtration (F t ) 0≤t≤T and satisfying ) holds Lebesgue-almost everywhere in Λ for all t ∈ [0, T ], then a stable limit theorem for the stochastic integrals holds as δ → 0: with an independent scalar Brownian motion (B(t), 0 ≤ t ≤ T ) (on an extension of the original filtered probability space).
The main point of this result is that the limiting Brownian motion B becomes independent because the support of Y δ (t) shrinks asymptotically to the point x 0 .Here we shall apply the proposition with Y δ (t) = δX ∆ δ,x0 (t)σ(X(t))K δ,x0 .Our first main result is that the additive noise estimator θδ satisfies a stable central limit theorem with rate δ.Theorem 3.3.Grant Assumptions (S) and (X).Then the ANE θδ satisfies on the event as δ → 0, where Z ∼ N (0, 1) is independent of F T .
Proof.The detailed proof is deferred to Section 5.4.
This result establishes a very desirable robustness property of the ANE θδ : Even though it was designed for estimation in the stochastic heat equation with additive noise, the ANE still converges with the same rate δ to the true parameter under multiplicative noise.This is analogous to the ordinary least squares estimator in linear regression with heteroskedastic noise, which still attains optimal rates, yet loses in the variance due to the variability in noise levels.An efficient regression estimator is obtained by a noise-level weighted least squares method, which provides an analogy for our next estimators.

The multiplicative noise estimator
We aim at improving the ANE by adjusting the estimator in such a way that the denominator contains already the quadratic variation of the martingale part in the numerator.To that end, we need to incorporate the term ∥σ(X(t))K δ,x0 ∥ 2 that is not observed directly, but is still attainable from the data.The quadratic variation of the observed semi-martingale (X δ,x0 (t), 0 ≤ t ≤ T ) equals So we have access to ∥σ(X(t))K δ,x0 ∥ 2 by differentiation of the realized quadratic variation.For discrete time data, sampled at high-frequency, some spot volatility estimators from the field of mathematical finance can be used to access this term, see Section 4 below.This way we obtain a second estimator, taking into account the multiplicative noise in the stochastic heat equation.
Definition 3.4.The multiplicative noise estimator (MNE) θδ of the parameter ϑ > 0 is defined as Let us remark that the MNE θδ can also be derived like the ANE θδ in [3], maximising a corresponding pseudo-likelihood in the multiplicative noise case.An alternative interpretation is that we regress the increment dX δ,x0 (t) on X ∆ δ,x0 (t) and weight it by the inverse squared noise level ∥σ(X(t))K δ,x0 ∥ −2 , exactly as in weighted least squares for regression.Since this is done under the correct model specification, we expect better estimation properties.
Then as δ → 0 Proof.The proof is deferred to Section 5.4.

The stabilised multiplicative noise estimator
The lower bound σ > 0 on σ(•) required for the MNE θδ can be restrictive.For instance, when the random field X(t, x) shall not take negative values, models usually require that lim x↓0 σ(x) = 0. To cover this case as well, we stabilise the denominators in the integrands of equation (3.5) by adding a number ε 2 δ which tends to zero slowly as δ → 0. Definition 3.6.Let ε δ = ε(δ) be a real function satisfying for any η > 0 as δ → 0. Then the stabilised multiplicative noise estimator (SMNE) ϑ ⋆ δ of the parameter ϑ > 0 is defined as Condition (3.8) says that ε δ tends to zero more slowly than any polynomial.It is satisfied for ε δ = 1 log(δ −1 ) .To analyse the asymptotic properties of the SMNE ϑ ⋆ δ , we need to strengthen Assumptions (S) and (X) slightly.

Assumption (S'
).The function σ(•) is β σ -Hölder continuous, i.e., for some Assumption (X').Assumption (X) is satisfied and moreover the solution X is in quadratic mean β x -Hölder continuous in the space variable and β t -Hölder continuous in the time variable, i.e., for some is Lipschitz continuous and the initial condition X 0 is continuous, then standard contraction arguments for the stochastic convolution and the regularity of the Green function for the heat equation yield Assumption (X), see e.g.[5].Even Assumption (X') holds with β x = 1/2 and β t = 1/4, provided σ(•) is Lipschitz continuous and the initial condition X 0 is 1/2-Hölder continuous.In fact, standard proofs for pathwise Hölder regularity go via the Kolmogorov-Chentsov theorem and thus establish (3.10), compare Theorem 2.1 in [18] or Corollary 3.4 in [20] for a slightly more involved case on an unbounded domain.
The intriguing questions of weak existence, regularity and pathwise uniqueness for the stochastic heat equation with β σ -Hölder continuous multiplicative noise σ(•) have so far only found partial answers.We refer to Theorem 1.3 in [17], which yields our Assumption (X) in case β σ > 3/4 in case of an unbounded domain.For their continuity result the authors assert that the results in [19], formulated for coloured noise in space, work analogously for the spacetime white noise case.Equations ( 10) and ( 19) in [19] then establish Hölder regularity of X in the sense of Assumption (X').
We turn to the analysis of the stabilised multiplicative noise estimator.The error decomposition for the SMNE ϑ ⋆ δ follows from (3.9) and (2.5): The term I ⋆ δ is the quadratic variation of the martingale part M ⋆ δ .The limits of I ⋆ δ and J ⋆ δ for δ → 0 involve a (in general random) time length T ⋆ during which σ(X(t, x 0 )) does not vanish, compare Proposition 5.10 below.So, we use again the stable limit theorem of Proposition 3.2 and derive a central limit theorem for the SMNE ϑ ⋆ δ with rate δ, without assuming a lower bound on σ(•).
Proof.The proof is deferred to Section 5.4.
Remark 3.9.From the series of inequalities we infer that the (conditional) asymptotic variance of the SMNE lies between those of the ANE and the MNE.Remember, however, that the asymptotics for the MNE were derived under the condition σ > 0, implying T ⋆ = T .The extreme case σ(•) ≡ 0 leads to the deterministic heat equation, which for the initial condition X 0 = 0 remains zero all the time and does not allow for inference on ϑ.This type of degeneracy is excluded for the SMNE by the condition T ⋆ > 0.

Confidence intervals
The asymptotic (mixed) normality of the three estimators allows us to prescribe asymptotic confidence intervals for the parameter ϑ.The asymptotic conditional variances depend on quantities unknown to the statistician.Yet, in all three error decompositions (3.2), (3.6) and (3.11) it is shown in the proofs (see Section 5.4 for the details) that the martingale term divided by the square root of its quadratic variation is asymptotically standard Gaussian.Dividing each error decomposition by the respective second factor on the right-hand side directly gives an asymptotic confidence statement.
Note that the confidence intervals only rely on the observation processes (X ∆ δ,x0 (t), 0 ≤ t ≤ T ), (X δ,x0 (t), 0 ≤ t ≤ T ) and the quadratic variation of the latter.Even the kernel K and the resolution level δ need not be known.In the next section we shall see how the estimation methods can be implemented when only data is available that is discretely sampled in time.

Implementation and simulation results
We illustrate the main results in a setting similar to the experimental setup in [1], where the diffusivity parameter ϑ was estimated in a concrete stochastic We have chosen σ 2 (•) to have Hölder regularity 0.8 in line with Example 3.7 and not to vanish completely at zero so that all three estimators are applicable.σ 3 (•) generates strong noise level fluctuations so that the quality of the estimators should differ significantly.
An approximate solution is computed on a regular time-space grid {(t j , y k ) : t j = T j/N, y k = Lk/M, j = 0, . . ., N, k = 0, . . ., M } with N = 48 000 and M = 800 by the Euler-Maruyama scheme.For the drift part, we use the finite difference approximation of ∆ that is applied implicitly, while σ(•) in the stochastic term is applied to the current state of the solution explicitly, compare Algorithm 10.8 in [16].The mesh sizes fulfill T /N ≍ (L/M ) 2 , ensuring the Courant-Friedrichs-Lewy (CFL) condition for stable simulations [16].Heat maps for typical realisations with multiplicative noise σ 2 (X(t)) and σ 3 (X(t)) are displayed in Figure 1.Under σ 2 (•) we see that fluctuations are larger for higher temperature levels, while at the boundary it cools down to zero almost deterministically.Under σ 3 (•) excitations by strong noise at the interface values 2 and 4 are counteracted by the diffusion, which leads to almost noiseless inner regions with strong fluctuations of the interfaces in time.The spatial gradient at the interfaces is very large, which is no numerical artefact, but due to expulsion by noise.
As in [1] we employ the smooth compactly supported kernel The term Y (t) := ∥σ(X(t))K δ,x0 ∥ 2 is accessed by the following procedure.In view of (3.4), Y (t) presents the spot squared volatility of X δ,x0 at time t, which we estimate by i.e., by taking the average disintegrated realised quadratic variation over the past D = 800 values (= 0.5 time units).It is a kernel type estimator of spot squared volatility that follows classical methods, see e.g.[12], Section 2, for a description and further references.The one-sided estimation kernel is employed so that only historical data are used in the construction and the averaging acts as a smoothing, putting the same weights on the past D values.Finally, we choose the stabilising value ε 2 δ = 0.001 log(10/δ) such that it satisfies condition (3.8) and lies within the range of typical values of ∥σ(X(•))K δ,x0 ∥ 2 .The possible issue could be that if the term ε 2 δ is much smaller than ∥σ(X(•))K δ,x0 ∥ 2 , the SMNE would practically become the MNE.On the other hand, if the term ε 2 δ dominated ∥σ(X(•))K δ,x0 ∥ 2 , then the SMNE would practically coincide with the ANE.So, in practice we recommend to estimate the spot volatility first and then to adjust ε 2 δ accordingly.Figure 2 displays simulation results for the estimators of the parameter ϑ obtained after 1 000 Monte Carlo runs for each of the functions σ 1 , σ 2 and σ 3 .The red lines in the histograms indicate the asymptotic distribution, obtained as a mixture of 1 000 Gaussian densities that (individually, for each run) follow the theoretical results established in Theorems 3.3, 3.5 and 3.8.Monte Carlo mean and standard deviation for every case are stated in Table 1.
In the additive noise case σ 1 all three estimators perform similarly well.In this case we have equalities in (3.13) and the resulting asymptotic distributions coincide.In the "Hölder" multiplicative noise case σ 2 , the estimator ANE performs slightly worse than the two alternatives.Since σ 2 (•) ≥ 0.01 > 0, we have T ⋆ = T and the estimators MNE and SMNE deliver similar results.
For σ 3 the histogram of the ANE in Figure 2 (bottom, left) is much more spread out, but has not yet entered the asymptotic regime with a very flat asymptotic density.There are, however, quite a few outliers (12.6 %) outside the interval [0, 0.1], which are not shown and which are caught pretty well by the tails of the asymptotic density.Note also that the corresponding empirical standard deviation in Table 1 is very high with about half the length of the interval [0, 0.1].The estimators MNE and SMNE give a significant improvement here with an error distribution that is almost unchanged with respect to the cases σ 1 and σ 2 .It is worth noting that the assumption σ > 0 from Theorem 3.5 for the MNE is violated by σ 3 and we also had T ⋆ < T , but with a minor difference only.In the discrete numerical setting we use the threshold 10 −6 to determine whether σ(X(t, x 0 )) is zero or not.
Simulation results for varying δ confirm the convergence rate δ as δ → 0. Figure 3 shows a log 10 -log 10 plot of root mean squared estimation errors for the estimators ANE θδ and SMNE ϑ ⋆ δ obtained after 100 Monte Carlo simulations for each δ based on the multiplicative noise σ 2 and σ 3 .The estimator MNE θδ is omitted here, because under the noise σ 2 the differences between MNE and SMNE are minimal and the assumption σ > 0 for the MNE is again violated by σ 3 .The estimation errors are significantly smaller in the "Hölder" multiplicative noise case σ 2 and the SMNE provides a very substantial improvement in the σ 3 case.The errors are very well aligned with the asymptotic standard error as predicted by Theorems 3.3 and 3.8.The reference line with slope 1 is added to compare with the theoretical convergence rate δ.Note that the spatial discretisation for reliable simulations must always be much finer than δ.In our setup with δ = 0.6 and L/M = 0.025, the localised kernel K δ,x0 was evaluated discretely on 48 grid points.Further unreported simulations show that the performance of the estimators is not influenced by the location of the central observation point x 0 , unless x 0 is located very close to the boundary.In fact, if several local measurements (localised around points {x j 0 : j = 1, . . .J}) are available, it is possible to combine local estimators, see [1], where such an approach is explained and used.

Mean (SD
In conclusion, the two newly proposed estimators MNE and SMNE performed as well as the ANE or even better than the ANE and their asymptotic distribution matches the results obtained in Section 3. The ANE provides good estimation accuracy also under multiplicative noise, but its accuracy suffers under strongly varying σ(•).

Proofs
First we shall establish all results under the additional condition (5.1) Using the continuity of the solution X(t, x), we shall then get rid of this assumption in the last step of the proofs of central limit theorems in Section 5.4.
Proof.For any t ∈ [0, T ], we have by the continuity of σ and X, provided by Assumptions (S) and (X), Here the dominated convergence theorem is applied with integrable majorant σ 2 K 2 (•).
For α = 2 we combine Proposition 3.5(i) in [3] and the second inequality from Lemma A.2(iii) in [3] such that R) has compact support and is fixed throughout.
In the sequel, we need a uniform bound on the second and fourth centered moment of X ∆ δ,x0 (t).
The second part follows via Jensen's inequality.
. Moreover, for any z, y ∈ L 2 (Λ) and δ ∈ (0, 1) we have where we used |A 2 − B 2 | = |A − B||A + B| together with the upper bound 2σ∥K∥ L 2 (R) in the first inequality and then we followed up with the reverse triangle inequality. For . Moreover, for any z, y ∈ L 2 (Λ) and δ ∈ (0, 1) we have Therefore the proof is finished as in the previous case. (ii).For ∥ and the reverse triangle inequality to obtain Similarly to the previous case, algebraic calculations yield which finishes the proof. Introduce (5.4) In the following proposition we present different expressions that are equal to L δ up to terms that are of lower order than δ −2 .This is the major ingredient for the proofs of the main results, noting that the techniques developed in [3] cannot be used here due to the multiplicative noise structure.
Proposition 5.7.Grant Assumptions (S) with (5.1), (X) and let f δ satisfy condition (F1) or grant Assumptions (S') with (5.1), (X') with (3.8) and let f δ satisfy condition (F2).Then L δ from (5.4) equals up to additive terms of order o P (δ −2 ) for δ → 0: Remark 5.8.The overall idea is to achieve the representation L (vii) δ via slight consecutive alterations.In point (i) we shorten the outer integral to the interval [δ, T ], in (ii), we present a slight time shift of the solution in the functional, i.e., f δ (X(t − δ)).In point (iii) the function σ(X(s, •)) is fixed in the space-point x 0 , in (iv) the stochastic integral is shortened, in (v) the function σ(X(s, x 0 )) is fixed at the time-point t − δ.In (vi) the expectation of the squared stochastic integral is implemented via conditional independence.Finally, in (vii) the expectation is approximated and the integral extended again.
Proof.We present the proof with Assumptions (S') with (5.1), (X') with (3.8) and f δ satisfying condition (F2).The other case is analogous and much simpler, mostly because it does not use the function ε δ at all.The proof of (ii) is shown for both cases.The order O(ε −4 δ ) for |f δ | is used frequently as the first step of the proof. (i).Compute using Lemma 5.5.Therefore, the remainder term is of order o P (δ −2 ) due to ε −4 δ δ → 0 by (3.8).
(ii).By the Cauchy-Schwarz inequality, we have ) so that the second factor is of order O P (δ −2 ).Hence, suffices to establish When we consider Assumptions (S'), (X'), condition (5.1) and f δ satisfying condition (F2), we obtain by the Hölder continuity of σ and X(•, x) and by ∥K δ,x0 ∥ = ∥K∥ L 2 (R) .This upper bound converges to zero by (3.8).When we consider Assumptions (S), (X), condition (5.1) and f δ satisfying condition (F1), we have The integrand in (5.7) converges to zero by the continuity of X and σ (i.e., Assumptions (X) and (S)).The integrable majorant 4σ 2 K 2 (•) serves for the dy-integral in (5.7) as well as for the dt-integral in (5.6).The proof that the second factor converges almost surely to zero is accomplished by the dominated convergence theorem.
For the term with G − we have from Assumptions (S') and (X') that E(σ(X(t − v, x 0 )) − σ(X(t − δ, x 0 ))) 2 ≲ δ 2βtβσ uniformly over t and v.The same argument thus gives here and compute We analyse the integral on two sets: (a) t > u + δ and (b) u + δ > t > u.
(a) t > u + δ.We condition on F t−δ and obtain using that H(t) is independent of F t−δ with E H(t) = 0 and that the other factors are F t−δ -measurable.(b) u + δ > t > u.In this case, we use the upper bounds for σ and f δ and bound (5.15) (up to a positive constant) by E H2 (t), (5.16)where the last step involves the Cauchy-Schwarz inequality.The analogous calculations to (5.3) yield (3.8).(vii).By translating the integrand in L (vii) δ and by Itô's isometry we have By the uniforms bounds on f δ and σ the last term is of order O(δ −2 δε −4 δ ).Next, we compute by the scaling properties in Lemma 5.1 and by partial integration Now, bounding the scalar product by the Cauchy-Schwarz inequality and Lemma 5.4 with α = 0, we see that it is of order O(δ 3/4 ).Using the upper bounds for f δ and σ again, we thus obtain

Asymptotics of quadratic variations and related terms
While Proposition 5.7 has treated the centered process X ∆ δ,x0 (t) − E X ∆ δ,x0 (t), we shall now consider the terms (5.17) Having achieved the representation L (vii) δ , we are now ready to determine the limit of δ 2 Lδ as δ → 0.
(ii).We use The integrable majorant for the dt-integral over f δ (X(t))σ 2 (X(t, x 0 )) can be taken as (up to a multiplicative constant).
(iv).We use f δ (X(t)) = This gives the uniform majorant in δ and t for sufficiently small δ due to (3.8).Next, we determine the pointwise limit of the L 1 (P)-distance.In view of (5.19)  stably − −−− → Z holds on {T ⋆ > 0} with Z ∼ N (0, 1) independent of the σ-algebra F T .The proof is concluded by applying Slutsky's lemma and approximating possibly unbounded σ by the truncated versions σ R .

A stable limit theorem for cylindrical Brownian martingales
Let H be a separable Hilbert space and (e k ) k≥1 a complete orthonormal system in H. Let (W k (t), t ≥ 0) k≥1 be a sequence of independent real-valued standard Brownian motions.Then W (t) = k≥1 W k (t)e k is an H-valued cylindrical Brownian motion (e.g., Proposition 4.11 in [10]).Consider the filtered probability space (Ω, F , (F t ) t≥0 , P), on which (W k (t), t ≥ 0) k≥1 are defined and where the Brownian filtration (F t ) t≥0 is the filtration generated by (W k (t), t ≥ 0) k≥1 and augmented by P-null sets.
We start with a Hilbert space-valued Brownian martingale representation theorem, which follows by approximation from the finite-dimensional version, but does not seem readily available in the literature.Proposition 6.1.Let (M (t), t ≥ 0) be a square-integrable real-valued martingale with respect to (F t ) t≥0 and with càdlàg paths, M (0) = 0. Then there exist progressively measurable processes (F k (t), t ≥ 0) k≥1 satisfying ) t≥0 generated by (W k (t), t ≥ 0) 1≤k≤K and consider By the tower property for s < t, we have and another application of the tower property yields We conclude that (M (K) (t), t ≥ 0) forms an L 2 (P)-martingale with respect to the K-dimensional Brownian filtration (F (K) t ) t≥0 .By standard martingale theory (e.g., Theorem 1.3.13 in [14]) we may choose a càdlàg version of (M (K) (t), t ≥ 0), which we shall do henceforth.Theorem 3.4.15 in [14] therefore shows that there are (F k (t), t ≥ 0) 1≤k≤K satisfying K k=1 T 0 EF 2 k (t) dt < ∞ for all T > 0 and P-a.s.
The uniqueness result of that theorem also shows that for each K the F k , k = 1, . . ., K, can be chosen to not depend on K because by independence of W K from (W k , 1 ≤ k ≤ K − 1), K ≥ 2, we have Since F t is generated by K≥1 F (K) t , the L 2 -martingale convergence theorem gives lim K→∞ M (K) (t) = M (t) in L 2 (P)-convergence for every t ≥ 0. Hence, also K k=1 t 0 F k (s) dW k (s) converges in L 2 (P) for K → ∞.By Itô's isometry this shows that the L 2 (P)-norms converge: s)⟩ is then well defined as an element of L 2 (P).Moreover, it equals the limit of M (K) (t) whence ⟨F (s), dW (s)⟩ holds P-a.s. for each fixed t ≥ 0. Using the càdlàg path versions on each side, this entails equality for all t ≥ 0 with probability one.Theorem 6.2.Let (Y δ (t), t ≥ 0) for δ > 0 be progressively measurable Hvalued processes on (Ω, F , (F t ) t≥0 , P) with T 0 ⟨Y δ (t), F (t)⟩ dt P → 0 as δ → 0 for all progressively measurable H-valued processes (F (t), t ≥ 0).
Then the following stable limit theorem for stochastic integrals holds: Proof.Since M δ (T ) := T 0 ⟨Y δ (t), dW (t)⟩ is a continuous martingale with quadratic variation C δ (T ) = T 0 ∥Y δ (t)∥ 2 dt, we can apply Theorem IX.7.3(b) in [13] with the trivial processes Z t = 0, B t = 0 (in that Theorem), so that it remains to check for all T :

Table 1 :
Monte Carlo mean and standard deviation of estimators for different σ(•).