Edinburgh Research Explorer Convergence of tamed Euler schemes for a class of stochastic evolution equations

. We prove stability and convergence of a full discretization for a class of stochastic evolution equations with super-linearly growing operators appearing in the drift term. This is done by using the recently developed tamed Euler method, which employs a fully explicit time stepping, coupled with a Galerkin scheme for the spatial discretization.


Introduction
In this paper we investigate the convergence of full discretizations, explicit in time, of stochastic evolution equations with the drift term governed by a super-linearly growing operator. When the operator appearing in the drift term grows at most linearly then the classical explicit Euler scheme applied to stochastic evolution equations is convergent (when coupled appropriately with the spatial discretization), see, for example, Gyöngy and Millet [6]. If the operator appearing in the drift term grows faster than linearly then one would, in general, not expect the explicit Euler scheme to be convergent (this is the case even in the setting of fully deterministic evolution equations). Instead, one would typically consider the implicit Euler scheme which is convergent in this situation (see, for example, Gyöngy and Millet [6]). The price one pays is the increased computational effort required at each time step of the numerical scheme.
Hutzenthaler, Jentzen and Kloeden [9] have observed that the absolute moments of explicit Euler approximations for stochastic differential equations with superlinearly growing coefficients may diverge to infinity at finite time. This led to development of "tamed" Euler schemes for stochastic differential equations. This was pioneered in Hutzenthaler, Jentzen and Kloeden [10] and, using different techniques, in Sabanis [16]. A taming-like technique in the form of truncation has been proposed by Roberts and Tweedie [14] in the context of Markov chain Monte Carlo methods. Further work on explicit numerical approximations of stochastic differential equations with super-linearly growing coefficients can be found in Tretyakov and Zhang [19], Hutzenthaler and Jentzen [12], Sabanis [17] as well Dareiotis, Kumar and Sabanis [18].
Moreover Hutzenthaler, Jentzen and Kloeden [11] have demonstrated that to apply multilevel Monte Carlo methods (see Heinrich [7], [8] and Giles [5]) to stochastic differential equations with super-linearly growing coefficients one must "tame" the explicit Euler scheme. In this paper we use the idea of "taming" to devise a new type of a convergent explicit scheme for a class of stochastic evolution equations with super-linearly growing operators in the drift term.
The article is organised as follows. In Section 2 we introduce the numerical scheme, give the precise assumptions and state the main result in Theorem 2.3. Essential a priori estimates are proved in Section 3. In Section 4 we first use the a priori estimates and a compactness argument to extract weakly convergent subsequences and limits of the approximation. The remaining step is to identify the weak limit of the approximation of the nonlinear term with the nonlinear term in the equation. This is done using a monotonicity argument in Section 4 where Theorem 2.3 is finally proved. In Section 5 we provide examples of stochastic partial differential equations where the numerical scheme can be applied.

Main results
Let T > 0. Let (Ω, F , P) be a probability space and let (F t ) t∈[0,T ] be a filtration such that F 0 contains all the P-null sets of F .
Let K > 0 and p ∈ [2, ∞) be given constants. Let p * := p/(p − 1). For a reflexive, separable Banach space (X, · X ) let X * and · X * denote its dual space and the norm on the dual space respectively. For f ∈ X * and v ∈ X we use f, v to denote the duality pairing. By L p (0, T ; X) we denote the Lebesgue-Bochner space of equivalence classes of measurable functions with values in X that satisfy By L p (Ω; X) we denote the Lebesgue-Bochner space of random variables with values in X and such that the norm is finite. Finally by L p (X) we denote the Lebesgue-Bochner space of dt × Pequivalence classes of (F t ) t∈[0,T ] -adapted and X-valued stochastic process that satisfy We say that an operator C : X ×Ω → X * is measurable with respect to some G ⊆ F if for any v, w ∈ X the real-valued random variable Cv, w is G-measurable.
We assume that, with respect to (F t ) t∈[0,T ] , (W t ) t∈[0,T ] is a cylindrical Q-Wiener process with Q = I on a separable Hilbert space (U, (·, ·) U , | · | U ). We assume that there are (V 1 , · V1 ) and (V 2 , · V2 ), separable and reflexive Banach spaces that are densely and continuously embedded in H, where (H, (·, ·), | · |) is a Hilbert space identified with its dual. We thus have two Gelfand triples is the space of Hilbert-Schmidt operators from U to H.
Let V := V 1 ∩ V 2 and let the norm in V be given by · := · V1 + · V2 . Assume that V is separable and dense in both V 1 and V 2 . Using Gajewski, Gröger and Zacharias [4, Kapitel I, Satz 5.13] one observes that the dual V * of V can be identified with We consider stochastic evolution equations of the form where u(0) = u 0 with u 0 a given H-valued and F 0 -measurable random variable. Let A := A 1 + A 2 and B := B 1 + B 2 . The operator A is defined on V × Ω with values in V * and the operator B is defined on V × Ω with values in L 2 (U, H). Then we can write (2.1) as (1.1).
We impose the following assumptions on the operators.
Moreover assume that the following conditions hold. Monotonicity: Coercivity: there is µ > 0 such that
We now define what is meant by solution of (1.1). Definition 2.1 (Solution). Let u 0 be an F 0 -measurable H-valued random variable. We say that a continuous, H-valued and (F t ) t∈[0,T ] -adapted process u is a solution to (1.1) if u is dt × P almost everywhere V -valued, if u ∈ L 2 (V 1 ) ∩ L p (V 2 ) and if for every t ∈ [0, T ] and every v ∈ V , almost surely, To the best knowledge of the authors, existence and uniqueness has not been proved for this class of stochastic evolution equations. Pardoux [13] considers the situation where the stochastic evolution equation is governed by a sum of monotone, coercive and hemicontinuous operators satisfying certain growth condition. However the operator A 2 in our case only satisfies a type of "degenerate" coercivity condition. Hence the existence theorem from Pardoux [13] does not apply. We prove that a solution to (1.1) must be unique in Theorem 2.2 and we prove existence of the solution in Theorem 2.3. We prove Theorem 2.2 in Section 4. Let us now describe the discretization scheme for the stochastic evolution equation (2.1). For the space discretization let (V m ) m∈N be a Galerkin scheme for V . To be precise we assume that V m ⊆ V are finite dimensional spaces with the dimension of V m equal to m. We further assume that V m ⊆ V m+1 for all m ∈ N and that (this is referred to as the limited completeness of the Galerkin scheme). We will need the following projection operators.
Assumption 2. For any m ∈ N let Π m : V * → V m satisfy the following: There is a constant, depending on m and denoted by c(m), such that In applications this assumption is easily satisfied. In particular if {ϕ j ∈ V : j = 1, 2, . . .} is an orthonormal basis in H then taking V m := span{ϕ 1 , . . . , ϕ m } is a Galerkin scheme for V . Taking Π m f := m j=1 f, ϕ j ϕ j satisfies the first three conditions in Assumption 2. Moreover, the following holds where c(m) := m j=1 ϕ j 2 V . Thus the fourth condition in Assumption 2 is also satisfied. Let {χ i } i∈N be an orthonormal basis of U . Fix k ∈ N and define For the time discretization take n ∈ N, let τ n := T /n and define the grid points on an equidistant grid as t n i := τ n i, i = 0, 1, . . . , n. Further consider some sequence ((n ℓ , m ℓ , k ℓ )) ℓ∈N such that n ℓ , m ℓ and k ℓ all go to infinity as ℓ → ∞.
Let c denote a generic positive constant that may depend on T , on the constants arising in the continuous embeddings V i ֒→ H ֒→ V * i , i = 1, 2 and on the constants arising in Assumptions 1 and 3 but that is always independent of the discretization parameters m, k and n. Define . . , n ℓ −1 and κ n ℓ (T ) = T . Fix some ℓ ∈ N (and hence m ℓ , n ℓ and k ℓ ). Let u ℓ (0) be a V m ℓ valued F 0 -measurable approximation of u 0 . For example we can take u ℓ (0) := Π m ℓ u 0 but other approximations are possible. For t > 0 we define a process u ℓ by where we use the "tamed" operator A 2,ℓ defined by for any v ∈ V 2 . We will use the following notation:ū ℓ (t) := u ℓ (κ n ℓ (t)) and a ℓ (v) : In particular at the time-grid points we have, for i = 0, 1, . . . , n ℓ − 1, We list below the properies which are satisfied by the tamed operator A 2,ℓ . These are consequences of its structure and the assumed properties of A 2 . For brevity let, and also, using the Growth assumption on A 2 , Furthermore, using the Coercivity assumption on (2.8) Thus the weaker coercivity assumption that has been made about A 2 is retained. Consider, for a moment, that A 2 satisfies the "usual" coercivity condition We see that in this case the best coercivity we can get from this for A 2,ℓ is again only (2.8). Hence to obtain the necessary a priori estimates we will need an interpolation inequality between V 2 and V 1 with H. Assumption 3. There are constants λ ∈ [0, 2/p) and Λ > 0 such that for any Note that in order to overcome the difficulty with coercivity it would suffice to have Assumption 3 satisfied with λ ∈ [0, 2/p]. However monotonicity of A 2 is not preserved by taming. To overcome this we will need to show that . To achieve this we use the fact that λ ∈ [0, 2/p) in Lemma 4.3 and the following observation: Thus we see that Assumption 3 allows us to control the approximate solution in the L p(1+η) ((0, T ) × Ω; V 2 ) norm, provided that we can control the approximate solution in the norms of L 2 (Ω; L 2 (0, T ; V 1 )) and L 2ρ (Ω; L ∞ (0, T ; H)).
Let us take q 0 := max(4, 2ρ). Now we can state the main result of this paper.
Theorem 2.3. Let Assumptions 1, 2 and 3 be satisfied. Let u 0 ∈ L q0 (Ω; H) and let In Section 5 we provide examples of stochastic partial differential equations where Theorem 2.3 can be applied. We also compute c(m) in case of the spectral Galerkin method to make the implications of the space-time coupling constraint more explicit. The crucial point is that the requirement is no more onerous than in the case of equations with operators growing at most linearly.

A priori estimates
We start with an important observation that allows us to use standard results on bounds of stochastic integrals driven by finite dimensional Wiener processes.
One applies Itô's formula to (2.4) to obtain which can be rewritten as in order to apply the coercivity assumption so as to obtain the a priori estimates for the discretized equation.
First we concentrate on the term that arises from the "correction" that one has to make to use the coercivity assumption due to the use of an explicit scheme.
Lemma 3.2. Let the Growth condition in Assumption 1 be satisfied. Let Assumption 2 hold. Let q ≥ 1 be given. Then and Proof. From (2.2) it is clear that Applying Hölder's inequality yields Using Assumption 2 and (2.6) one obtains The Growth assumption on A 1 implies that Using Hölder's inequality leads to ds.
Due to Remark 3.1 and the Growth assumption on B one observes that This implies (3.2). Moreover Assumption 2 and (2.6) imply that Applying the Growth assumption on A 1 yields (3.5) Using (3.2) in (3.5) concludes the proof.

Young's inequality and the Growth assumption on B imply that
Applying this in (3.7) leads to

Application of Gronwall's lemma yields
Now we use Theorem 3.3 and Assumption 3 to obtain the remaining required estimates.

Convergence
Having obtained the required a priori estimates we can use compactness arguments to extract weakly convergent subsequences of the approximation.

Lemma 4.1. Let the Growth and Coercivity conditions in Assumption 1 hold. Let Assumptions 2 and 3 be satisfied. Let
Then there is a subsequence of the sequence ℓ, which we denote ℓ ′ , and Proof. The sequence (ū ℓ ) is bounded in L 2 (V 1 ) due to Theorem 3.3 and in L p (V 2 ) due to Corollary 3.4. The sequences (A 1ūℓ ), (A 2,ℓūℓ ) and (Bū ℓ ) are bounded in L 2 (V * 1 ), L p * (V * 2 ) and in L 2 (L 2 (U, H)) respectively, due to Corollary 3.4. Finally the sequence (u ℓ (T )) is bounded in L q0 (Ω; H) due to Theorem 3.3.
Since it is assumed that V 1 , V 2 are reflexive, it follows that L 2 (V 1 ) and L p (V 2 ) are reflexive. A bounded sequence in a reflexive Banach space must have a weakly convergent subsequence (see e.g. Brézis [2,Theorem 3.18]). Applying this to the sequences in question concludes the proof of the lemma.  and almost surelyũ In the rest of this paper we will write u instead ofũ for notational simplicity.
Proof. Fix M ∈ N. Let ϕ be a V M -valued adapted stochastic process such that |ϕ(t)| < M for all t ∈ [0, T ] and ω ∈ Ω. For g ∈ U letΠ m g := m j=1 (χ j , g) U χ j and note that for any v ∈ V one has BvΠ m ∈ L 2 (U, H). From (2.2) one observes that The operator G is linear and bounded and as such it is weakly-weakly continuous. This operator H is clearly linear. Furthermore, due to Itô's isometry, (L2(U,H)) .
Thus the operator H is also bounded. It follows that H is also weakly-weakly continuous. Therefore, taking the limit as ℓ ′ → ∞ and using Lemma 4.1, one obtains This holds for any ϕ as specified at the beginning of the proof. By letting M → ∞ and using the limited completeness of the Galerkin scheme it follows that this also holds for any ϕ ∈ L p (V ). Thus holds for almost all (t, ω) ∈ (0, T ) × Ω.
Let ϕ be a V M -valued and F T -measurable random variable such that E ϕ 2 V < ∞. Setting t = T in (4.3) and taking the expectation yields Let ℓ ′ → ∞. The weak-weak continuity of the operators G and H, together with Lemma 4.1, implies that (4.5) Letting M → ∞ and again using the limited completeness of the Galerkin scheme shows that the above equality holds for any F T -measurable ϕ ∈ L 2 (Ω; V ). If one now applies Itô's formula to (4.4) then one obtains an adapted processũ with paths in C([0, T ]; H) that is equal to u almost surely. Furthermore, for any ϕ ∈ L 2 (Ω; V ) and due to continuity ofũ, Thus ξ =ũ(T ). This together with (4.5) implies (4.2).
All that remains to be done to prove Theorem 2.3 is to identify a ∞ with Au and b ∞ with Bu and to show strong convergence of u ℓ (T ) to u(T ). To that end we would like to use monotonicity of A. In order to overcome the difficulty arising from the fact that the tamed operator A 2,ℓ does not preserve the monotonicity property of A 2 we need the following lemma.

Lemma 4.3. Let the Growth and Coercivity conditions in Assumption 1 hold. Let Assumptions 2 and 3 be satisfied. Let
Proof. Consider some M > 0. Recall that T ℓ is given by (2.5). Then It is observed that Recall that due to Corollary 3.4 one knows that ℓ∈N is uniformly integrable on (0, T ) × Ω with respect to dt × P . Hence for any ǫ > 0 there exists M such that I 2,ℓ,M < ǫ/2 for all ℓ. Finally, since c(m ℓ ) n ℓ → 0 as ℓ → ∞, one can choose ℓ large such that I 1,ℓ,M < ǫ/2.
We now prove Theorem 2.2. This is needed to later show that the whole sequence of approximations converges rather than just a subsequence.
Proof of Theorem 2.2. Assume that u 1 and u 2 are two distinct solutions to (1.1) such that u 1 (0) = u 2 (0) = u 0 . One would now like to apply Itô's formula for the square of the norm from Pardoux [13,Chapitre 2,Theoreme 5.2]. To that end one immediately observes that u 1 − u 2 ∈ L 2 (V 1 ) ∩ L p (V 2 ) and that u 1 (0) − u 2 (0) = 0 ∈ L 2 (Ω; H). Moreover . Using the Growth assumption on A 1 one observes that Also, using the Growth assumption on A 2 one obtains . Finally, using the Growth assumption on B one deduces that Bu 1 − Bu 2 ∈ L 2 (L 2 (U, H)). Hence the afromentioned Itô's formula for the square of the norm can be applied, yielding One then observes, due to the monotonicity of A : V × Ω → V * , that and hence M is non-negative. It is also a real-valued continuous local martingale, and thus a supermartingale. Furthermore it starts from 0 and thus, almost surely, M (t) = 0 for all t ∈ [0, T ]. One thus concludes that u 1 (t) = u 2 (t) for all t ∈ [0, T ] almost surely.
Finally we can prove Theorem 2.3.

Proof of Theorem 2.3. Recall that
Applying Itô's formula to the scheme (2.2) and taking expectations yields Using Hölder's inequality results in Using Assumption 2 and Corollary 3.4 yields Thus, and one may proceed with a monotonicity argument. Let w ∈ L p (V ). Then

L2(U,H)
+ Bū ℓ ′ (s)) − Bw(s) 2 L2(U,H) ds + c(c(m ℓ ′ )τ n ℓ ′ ) 1/2 . Using the Monotonicity assumption on A one obtains  Applying Itô's formula to (4.1) and taking expectations yields Subtracting this from (4.7) one arrives at (4.9) Note that so far w was arbitrary. It will now be used to identify the nonlinear terms. First, one takes w = u and observes that, Next, one sets w = u + ǫz with ǫ > 0 and z ∈ L p (V ) in (4.9). Dividing by ǫ > 0 leads to Using hemicontinuity of A while letting ǫ → 0 results in This holds for an arbitrary z ∈ L p (V ) and hence, in particular, for −z. Thus one obtains that a ∞ = Au.
Due to Theorem 2.2, the solution u to (1.1) is unique. Thus the whole sequences of approximations converges rather than just the subsequence denoted by ℓ ′ . Thus, due to (4.8), From Lemma 4.1 and Lemma 4.2, one already knows that u ℓ (T ) ⇀ u(T ) in L 2 (Ω; H). This is a uniformly convex space (as it is a Hilbert space). Thus one concludes that u ℓ (T ) → u(T ) in L 2 (Ω; H). For this see, e.g., Brézis [2, Proposition 3.32].

Examples
In with appropriate boundary and initial conditions. The domain D is assumed to be a bounded Lipschitz domain in R 2 . With Dirichlet boundary conditions we would take V 1 = H 2 0 (D) and V 2 = L p (D) with p ∈ [2, 6). The third example is the spatially extended stochastic FitzHugh-Nagumo system for signal propagation in nerve cells (originally stated by FitzHugh [3] as a system of ordinary differential equations, see Bonaccorsi and Mastrogiacomo [1] for mathematical analysis of the spatially extended stochastic version): together with appropriate initial data for u and v as well as homogeneous Neumann boundary conditions for u only. In this situation V 1 = H 1 ((0, 1)) × L 2 ((0, 1)) while V 2 = L 4 ((0, 1)) × L 2 ((0, 1)).
We now provide estimates on the constant c(m) in the particular case when D = (0, π) 2 ⊂ R 2 and we use a spectral Galerkin method to construct the spaces V m .
We also note that if D = (0, π) d then an analogous construction of V m would lead to the conclusion that we need n ℓ = ⌊m d+δ l ⌋ for some δ > 0. Crucially we see that the space-time coupling requirement is no more onerous than in the case of equations with operators growing at most linearly.