A Break of the Complexity of the Numerical Approximation of

arXiv:1001.2751v3 [math.NA] 28 Jul 2010

A Break of the Complexity of the Numerical Approximation of Nonlinear SPDEs with Multiplicative Noise Arnulf Jentzen∗ and Michael Röckner† July 29, 2010

Abstract A new algorithm for simulating stochastic partial differential equations (SPDEs) of evolutionary type, which is in some sense an infinite dimensional analog of Milstein’s scheme for finite dimensional stochastic ordinary differential equations (SODEs), is introduced and analyzed in this article. The Milstein scheme is known to be impressively efficient for scalar one-dimensional SODEs but only for some special multidimensional SODEs due to difficult simulations of iterated stochastic integrals in the general multidimensional SODE case. It is a key observation of this article that, in contrast to what one may expect, its infinite dimensional counterpart introduced here is very easy to simulate and this, therefore, leads to a break of the complexity (number of computational operations and random variables needed to compute the scheme) in comparison to previously considered algorithms for simulating nonlinear SPDEs with multiplicative trace class noise. The analysis is supported by numerical results for a stochastic heat equation, stochastic reaction diffusion equations and a stochastic Burgers equation showing significant computational savings. ∗

Arnulf Jentzen, Faculty of Mathematics, Bielefeld University, Germany ([email protected]) † Michael R¨ ockner, Faculty of Mathematics, Bielefeld University, Germany ([email protected]) and Department of Mathematics and Statistics, Purdue University, USA ([email protected])

1

1

Introduction

Stochastic partial differential equations (SPDEs) of evolutionary type are a fundamental instrument for modelling all kinds of dynamics with stochastic influence in nature or in man-made complex systems. Since explicit solutions of such equations are usually not available, it is a very active research topic in the last two decades to solve SPDEs approximatively. The key difficulty in this research field is the high computational complexity (in comparison to solve deterministic partial and stochastic ordinary differential equations approximatively) needed to compute an approximation of the solution of a SPDE. The high computational complexity is a consequence of the need of discretizing the continued time interval, the infinite dimensional state space of the SPDE and the infinite dimensional driving noise process. In this article a new numerical method for SPDEs with reduced computational complexity in comparison to previously considered algorithms for simulating nonlinear SPDEs with multiplicative trace class noise is introduced and analyzed. This method is in a sense an infinite dimensional analog of Milstein’s scheme for finite dimensional stochastic ordinary differential equations (SODEs). The Milstein scheme is known to be impressively efficient for scalar one-dimensional SODEs but only for some special multidimensional SODEs due to difficult simulations of iterated stochastic integrals in the general multidimensional SODE case. In particular, this suggests that there is no hope to expect that an infinite dimensional analog of Milstein’s scheme can be simulated efficiently and even less to expect that such a method yields a break of the computational complexity for approximating SPDEs. However, the main contribution of this article is to derive an infinite dimensional analog of Milstein’s scheme that can be simulated very easily by exploiting that the infinite dimensional SPDE state space is a function space on which the noise acts multiplicatively. In contrast to Milstein’s scheme for SODEs, an additional ingredient of its infinite dimensional counterpart introduced here is a mollifying exponential term approximating the dominant linear operator of the SPDE. We also mention that in the case of a linear SPDE, the splitting-up method as considered by I. Gyöngy and N. Krylov in [20] already yields a break of the computational complexity and refer to Section 4.3 for a detailed comparison of the splitting-up method and our algorithm in this article. In the rest of this introductory section we first review Milstein’s scheme for SODEs, then reconsider a standard method for solving SPDEs approximatively and finally, introduce our algorithm for simulating SPDEs. Let T ∈ (0, ∞) be a real number, let (Ω, F , P) be a probability space and let w = (w 1, . . . , w m ) : [0, T ] × Ω → Rm with m ∈ N := {1, 2, . . .} be a m-dimensional standard Brownian motion with respect to a normal filtration 2

(Ft )t∈[0,T ] . Moreover, let d ∈ N, x0 ∈ Rd and let µ = (µ1 , . . . , µd ) : Rd → Rd and σ = (σi,j ) i∈{1,...,d} : Rd → Rd×m be two appropriate smooth and regular j∈{1,...,m}

functions with globally bounded derivatives (see, e.g., Theorem 10.3.5 in P. E. Kloeden and E. Platen [36] for details). The stochastic ordinary differential equation dXt = µ(Xt ) dt + σ(Xt ) dwt , X0 = x0 (1) for all t ∈ [0, T ] then admits a unique solution. More precisely, there exists an up to indistinguishability unique adapted stochastic process X : [0, T ] × Ω → Rd with continuous sample paths which satisfies Z t Z t Xt = x0 + µ(Xs ) ds + σ(Xs ) dws (2) 0

= x0 +

Z

t

µ(Xs ) ds + 0

0 m XZ t i=1

0

σi (Xs ) dwsi

P-a.s.

for all t ∈ [0, T ]. Here σi : Rd → Rd is given by σi (x) = (σ1,i (x), . . . , σd,i (x)) for all x ∈ Rd and all i ∈ {1, . . . , m}. Milstein’s method (see, e.g., (3.3) in Section 10.3 in P. E. Kloeden and E. Platen [36] and also G. N. Milstein’s original article [44]) applied to the SODE (1) is then given by F /B(Rd )measurable mappings ynN : Ω → Rd , n ∈ {0, 1, . . . , N}, N ∈ N, with y0N = x0 and N yn+1

=

ynN

m X T N N i i + · µ(yn ) + σi (yn ) · w (n+1)T − w nT N N N i=1 Z (n+1)T Z s d m X X N ∂ N N σi (yn ) · σk,j (yn ) · dwuj dwsi (3) + nT nT ∂xk i,j=1 k=1 N N

P-a.s. for all n ∈ {0, 1, . . . , N −1} and all N ∈ N. Although Milstein’s scheme is known to converge significantly faster than many other methods such as the Euler-Maruyama scheme, it is only of limited use due to difficult simulations R (n+1)T R s j i of the iterated stochastic integrals nT N nT dwu dws for i, j ∈ {1, . . . , m} N

N

with i 6= j, n ∈ {0, 1, . . . N − 1} and N ∈ N in (3). In the special situation of so called commutative noise (see (3.13) in Section 10.3 in [36]), i.e. d d X X ∂ ∂ σi (x) · σk,j (x) = σj (x) · σk,i (x) ∂xk ∂xk k=1 k=1

(4)

for all x ∈ Rd and all i, j ∈ {1, . . . , m}, the Milstein scheme can be simplified and complicated iterated stochastic integrals in (3) can be avoided. More 3

precisely, in the case (4), Milstein’s scheme (3) reduces to m X T · µ(ynN ) + σi (ynN ) · w i(n+1)T − w inT N N N i=1 m d 1 XX ∂ j j i N N i σi (yn ) · σk,j (yn ) · w (n+1)T − w nT · w (n+1)T − w nT + N 2 i,j=1 k=1 ∂xk N N N m d T XX ∂ − σi (ynN ) · σk,i (ynN ) (5) 2N i=1 k=1 ∂xk N yn+1 = ynN +

P-a.s. for all n ∈ {0, 1, . . . , N − 1} and all N ∈ N (see (3.16) in Section 10.3 in [36]). For instance, in the case d = m = 1, condition (4) is obviously fulfilled and the Milstein scheme (5) can then be written as N yn+1 = ynN +

T · µ(ynN ) + σ(ynN ) · w (n+1)T − w nT N N N 2 T 1 ′ N N + · σ (yn ) · σ(yn ) · (6) w (n+1)T − w nT − N N 2 N

P-a.s. for all n ∈ {0, 1, . . . , N − 1} and all N ∈ N (see (3.1) in Section 10.3 in [36]). Of course, (6) can be simulated very efficiently. However, (4) is in the case of a multidimensional SODE seldom fulfilled and even if it is fulfilled, Milstein’s method (5) becomes less useful if d, m ∈ N are large. For example, if d = m = 20 holds, then the middle term in (5) contains 203 = 8000 summands. So, more than 8000 additional arithmetic operations N are needed to compute yn+1 from ynN for n ∈ {0, 1, . . . , N − 1}, N ∈ N via (5) in the case d = m = 20 in general which makes Milstein’s scheme less efficient. This suggests that there is no hope to expect that an infinite dimensional analog of Milstein’s method can be simulated efficiently in the case of infinite dimensional state spaces such as L2 ((0, 1), R) instead of Rd and Rm respectively. The purpose of this article is to demonstrate that this is not true. More precisely, an infinite dimensional analog of Milstein’s scheme that can be simulated very easily is derived and analyzed here. This leads to a break of the complexity (number of computational operations and random variables needed to compute the scheme) in comparison to previously considered algorithms for simulating nonlinear SPDEs with multiplicative trace class noise which is illustrated in the following. Let H = L2 ((0, 1), R) be the R-Hilbert space of equivalence classes of B((0, 1))/B(R)-measurable and Lebesgue square integrable functions from (0, 1) to R and let f, b : (0, 1) × R → R be two appropriate smooth and 4

regular functions with globally bounded derivatives (see (30) and (31) for details). As usual we do not distinguish between a B((0, 1))/B(R)-measurable and Lebesgue square integrable function from (0, 1) to R and its equivalence class in H. Moreover, let κ ∈ (0, ∞) be a real number, let ξ : [0, 1] → R with ξ(0) = ξ(1) = 0 be a smooth function and let W : [0, T ] × Ω → H be a standard Q-Wiener process with respect to Ft , t ∈ [0, T ], with a trace class operator Q : H → H (see, for instance, Definition 2.1.12 in [50]). It is a classical result (see, e.g., Proposition 2.1.5 in [50]) that the covariance operator Q : H → H of the Wiener process W : [0, T ] × Ω → H has an orthonormal basis gj ∈ H, j ∈ N, of eigenfunctions with summable eigenvalues µj ∈ [0, ∞), j ∈ N.√ In order to have a more concrete example, we consider the choice gj (x) = 2 sin(jπx) and µj = j12 for all x ∈ (0, 1) and all j ∈ N in the following and refer to Section 2 for our general setting and to Section 4 for further possible examples. Then we consider the SPDE ∂2 (7) dXt (x) = κ 2 Xt (x) + f (x, Xt (x)) dt + b(x, Xt (x)) dWt (x) ∂x with Xt (0) = Xt (1) = 0 and X0 (x) = ξ(x) for x ∈ (0, 1) and t ∈ [0, T ] on H. Under the assumptions above the SPDE (7) has a unique mild solution. Specifically, there exists an up to indistinguishability unique adapted stochastic process X : [0, T ] × Ω → H with continuous sample path which satisfies Z t Z t At A(t−s) Xt = e ξ + e F (Xs ) ds + eA(t−s) B(Xs ) dWs P-a.s. (8) 0

0

for all t ∈ [0, T ] where A : D(A) ⊂ H → H is the Laplacian with Dirichlet boundary conditions times the constant κ ∈ (0, ∞) and where F : H → H and B : H → HS(U0 , H) are given by (F (v))(x) = f (x, v(x)) and (B(v)u)(x) = b(x, v(x)) · u(x) for Dall x ∈ (0, 1),E v ∈ H and all u ∈ U0 . 1 1 1 Here U0 = Q 2 (H) with hv, wiU0 = Q− 2 v, Q− 2 w for all v, w ∈ U0 is the H

1

image R-Hilbert space of Q 2 (see Appendix C in [50]). Then our goal is to solve the strong approximation problem (see Section 9.3 in [36]) of the SPDE (7). More precisely, we want to compute a F /B(H)-measurable numerical approximation Y : Ω → H such that Z 1 12 2 0 with the least possible computational effort (number of computational operations and independent standard normal random variables needed to compute Y : Ω → H). A computational operation is 5

here an arithmetic operation (addition, subtraction, multiplication, division), a trigonometric operation (sine, cosine) or an evaluation of f : (0, 1)×R → R, b : (0, 1) × R → R or the exponential function. In order to be able to simulate such a numerical approximation on a computer both the time interval [0, T ] and the infinite dimensional space H = L2 ((0, 1), R) have to be discretized. While for temporal discretizations the linear implicit Euler scheme (see [7, 8, 18, 25, 26, 27, 57, 58, 60]) and the linear implicit Crank-Nicolson scheme (see [25, 26, 57, 58]) are often used, spatial discretizations are usually achieved with finite elements (see [1, 2, 7, 8, 13, 27, 35, 38, 39, 42, 58, 60]), finite differences (see [16, 24, 43, 49, 52, 53, 54, 55, 57, 59]) and spectral Galerkin methods (see [15, 26, 29, 31, 37, 40, 41, 46, 48]). For instance, the linear implicit Euler scheme combined with spectral Galerkin methods which we denote by F /B(H)-measurable mappings ZnN : Ω → H, n ∈ {0, 1, . . . , N 3 }, N ∈ N, is given by Z0N = PN (ξ) and −1 T T N N N N N N Zn+1 = PN I − 3 A Zn + 3 · f (·, Zn ) + b(·, Zn ) · W (n+1)T − W nT N N N3 N3 (10) P-a.s. for all n ∈ {0, 1, . . . , N 3 − 1} and all N ∈ N. Here the bounded linear operators PN : H → H, N ∈ N, and the Wiener processes W N : [0, T ] × Ω → H, N ∈ N, are given by (PN (v))(x) =

N X

2 sin(nπx)

Z

1

sin(nπy) v(y) dy

0

n=1

for all x ∈ (0, 1), v ∈ H, N ∈ N and by WtN (ω) = PN (Wt (ω)) for all t ∈ [0, T ], ω ∈ Ω, N ∈ N. Moreover, we use the notations v · w : (0, 1) → R, v 2 : (0, 1) → R and ϕ(·, v) : (0, 1) → R given by v · w (x) = v(x) · w(x),

2 v 2 (x) = v(x) ,

ϕ(·, v) (x) = ϕ(x, v(x))

for all x ∈ (0, 1) and all functions v, w : (0, 1) → R, ϕ : (0, 1) × R → R here and below. In (10) the infinite dimensional R-Hilbert space H is projected down to the N-dimensional R-Hilbert space PN (H) with N ∈ N and the infinite dimensional Wiener process W : [0, T ] × Ω → H is approximated by the finite dimensional processes W N : [0, T ] × Ω → H, N ∈ N, for the spatial discretization. For the temporal discretization in the scheme ZnN , n ∈ {0, 1, . . . , N 3 }, above the time interval [0, T ] is divided into N 3 subintervals, i.e. N 3 time steps are used, for N ∈ N. The exact solution X : [0, T ] × Ω → H of the SPDE (7) has values in D((−A)γ ) and satisfies 6

Ek(−A)γ XT k2H < ∞ for all γ ∈ (0, 43 ) (see Section 4.3 in [34]). This shows E kXT − PN (XT )k2H

12

21

(−A)−γ (I − PN ) L(H) 1 2 ≤ E k(−A)γ XT kH 2 1 + κ−1 N −2γ < ∞

≤ E k(−A)γ XT k2H

for all N ∈ N and all γ ∈ (0, 43 ). So, PN (XT ) converges in the root mean square sense to XT with order 23 − as N goes to infinity. (For a real number δ ∈ (0, ∞), we write δ− for the convergence order if the convergence order is higher than δ − r for every arbitrarily small r ∈ (0, δ).) Additionally, the solution process of the SPDE (7) is known to be 21 -Hölder continuous in the root mean square sense (see, for instance, Theorem 1 in [34]) and therefore, the linear implicit Euler scheme converges temporally in the root mean square to the exact solution of the SPDE (7) with order 12 (see, e.g., Theorem 1.1 in [60]). Combining the convergence rate 23 − for the spatial discretization and the convergence rate 21 for the temporal discretization indicates that it is asymptotically optimal to use the cubic number N 3 of time steps in the linear implicit Euler scheme ZnN , n ∈ {0, 1, . . . , N 3 }, above. We now review how efficiently the numerical method (10) solves the strong approximation problem (9) of the SPDE (7). Standard results in the literature (see, for instance, Theorem 2.1 in [26]) yield the existence of real numbers Cr > 0, r ∈ (0, 23 ), such that Z E

0

1

12 3 XT (x) − Z N3 (x) 2 dx ≤ Cr · N (r− 2 ) N

(11)

holds for all N ∈ N and all arbitrarily small r ∈ (0, 32 ). The linear implicit Euler approximation ZNN3 thus converges in the root mean square sense to XT with order 23 − as N goes to infinity. Moreover, since PN (H) is N-dimensional and since N 3 time steps are used in (10), O(N 4 log(N)) computational operations and random variables are needed to compute ZNN3 . The logarithmic term in O(N 4 log(N)) arises due to computing the nonlinearities f and b with fast Fourier transform where aliasing errors are neglected here and below. Combining the computational effort O(N 4 log(N)) and the convergence order 23 − 8 in (11) shows that the linear implicit Euler scheme needs about O(ε− 3 ) computational operations and independent standard normal random variables to achieve the desired precision ε > 0 in (9). In fact, we have demonstrated that 8 Euler’s method (10) needs O(ε−( 3 +r) ) computational operations and random variables to solve (9) for every arbitrarily small r ∈ (0, ∞) but for simplicity 8 we write about O(ε− 3 ) computational operations and random variables here and below. 7

Having reviewed Euler’s method (10), we now derive our infinite dimensional analog of Milstein’s scheme. In the finite dimensional SODE case, the Milstein scheme (3) is derived by applying Itô’s formula to the integrand process σ(Xt ), t ∈ [0, T ], in (2). This approach is based on the fact that the diffusion coefficient σ is a smooth test function and that the solution process of (1) is a Itô process. This strategy is not directly available in infinite dimensions since (7) does in general not admit a strong solution to which Itô’s formula could be applied. Recently, in [32] in the case of additive noise and in [30] in the general case, this problem has been overcome by first applying Taylor’s formula in Banach spaces to the diffusion coefficient B in the mild integral equation (8) and by then inserting a lower order approximation recursively (see Section 4.3 in [30]). More formally, using F (Xs ) ≈ F (X0 ) and B(Xs ) ≈ B(X0 ) + B ′ (X0 )(Xs − X0 ) for s ∈ [0, T ] in (8) shows Z t Z t At A(t−s) Xt ≈ e ξ + e F (X0 ) ds + eA(t−s) B(X0 ) dWs 0 Z t 0 + eA(t−s) B ′ (X0 )(Xs − X0 ) dWs 0 Z t Z t At ′ X0 + t · F (X0 ) + ≈e B(X0 ) dWs + B (X0 )(Xs − X0 ) dWs 0

0

for t ∈ [0, T ]. The estimate Xs ≈ X0 +

Rs 0

B(X0 ) dWu for s ∈ [0, T ] then gives

Z s Z t Z t ′ X0 + t · F (X0 ) + Xt ≈ e B(X0 ) dWs + B (X0 ) B(X0 ) dWu dWs At

0

0

0

(12) for t ∈ [0, T ]. Using Itô’s formula this temporal approximation has already been obtained in (1.12) in [45] under additional smoothness assumptions of the driving noise process of the SPDE (8) (see Assumption C in [45]) which guarantee the existence of a strong solution and thus allow the application of Itô’s formula. Combining the temporal approximation (12) and the spatial discretization in (10) indicates the numerical scheme with F /B(H)measurable mappings YnN : Ω → H, n ∈ {0, 1, . . . , N 2 }, N ∈ N, given by Y0N = PN (ξ) and N Yn+1

A

= PN e

T N2

YnN

T N N N N + 2 · F (Yn ) + B(Yn ) W (n+1)T − W nT N N2 N2 ! ! Z s Z (n+1)T N2 (13) B ′ (YnN ) B(YnN ) dWuN dWsN + nT N2

nT N2

8

P-a.s. for all n ∈ {0, 1, . . . , N 2 − 1} and all N ∈ N. Now, we are at a stage similar to the finite dimensional case (3): a higher order method seems to be derived which nevertheless seems to be of limited use due to the iterated high dimensional stochastic integral in (13). However, a key observation of this article is the following formula ! Z (n+1)T Z N2

nT N2

1 = 2

s

B

′

(YnN )

nT N2

dWsN

B(YnN ) dWuN

∂ b (·, YnN ) · b(·, YnN ) · ∂y

N

N

W (n+1)T − W nT N2

N2

2

! N T X 2 µi (gi ) − 2 N i=1 (14)

P-a.s. for all n ∈ {0, 1, . . . , N 2 − 1} and all N ∈ N (see Subsection 6.7 for the proof of the iterated integral identity (14) and see below for a heuristic explanation of this fact). So, the iterated high dimensional stochastic integral ∂ in (13) reduces to a simple product of functions. The function ∂y b : (0, 1) × ∂ R → R is here the partial derivative ( ∂y b)(x, y) for x ∈ (0, 1) and y ∈ R. Using (14) the numerical scheme (13) thus reduces to T N N N N = PN e + 2 · f (·, Yn ) + b(·, Yn ) · W (n+1)T − W nT N N2 N2 !! 2 N T X 1 ∂ N 2 N N N − 2 + µi (gi ) b (·, Yn ) · b(·, Yn ) · W (n+1)T − W nT 2 ∂y N i=1 N2 N2 (15) N Yn+1

A

T N2

YnN

P-a.s. for all n ∈ {0, 1, . . . , N 2 − 1} and all N ∈ N. It can be seen that only increments of the finite dimensional Wiener processes W N : [0, T ] × Ω → H, N ∈ N, are used in (15). Note that, as in the case of (10), the infinite dimensional R-Hilbert space H is projected down to the N-dimensional RHilbert space PN (H) with N ∈ N and the infinite dimensional Wiener process W : [0, T ] × Ω → H is approximated by the finite dimensional Wiener processes W N : [0, T ] × Ω → H, N ∈ N, for the spatial discretization in (15). For the temporal discretization in the scheme YnN , n ∈ {0, 1, . . . , N 2 }, above the time interval [0, T ] is divided into N 2 subintervals, i.e. N 2 instead of N 3 time steps are used in (15), for N ∈ N. In the following we explain why it is crucial to use N 2 time steps in (15) instead of N 3 time steps in the case of the linear implicit Euler scheme (10). More formally, we now illustrate how efficiently the method (15) solves the strong approximation problem (9) of the SPDE (7). Theorem 1 (see 9

Section 3 below) gives the existence of real numbers Cr > 0, r ∈ (0, 32 ), such that Z 1 12 2 3 N E (16) XT (x) − YN 2 (x) dx ≤ Cr · N (r− 2 ) 0

holds for all N ∈ N and all arbitrarily small r ∈ (0, 32 ). The approximation YNN2 thus converges in the root mean square sense to XT with order 23 − as N goes to infinity. The expression ! N 2 1 ∂ T X N 2 N N N (17) µi (gi ) b (·, Yn ) · b(·, Yn ) · W (n+1)T − W nT − 2 N 2 ∂y N i=1 N for n ∈ {0, 1, . . . , N 2 − 1} and N ∈ N in (15) contains additional information of the solution process of (7) and this allows us to use less time steps, N 2 in (15) instead of N 3 in (10), to achieve the same convergence rate as the linear implicit Euler scheme (10) (compare (11) and (16)). Nonetheless, (17) and hence the Pnumerical2 method (15) can be simulated very easily. The function NT2 N i=1 µi (gi ) in (17) can be computed once in ad2 vance for P which O(N ) computational operations are needed. Having com2 puted NT2 N i=1 µi (gi ) , O(N log(N)) further computational operations and random variables are needed to compute (17) from YnN for one fixed n ∈ {0, 1, . . . , N 2 − 1} by using fast Fourier transform. Since O(N log(N)) computational operations and random variables are needed for one time step and since N 2 time steps are used in (15), O(N 3 log(N)) computational operations and random variables are needed to compute YNN2 . Combining the computational effort O(N 3 log(N)) and the convergence order 23 − in (16) shows that the numerical method (15) needs about O(ε−2 ) computational operations and independent standard normal random variables to achieve the desired precision ε > 0 in (9). To sum up, the algorithm (15) reduces the complexity 8 of the problem (9) of the SPDE (7) from about O(ε− 3 ) to about O(ε−2 ). 8 However, the complexity rates O(ε− 3 ) and O(ε−2) are both asymptotic results as ε > 0 tends to zero. Therefore, from a practical point of view, one may ask whether the algorithm (15) solves the strong approximation problem (9) more efficiently than the linear implicit Euler scheme (10) for a given concrete ε > 0 and a given example of the form (7). In order to analyze this question we compare both methods in the case of a simple stochastic 1 reaction diffusion equation. More formally, let κ = 100 , let ξ : [0, 1] → R be given by ξ(x) = 0 for all x ∈ [0, 1] and suppose that f, b : (0, 1) × R → R are 1−y given by f (x, y) = 1 − y and b(x, y) = 1+y 2 for all x ∈ (0, 1), y ∈ R. The

10

SPDE (7) thus reduces to 1 ∂2 1 − Xt (x) dXt (x) = Xt (x) + 1 − Xt (x) dt + dWt (x) 2 100 ∂x 1 + Xt (x)2

(18)

with Xt (0) = Xt (1) = 0 and X0 = 0 for x ∈ (0, 1) and t ∈ [0, 1] (see also Section 4.1 for more details concerning this example). Additionally, assume that (9) for the SPDE (18) should be solved with the precision of say three 1 decimals, i.e. with the precision ε = 1000 in (9). In Figure 1 the approximation error in the sense of (9) of the linear implicit Euler approximation ZNN3 (see (10)) and of the approximation YNN2 (see (15)) is plotted against the precise number of independent standard normal random variables needed to compute the corresponding approximation for N ∈ {2, 4, 8, 16, 32, 64, 128}: 128 It turns out that Z128 3 in the case of the linear implicit Euler scheme (10) 128 and that Y1282 in the case of the algorithm (15) achieve the desired precision 1 128 ε = 1000 in (9) for the SPDE (18). The Matlab codes for simulating Z128 3 128 via (10) and Y128 2 via (15) for the SPDE (18) are presented below in Figure 2 and Figure 3 respectively. The differences of the codes and the additional code needed for the new algorithm (15) are printed bold in Figure 3. The Matlab code in Figure 2 requires on an Intel Pentium D a CPU time of about 15 minutes and 25.03 seconds (925.03 seconds) while the code in Figure 3 requires a CPU time of about 8.93 seconds to be evaluated on the same computer. So, the algorithm (15) is for the SPDE (18) more than hundred times faster than the linear implicit Euler scheme (10) in order to achieve a precision of three decimals in (9). To sum up, the four code changes (line 1, lines 3-6, line 10 and line 11) in Figure 3 enables us to simulate (18) with the precision of three decimals in about 9 seconds instead of about 15 minutes. Further numerical examples for the algorithm (15) can be found in Section 4 and Section 5. Having illustrated the efficiency of the method (15), we now take a short look into the literature of numerical analysis for SPDEs. First, it should be mentioned that any combination of finite elements, finite differences or spectral Galerkin methods for the spatial discretization and the linear implicit Euler scheme or also the linear implicit Crank-Nicolson scheme for the temporal 8 discretization do not reduce the complexity O(ε− 3 ) of the problem (9) of the SPDE (7). However, in the case of a linear SPDE, the splitting-up method as considered by I. Gyöngy and N. Krylov in [20] (see also [3, 4, 12, 19, 21, 22, 23] and the references therein) converges with a higher temporal order and therefore breaks the computational complexity in comparison to the linear implicit Euler scheme. The key idea of the splitting-up method is to split the considered SPDE into appropriate subequations that are easier to solve than the original SPDE, e.g., that can be solved explicitly. The splitting-up method 11

0

10

Linear implicit Euler scheme New algorithm in this article Order lines 1/2, 3/8 Approximation error 1/1000

−1

Approximation error

10

−2

10

−3

10

−4

10

0

10

2

4

6

8

10 10 10 10 Precise number of used random variables

10

10

Figure 1: SPDE (18): Approximation error in the sense of (9) of the linear implicit Euler approximation ZNN3 (see (10)) and of the approximation YNN2 (see (15)) against the precise number of independent standard normal random variables needed to compute the corresponding approximation for N ∈ {2, 4, 8, 16, 32, 64, 128}. thus essentially depends on the simplicity of the splitted subequations and can therefore in general not be used efficiently for nonlinear SPDEs which are investigated in this article. Nonetheless, in the case of a linear SPDE as in I. Gyöngy and N. Krylov’s article [20], the splitting-up method and the scheme in this article converge with the same complexity rate (see Section 4.3 for a more detailed comparison of the splitting-up method and the algorithm in this article). Additionally, in the case f = 0 in (7), T. M¨ uller-Gronbach and K. Ritter invented a new scheme which reduces the number of random 8 variables needed for solving a similar problem as (9) from about O(ε− 3 ) to about O(ε−2) (see [46] and also [47]). Nonetheless, the number of computational operations needed and thus the overall computational complexity could not be reduced by their algorithm. Moreover, Milstein type schemes for SPDEs have been considered in [5, 15, 37, 45]. In [15], W. Grecksch 12

1 2 3 4 5 6 7 8 9

N = 1 2 8 ; M = Nˆ 3 ; A = −p i ˆ 2 ∗ ( 1 :N) . ˆ 2 / 1 0 0 ; Y = z e r o s ( 1 ,N ) ; mu = ( 1 : N) . ˆ − 2 ; f = @( x ) 1−x ; b = @( x ) (1−x ) . / ( 1 + x . ˆ 2 ) ; f o r m=1:M y = d s t (Y) ∗ s q r t ( 2 ) ; dW = d s t ( randn ( 1 ,N) . ∗ s q r t (mu∗2/M) ) ; y = y + f ( y )/M + b ( y ) . ∗dW; Y = i d s t ( y ) / s q r t ( 2 ) . / ( 1 − A/M ) ; end p l o t ( ( 0 :N+1)/(N+1) , [ 0 , d s t (Y)∗ s q r t ( 2 ) , 0 ] ) ;

Figure 2: Matlab code for simulating the linear implicit Euler approximation ZNN3 with N = 128 (see (10)) for the SPDE (18). and P. E. Kloeden proposed a Milstein like scheme for a SPDE driven by a scalar one-dimensional Brownian motion (see also [37]). In view of (6), their Milstein scheme can be simulated efficiently since the driving noise process is one-dimensional. Furthermore, in the case of a linear SPDE, P.-L. Chow et al. constructed in the interesting article [5] a scheme similar to (13) but with an additional term. (The additional term may be useful for decreasing the error constant but turns out not to be needed in order to achieve the higher approximation order due to Theorem 1 here.) In order to simulate the iterated stochastic integral in their scheme, they then suggest to omit the summands in the double sum in (5) for which i 6= j holds (see (2.8) in [5]). Their idea thus yields a scheme that can be simulated very efficiently but does in general not converge with a higher order anymore except for a linear SPDE driven by a scalar one-dimensional Brownian motion. Finally, based on Itô’s formula, Y. S. Mishura and G. M. Shevchenko proposed in [45] the temporal approximation (12) under additonal smoothness assumptions of the driving noise process of the SPDE (8) which garantuee the existence of a strong solution and thus allow the application of Itô’s formula (see Assumption C in [45]). The simulation of the iterated stochastic integrals in their numerical approximation remained an open question (see Remark 1.1 in [45]). To sum up, to the best of our knowledge the computational complexity 8 barrier O(ε− 3 ) for the problem (9) of the SPDE (7) has not been broken by any algorithm proposed in the literature yet. It is in some sense amazing that the Milstein type scheme (13) reduces to the simple algorithm (15) in the infinite dimensional SPDE setting (7) which is the key observation in order to beat the computational complexity 8 barrier O(ε− 3 ). The reason for this simplification is that B(v) ∈ HS(U0 , H) for v ∈ H does not act as an arbitrary linear operator on the infinite di13

1 2 3 4 5 6 7 8 9 10 11 12 13

N = 1 2 8 ; M = Nˆ2 ; A = −p i ˆ 2 ∗ ( 1 :N) . ˆ 2 / 1 0 0 ; Y = z e r o s ( 1 ,N ) ; mu = ( 1 : N) . ˆ − 2 ; f = @( x ) 1−x ; b = @( x ) (1−x ) . / ( 1 + x . ˆ 2 ) ; bb = @(x) (1−x ) . ∗ ( x.ˆ2−2∗x−1)/2./(1+x . ˆ 2 ) . ˆ 3 ; g = zeros (1 ,N) ; for n=1:N g = g+2∗sin (n∗( 1:N)/(N+1)∗pi ).ˆ2∗mu(n)/M; end f o r m=1:M y = d s t (Y) ∗ s q r t ( 2 ) ; dW = d s t ( randn ( 1 ,N) . ∗ s q r t (mu∗2/M) ) ; y = y + f ( y )/M + b ( y ) . ∗dW + bb(y ) . ∗ (dW.ˆ2 − g) ; Y = exp( A/M ) . ∗ i d s t ( y ) / s q r t ( 2 ) ; end p l o t ( ( 0 :N+1)/(N+1) , [ 0 , d s t (Y)∗ s q r t ( 2 ) , 0 ] ) ;

Figure 3: Matlab code for simulating the approximation YNN2 with N = 128 (see (15)) for the SPDE (18). mensional vector space U0 but as a multiplication operator on the function space U0 ⊂ L2 ((0, 1), R). First, this ensures that B : H → HS(U0 , H) naturally falls into the infinite dimensional analog of the commutative noise case (4) although b : (0, 1) × R → R and hence B : H → HS(U0 , H) are possibly nonlinear mappings (see (34) for details). Second, in the infinite dimensional commutative noise setting, the acting of B(v) for v ∈ H as a multiplication operator assures that the infinite dimensional analog of (5) even simplifies in infinite dimensions to (15). These two facts guarantee that the method (13) reduces to the simple algorithm (15) which is in a sense the main contribution of this article. To sum up, due to the reduced complexity, the algorithm (15) provides an in comparison to previously considered numerical methods impressive instrument for simulating nonlinear stochastic partial differential equations with multiplicative trace class noise. The rest of this article is organized as follows. In Section 2 the setting and the assumptions used are formulated. The numerical method and its convergence result (Theorem 1) are presented in Section 3. In Section 4 several examples of Theorem 1 including a stochastic heat equation and stochastic reaction diffusion equations are considered. Although our setting in Section 2 uses the standard global Lipschitz assumptions on the nonlinear coefficients of the SPDE, we demonstrate the efficiency of our method numerically for a stochastic Burgers equation with a non-globally Lipschitz nonlinearity in Section 5. The proof of Theorem 1 is postponed to Section 6. Finally, we would like to add some concluding remarks. There are a num-

14

ber of directions for further research arising from this work. One is to analyze whether the exponential term in (15) can be replaced by a simpler mollifier such as (I − NT2 A)−1 for N ∈ N. This would make the scheme even simpler to simulate. A second direction is to combine the temporal approximation in (15) with other spatial discretizations such as finite elements. This makes it possible to handle more complicated multidimensional domains on which the eigenfunctions of the Laplacian are not known explicitly. A third direction is to reduce the Lipschitz assumptions in Section 2 in order to handle the stochastic Burgers equation in Section 5 and further SPDEs with non-globally Lipschitz nonlinearities such as the stochastic porous medium equation and hyperbolic SPDEs. Finally, a combination of Giles’ multilevel Monte Carlo approach in [14] with the method in this article should yield a break of the computational complexity in comparison to previously considered algorithms for solving the weak approximation problem (see, e.g., Section 9.4 in [36]) of the SPDE (7).

2

Setting and assumptions

Throughout this article suppose that the following setting and the following assumptions are fulfilled. Fix T ∈ (0, ∞), let (Ω, F , P) be a probability space with a normal filtration (Ft )t∈[0,T ] and let (H, h·, ·iH , k·kH ) and (U, h·, ·iU , k·kU ) be two separable R-Hilbert spaces. Moreover, let Q : U → U be a trace class operator and let W : [0, T ] × Ω → U be a standard Q-Wiener process with respect to (Ft )t∈[0,T ] . Assumption 1 (Linear operator A). Let I be a finite or countable set and let (λi )i∈I ⊂ (0, ∞) be a family of real numbers with inf i∈I λi ∈ (0, ∞). Moreover, let (ei )i∈I be an orthonormal basis of H and let A : D(A) ⊂ H → H be a linear operator with X Av = −λi hei , viH ei (19) i∈I

P for every v ∈ D(A) and with D(A) = w ∈ H i∈I |λi |2 |hei , wiH |2 < ∞ .

By Vr := D ((−A)r ) equipped with the norm kvkVr := k(−A)r vkH for all v ∈ Vr and all r ∈ [0, ∞) we denote the R-Hilbert spaces of domains of fractional powers of the linear operator −A : D(A) ⊂ H → H. Assumption 2 (Drift term F ). Let β ∈ [0, 1) be a real number and let F : Vβ → H be a twice continuously Fréchet differentiable mapping with supv∈Vβ kF ′ (v)kL(H) < ∞ and supv∈Vβ kF ′′ (v)kL(2) (Vβ ,H) < ∞. 15

In order to formulate the assumption on the diffusion coefficient of our SPDE, we denote by U0 , h·, ·iU0 , k·kU0 the separable R-Hilbert space U0 := D 1 E 1 1 Q 2 (U) with hv, wiU0 = Q− 2 v, Q− 2 w for all v, w ∈ U0 (see, for example, U

Section 2.3.2 in [50]). For an arbitrary bounded linear operator S ∈ L(U), we denote by S −1 : im(S) ⊂ U → U the pseudo inverse of S (see, for instance, Appendix C in [50]).

Assumption 3 (Diffusion term B). Let B : Vβ → HS(U0 , H) be a twice continuously Fréchet differentiable mapping with supv∈Vβ kB ′ (v)kL(H,HS(U0 ,H)) < ∞ and supv∈Vβ kB ′′ (v)kL(2) (Vβ ,HS(U0 ,H)) < ∞. Moreover, let α, c ∈ (0, ∞), δ, ϑ ∈ (0, 21 ), γ ∈ [max(δ, β), δ + 21 ) be real numbers, let B(Vδ ) ⊂ HS(U0 , Vδ ) and suppose that kB(u)kHS(U0 ,Vδ ) ≤ c 1 + kukVδ , (20) kB ′ (v) B(v) − B ′ (w) B(w)kHS (2) (U0 ,H) ≤ c kv − wkH ,

−ϑ ≤ c 1 + kvkVγ

(−A) B(v)Q−α HS(U0 ,H)

(21)

(22)

holds for every u ∈ Vδ and every v, w ∈ Vγ . Finally, let the bilinear HilbertSchmidt operator B ′ (v)B(v) ∈ HS (2) (U0 , H) be symmetric for every v ∈ Vβ . The operator B ′ (v)B(v) : U0 × U0 → H given by (B ′ (v)B(v)) (u, u ˜) = {B ′ (v) (B(v)u)} (˜ u) for all u, u ˜ ∈ U0 is a bilinear Hilbert-Schmidt operator in HS (2) (U0 , H) ∼ = HS(U0 ⊗ U0 , H) for every v ∈ Vβ . Therefore, condition (21) means that the mapping Vγ → HS (2) (U0 , H),

v 7→ B ′ (v)B(v),

v ∈ Vγ ,

is globally Lipschitz continuous with respect to k·kH and k·kHS (2) (U0 ,H) . We also would like to point out that the assumed symmetry of B ′ (v)B(v) ∈ HS (2) (U0 , H) for all v ∈ Vβ is the abstract (infinite dimensional) analog of (4). More formally, if H = Rd , U = Rm and Q = I with d, m ∈ N holds, then the symmetry of B ′ (v)B(v) ∈ HS (2) (Rm , Rd ) reduces to (4) (with σ replaced by B). Although (4) is seldom fulfilled for finite dimensional SODEs, the symmetry of B ′ (v)B(v) ∈ HS (2) (U0 , H) for all v ∈ Vβ is naturally met in the case of multiplicative noise on infinite dimensional function spaces as we will demonstrate in Sections 4 and 5. Finally, we emphasize that we assumed in no way that the linear operators A : D(A) ⊂ H → H and Q : U → U are simultaneously diagonalizable. 16

Assumption 4 (Initial value ξ). Let ξ : Ω → Vγ be a F0 /B (Vγ )-measurable mapping with E kξk4Vγ < ∞. These assumptions suffice to ensure the existence of a unique solution of the SPDE (23). Proposition 1 (Existence of the solution). Let Assumptions 1-4 in Section 2 be fulfilled. Then there exists an up to modifications unique predictable stochastic process X : [0, T ] × Ω → Vγ which fulfills supt∈[0,T ] E kXt k4Vγ < ∞, supt∈[0,T ] E kB(Xt )k4HS(U0 ,Vδ ) < ∞ and Z t Z t At A(t−s) Xt = e ξ + e F (Xs ) ds + eA(t−s) B(Xs ) dWs P-a.s. (23) 0

0

for all t ∈ [0, T ]. Moreover, we have sup t1 ,t2 ∈[0,T ] t1 6=t2

E kXt2 − Xt1 k4Vr

1

41

|t2 − t1 |min(γ−r, 2 )

R = 0 ∈ [0, ∞]

is used for all B((0, 1)2 )/B(R)-measurable mappings v : (0, 1)2 → R in this subsection. Then we obtain X |hei , viH | kei kC((0,1)2 ,R) kvkL∞ ((0,1)2 ,R) ≤ i∈N2

≤2

X

i∈N2

(λi )−2r

! 21

X

i∈N2

(λi )2r |hei , viH |2

32

! 21

=2

X

i∈N2

(λi )−2r

! 12

kvkVr (51)

for all v ∈ Vr and all r ∈ ( 21 , ∞). Moreover, we have

−ϑ

(−A) B(v)Q−α HS(U0 ,H)

1 1

= (−A)−ϑ B(v)Q( 2 −α) = (−A)−ϑ B(v)Q( 2 −α) HS(H) S2 (H)

1

≤ (−A)−ϑ kB(v)kL(H) Q( 2 −α) S 1 (H)

S

α

X

≤

(λi )

ϑ −α

i∈N2

!α

kb(·, v)kL∞ ((0,1)2 ,R)

2 (1−2α)

X

(H)

( 1 −α) 2 µj 2 (1−2α)

j∈J

! (1−2α) 2

and using (51) shows

−ϑ −α

(−A) B(v)Q HS(U0 ,H) !α X 1 ϑ −α q kvkL∞ ((0,1)2 ,R) + q (Tr(Q))( 2 −α) (λi ) ≤ i∈N2

≤ q (1 + Tr(Q)) ≤ 2q (1 + Tr(Q))

X

(λi )

ϑ −α

i∈N2

X

i∈N2

(λi )

!α

ϑ −α

1 + kvkL∞ ((0,1)2 ,R)

!α

1+

X

i∈N2

(λi )−2γ

!

1 + kvkVγ < ∞

for every ϑ ∈ (α, 21 ), α ∈ (0, 12 ), v ∈ Vγ and every γ ∈ ( 21 , 1). Inequality (22) thus holds for all α ∈ (0, 21 ) and all γ ∈ ( 21 , 1) here. This finally shows that Assumptions 1-4 are fulfilled for the SPDE (50) for all α ∈ (0, 12 ), β = 25 and all γ ∈ ( 21 , 1). Theorem 1 therefore yields the existence of real numbers Cr ∈ (0, ∞), r ∈ (0, 1), such that Z E

1 0

Z

1

0

21 2 N,M,K (x1 , x2 ) dx1 dx2 XT (x1 , x2 ) − YM

≤ Cr N (r−2) + K (r−2) + M (r−1)

(52)

holds for all N, M, K ∈ N and all arbitrarily small r ∈ (0, 1). In order to balance the error terms on the right hand side of (52) we choose M = N 2 = K 2 in (52) and obtain the existence of real numbers Cr ∈ (0, ∞), r ∈ (0, 2), such that Z 1Z 1 12 2 N,N 2 ,N ≤ Cr · N (r−2) (53) E (x1 , x2 ) dx1 dx2 XT (x1 , x2 ) − YN 2 0

0

33

holds for all N ∈ N and all arbitrarily small r ∈ (0, 2). The approximation 2 ,N YNN,N thus converges in the root mean square sense to XT with order 2 2 2− as N goes to infinity. The numerical approximations YnN,N ,N : Ω → H, 2 n ∈ {0, 1, . . . , N 2 }, N ∈ N, (see (27) and (37)) are here given by Y0N,N ,N = x0 and N,N Yn+1

2 ,N

T

PN eA N 2

=

(54) !

i 2 h T X 1 2 2 2 2 · YnN,N ,N µ (g ) ∆WnN ,N − 1 + ∆WnN ,N + j j 2 2 2N j∈J K

for all n ∈ {0, 1, . . . , N 2 − 1} and all N ∈ N. Since PN (H) ⊂ H in N 2 2 ,N dimensional here and since N 2 time steps are used to simulate YNN,N , 2 O(N 4 log(N)) computational operations and random variables are needed to 2 ,N . Combining the computational effort O(N 4 log(N)) and simulate YNN,N 2 the convergence order 2− in (53) shows that the algorithm (54) in this article needs about O(ε−2) computational operations and random variables to achieve a root mean square precision ε > 0. The linear implicit Euler scheme combined with spectral Galerkin methods which we denote by F /B(H)-measurable mappings ZnN : Ω → H, n ∈ {0, 1, . . . , N 4 }, N ∈ N, is given by Z0N = x0 and −1 h i T N N 4 ,N N Zn+1 = PN I − 4 A 1 + ∆Wn · Zn (55) N for all n ∈ {0, 1, . . . , N 4 − 1} and all N ∈ N here. Moreover, since the SPDE (50) is linear here, the splitting-up method in [20] can be used in order to solve (50) approximatively. The key idea of the splitting-up approach is to split the SPDE (50) into the explicit solvable subequations 2 ∂2 ∂ 1 ˜ ˜ t |∂(0,1)2 ≡ 0 (56) ˜ + Xt (x1 , x2 ) dt, X dXt (x1 , x2 ) = 50 ∂x21 ∂x22 and

˜˜ (x , x ) = X ˜˜ (x , x ) dW (x , x ) dX t 1 2 t 1 2 t 1 2

(57)

˜˜ : [0, T ]×Ω → ˜ X for t ∈ [0, 1] and x1 , x2 ∈ (0, 1). For the solution processes X, P ˜˜ ˜˜ = e(Wt − 2t j∈J µj (gj )2 ) · X ˜ t = eAt X ˜ 0 and X H of (56) and (57) we obtain X 0 t P-a.s. for all t ∈ [0, 1]. This suggests the splitting-up approximation P µj (gj )2 ) Wt − 2t At ( j∈J · X0 Xt ≈ e e 34

for t ∈ [0, 1] where X : [0, T ] × Ω → H is the solution process of the SPDE (50). The resulting splitting-up method which we denote by F /B(H)measurable mappings ZñN : Ω → H, n ∈ {0, 1, . . . , N 2 }, N ∈ N, is then given by Z˜0N = x0 and ! N 2 ,N T P 2 µ (g ) ∆W − T j j n j∈JK 2N 2 · Z˜ N (58) Z˜ N = PN eA N 2 e n

n+1

for all n ∈ {0, 1, . . . , N 2 − 1} and all N ∈ N. We remark that I. Göngy and N. Krylov considered in [20] temporal splitting-up approximations and we added an appropriate spatial approximation here in order to compare the splitting-up method with the algorithm (54) in this article. Using the Talyor 2 approximation ex ≈ 1 + x + x2 for all x ∈ R then yields

e

∆WnN

2 ,N

−

T 2N 2

P

≈ 1 + ∆WnN

2 ,N

≈ 1 + ∆WnN

2 ,N

j∈JK

µj (gj )2

!2 X 1 T T X 2 µj (gj )2 + µj (gj )2 ∆WnN ,N − − 2 2 2N j∈J 2 2N j∈J K K X 2 1 T 2 + ∆WnN ,N − µj (gj )2 (59) 2 2N 2 j∈J K

for all n ∈ {0, 1, . . . , N 2 − 1} and all N ∈ N. Using approximation (59) in (58) finally shows ! i h X 2 T T 1 2 2 N µj (gj )2 · ZñN ∆WnN ,N − Zñ+1 ≈ PN eA N 2 1 + ∆WnN ,N + 2 2 2N j∈J K

for all n ∈ {0, 1, . . . , N 2 − 1} and all N ∈ N which is nothing else than the 2 recursion for YnN,N ,N , n ∈ {0, 1, . . . , N 2 }, N ∈ N in (54). So, in the case of the linear SPDE (50) an alternative way for deriving the algorithm (54) in this article is to apply an appropriate Taylor approximation for the exponential function (see (59) for details) to the splitting-up method (58). More results for the splitting-up method can be found in [3, 4, 12, 19, 20, 21, 22, 23] and the references therein. 1 In Figure 5 the root mean square approximation error (EkXT −ZNN4 k2H ) 2 of the linear implicit Euler approximation ZNN4 (see (55)), the root mean square 2 ,N 2 ,N 1 approximation error (EkXT − YNN,N k2H ) 2 of the approximation YNN,N in 2 2 this article (see (54)) and the root mean square approximation error (EkXT − 1 ZÑN2 k2H ) 2 of the splitting-up approximation ZÑN2 (see (58)) is plotted against the precise number of independent standard normal random variables needed 35

to compute the corresponding approximation for N ∈ {2, 4, 8, 16, 32}: It 32 6 turns out that Z32 = 1 073 741 824 random variables) in the case of 4 (32 2 the linear implicit Euler scheme (55), that Y3232,32 ,32 (324 = 1 048 576 random 32 variables) in the case of the algorithm (54) in this article and that Z˜32 2 (324 = 1 048 576 random variables) in the case of the splitting-up method (58) 1 achieve a root mean square precision ε = 1000 for the SPDE (50). The 2 ,32 32,32 32 32 via (54) and Z˜32 Matlab codes for simulating Z32 2 via 4 via (55), Y32 (58) are presented below in Figures 6, 7 and 8 respectively. The Matlab code in Figure 6 requires on an Intel Pentium D a CPU time of about 29 minutes and 57.06 seconds (1797.06 seconds), the code in Figure 7 requires a CPU time of about 1.99 seconds while the code in Figure 8 requires a CPU time of about 2.10 seconds to be evaluated on the same computer. Finally, we conclude that for linear SPDEs the splitting-up method converges with the same order as the algorithm (27) in this article. For SPDEs with nonlinear diffusion coefficients the splitting-up method can in general not be used efficiently anymore since the splitted subequations do in general not simplify in comparison to the original considered SPDE. In contrast to the splitting-up method the algorithm (27) works quite well for nonlinear equations as demonstrated in Section 1, Subsection 4.1, Subsection 4.2 and Section 5.

5

A further numerical example

Although the setting in Section 2 requires the nonlinear coefficients F and B of the SPDE (23) to be globally Lipschitz continuous, we strongly believe that our method (27) produces efficient results for SPDEs with non-globally Lipschitz nonlinearities. To substantiate this claim, we apply in this section the algorithm (27) and the linear implicit Euler scheme to a stochastic Burgers equation whose nonlinear drift term F is quadratic and therefore not globally Lipschitz continuous anymore. More formally, we consider the SPDE ∂ 1 ∂2 Xt (x) − Xt (x) Xt (x) dt + Xt (x) dWt (60) dXt (x) = 100 ∂x2 ∂x √

with Xt (0) = Xt (1) = 0 and X0 (x) = 3 5 2 (sin(πx) + sin(2πx)) for x ∈ (0, 1) and t ∈ [0, T ] on H = L2 ((0, 1), R) where T = 1 and where W : [0, T ] × Ω → H is a standard Q-Wiener with Rthe covariance operator Q : H → H P∞ process 1 −3 given by (Qv) (x) = n=1 n sin(nπx) 0 sin(nπy) v(y) dy for all x ∈ (0, 1) 36

0

10

Linear implicit Euler scheme New algorithm in this article Splitting−up method Order lines 1/2, 1/3 Approximation error 1/1000

−1

Approximation error

10

−2

10

−3

10

−4

10

0

10

2

10

4

6

8

10 10 10 Precise number of used random variables

10

10

Figure 5: SPDE (50): Root mean square approximation error (EkXT − 1 ZNN4 k2H ) 2 of the linear implicit Euler approximation ZNN4 (see (55)), root 2 ,N 1 k2H ) 2 of the approximation mean square approximation error (EkXT −YNN,N 2 2 ,N YNN,N in this article (see (54)) and root mean square approximation error 2 1 (EkXT − ZÑN2 k2H ) 2 of the splitting-up approximation ZÑN2 (see (58)) against the precise number of independent standard normal random variables needed to compute the corresponding approximation for N ∈ {2, 4, 8, 16, 32}.

37

1 2 3 4 5 6 7 8 9 10 11

N = 3 2 ; M = Nˆ 4 ; Y = z e r o s (N,N ) ; Y( 1 , 1 ) = 1 ; [ n1 , n2 ] = m es h gr id ( 1 :N ) ; A = −(n1 . ˆ 2 + n2 . ˆ 2 ) ∗ p i ˆ 2 / 5 0 ; mu = 1 . / ( n1 . ˆ 4 + n2 . ˆ 4 ) ; f o r m=1:M y = dst ( dst ( Y ) ’ ) ’ ∗ 2; dW = d s t ( d s t ( randn (N,N) . ∗ s q r t (mu/M) ) ’ ) ’ ∗ 2 ; y = (1 + dW) . ∗ y ; Y = i d s t ( i d s t ( y ’ ) ’ ) / 2 . / ( 1 − A/M ) ; end s u r f ( n1 /(N+1) , n2 /(N+1) , d s t ( d s t ( Y ) ’ ) ’ ∗ 2 ) ;

Figure 6: Matlab code for simulating the linear implicit Euler approximation ZNN4 with N = 32 (see (55)) for the SPDE (46).

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

N = 3 2 ; M = Nˆ2; Y = z e r o s (N,N ) ; Y( 1 , 1 ) = 1 ; [ n1 , n2 ] = m es h gr id ( 1 :N ) ; A = −(n1 . ˆ 2 + n2 . ˆ 2 ) ∗ p i ˆ 2 / 5 0 ; mu = 1 . / ( n1 . ˆ 4 + n2 . ˆ 4 ) ; g = zeros (N,N) ; grid = ( 1 :N)/(N+1); for m1=1:N for m2=1:N g = g+4∗sin (m1∗grid ’∗ pi ).ˆ2∗ sin (m2∗grid∗pi ) . ˆ 2 . ∗mu(m1,m2)/M; end end f o r m=1:M y = dst ( dst ( Y ) ’ ) ’ ∗ 2; dW = d s t ( d s t ( randn (N,N) . ∗ s q r t (mu/M) ) ’ ) ’ ∗ 2 ; y = (1 + dW + (dW.ˆ2 − g)/2) .∗ y ; Y = exp( A/M ) . ∗ i d s t ( i d s t ( y ’ ) ’ ) / 2 ; end s u r f ( n1 /(N+1) , n2 /(N+1) , d s t ( d s t ( Y ) ’ ) ’ ∗ 2 ) ;

Figure 7: Matlab code for simulating the approximation YNN,N 2 article with N = 32 (see (54)) for the SPDE (46).

38

2 ,N

in this

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

N = 3 2 ; M = Nˆ 2 ; Y = z e r o s (N,N ) ; Y( 1 , 1 ) = 1 ; [ n1 , n2 ] = m es h gr id ( 1 :N ) ; A = −(n1 . ˆ 2 + n2 . ˆ 2 ) ∗ p i ˆ 2 / 5 0 ; mu = 1 . / ( n1 . ˆ 4 + n2 . ˆ 4 ) ; g = z e r o s (N,N ) ; g r i d = ( 1 : N) / (N+1); f o r m1=1:N f o r m2=1:N g = g+4∗ s i n (m1∗ g r i d ’ ∗ p i ) . ˆ 2 ∗ s i n (m2∗ g r i d ∗ p i ) . ˆ 2 . ∗ mu(m1, m2)/M; end end f o r m=1:M y = dst ( dst ( Y ) ’ ) ’ ∗ 2; dW = d s t ( d s t ( randn (N,N) . ∗ s q r t (mu/M) ) ’ ) ’ ∗ 2 ; y = exp (dW − g / 2 ) . ∗ y ; Y = exp ( A/M ) . ∗ i d s t ( i d s t ( y ’ ) ’ ) / 2 ; end s u r f ( n1 /(N+1) , n2 /(N+1) , d s t ( d s t ( Y ) ’ ) ’ ∗ 2 ) ;

Figure 8: Matlab code for simulating the splitting-up approximation ZÑN2 with N = 32 (see (58)) for the SPDE (46). and all v√∈ H. The family gj ∈ H, j ∈ J , of functions with J = N and gj (x) = 2 sin(jπx) for all x ∈ (0, 1), j ∈ N, thus represents an orthonormal basis of eigenfunctions of Q : H → H. The corresponding eigenvalues (µj )j∈J satisfy µj = j12 for all j ∈ N and we choose JK = {1, 2, . . . , K} for all K ∈ N here. Theoretical results for the stochastic Burgers equation including existence and uniqueness results can be found in [6, 10, 11], for instance. The method (27) with N 2 = K 2 = M applied to the SPDE (60) is given 2 by Y0N,N ,N = PN (X0 ) and N,N 2 ,N Yn+1

A

= PN e

N,N 2 ,N

+ Yn

T N2

2 YnN,N ,N

T 2 − 2 · YnN,N ,N · N

∂ N,N 2 ,N Y ∂x n

N 2 1 T X 2 N 2 ,N N 2 ,N · ∆Wn + ∆Wn µ (g ) − j j 2 2N 2 j=1

!

(61)

for all n ∈ {0, 1, . . . , N 2 − 1} and all N ∈ N here. The linear implicit Euler scheme applied to the SPDE (60) which we denote by F /B(H)-measurable mappings ZnN : Ω → H, n ∈ {0, 1, . . . , N 3 },

39

N ∈ N, is given by Z0N = PN (X0 ) and A T3 N

N Zn+1 = PN e

T ZnN − 3 · ZnN · N

∂ N Z ∂x n

N 3 ,N

+ ZnN · ∆Wn

!

(62)

for all n ∈ {0, 1, . . . , N 3 − 1} and all N ∈ N here. Recently, it has been shown in [28] that Euler’s method fails to converge in the root mean square sense to the exact solution of a SODE with a superlinearly growing drift coefficient of the form (60) (see Theorem 2 in [28] for the precise statement of this result) although pathwise convergence often holds (see, e.g., [17, 33] for SODEs and [18, 29, 49] for SPDEs). That is the reason why we consider the pathwise instead of the root mean square approximation error in this section. More precisely, in Figure 9 the pathwise approximation error kXT (ω) − ZNN3 (ω)k2H of the linear implicit Euler approximation ZNN3 2 ,N (see (62)) and the pathwise approximation error kXT (ω) − YNN,N (ω)kH of 2 N,N 2 ,N the approximation YN 2 in this article (see (61)) is plotted against the precise number of independent standard normal random variables needed to compute the corresponding approximation for N ∈ {2, 4, 8, 16, 32, 64, 128} and one random ω ∈ Ω.

6

Proof of Theorem 1

The notation kZkLp (Ω;E)

p1 p := E kZkE ∈ [0, ∞]

is used throughout this section for an R-Banach space (E, k·kE ), a F /B(E)measurable mapping Z : Ω → E and a real number p ∈ [1, ∞). We also use the following simple lemma (see, e.g., Lemma 1 in [34] and also Theorem 37.5 in [56]). Lemma 1. Let Assumptions 1-4 in Section 2 be fulfilled. Then we have

(−tA)−r eAt − I

(−tA)r eAt ≤1 ≤ 1 and L(H) L(H) for every t ∈ (0, ∞) and every r ∈ [0, 1].

In the following Theorem 1 is established. First of all, note that the exact

40

−1

10

Linear implicit Euler scheme New algorithm in this article Order lines 1/2, 3/8 −2

Approximation error

10

−3

10

−4

10

−5

10

0

10

2

4

6

8

10 10 10 10 Precise number of used random variables

10

10

Figure 9: SPDE (60): Pathwise approximation error kXT (ω) − ZNN3 (ω)k2H of the linear implicit Euler approximation ZNN3 (see (62)) and pathwise approx2 ,N 2 ,N in this (ω)kH of the approximation YNN,N imation error kXT (ω) − YNN,N 2 2 article (see (61)) against the precise number of independent standard normal random variables needed to compute the corresponding approximation for N ∈ {2, 4, 8, 16, 32, 64, 128} and one random ω ∈ Ω.

41

solution of the SPDE (23) satisfies Amh

Xmh = e

= eAmh ξ +

ξ+

Z

mh A(mh−s)

e

0 m−1 X Z (l+1)h

F (Xs ) ds +

eA(mh−s) F (Xs ) ds +

lh

l=0

Z

mh

eA(mh−s) B(Xs ) dWs

0 m−1 X Z (l+1)h

eA(mh−s) B(Xs ) dWs

lh

l=0

(63)

P-a.s. for every m ∈ {0, 1, . . . , M} and every M ∈ N. Here and below h is T the time stepsize h = hM = M with M ∈ N. In particular, (63) shows ! m−1 X Z (l+1)h eA(mh−s) F (Xs ) ds PN (Xmh ) = eAmh PN (ξ) + PN lh

l=0

+ PN

m−1 X Z (l+1)h

A(mh−s)

e

B(Xs ) dWs

lh

l=0

!

(64)

P-a.s. for every m ∈ {0, 1, . . . , M} and every N, M ∈ N. In order to estimate the difference of the exact solution (63) and the numerical solution (27) we rewrite the numerical method (27) in some sense. More precisely, the identity 1 ′ N,M,K N,M,K M,K B Ym B Ym ∆Wm ∆WmM,K 2 T X N,M,K ′ N,M,K − B Ym gj gj µj B Ym 2M j∈J K

µj 6=0

=

Z

(m+1)T M

B

mT M

′

YmN,M,K

Z

s

B mT M

YmN,M,K

dWsK (65)

dWuK

P-a.s. holds for all m ∈ {0, 1, . . . , M − 1} and all N, M, K ∈ N. The proof of (65) can be found in Subsection 6.7. Using (65) shows that the numerical solution (27) fulfills N,M,K Ym+1

T AM

= PN e

+

Z

Z (m+1)T M T N,M,K + B YmN,M,K dWsK · F Ym + mT M M ! Z s ′ N,M,K N,M,K K K B Ym B Ym dWu dWs (66)

YmN,M,K (m+1)T M

mT M

mT M

42

P-a.s. for every m ∈ {0, 1, . . . , M − 1} and every N, M, K ∈ N. Therefore, the numerical solution (27) satisfies ! m−1 X Z (l+1)h N,M,K A(m−l)h N,M,K Amh ds e F Yl Ym =e PN (ξ) + PN l=0

+ PN + PN

m−1 X Z (l+1)h

l=0 lh m−1 X Z (l+1)h l=0

lh

lh

N,M,K A(m−l)h dWsK e B Yl

eA(m−l)h B ′ YlN,M,K

Z

!

s

B lh

(67)

YlN,M,K

dWuK dWsK

!

P-a.s. for every m ∈ {0, 1, . . . , M} and every N, M, K ∈ N. In order to ap

2 praise E Xmh − YmN,M,K H for m ∈ {0, 1, . . . , M} and N, M, K ∈ N, we deN,M,K fine the F /B(H)-measurable mappings Zm : Ω → H, m ∈ {0, 1, . . . , M}, N, M, K ∈ N, by ! m−1 X Z (l+1)h N,M,K eA(m−l)h F (Xlh ) ds Zm := eAmh PN (ξ) + PN l=0

+ PN

m−1 X Z (l+1)h lh

+ PN

l=0 m−1 X Z (l+1)h l=0

lh

eA(m−l)h B(Xlh ) dWsK eA(m−l)h B ′ (Xlh )

lh

Z

!

s lh

(68)

B(Xlh ) dWuK dWsK

!

P-a.s. for every m ∈ {0, 1, . . . , M} and every N, M, K ∈ N. The inequality (69) (a1 + . . . + an )2 ≤ n (a1 )2 + . . . + (an )2

for every a1 , . . . , an ∈ R and every n ∈ N then shows

2 E Xmh − YmN,M,K H ≤ 3 · E kXmh − PN (Xmh )k2H

N,M,K

N,M,K 2 N,M,K 2

Z (70) − Y + 3 · E + 3 · E PN (Xmh ) − Zm m m H H

for every m ∈ {0, 1, . . . , M} and every N, M, K ∈ N. In order to esti

N,M,K 2 mate the expressions E kXmh − PN (Xmh )k2H , E PN (Xmh ) − Zm and H

N,M,K 2 − YmN,M,K H for m ∈ {0, 1, . . . , M} and N, M, K ∈ N, the real E Zm number R ∈ (0, ∞) satisfying E kB(Xt )k2HS(U0 ,Vδ ) ≤ R,

kF ′ (v)kL(H) ≤ R, 43

kF ′′ (v)kL(2) (Vβ ,H) ≤ R,

E kF (Xt )k2H ≤ R, kB ′ (v)kL(H,HS(U0 ,H)) ≤ R, kB ′′ (v)kL(2) (Vβ ,HS(U0 ,H)) ≤ R, 2

E k(−A)γ Xt kH = E kXt k2Vγ ≤ R, E kXt2 − Xt1 k4Vβ ≤ R |t2 − t1 |min(4(γ−β),2) , c+

1 1 1 + + + T + A−1 L(H) ≤ R (1 − γ) (1 − 2ϑ) (1 − 2δ)

for every v ∈ Vβ and every t, t1 , t2 ∈ [0, T ] is used throughout this proof. Due to Assumptions 1-4 in Section 2 and Proposition 1 such a real number indeed exists. For the spatial discretization error E kXmh − PN (Xmh )k2H we then obtain E kXmh − PN (Xmh )k2H

2 = E k(I − PN ) Xmh k2H = E (−A)−γ (I − PN ) (−A)γ Xmh H

2

≤ (−A)−γ (I − PN ) L(H) E kXmh k2Vγ ≤ R (rN )2

(71)

for every m ∈ {0, 1, . . . , M} and every N, M ∈ N where here and below the real numbers (rN )N ∈N ⊂ R are given by

rN := (−A)−γ (I − PN ) L(H) =

inf λi

i∈I\IN

−γ

for every N ∈ N. The rest of this proof is then divided into six parts. In the first part (see Subsection 6.1) we establish

2 Z (l+1)h

m−1

X 36R8

A(mh−s) A(m−l)h E e F (Xs ) − e F (Xlh ) ds ≤ min(4(γ−β),2γ)

M l=0 lh H (72) for every m ∈ {0, 1, . . . , M} and every M ∈ N. We show

2

2α Z (l+1)h

m−1 X

5 A(mh−s) K sup µj e B(Xs ) d Ws − Ws ≤ 4R (73) E

j∈J \JK lh l=0

H

for every m ∈ {0, 1, . . . , M} and every M, K ∈ N in the second part (see Subsection 6.2) and

2 Z (l+1)h

m−1

X 3R4

E eA(mh−s) − eA(m−l)h B(Xs ) dWsK ≤ (1+2δ)

M lh l=0

H

44

(74)

for every m ∈ {0, 1, . . . , M} and every M, K ∈ N in the third part (see Subsection 6.3). The fourth part (see Subsection 6.4) gives

Z (l+1)h Z 1

m−1

X A(m−l)h e B ′′ (Xlh + r (Xs − Xlh )) (Xs − Xlh , Xs − Xlh ) E

0 l=0 lh

2

R6 K · (1 − r) dr dWs ≤ min(4(γ−β),2) (75)

M H

and in the fifth part (see Subsection 6.5) we obtain

2

Z (l+1)h Z s

m−1

X A(m−l)h ′ K K e B (Xlh ) Xs − Xlh − B(Xlh ) dWu dWs E

lh l=0 lh H 2α 13 20R ≤ min(4(γ−β),2γ) + 20R11 sup µj (76) M j∈J \JK for every m ∈ {0, 1, . . . , M} and ∈ N. The

every M, K N,M,K

2 inequalities (72)-(76)

for m ∈ {0, 1, . . . , M} are used below to estimate E PN (Xmh ) − Zm H and N, M, K ∈ N in (70). In the sixth part (see Subsection 6.6) we estimate ! m−1

2 4 X

N,M,K 9R 2

(77) E Xlh − YlN,M,K − YmN,M,K H ≤ E Zm H M l=0

for every m ∈ {0, 1, . . . , M} and every N, M, K ∈ N by using the global Lipschitz continuity of the coefficients F : Vβ → H (see Assumption 2) and B : Vβ → HS(U0, H) (see Assumption 3). Combining (70), (71) and (77) then yields

2 E Xmh − YmN,M,K H ≤ 3R (rN )2

27R4 N,M,K 2 + + 3 · E PN (Xmh ) − Zm H M

m−1 X l=0

2

N,M,K E Xlh − Yl

H

!

for every m ∈ {0, 1, . . . , M} and every N, M, K ∈ N. Hence, (64), (68) and

45

(69) show

2

E Xmh − YmN,M,K H

! 2 m−1

X Z (l+1)h

A(mh−s) A(m−l)h e F (Xs ) − e F (Xlh ) ds ≤ 9 · E PN

l=0 lh H

! 2 Z m−1

(l+1)h X

eA(mh−s) B(Xs ) d Ws − WsK + 9 · E PN

l=0 lh H

Z m−1

X (l+1)h

eA(mh−s) B(Xs ) − eA(m−l)h B(Xlh ) dWsK + 9 · E PN

l=0 lh ! 2 Z s m−1

X Z (l+1)h

− eA(m−l)h B ′ (Xlh ) B(Xlh ) dWuK dWsK

lh l=0 lh H ! m−1

4 X 27R

2 + 3R (rN )2 + E Xlh − YlN,M,K H M l=0

and using kPN (v)kH ≤ kvkH for all v ∈ H additionally gives

2

E Xmh − YmN,M,K H

2

Z (l+1)h

m−1

X eA(mh−s) F (Xs ) − eA(m−l)h F (Xlh ) ds ≤ 9 · E

l=0 lh H

2

Z

m−1 (l+1)h

X eA(mh−s) B(Xs ) d Ws − WsK + 9 · E

l=0 lh H

Z (l+1)h

m−1 X

eA(mh−s) B(Xs ) − eA(m−l)h B(Xlh ) dWsK + 9 · E

l=0 lh

2 Z s m−1

X Z (l+1)h A(m−l)h ′ K K e B (Xlh ) B(Xlh ) dWu dWs −

lh l=0 lh H ! m−1

27R4 X

2 + 3R (rN )2 + E Xlh − YlN,M,K M H l=0

for every m ∈ {0, 1, . . . , M} and every N, M, K ∈ N. Therefore, (72) and

46

(73) yield

2

E Xmh − YmN,M,K H ≤

324R8 M min(4(γ−β),2γ)

+ 36R

5

sup µj j∈J \JK

2α

2 Z (l+1)h

m−1

X

A(mh−s) A(m−l)h K + 18 · E e −e B(Xs ) dWs

l=0 lh H

Z (l+1)h

m−1 X

eA(m−l)h (B(Xs ) − B(Xlh )) dWsK + 18 · E

l=0 lh

2 Z s m−1

X Z (l+1)h A(m−l)h ′ K K e B (Xlh ) B(Xlh ) dWu dWs −

lh l=0 lh H ! m−1

27R4 X

2 + 3R (rN )2 + E Xlh − YlN,M,K M H l=0

and (74) shows

2α

324R8 54R4 N,M,K 2 5

E Xmh − Ym ≤ sup + 36R µ + j H M min(4(γ−β),2γ) M (1+2δ) j∈J \JK

Z (l+1)h

m−1

X eA(m−l)h (B(Xs ) − B(Xlh )) dWsK + 18 · E

l=0 lh

2 Z s m−1

X Z (l+1)h

eA(m−l)h B ′ (Xlh ) B(Xlh ) dWuK dWsK −

lh l=0 lh H ! m−1

4 X 27R

2 + 3R (rN )2 + E Xlh − YlN,M,K H M l=0

for every m ∈ {0, 1, . . . , M} and every N, M, K ∈ N. The fact

B(Xs ) − B(Xlh ) = B ′ (Xlh )(Xs − Xlh ) Z 1 + B ′′ (Xlh + r(Xs − Xlh ))(Xs − Xlh , Xs − Xlh )(1 − r) dr 0

47

for every s ∈ [lh, (l +1)h], l ∈ {0, 1, . . . , M − 1} and every M ∈ N then yields 2α

324R8 54R4 N,M,K 2 5

≤ E Xmh − Ym sup + 36R µ + j H M min(4(γ−β),2γ) M (1+2δ) j∈J \JK

2 Z (l+1)h Z s

m−1

X

+ 36 · E eA(m−l)h B ′ (Xlh ) Xs − Xlh − B(Xlh ) dWuK dWsK

lh l=0 lh H

m−1 Z Z 1

X (l+1)h

eA(m−l)h B ′′ (Xlh + r (Xs − Xlh )) Xs − Xlh , + 36 · E

0 l=0 lh

2

K Xs − Xlh (1 − r) dr dWs

H ! m−1

4 X 2 27R

E Xlh − YlN,M,K + 3R (rN )2 + M H l=0

for every m ∈ {0, 1, . . . , M − 1} and every N, M, K ∈ N. Therefore, (75) and (76) give 2α 8

2

324R 54R4 N,M,K 11

≤ E Xmh − Ym sup + 756R µ + j H M min(4(γ−β),2γ) M (1+2δ) j∈J \JK +

720R13

+

36R6

M min(4(γ−β),2γ) M min(4(γ−β),2) ! m−1

2 4 X 27R

+ 3R (rN )2 + E Xlh − YlN,M,K M H l=0

and hence

2

1 E Xmh − YmN,M,K H ≤ 324R8 + 54R4 + 720R13 + 36R6 M min(4(γ−β),2γ) ! 2α m−1

2 27R4 X 2 N,M,K 11 E Xlh − Yl sup µj + 756R + 3R (rN ) +

H M j∈J \JK l=0

for every m ∈ {0, 1, . . . , M} and every N, M, K ∈ N. Gronwall’s lemma thus shows

2

(324R8 + 54R4 + 720R13 + 36R6 ) 4 E Xmh − YmN,M,K H ≤ e27R M min(4(γ−β),2γ) ! 2α sup µj + 756R11 + 3R (rN )2 j∈J \JK

48

and hence

2

E Xmh − YmN,M,K H

4

≤ 1134R13 e27R

(rN )2 +

sup µj j∈J \JK

2α

+ M − min(4(γ−β),2γ)

!

for every m ∈ {0, 1, . . . , M} and every N, M, K ∈ N. Finally, we obtain

2 21 E Xmh − YmN,M,K H ! −γ α 4 ≤ 34R7 e14R inf λi + sup µj + M − min(2(γ−β),γ) i∈I\IN

20R4

≤e

inf λi

i∈I\IN

−γ

j∈J \JK

+

sup µj j∈J \JK

α

+ M − min(2(γ−β),γ)

!

for every m ∈ {0, 1, . . . , M} and every N, M, K ∈ N.

6.1

Temporal discretization error: Proof of (72)

First of all, we have

Z (l+1)h

m−1

X A(mh−s) A(m−l)h e F (Xs ) − e F (Xlh ) ds

lh l=0

≤

L2 (Ω;H)

m−1 XZ l=0

(l+1)h

lh

A(mh−s)

e − eA(m−l)h F (Xs ) L2 (Ω;H) ds

Z (l+1)h

m−1 X

A(m−l)h e (F (Xs ) − F (Xlh )) ds +

lh l=0

L2 (Ω;H)

for every m ∈ {0, 1, . . . , M} and every M ∈ N. Using

F (Xs ) − F (Xlh ) = F ′ (Xlh ) (Xs − Xlh ) Z 1 + F ′′ (Xlh + r (Xs − Xlh )) (Xs − Xlh , Xs − Xlh ) (1 − r) dr 0

49

for all s ∈ [lh, (l + 1)h], l ∈ {0, 1, . . . , M − 1} and all M ∈ N then shows

Z (l+1)h

m−1

X

eA(mh−s) F (Xs ) − eA(m−l)h F (Xlh ) ds

2 lh l=0

≤

L (Ω;H)

m−1 X Z (l+1)h lh

l=0

A(mh−s)

e − eA(m−l)h L(H) kF (Xs )kL2 (Ω;H) ds

m−1 Z

X (l+1)h

eA(m−l)h F ′ (Xlh ) (Xs − Xlh ) ds +

lh l=0

+

L2 (Ω;H)

m−1 X Z (l+1)h l=0

lh

Z

1

0

kF ′′ (Xlh + r (Xs − Xlh )) (Xs − Xlh , Xs − Xlh )kL2 (Ω;H) (1 − r) dr ds

and hence

Z (l+1)h

m−1

X


lh l=0

L2 (Ω;H)

≤ R 2h +

m−2 X Z (l+1)h lh

A(mh−s)

e − eA(m−l)h

L(H)

l=0

Z (l+1)h

m−1 X

eA(m−l)h F ′ (Xlh ) (Xs − Xlh ) ds +

lh l=0

+

L2 (Ω;H)

m−1 X Z (l+1)h l=0

ds

!

lh

Z

1

0

2

R kXs − Xlh kVβ

L2 (Ω;R)

(1 − r) dr ds

for every m ∈ {0, 1, . . . , M} and every M ∈ N. Therefore, we obtain

Z (l+1)h

m−1 X

A(mh−s) A(m−l)h e F (Xs ) − e F (Xlh ) ds

2

lh l=0

≤ R 2h +

L (Ω;H)

m−2 X Z (l+1)h lh

−1 A(s−lh)

A(mh−s)

A

Ae

e − I ds L(H) L(H)

m−1 Z l=0

X (l+1)h

A(m−l)h ′ e F (Xlh ) (Xs − Xlh ) ds +

2

l=0 lh L (Ω;H) ! Z m−1 1 (l+1)h R X 2 + E kXs − Xlh k4Vβ ds 2 l=0 lh 50

!

and Lemma 1 gives

Z (l+1)h

m−1


2

l=0 lh L (Ω;H) ! Z m−2 (l+1)h X (s − lh) ≤ R 2h + ds (mh − s) l=0 lh m−1 X Z (l+1)h

eA(m−l)h F ′ (Xlh ) eA(s−lh) − I Xlh 2 ds + L (Ω;H) +

l=0 lh m−1 X Z (l+1)h

Z s

A(m−l)h ′

A(s−u)

e

F (X ) e F (X ) du lh u

lh

l=0 Z m−1

X (l+1)h

+

eA(m−l)h F ′ (Xlh )

lh

l=0

R + 2

lh

m−1 X Z (l+1)h l=0

lh

Z

s

eA(s−u) B(Xu ) dWu

lh

1 min(4(γ−β),2) 2 R (s − lh) ds

ds

L2 (Ω;H)

!

ds

L2 (Ω;H)

for every m ∈ {0, 1, . . . , M} and every M ∈ N. This shows

Z (l+1)h

m−1 X

A(mh−s) A(m−l)h e F (X ) − e F (X ) ds

s lh

2

l=0 lh L (Ω;H) ! Z m−2 X (l+1)h (s − lh) ≤ R 2h + ds (m − l − 1)h lh l=0 m−1 X Z (l+1)h

F ′ (Xlh ) eA(s−lh) − I Xlh 2 ds + L (Ω;H) +

l=0 lh m−1 X Z (l+1)h l=0

lh

Z s

′

A(s−u)

F (Xlh ) e F (Xu ) du

lh

ds

L2 (Ω;H)

1

Z Z s

(l+1)h

2  2

+ E eA(m−l)h F ′ (Xlh ) eA(s−u) B(Xu ) dWu ds 

lh

 lh l=0 H ! m−1 R2 X (1+min(2(γ−β),1)) + h 2 l=0  m−1 X

51

and

Z (l+1)h

m−1

X


2 l=0 lh L (Ω;H) ! m−2 X h 1 ≤ R 2h + + R2 T hmin(2(γ−β),1) 2(m − l − 1) 2 l=0 ! m−1 X Z (l+1)h

eA(s−lh) − I Xlh 2 ds +R L (Ω;H) l=0 lh m−1 X Z (l+1)h

+R

l=0

lh

(m−1 Z X √ + h

Z s

A(s−u)

e F (Xu ) du

lh

(l+1)h

lh

l=0

ds

L2 (Ω;H)

!

Z s 2 ) 12

′

E eA(s−u) B(Xu ) dWu

F (Xlh )

ds lh

H

for every m ∈ {0, 1, . . . , M} and every M ∈ N. Hence, we have

Z (l+1)h

m−1

X


2 l=0 lh L (Ω;H) !! m−1 h X1 1 ≤ R 2h + + R3 hmin(2(γ−β),1) 2 l=1 l 2 ! m−1 X Z (l+1)h −γ γ

(−A) eA(s−lh) − I L(H) k(−A) Xlh kL2 (Ω;H) ds +R l=0 lh m−1 X Z (l+1)h

+R

l=0

√

+R h

lh

(m−1 Z X l=0

Z

s

lh

(l+1)h

lh

kF (Xu )kL2 (Ω;H) du ds

!

Z s

2 ) 12

E eA(s−u) B(Xu ) dWu

ds lh

H

52

and Lemma 1 shows

Z (l+1)h

m−1


2

l=0 lh L (Ω;H) h 1 ≤ R 2h + (1 + ln(M)) + R4 M − min(2(γ−β),1) 2 2 ! Z m−1 (l+1)h X 1 γ 2 (s − lh) ds + R2 Mh2 +R 2 l=0 lh ) 12 (m−1 Z X (l+1)h Z s √

2 +R h E eA(s−u) B(Xu ) du ds l=0

lh

HS(U0 ,H)

lh

for every m ∈ {0, 1, . . . , M} and every M ∈ N. Therefore, we obtain

Z (l+1)h

m−1 X


2

l=0 lh L (Ω;H) 5 1 1 1 ≤ Rh + ln(M) + R4 M − min(2(γ−β),1) + R2 Mh(1+γ) + R3 h 2 2 2 2 1 ) (m−1 Z 2 X (l+1)h Z s √ E kB(Xu )k2HS(U0 ,H) du ds +R h l=0

lh

lh

and hence

m−1 Z

X (l+1)h


2

l=0 lh L (Ω;H) 1 1 5 1 + ln(M) + R4 M − min(2(γ−β),1) + R4 M −γ + R4 M −1 ≤ R2 M −1 2 2 2 2 ) 21 (m−1 Z

2

2 X (l+1)h Z s √

−δ du ds E (−A)δ B(Xu ) +R h

(−A) l=0

lh

lh

HS(U0 ,H)

L(H)

for every m ∈ {0, 1, . . . , M} and every M ∈ N. This yields

Z (l+1)h

m−1 X


2

lh l=0

L (Ω;H)

5 1 1 ≤ R2 M −1 (1 + ln(M)) + R4 M − min(2(γ−β),1) + R4 M −γ + R4 M −1 2 2 2 21 √ 1 2 2 Mh +R h 2 53

for every m ∈ {0, 1, . . . , M} and every M ∈ N. The estimate 1 + ln(x) = 1 +

Z

1

x

1 ds ≤ 1 + s

Z

1

x

s=x sr ds = 1 + s(1−r) r s=1 r xr (1 − r) xr (x − 1) = − ≤ =1+ r r r r 1

for every r ∈ (0, 1] and every x ∈ [1, ∞) then shows

Z (l+1)h

m−1


lh l=0

L2 (Ω;H)

√ 1 5 M (1−γ) R4 R4 R4 2 ≤ R2 + + + + R h (T h) 2 min(2(γ−β),1) γ 2 M (1 − γ) 2M M 2M 4 4 4 4 4 R R R R 5R + + γ + + ≤ γ min(2(γ−β),1) 2M 2M M 2M M

and finally

Z (l+1)h

m−1 X


2

l=0 lh L (Ω;H) 5 1 1 R4 6R4 ≤ + +1+ +1 ≤ min(2(γ−β),γ) 2 2 2 M min(2(γ−β),γ) M for every m ∈ {0, 1, . . . , M} and every M ∈ N.

6.2

Noise discretization error: Proof of (73)

In this subsection we have

Z t

2 A(t−u) K

E e B(Xu ) d Wu − Wu s H

2

X Z t

A(t−u)

e B(X )g dhg , W i = E u j j u U

j∈J \JK s

µj 6=0 Z t H X

2

A(t−u) µj B(Xu )gj H du E e = j∈J \JK µj 6=0

s

54

and hence

Z t

2 A(t−u) K

E e B(Xu ) d Wu − Wu s H Z t X

2

A(t−u) −α α µj B(Xu )Q (Q gj ) H du E e = s

j∈J \JK µj 6=0

X

=

(µj )

(1+2α)

t

Z

s

j∈J \JK µj 6=0

2 E eA(t−u) B(Xu )Q−α gj H du

for every s, t ∈ [0, T ] with s ≤ t and every K ∈ N. This shows

Z t

2 A(t−u) K E B(Xu ) d Wu − Wu

e

s H   2α Z t

A(t−u)

  X −α 2

e   µ E B(X )Q g ≤ sup µj du j u j H   j∈J \JK

≤

s

j∈J \JK µj 6=0

sup µj j∈J \JK

2α

X j∈J

µj

t

Z

s

and

2

E eA(t−u) B(Xu )Q−α gj H du

!

Z t

2 A(t−u) K B(Xu ) d Wu − Wu E

e s H 2α Z t

A(t−u)

2 −α ≤ sup µj B(Xu )Q HS(U0 ,H) du E e j∈J \JK

≤

s

sup µj

j∈J \JK

2α Z

t

s

(t − u)

−2ϑ

2

E (−A)−ϑ B(Xu )Q−α

HS(U0 ,H)

du

for every s, t ∈ [0, T ] with s ≤ t and every K ∈ N. Therefore, we obtain

Z t

2

A(t−u) K E e B(X ) d W − W u u u

s

≤c

2

H

≤ 2c

2

sup µj

j∈J \JK

2α Z

sup µj j∈J \JK

t

(t − u)

s

2α Z

s

t

−2ϑ

E

−2ϑ

(t − u) 55

1 + kXu kVγ

1+

E kXu k2Vγ

2

du

du

and

Z t

2 A(t−u) K

E e B(Xu ) d Wu − Wu s H 2α Z t −2ϑ 3 sup µj ≤ 4R (t − u) du j∈J \JK

= 4R3

sup µj j∈J \JK

s

2α

Z

(t−s)

u−2ϑ du

0

!

for every s, t ∈ [0, T ] with s ≤ t and every K ∈ N. Hence, we have

Z t

2 A(t−u) K E B(Xu ) d Wu − Wu

e

s H 2α (1−2ϑ) u=(t−s) u sup µj ≤ 4R3 (1 − 2ϑ) u=0 j∈J \JK 2α sup µj ≤ 4R4 (t − s)(1−2ϑ) j∈J \JK

and finally

Z t

2α

2 A(t−u) K 5 E B(Xu ) d Wu − Wu sup µj

e

≤ 4R s

(78)

j∈J \JK

H

for every s, t ∈ [0, T ] with s ≤ t and every K ∈ N. In particular, we obtain

2

Z (l+1)h

m−1

X A(mh−s) K e B(Xs ) d Ws − Ws E

l=0 lh H

Z mh

2α

2 A(mh−s) K 5

sup µj = E e B(Xs ) d Ws − Ws ≤ 4R 0

H

j∈J \JK

for every m ∈ {0, 1, . . . , M} and every M, K ∈ N which shows (73).

56

6.3


Here we have

2

Z (l+1)h

m−1 X

eA(mh−s) − eA(m−l)h B(Xs ) dWsK E

l=0 lh H Z m−1 (l+1)h X

2

E eA(mh−s) − eA(m−l)h B(Xs ) HS(U0 ,H) ds ≤ ≤

l=0 lh m−1 X Z (l+1)h l=0

2

−δ A(mh−s) A(m−l)h −e

(−A) e

lh

2

δ E (−A) B(Xs )

and hence

2

Z (l+1)h

m−1

X A(mh−s) A(m−l)h K e −e B(Xs ) dWs E

l=0 lh H m−1

X Z (l+1)h 2

−δ eA(mh−s) − eA(m−l)h ≤R

(−A) ≤R

+R

Z

L(H)

lh

l=0 mh

(m−1)h

2

−δ A(mh−s) Ah e −e

(−A)

L(H)

m−2 X Z (l+1)h l=0

lh

(−A)−1

ds

HS(U0 ,H)

L(H)

ds

!

ds

2

(1−δ) A(mh−s) (−A) e

L(H)

2 eA(s−lh) − I

ds

L(H)

for every m ∈ {0, 1, . . . , M} and every M, K ∈ N. Therefore, we obtain

2

m−1 Z

X (l+1)h


l=0 lh H Z mh

2

−δ ds eA(s−(m−1)h) − I ≤R

(−A) L(H) (m−1)h ! m−2

2 X Z (l+1)h

(1−δ) A(mh−s) ds e + Rh2

(−A)

l=0

L(H)

lh

57

!

and

2

Z (l+1)h

m−1

X A(mh−s) A(m−l)h K e −e B(Xs ) dWs E

l=0 lh H ! Z (l+1)h Z mh m−2 X (mh − s)2(δ−1) ds ≤R (s − (m − 1)h)2δ ds + Rh2 (m−1)h

≤ Rh(1+2δ) + Rh3

l=0

m−2 X l=0

lh

(m − l − 1)2(δ−1) h2(δ−1)

!

for every m ∈ {0, 1, . . . , M} and every M, K ∈ N. This shows

2

Z (l+1)h

m−1 X


l=0 lh H ! ! ∞ m−1 X X l2(δ−1) ≤ Rh(1+2δ) + Rh(1+2δ) l2(δ−1) ≤ Rh(1+2δ) 2 + l=2

l=1

Z (1+2δ) 2+ ≤ Rh

∞

1

s2(δ−1) ds

and finally

2

Z (l+1)h

m−1 X


l=0 lh (2δ−1) s=∞ H s 1 (1+2δ) (1+2δ) 2+ ≤ Rh 2+ = Rh (2δ − 1) s=1 (1 − 2δ) 4 3R ≤ 3R2 h(1+2δ) ≤ (1+2δ) M for every m ∈ {0, 1, . . . , M} and every M, K ∈ N.

58

6.4


We have

Z (l+1)h Z 1

m−1


lh 0 l=0

2

· (1 − r) dr dWsK

H

≤

m−1 X Z (l+1)hZ 1 l=0

lh

0

2

E B ′′ (Xlh +r (Xs −Xlh ))(Xs −Xlh , Xs −Xlh ) HS(U0 ,H) dr ds

for every m ∈ {0, 1, . . . , M} and every M, K ∈ N. Therefore, we obtain

Z (l+1)h Z 1

m−1


lh 0 l=0

≤

m−1 X Z (l+1)h

E

l=0

lh

2 2

ds = R2 R Xs − Xlh V β

2

K · (1 − r) dr dWs

H ! m−1 X Z (l+1)h

4 E Xs − Xlh ds l=0

Vβ

lh

and hence

Z (l+1)h Z 1

m−1

X A(m−l)h E e B ′′ (Xlh + r (Xs − Xlh )) (Xs − Xlh , Xs − Xlh )

lh 0 l=0

≤R

2

m−1 X Z (l+1)h l=0

lh

R (s − lh)

min(4(γ−β),2)

ds

!

2

· (1 − r) dr dWsK

!H m−1 X h(1+min(4(γ−β),2))

≤ R3

≤ R3 Mh(1+min(4(γ−β),2)) = R3 T hmin(4(γ−β),2) ≤

l=0 6

R

M min(4(γ−β),2)

for every m ∈ {0, 1, . . . , M} and every M, K ∈ N.

6.5


In order to show (76), we first estimate

2 Z t

K

X − X − E B(X ) dW t s s u

s

59

H

for all s, t ∈ [0, T ] with s ≤ t and all K ∈ N. More precisely, we have

2 Z t

A(t−s)

2 K

− I Xs H E B(Xs ) dWu

Xt − Xs −

≤5·E e s H

Z t

2

Z t

2 A(t−u) A(t−u) K

+ 5· E F (Xu ) du B(Xu ) d Wu − Wu

e

+ 5 · E e

s

s

H

H

Z t

2

Z t

2

K A(t−u) K

+5·E − I B(Xu ) dWu

e

+5·E (B(Xu ) − B(Xs )) dWu s

H

s

H

and using (78) shows

2 Z t

2 2 −γ A(t−s) K

≤ 5 (−A) e − I Ek(−A)γ Xs kH X −X − E B(X ) dW t s s u

L(H) s

+ 5 (t − s) Z

t

H

Z

t

s

2α

A(t−u)

2 5 sup µj E e F (Xu ) H du + 20R j∈J \JK

2

E eA(t−u) − I B(Xu ) HS(U0 ,H) du Zs t 2 +5 E kB(Xu ) − B(Xs )kHS(U0 ,H) du

+5

s


2

Z t

K

B(Xs ) dWu E Xt − Xs − s H Z t 2α 2γ 2 5 sup µj ≤ 5R (t − s) + 5 (t − s) E kF (Xu )kH du + 20R j∈J \JK

s

Z t

2

2

δ −δ A(t−u) du E (−A) B(Xu ) +5 e −I

(−A) HS(U0 ,H) L(H) s Z t 2 2 + 5R E kXu − Xs kH du s

60

and

2 Z t

K

E Xt − Xs − B(Xs ) dWu s

≤ 5R (t − s)

2γ

H

2

+ 5R (t − s) + 20R

t

5

sup µj j∈J \JK

2α

2

δ du +5 (t − u) E (−A) B(Xu ) HS(U0 ,H) s Z t

2

2 −β 2 E kXu − Xs kVβ du + 5R

(−A) Z

2δ

s

L(H)


2 Z t

K

≤ 5R (t − s)2γ + 5R (t − s)2 X − X − E B(X ) dW t s s u

s H 2α Z t 2δ 5 sup µj + 20R + 5R (t − u) du j∈J \JK

s

+ 5R

4

Z

t

s

E kXu −

Xs k2Vβ

du

and

2 2α Z t

2γ 3 5 K

sup µj E Xt − Xs − B(Xs ) dWu ≤ 10R (t − s) + 20R j∈J \JK s H Z t (1+2δ) 2 4 + 5R (t − s) + 5R E kXu − Xs kVβ du s


2

2α Z t

2γ 3 5 K

sup µj E Xt − Xs − B(Xs ) dWu ≤ 15R (t − s) + 20R j∈J \JK s H Z t 21 4 4 E kXu − Xs kVβ du + 5R s

and

2 2α Z t

2γ 3 5 K sup µj E B(Xs ) dWu

≤ 15R (t − s) + 20R

Xt − Xs − j∈J \JK s H Z t 1 min(4(γ−β),2) 2 4 du R (u − s) + 5R s

61

and hence

2 2α Z t

2γ 3 5 K sup µj E B(Xs ) dWu

Xt − Xs −

≤ 15R (t − s) + 20R j∈J \JK s H Z t min(2(γ−β),1) 5 + 5R (u − s) du s


2 Z t

K

E X − X − B(X ) dW t s s u

s H 2α 2γ 3 5 sup µj ≤ 15R (t − s) + 20R + 5R5 (t − s)(1+min(2(γ−β),1)) j∈J \JK

3

≤ 15R (t − s)

2γ

+ 20R

5

sup µj j∈J \JK

2α

+ 5R6 (t − s)min(4(γ−β),2)

and finally

2

Z t

K

B(Xs ) dWu E Xt − Xs − s

H

8

≤ 20R (t − s)

min(4(γ−β),2γ)

+ 20R

5

sup µj j∈J \JK

2α

(79)

for every s, t ∈ [0, T ] with s ≤ t and every K ∈ N. Now we prove (76). To this end note that

2

Z (l+1)h Z s

m−1 X

eA(m−l)h B ′ (Xlh ) Xs − Xlh − B(Xlh ) dWuK dWsK E

lh l=0 lh H

Z Z m−1 2 (l+1)h s X

′

ds E B(Xlh ) dWuK ≤

B (Xlh ) Xs − Xlh −

l=0

≤ R2

lh

lh

m−1 XZ l=0

HS(U0 ,H)

2 ! Z s (l+1)h

K E X − X − B(X ) dW s lh lh u

ds

lh

lh

62

H

holds for every m ∈ {0, 1, . . . , M} and every M, K ∈ N. Hence, (79) yields

2

Z (l+1)h Z s

m−1 X


lh l=0 lh ! ! H Z m−1 2α (l+1)h X (s − lh)min(4(γ−β),2γ) + sup µj ds ≤ 20R10 ≤ 20R

10

l=0 lh m−1 X Z (l+1)h l=0

lh

j∈J \JK

(s − lh)

min(4(γ−β),2γ)

ds

!

10

+ 20R T

sup µj j∈J \JK

2α

and therefore

2

m−1 Z Z s

X (l+1)h


lh l=0 lh H 2α sup µj ≤ 20R10 Mh(1+min(4(γ−β),2γ)) + 20R11 j∈J \JK

11 min(4(γ−β),2γ)

≤ 20R h

+ 20R

11

sup µj j∈J \JK

2α

and finally

2

Z (l+1)h Z s

m−1

X eA(m−l)h B ′ (Xlh ) Xs − Xlh − B(Xlh ) dWuK dWsK E

lh l=0 lh H 2α 13 20R ≤ min(4(γ−β),2γ) + 20R11 sup µj M j∈J \JK for every m ∈ {0, 1, . . . , M} and every M, K ∈ N.

63

6.6

Lipschitz estimates: Proof of (77)

2

N,M,K Before we estimate E Zm − YmN,M,K H for m ∈ {0, 1, . . . , M} and for N, M, K ∈ N, we need some preparations. More precisely, we have

2 Z s Z s

′

N,M,K N,M,K K ′ K

E B (Xlh ) B(Xlh ) dWu − B Yl B Yl dWu lh lh HS(U0 ,H)  

Z

 X s

′  B(Xlh ) gj dhgj , Wu iU  = E B (Xlh )  

lh j∈JK µj 6=0



2 Z

X s 

N,M,K N,M,K  ′  gj dhgj , Wu iU  B Yl − B Yl 



j∈JK µj 6=0

and hence

lh

HS(U0 ,H)

2

Z s Z s

′ N,M,K N,M,K K K ′

dWu E B (Xlh ) B Yl B(Xlh ) dWu − B Yl lh lh HS(U0 ,H)  

 X

 B(X ) g hg , W − W i = E B ′ (Xlh )  lh j j s lh U  



j∈JK µj 6=0



2

X 

N,M,K N,M,K ′   gj hgj , Ws − Wlh iU  B Yl − B Yl 

j∈JK µj 6=0

and

HS(U0 ,H)

2 Z s Z s

′

N,M,K N,M,K K K ′ E dWu B Yl B(Xlh ) dWu − B Yl

B (Xlh )

lh lh HS(U0 ,H)

X

B ′ (Xlh ) (B(Xlh ) gj ) = E

j∈JK µj 6=0

2

hgj , Ws − Wlh iU − B ′ YlN,M,K B YlN,M,K gj

HS(U0 ,H)

64

for every s ∈ [lh, (l + 1)h], l ∈ {0, 1, . . . , M − 1} and every M, K ∈ N. Moreover, we have

2

Z s Z s


dW B Y E B (X ) Y B(X ) dW − B lh lh u u l l

lh lh HS(U0 ,H)

X

E B ′ (Xlh ) (B(Xlh ) gj ) =

j∈JK µj 6=0

2

hgj , Ws − Wlh iU − B ′ YlN,M,K B YlN,M,K gj

and

HS(U0 ,H)

2

Z s Z s


B (X ) dW Y E B Y B(X ) dW − B lh lh u u l l

lh lh HS(U0 ,H) 2 X

′ E B (Xlh ) (B(Xlh ) gj ) − B ′ YlN,M,K B YlN,M,K gj = HS(U0 ,H)

j∈JK µj 6=0

2 · E hgj , Ws − Wlh iU

for every s ∈ [lh, (l + 1)h], l ∈ {0, 1, . . . , M − 1} and every M, K ∈ N. This shows

2

Z s Z s


dWu B Yl E B (Xlh ) B(Xlh ) dWu − B Yl lh

lh

=

X

j∈JK µj 6=0

HS(U0 ,H)

2

′ N,M,K N,M,K ′ gj B Yl µj · E B (Xlh )(B(Xlh ) gj ) − B Yl

HS(U0 ,H)

· (s − lh)

and therefore

2

Z s Z s

′ N,M,K N,M,K K ′ K

dW B Y E B (X ) Y B(X ) dW − B lh lh u u l l

lh

lh

≤

X

HS(U0 ,H)

2

µj µk E B ′ (Xlh )(B(Xlh ) gj ) gk − B ′ YlN,M,K B YlN,M,K gj gk

j,k∈J µj ,µk 6=0

· (s − lh)

65

H

for every s ∈ [lh, (l + 1)h], l ∈ {0, 1, . . . , M − 1} and every M, K ∈ N. Hence, we obtain

2

Z s Z s


dWu B Yl E B (Xlh ) B(Xlh ) dWu − B Yl lh lh HS(U0 ,H)

2

= (s − lh) · E B ′ (Xlh ) B(Xlh ) − B ′ YlN,M,K B YlN,M,K (2) HS

(U0 ,H)

and finally

2

Z s Z s

′ N,M,K N,M,K K K ′ dW B Y Y E B (X ) B(X ) dW − B lh lh u u l l

lh lh HS(U0 ,H)

2

2

(80) ≤ c2 · (s − lh) · E Xlh − YlN,M,K ≤ R3 · E Xlh − YlN,M,K H

H

for every s ∈ [lh, (l + 1)h], l ∈ {0, 1, . . . , M − 1} and every M, K ∈ N. Additionally, (69) shows

N,M,K

2 E Zm − YmN,M,K H

! 2 Z (l+1)h m−1

X

≤ 3 · E PN eA(m−l)h F (Xlh ) − F YlN,M,K ds

l=0 lh H

! 2 Z m−1

(l+1)h X

eA(m−l)h B(Xlh ) − B YlN,M,K dWsK + 3 · E PN

l=0 lh H

Z Z m−1

(l+1)h s X

eA(m−l)h B ′ (Xlh ) B(Xlh ) dWuK + 3 · E PN

lh l=0 lh ! 2

Z s

B YlN,M,K dWuK dWsK − B ′ YlN,M,K

lh

H

66

and hence

2

N,M,K E Zm − YmN,M,K H

m−1 Z

X (l+1)h 2

eA(m−l)h F (Xlh ) − F YlN,M,K ds ≤ 3 · E

l=0 lh H

2

Z m−1

X (l+1)h

eA(m−l)h B(Xlh ) − B YlN,M,K dWsK + 3 · E

l=0 lh H

Z (l+1)h Z s

m−1 X

eA(m−l)h B ′ (Xlh ) B(Xlh ) dWuK + 3 · E

lh l=0 lh

2

Z s N,M,K N,M,K K K ′ dWu dWs B Yl − B Yl

lh

H

for every m ∈ {0, 1, . . . , M} and every N, M, K ∈ N. Moreover, we have

2

N,M,K − YmN,M,K H E Zm ! m−1 2 X

N,M,K E eA(m−l)h F (Xlh ) − F Yl ≤ 3Mh2

+3

+3

l=0 m−1 X Z (l+1)h

l=0 lh m−1 X Z (l+1)h lh

l=0

H

2

E eA(m−l)h B(Xlh ) − B YlN,M,K

HS(U0 ,H)

ds

!

Z s

′

E B (Xlh ) B(Xlh ) dWuK lh

2 Z s

N,M,K N,M,K K ′ dWu B Yl − B Yl lh

ds

HS(U0 ,H)

and due to (80) we obtain

m−1 2 X

N,M,K

N,M,K N,M,K 2

E F (Xlh ) − F Yl − Ym ≤ 3T h E Zm

H

H

l=0

m−1 X

+ 3h

3

+ 3R h

2

N,M,K E B(Xlh ) − B Yl

l=0 m−1 X l=0

HS(U0 ,H)

2

N,M,K E Xlh − Yl

H

67

!

!

!

!

for every m ∈ {0, 1, . . . , M} and every N, M, K ∈ N. Finally, we obtain ! m−1

2 X

N,M,K

2

E Xlh − Y N,M,K E Z − Y N,M,K ≤ 9R3 h m

m

l

H

9R4 ≤ M

l=0 m−1 X l=0

H

2

E Xlh − YlN,M,K

H

!

for every m ∈ {0, 1, . . . , M} and every N, M, K ∈ N.

6.7

Iterated integral identity: Proof of (65)

First of all, we have Z Z (m+1)T M ′ N,M,K B Ym mT M

X Z

=

j,k∈JK µj ,µk 6=0

X

=

B

(m+1)T M

B

′

mT M

′

YmN,M,K

j,k∈JK µj ,µk 6=0

s

B mT M

YmN,M,K

mT M

=

B

′

YmN,M,K

j∈JK µj 6=0

+

X

j,k∈JK µj ,µk 6=0 j6=k

B

′

YmN,M,K

Z

s

B mT M

dWuK

B mT M

B

YmN,M,K

gk gj ·

gk dhgk , Wu iU gj dhgj , Ws iU Z

mT M

dWuK

Z N,M,K B Ym gj gj ·

(m+1)T M

s

YmN,M,K

dWsK

YmN,M,K

Z N,M,K B Ym gk gj ·

P-a.s. and hence Z Z (m+1)T M ′ N,M,K B Ym X

YmN,M,K

Z

Z

dhgk , Wu iU dhgj , Ws iU

s

dhgj , Wu iU dhgj , Ws iU

mT M

(m+1)T M mT M

mT M

dWsK

(m+1)T M mT M

s

Z

s mT M

dhgk , Wu iU dhgj , Ws iU

P-a.s. for all m ∈ {0, 1, . . . , M − 1} and all N, M, K ∈ N. Moreover, since the bilinear operator B ′ YmN,M,K B YmN,M,K ∈ HS (2) (U0 , H) is symmetric (see Assumption 3) and since Z (m+1)T Z s M 2 T µj 1

M,K gj , ∆Wm − dhgj , Wu iU dhgj , Ws iU = U mT mT 2 M M M 68

P-a.s. holds for all j ∈ JK , m ∈ {0, 1, . . . , M − 1} and all N, M, K ∈ N (see (3.6) in Section 10.3 in [36]), we obtain Z

(m+1)T M

B

′

YmN,M,K

mT M

Z

s

B mT M

YmN,M,K

dWuK

dWsK

2 T µj 1 X ′ N,M,K N,M,K M,K B Ym B Ym gj gj gj , ∆Wm = − U 2 j∈J M K

µj 6=0

1 X ′ N,M,K N,M,K + B Ym B Ym gk gj 2 j,k∈J K

µj ,µk 6=0 j6=k

Z

·

(m+1)T M

Z

mT M

s mT M

dhgk , Wu iU dhgj , Ws iU +

Z

(m+1)T M

Z

mT M

s mT M

dhgj , Wu iU dhgk , Ws iU

!

P-a.s. for all m ∈ {0, 1, . . . , M − 1} and all N, M, K ∈ N. The fact Z

(m+1)T M

Z

mT M

s mT M

dhgk , Wu iU dhgj , Ws iU +

Z

(m+1)T M mT M

Z

s mT M

dhgj , Wu iU dhgk , Ws iU

= gk , ∆WmM,K U gj , ∆WmM,K U

P-a.s.

for all j ∈ JK , m ∈ {0, 1, . . . , M − 1} and all N, M, K ∈ N (see (3.15) in Section 10.3 in [36]) then yields Z

(m+1)T M

B

′

mT M

YmN,M,K

Z

s

B mT M

YmN,M,K

dWuK

dWsK

1 X ′ N,M,K N,M,K B Ym B Ym gk gj gk , ∆WmM,K U gj , ∆WmM,K U = 2 j,k∈J K

µj ,µk 6=0

T X ′ N,M,K N,M,K µj B Y m B Ym gj gj − 2M j∈J K

µj 6=0

1 ′ N,M,K N,M,K M,K = B Ym ∆WmM,K B Ym ∆Wm 2 T X N,M,K ′ N,M,K − B Ym gj gj µj B Y m 2M j∈J

P-a.s.

K

µj 6=0

for all m ∈ {0, 1, . . . , M − 1} and all N, M, K ∈ N which shows (65). 69

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