Abstract
Higher order numerical schemes for stochastic partial differential equations that do not possess commutative noise require the simulation of iterated stochastic integrals. In this work, we extend the algorithms derived by Kloeden et al. (Stoch Anal Appl 10(4):431–441, 1992. https://doi.org/10.1080/07362999208809281) and by Wiktorsson (Ann Appl Probab 11(2):470–487, 2001. https://doi.org/10.1214/aoap/1015345301) for the approximation of two-times iterated stochastic integrals involved in numerical schemes for finite dimensional stochastic ordinary differential equations to an infinite dimensional setting. These methods clear the way for new types of approximation schemes for SPDEs without commutative noise. Precisely, we analyze two algorithms to approximate two-times iterated integrals with respect to an infinite dimensional Q-Wiener process in case of a trace class operator Q given the increments of the Q-Wiener process. Error estimates in the mean-square sense are derived and discussed for both methods. In contrast to the finite dimensional setting, which is contained as a special case, the optimal approximation algorithm cannot be uniquely determined but is dependent on the covariance operator Q. This difference arises as the stochastic process is of infinite dimension.
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1 Introduction
In order to obtain higher convergence rates for numerical schemes for stochastic differential equations, in general, we need to incorporate the information contained in iterated integrals. However, usually these integrals cannot be simulated directly. Therefore, we need to replace these terms by an approximation. We illustrate this statement in a finite dimensional setting first, although, we are concerned about the approximation of iterated Itô integrals in infinite dimensions in this work.
The numerical approximation of stochastic ordinary differential equations (SDEs) that do not possess commutative noise requires the simulation of iterated stochastic integrals to achieve a high order of convergence, see [3, 5, 9]. One example of such a higher order scheme is the Milstein scheme developed in [7], which we present below to illustrate the issue. For some fixed \(d, K\in \mathbb {N}\), we consider a d-dimensional SDE of type
with functions \(a :\mathbb {R}^d \rightarrow \mathbb {R}^d\), \(b^j=(b^{1,j},\ldots ,b^{d,j})^T :\mathbb {R}^d \rightarrow \mathbb {R}^d\), \(j\in \{1,\ldots ,K\}\) for all \(t\ge 0\) and initial value \(X_0=x_0 \in \mathbb {R}^d\). Moreover, \((\beta ^j_t)_{t\ge 0}\), \(j \in \{1,\ldots ,K\}\) denote independent real-valued Brownian motions. For some \(T>0\), we divide the time interval [0, T] into \(M\in \mathbb {N}\) equal time steps \(h = \frac{T}{M}\) and denote \(t_m =mh\) for \(m\in \{0,\ldots ,M\}\). The increments of the Brownian motion are given as \(\varDelta \beta ^j_m = \beta ^j_{t_{m+1}}-\beta ^j_{t_m}\) for all \(j\in \{1,\ldots ,K\}\) and \(m\in \{0,\ldots ,M-1\}\). Then, the Milstein scheme [7] reads as \(Y_0=x_0\) and
for \(m\in \{0,\ldots ,M-1\}\) using the notation \(Y_m=Y_{t_m}\). Under suitable assumptions, the following error estimate holds
see [3]. If SDE (1) does not possess commutative noise, see [3] for details, the Milstein scheme cannot be simplified and one has to approximate the iterated stochastic integrals involved in the method. We denote these iterated Itô integrals by
for some \(t\ge 0\), \(h>0\), and for all \(i,j \in \{1,\ldots ,K\}\), where \(K\in \mathbb {N}\) is the number of independent Brownian motions driving the SDE. The research by Kloeden, Platen and Wright [4] and by Wiktorsson [9] suggests different methods for an approximation of these integrals; the main ideas are outlined in Sect. 2. We denote by \(\bar{I}^{(D)}_{(i,j)}(h)\) the approximation of \(I_{(i,j)}(h)\) with the algorithm derived in [4] for \(i,j \in \{1,\ldots ,K\}\), \(D,K\in \mathbb {N}\), \(h>0\). In [4], the authors proved that for all \(i,j \in \{1,\ldots ,K\}\) and \(h>0\), it holds
where \(D\in \mathbb {N}\) denotes the index of the summand at which the series representation of the stochastic double integral is truncated to obtain the approximation \(\bar{I}^{(D)}_{(i,j)}(h)\). If we use the algorithm derived in [9] instead, we denote the approximation of \({I}_{(i,j)}(h)\) by \(\hat{I}^{(D)}_{(i,j)}(h)\) for all \(i,j \in \{1,\ldots ,K\}\), \(h>0\). This scheme employs the same series representation as proposed in [4] but incorporates an approximation of the truncated term additionally. The error resulting from this scheme is estimated as
where D is again the index of the summand at which the series is truncated to obtain the approximation and K is the number of independent Brownian motions, see [9]. Let h and K be fixed, then both approximations converge in the mean-square sense as D goes to infinity—with a different order of convergence, however. For the numerical approximation of SDEs, the integer D is determined such that the overall order of convergence in the time step is not distorted. For the Milstein scheme, for example, error estimate (2) is considered, that is, a strong order of convergence of 1 can be achieved. Therefore, \(D\ge \frac{C}{h}\) is chosen for the method derived in [4], whereas \(D\ge \frac{\sqrt{5K^2(K-1)}}{\sqrt{24\pi ^2h}}\) is selected for the algorithm developed in [9], see also [3, Cor. 10.6.5]. This shows that if we decrease the step size h, the value for D has to increase faster for the scheme developed in [4]. Note that the error estimate (4) depends on the number of Brownian motions K as well. As this number is fixed in the setting of finite dimensional SDEs, this factor is not crucial but simply a constant. Therefore, the algorithm proposed by Wiktorsson [9] is superior to the one derived in [4] in terms of the computational effort when a given order of convergence in the step size h is to be achieved.
The same issue arises in the context of higher order numerical schemes designed for infinite dimensional stochastic differential equations that need not have commutative noise—we also have to approximate the involved iterated stochastic integrals in order to implement the scheme. This time, however, the stochastic process is infinite dimensional, in general. In this work, we aim at devising numerical algorithms for the simulation of iterated integrals which arise, for example, in the approximation of the mild solution of stochastic partial differential equations (SPDEs) of type
where the commutativity condition from [2]
for all \(y\in H_{\beta }\), \(u,v\in U_0\) is not assumed to hold. Here, \(H_{\beta }=\mathcal {D}((-A)^{\beta })\) denotes the domain of the operator \((-A)^{\beta }\) which is a separable Hilbert space for some \(\beta \in [0,1)\). The operators A, F, B, and the initial value \(\xi \) are assumed to fulfill the conditions imposed for the existence of a unique mild solution, see [1], and are not specified further. The spaces are introduced in Sect. 2 and \((W_t)_{t\ge 0}\) denotes a Q-Wiener process taking values in some separable Hilbert space U for some trace class operator Q. In order to approximate the mild solution of SPDEs of type (5) with a higher order scheme, we need to simulate iterated stochastic integrals of the form
for \(t \ge 0\), \(h>0\), and some operators \(\varPsi \), \(\varPhi \) specified in Sect. 2. These terms arise if condition (6) is not fulfilled, for example, in the Milstein scheme for SPDEs [2]. In the Milstein scheme, it holds \(\varPsi = B'(Y_t)\) and \(\varPhi = B(Y_t)\) for some \(B :H \rightarrow L_{HS}(U_0,H)\) and an approximation \(Y_t \in H_{\beta }\) with \(t\ge 0\) and \(\beta \in [0,1)\), where \(L_{HS}(U_0,H)\) denotes the space of Hilbert–Schmidt operators from \(U_0\) to H. For more details, we refer to [2].
We want to emphasize that the algorithms developed for the approximation of iterated stochastic integrals in the setting of SDEs are designed for some fixed finite number K of driving Brownian motions and that the approximation error (4) even involves this number K as a constant. In contrast, when approximating the solution of SPDEs driven by an infinite dimensional Q-Wiener process, this number corresponds to the dimension of the finite-dimensional approximation subspace onto which the Q-Wiener process is projected. Thus, the dimension K of the approximation subspace has to increase, in general, to attain higher accuracy, i.e., K is not constant anymore; see the error estimate of the Milstein scheme for SPDEs in [2], for example. Therefore, this aspect has to be taken into account in order to identify an appropriate approximation algorithm. In the following, we derive two algorithms for the approximation of iterated integrals of type (7) based on the methods developed for the finite dimensional setting by Kloeden, Platen, and Wright [4] and by Wiktorsson [9] as well as on [5] for the infinite dimensional case. These algorithms allow for the first time to implement higher order schemes for SPDEs that do not possess commutative noise and include the algorithms that can be used for finite dimensional SDEs as a special case. We show that the method that is superior in the setting of an infinite dimensional Q-Wiener process cannot be uniquely determined in general but is dependent on the covariance operator Q. In the analysis of the approximation error, we need to incorporate the eigenvalues of the covariance operator Q. For the algorithm based on the approach by Kloeden, Platen, and Wright [4], we obtain a similar estimate as in (3) in the mean-square sense, see Corollary 1. For the method derived in the work of Wiktorsson [9], we can prove two differing error estimates for the case of infinite dimensional Q-Wiener processes by different means. One is the same, apart from constants, as estimate (4). Moreover, the fact that we integrate with respect to a Q-Wiener process with a trace class operator Q allows for an alternative proof to the one given in [9]. The result indicates a possibly superior convergence in K—this depends on the rate of decay of the eigenvalues of Q. Details can be found in Theorem 3 and Theorem 4.
2 Approximation of iterated stochastic integrals
Throughout this work, we fix the following setting and notation. Let H and U be separable real-valued Hilbert spaces. In the following, let \((\varOmega ,\mathcal {F},P,(\mathcal {F}_t)_{t\ge 0})\) be a probability space, let \((W_t)_{t \ge 0}\) denote a U-valued Q-Wiener process with respect to \((\mathcal {F}_t)_{t\ge 0}\) where Q is a trace class covariance operator, and let \(U_0 := Q^{{1}/{2}}U\). Let L(U, H) denote the space of linear and bounded operators mapping from U to H. We define \(L(U,H)_{U_0} :=\{T|_{U_0} : T\in L(U,H)\}\) which is a dense subset of the space of Hilbert–Schmidt operators \(L_{HS}(U_0,H)\) [8]. Moreover, we assume that the operators \(\varPhi \) and \(\varPsi \) in (7) fulfill
-
(A1)
\(\varPhi \in L(U,H)_{U_0}\) with \(\Vert \varPhi Q^{-\alpha }\Vert _{L_{HS}(U_0,H)}<C\),
-
(A2)
\(\varPsi \in L(H,L(Q^{-\alpha }U,H)_{U_0})\)
for some \(\alpha \in (0,\infty )\). The parameter \(\alpha \) determines the rate of convergence for the approximation of the Q-Wiener process, see Theorems 1 or 3. Note that assumption (A1), needed to prove the convergence of the approximation algorithms for iterated integrals in Theorems 1, 3, and 4, is less restrictive than the condition imposed on the operator B in SPDE (5) to obtain the error estimate for some numerical scheme to approximate its mild solution, e.g., in [6]. However, for the Milstein scheme in [2], assumption (A2) does not need to be fulfilled for the error analysis of the Milstein scheme to hold true.
If we are interested in the approximation of the mild solution of (5), a combination of the error estimate for a numerical scheme to obtain this process and the error from the approximation of the iterated integrals has to be analyzed. In this case, we impose the following assumptions instead
-
(B1)
\(\varPhi \in L(U,H)_{U_0}\),
-
(B2)
\(\varPsi \in L(H,L(U,H)_{U_0})\).
For the convergence results in this case, we refer to Corollary 1 and 2, which have to be combined with estimates on the respective numerical scheme. These weaker conditions are sufficient as in the proof the Q-Wiener process is approximated first and then the iterated integral is compared to the approximation.
Let \(Q \in L(U)\) be a nonnegative and symmetric trace class operator with eigenvalues \(\eta _j\) and corresponding eigenfunctions \(\tilde{e}_j\) for \(j \in \mathcal {J}\) where \(\mathcal {J}\) is some countable index set. The eigenfunctions \(\{\tilde{e}_j : j\in \mathcal {J}\}\) constitute an orthonormal basis of U, see [8, Prop. 2.1.5]. Then, for the Q-Wiener process \((W_t)_{t \ge 0}\), the following series representation holds, see [8, Prop. 2.1.10],
As the Q-Wiener process \((W_t)_{t\ge 0}\) is an infinite dimensional stochastic process, it has to be projected to some finite dimensional subspace by truncating the series (8) such that it can be simulated in a numerical scheme. For \(K \in \mathbb {N}\), we denote by \((W_t^K)_{t\ge 0}\) the projected Q-Wiener process, which is defined as
for some finite index set \(\mathcal {J}_K \subset \mathcal {J}\) with \(|\mathcal {J}_K|=K\). This expression allows to write the iterated integral with respect to the projected Q-Wiener process \((W_t^K)_{t\ge 0}\) for any \(t\ge 0\) and \(h>0\) as
with
for \(i,j \in \mathcal {J}_K\). Therefore, we aim at devising a method to approximate the iterated stochastic integrals \(I^Q_{(i,j)}(t,t+h)\) for all \(i,j\in \mathcal {J}_K\). Below, we introduce two such algorithms and analyze as well as discuss their convergence properties. For simplicity of notation, we assume, without loss of generality, \(\mathcal {J}_K = \{1,\ldots ,K\}\) with \(\eta _j \ne 0\) for \(j \in \mathcal {J}_K\) and denote \(I^Q_{(i,j)}(h)=I^Q_{(i,j)}(t,t+h)\).
2.1 Algorithm 1
In the following, we mainly adapt the method introduced by Kloeden, Platen, and Wright [4] to the setting of infinite dimensional stochastic processes. Here, we additionally have to take into account the error arising from the projection of the Q-Wiener process to a finite dimensional subspace.
For some \(t \ge 0\), the coefficients of the Q-Wiener process \(w_{t}^j := \langle W_t, \tilde{e}_j \rangle _U\) are independent real-valued random variables that are \(N(0, \eta _j \, t)\) distributed for \(j \in \mathcal {J}\). Thus, the increments \(\varDelta w_{h}^j := \langle W_{t+h}-W_t, \tilde{e}_j \rangle _U\) can be easily simulated since \(\varDelta w_{h}^j\) is \(N(0, \eta _j \, h)\) distributed for \(j \in \mathcal {J}\) and \(h \ge 0\). Our goal is to obtain an approximation of the iterated integrals \(I^Q_{(i,j)}(h)\) for all \(i,j \in \mathcal {J}_K\), \(K\in \mathbb {N}\), \(h>0\) given the realizations of the increments \(\varDelta w_{h}^j\) for \(j \in \mathcal {J}_K\). The following derivation of the approximation method follows the representation in [4] closely. Below, let \(K \in \mathbb {N}\) be arbitrarily fixed and let us introduce the scaled Brownian bridge process \((w_s^j-\frac{s}{h}w_h^j)_{0\le s\le h}\) for \(j \in \mathcal {J}_K\) and some \(h \in (0,T]\). We consider its series expansion
which converges in \(L^2(\varOmega )\). The coefficients are given by the following expressions
for all \(j \in \mathcal {J}_K\), \(r \in \mathbb {N}_0\), and all \(0\le s\le h \le T\), see also [4]. All coefficients \(a^j_r\) and \(b^j_r\) are independent and \(N(0,\tfrac{\eta _j h}{2 \pi ^2 r^2})\) distributed for \(r \in \mathbb {N}\) and \(j \in \mathcal {J}_K\) and it holds \(a_0^j = -2 \sum _{r=1}^{\infty } a_r^j\). In contrast to [4], the distributions of the coefficients additionally depend on the eigenvalues \(\eta _j\) of the covariance operator Q. In order to obtain an approximation of the scaled Brownian motion \((w_s^j)_{0\le s\le h}\) for some \(h \in (0,T]\), we truncate expression (10) at some integer \(R\in \mathbb {N}\) and define
In fact, we are interested in the integration with respect to this process. According to Wong and Zakai [10, 11], or [3, Ch. 6.1], an integral with respect to process (11) converges to a Stratonovich integral J(h) as \(R\rightarrow \infty \). We are, however, interested in the Itô stochastic integral. Following [3, p. 174], the Stratonovich integral \(J^Q_{(i,j)}(h)\) can be converted to an Itô integral \(I^Q_{(i,j)}(h)\), \(i,j \in \mathcal {J}_K\) according to
That is, \({I}^Q_{(i,j)}(h) = {J}^Q_{(i,j)}(h)\) for all \(i,j \in \mathcal {J}_K\) with \(i \ne j\). Moreover, we compute
directly for \( i \in \mathcal {J}_K\), see [3, p. 171]. This implies that we only have to approximate \({I}^Q_{(i,j)}(h)\) for \(i,j \in \mathcal {J}_K\) with \(i \ne j\). Thus, we obtain the desired approximation of the Itô stochastic integral directly by integrating with respect to process (11). Without loss of generality, let \(t=0\). By (10) and the fact that \(\int _0^h f(u) \, \mathrm {d} w_u^j = f(h) w_h^j - \int _0^h f'(u) w_u^j \, \mathrm {d}u\) for a continuously differentiable function \(f :[0,h] \rightarrow \mathbb {R}\), \(h > 0\), see [3, p. 89], we obtain the following expression for the iterated stochastic integrals for all \(i,j \in \mathcal {J}_K\), \(i \ne j\), and \(h>0\)
The expression \(a_0^j = \frac{2}{h} \int _0^h w_u^j \, \mathrm {d}u -w_h^j\) and the definition of \(a_r^j\), \(b_r^j\), \(r\in \mathbb {N}\) as well as in the last step \(a_0^j = -2 \sum _{r=1}^{\infty } a_r^j\), \(j \in \mathcal {J}_K\) yield
for all \(i,j \in \mathcal {J}_K\), \(i \ne j\), and \(h>0\). Expression (12) involves some scaled Lévy stochastic area integrals which are defined as
for all \( i,j \in \mathcal {J}_K\), \(i \ne j\), \(h>0\). We approximate these terms instead of the iterated stochastic integrals, as proposed in [4] and [9]. Due to the relations
P-a.s. for all \(i,j \in \mathcal {J}_K\), \(h>0\), see [9], it is sufficient to simulate \(A_{(i,j)}^Q(h)\) for \(i,j \in \mathcal {J}_K\) with \(i<j\). By the distributional properties of \(a_r^i\) and \(b_r^i\) for \(r \in \mathbb {N}_0\), \(i \in \mathcal {J}_K\), we write
for all \( i,j \in \mathcal {J}_K\), \(i \ne j\), \(h>0\) and \(A^Q(h) = \big ( A_{(i,j)}^Q(h)\big )_{1 \le i, j \le K}\) in order to relate to the derivation in [9]. This representation entails the random variables \(U_{ri}^Q \sim N(0,\eta _i)\), \(Z_{ri}^Q \sim N(0,\eta _i)\), and \(\varDelta w_h^i \sim N(0,\eta _i \, h)\) that are all independent for \(i \in \mathcal {J}_K\), \(r \in \mathbb {N}\). As described above, we only need to approximate \(A_{(i,j)}^{Q}(h)\), \(h>0\) for \(i, j \in \mathcal {J}_K\) with \(i <j \), that is, we want to simulate
Therefore, we write
and introduce the selection matrix
which defines the integrals that have to be computed, compare to [9]. Note that \(I_K\) represents the identity matrix of size \(K\times K\). Further, we define the matrix
This allows to express the vector \(\tilde{A}^Q(h)\) as
with \(\varDelta w_h^Q =(\varDelta w_h^1, \dots , \varDelta w_h^K)^T\) and the random vectors \(U_r^Q = (U_{r1}^Q, \ldots , U_{rK}^Q)^T\) and \(Z_r^Q = (Z_{r1}^Q, \ldots , Z_{rK}^Q)^T\) that are independent and identically \(N(0_K,Q_K)\) distributed for all \(r \in \mathbb {N}\). As expression (18) contains an infinite sum, we need to truncate it in order to compute this vector. For some \(D \in \mathbb {N}\), this approximation is denoted as
and we specify the remainder
Let \(A^I(h) = Q_K^{-{1}/{2}} A^Q(h) Q_K^{-{1}/{2}}\) denote the matrix containing the standard Lévy stochastic area integrals that correspond to the case that \(Q_K=I_K\), i.e., \(\eta _j=1\) for all \(j \in \mathcal {J}_K\). Therewith, we obtain the relationship
where \(\tilde{A}^I(h) := H_K \text {vec}(A^I(h)^T)\) and where we employed
and the fact that we are interested in indices \(i,j \in \mathcal {J}_K\) with \(i<j\) only. We denote
which is of size \(L \times L\) with \(L = \frac{K(K-1)}{2}\), such that the vector of interest is given by
Now, we can represent the approximation \(\tilde{A}^{Q,(D)}(h)\) of \(\tilde{A}^{Q}(h)\) as
and the vector of truncation errors by \(\tilde{R}^{Q,(D)}(h) = \tilde{Q}_K \tilde{R}^{I,(D)}(h)\) where \(\tilde{A}^{I,(D)}(h)\) and \(\tilde{R}^{I,(D)}(h)\) denote, in analogy to (19) and (20), the truncated part of \(\tilde{A}^I(h)\) and its truncation error, respectively. Note that \(\tilde{A}^I(h)\) and especially \(\tilde{A}^{I,(D)}(h)\) correspond to the case where \(\eta _j=1\) for all \(j \in \mathcal {J}_K\), i.e., \(Q_K=I_K\) in (18) and (19), respectively. This also relates to the setting in [4] if \(\mathcal {J}\) is finite.
We summarize the representation above to formulate Algorithm 1 for some \(h>0\), \(t,t+h \in [0,T]\), and \(D,K\in \mathbb {N}\):
-
1.
For \(j \in \mathcal {J}_K\), simulate the Fourier coefficients \(\varDelta w_{h}^j = \langle W_{t+h}-W_t, \tilde{e}_j \rangle _U\)of the increment \(W_{t+h}-W_t\)with\(\varDelta w_h^Q = \big ( \varDelta w^1_h, \ldots , \varDelta w^K_h \big )^T\)as
$$\begin{aligned} \varDelta w_h^Q = \sqrt{h} \, Q_K^{{1}/{2}} V, \end{aligned}$$where \(V \sim N(0_K,I_K)\).
-
2.
Approximate \(\tilde{A}^Q(h)\)as
$$\begin{aligned} \tilde{A}^{Q,(D)}(h)=&H_K \big ( Q_K^{{1}/{2}} \otimes Q_K^{{1}/{2}} \big ) H_K^T\\&\times \frac{h}{2\pi } \sum _{r=1}^{D} \frac{1}{r} H_K \Big (U_{r} \otimes \Big ( Z_{r} -\sqrt{2} V \Big ) -\Big (Z_{r} - \sqrt{2} V \Big ) \otimes U_r \Big ), \end{aligned}$$where \(U_r, Z_r \sim N(0_K, I_K)\)are independent.
-
2.
Compute the approximation \( \text {vec}((\bar{I}^{Q,(D)}(h))^T)\)of\( \text {vec}((I^Q(h)^T)\)as
$$\begin{aligned} \text {vec}((\bar{I}^{Q,(D)}(h))^T) = \frac{\varDelta w_h^Q \otimes \varDelta w_h^Q - \text {vec}(h \, Q_K)}{2} + (I_{K^2} - S_K) H_K^T \tilde{A}^{Q,(D)}(h) \end{aligned}$$with \(S_K := \sum _{i=1}^K \mathbf {e}_i^T \otimes (I_K \otimes \mathbf {e}_i)\), where \(\mathbf {e}_i\)denotes the i-th unit vector.
We obtain the following error estimate for this approximation method; the mean-square error converges with order 1 / 2 in D while the convergence in K is determined by the operator Q. The first term results from the approximation of the Q-Wiener process by \((W_t^K)_{t\ge 0}\), whereas the second term is due to the approximation of the iterated integral with respect to this truncated process by Algorithm 1.
Theorem 1
(Convergence result for Algorithm 1) Assume that the trace class operator \(Q\in L(U)\) is nonnegative and symmetric with eigenvalues \(\{ \eta _j : j \in \mathcal {J}\}\). Further, let \(\varPhi \in L(U,H)_{U_0}\) with \(\Vert \varPhi Q^{-\alpha }\Vert _{L_{HS}(U_0,H)}<C\) for some \(C>0\), let \(\varPsi \in L(H,L(Q^{-\alpha }U,H)_{U_0})\) for some \(\alpha \in (0,\infty )\), i.e., (A1) and (A2) are fulfilled, and let \((W_t)_{t \ge 0}\) be a Q-Wiener process. Then, it holds
for some \(C>0\) and all \(h>0\), \(t,t+h\in [0,T]\), \(D, K \in \mathbb {N}\), and \(\mathcal {J}_K \subset \mathcal {J}\) with \(|\mathcal {J}_K|=K\).
Proof
For a proof, we refer to Sect. 3. \(\square \)
Note that in the convergence analysis of numerical schemes for SPDEs, we compare the approximation of the iterated stochastic integrals to integrals with respect to \((W_t^K)_{t\ge 0}\), \(K\in \mathbb {N}\), see the proofs in [2] and [6], for example. In this case, the analysis involves the error estimate stated in Corollary 1 below. We want to emphasize that this estimate is independent of the integer K.
Corollary 1
Assume that Q is a nonnegative and symmetric trace class operator and \((W_t)_{t \ge 0}\) is a Q-Wiener process. Furthermore, let \(\varPhi \in L(U,H)_{U_0}\) and \(\varPsi \in L(H,L(U,H)_{U_0})\), i.e., assumptions (B1) and (B2) are fulfilled. Then, it holds
for some \(C>0\) and all \(h>0\), \(t,t+h\in [0,T]\), \(D,K\in \mathbb {N}\), and \(\mathcal {J}_K \subset \mathcal {J}\) with \(|\mathcal {J}_K|=K\).
Proof
If we set \(\eta _i=0\) for all \(i \in \mathcal {J} {\setminus } \mathcal {J}_K\), the result follows directly from Theorem 1. \(\square \)
Next, we outline an alternative algorithm to approximate integrals of type (7). In contrast to the method presented above, the vector of tail sum \(\tilde{R}^{Q,(D)}(h)\) is approximated and included in the computation.
2.2 Algorithm 2
The following derivation is based on the scheme developed by Wiktorsson [9] for SDEs. In the finite dimensional setting, the error estimate (4) depends on the number of Brownian motions K additionally to the time step size h. This suggests that the computational cost involved in the simulation of the stochastic double integrals is much larger in the setting of SPDEs as the number of independent Brownian motions is, in general, not finite, see also expression (8). The eigenvalues of the Q-Wiener process are, however, not incorporated in the error estimate (4). For example, if we assume \(\eta _j = \mathcal {O}(j^{-\rho _Q})\), \(j\in \mathcal {J} \subset \mathbb {N}\), we obtain for \(\rho _Q\in (1,3)\) an improved error estimate which depends on the rate of decay of the eigenvalues instead of some fixed exponent of K. This results from the fact that we integrate with respect to a Q-Wiener process in our setting, where Q is a nonnegative, symmetric trace class operator. For \(\rho _Q\ge 3\), we can show that the exponent of K is bounded by 3.
As before, we truncate the series (18) at some integer \(D\in \mathbb {N}\) and obtain the approximation \(\tilde{A}^{Q,(D)}(h)\) in (19). The vector of tail sum \(\tilde{R}^{Q,(D)}(h)\) in (20), however, is not discarded but approximated by a multivariate normally distributed random vector instead, as described in [9] for \(Q_K = I_K\) and \(|\mathcal {J}|=K\). First, we determine the distribution of the tail sum; for \(r\in \mathbb {N}\), we compute the covariance matrix of
conditional on \(Z_r^Q\) and \(\varDelta w_h^Q\) with \(S_K = \sum _{i=1}^K \mathbf {e}_i^T \otimes (I_K \otimes \mathbf {e}_i)\), where \(\mathbf {e}_i\) denotes the i-th unit vector. Note that \(S_K(x \otimes y) = y \otimes x\) for any vectors \(x, y \in \mathbb {R}^{K}\) [9]. Considering that \(U_r^Q\) is a standard normally distributed vector, the covariance matrix results in
This expression can be reformulated without using the operator \(S_K\) by taking into account that
as
Analogously to [9], by taking the expectation, we define
Taking into consideration that
and that \(H_K \big ( \sum _{i=1}^K \big ( Q_K^{1/2} \mathrm {e}_i \big )^T \otimes \big ( I_K \otimes Q_K^{1/2} \mathrm {e}_i \big ) \big ) H_K^T = 0\), it follows that expression (23) can be rewritten as
The definitions above imply that, given \(Z^Q=(Z_r^Q)_{r \in \mathbb {N}}\) and \(\varDelta w_h^Q\), the vector of tail sum \(\tilde{R}^{Q,(D)}(h)\) is conditionally Gaussian distributed with the following parameters
for \(D \in \mathbb {N}\). Hence, given \(Z^Q\) and \(\varDelta w_h^Q\), we can approximate the tail sum by simulating a conditionally standard Gaussian random vector \({\varUpsilon ^{Q,(D)}}_{|Z^Q,\varDelta w_h^Q} \sim N(0_{L}, I_{L})\) defined as
and, therewith, obtain the vector of tail sum
It remains to examine how the covariance matrix evolves as \(D \rightarrow \infty \). For \(D \in \mathbb {N}\), we define the matrix
By the proof of Theorem 3 below, we get convergence in the following sense
where \(\Vert \cdot \Vert _F\) denotes the Frobenius norm. Thus, it follows
as \(D \rightarrow \infty \), see also [9].
Combining the above, we obtain an algorithm very similar to the one in [9], where steps 1, 2, and 4 equal Algorithm 1. Additionally, we approximate the vector of tail sum in step 3. For some \(h>0\), \(t,t+h \in [0,T]\), and \(D,K\in \mathbb {N}\), Algorithm 2 is defined as follows:
-
1.
For \(j \in \mathcal {J}_K\), simulate the Fourier coefficients \(\varDelta w_{h}^j = \langle W_{t+h}-W_t, \tilde{e}_j \rangle _U\)ofthe increment \(W_{t+h}-W_t\)with\(\varDelta w_h^Q = \big ( \varDelta w^1_h, \ldots , \varDelta w^K_h \big )^T\)as
$$\begin{aligned} \varDelta w_h^Q = \sqrt{h} \, Q_K^{{1}/{2}} V, \end{aligned}$$where \(V \sim N(0_K,I_K)\).
-
2.
Approximate \(\tilde{A}^Q(h)\)as
$$\begin{aligned} \tilde{A}^{Q,(D)}(h)&= H_K \big (Q_K^{{1}/{2}} \otimes Q_K^{{1}/{2}} \big ) H_K^T \\&\quad \times \frac{h}{2\pi } \sum _{r=1}^{D} \frac{1}{r} H_K \Big (U_{r} \otimes \Big ( Z_{r} -\sqrt{2} V \Big ) -\Big (Z_{r} - \sqrt{2} V \Big ) \otimes U_r \Big ), \end{aligned}$$where \(U_r, Z_r \sim N(0_K, I_K)\)are independent.
-
3.
Simulate \(\varUpsilon ^{Q,(D)} \sim N(0_{L},I_{L})\) and compute
$$\begin{aligned} \hat{A}^{Q,(D)}(h) = \tilde{A}^{Q,(D)}(h) + \frac{h}{2\pi } \bigg ( \sum _{r=D+1}^{\infty }\frac{1}{r^2} \bigg )^{\frac{1}{2}} \sqrt{\varSigma ^Q_{\infty }} \varUpsilon ^{Q,(D)} . \end{aligned}$$(27) -
4.
Compute the approximation \( \text {vec}((\hat{I}^{Q,(D)}(h))^T)\)of\( \text {vec}((I^Q(h)^T)\)as
$$\begin{aligned} \text {vec}((\hat{I}^{Q,(D)}(h))^T) = \frac{\varDelta w_h^Q \otimes \varDelta w_h^Q -\text {vec}(h Q_K)}{2} + (I_{K^2}-S_K) H_K^T \hat{A}^{Q,(D)}(h) \end{aligned}$$with \(S_K = \sum _{i=1}^K \mathbf {e}_i^T \otimes (I_K \otimes \mathbf {e}_i)\).
Note that the matrix \(\sqrt{\varSigma ^Q_{\infty }}\) in step 3 may be calculated by a Cholesky decomposition. However, the square root of \(\varSigma ^Q_{\infty }\) can be obtained in closed form and does not have to be computed numerically.
Theorem 2
Let \(\varSigma ^Q_{\infty }\) be defined as in (23) with \(\varDelta w_h^Q = \sqrt{h} \, Q_K^{{1}/{2}} V\) and let \(\varSigma _{\infty }^I\) be defined by (23) with \(Q_K=I_K\). Then, it holds
Proof
For a proof, we refer to Sect. 3. \(\square \)
Now, we analyze the error resulting from Algorithm 2. In the following theorem, the first term is the same as in the error estimate of Algorithm 1, see Theorem 1. Due to the second term, the approximation converges with order 1 in D, which is twice the order that Algorithm 1 attains. However, this expression is dependent on K as well. Note that \(\max _{j\in \mathcal {J}_K} \eta _j \le \max _{j\in \mathcal {J}} \eta _j \le {\text {tr}}Q < \infty \). Below, we state an alternative estimate—there, the exponent of K is not fixed but dependent on the eigenvalues \(\eta _j\), \(j\in \mathcal {J}_K\), see Theorem 4. The algorithm that is superior can only be determined in dependence on the operator Q.
Theorem 3
(Convergence result for Algorithm 2) Assume that the trace class operator Q is nonnegative and symmetric and \((W_t)_{t \ge 0}\) is a Q-Wiener process. Further, let \(\varPhi \in L(U,H)_{U_0}\) with \(\Vert \varPhi Q^{-\alpha } \Vert _{L_{HS}(U_0,H)}<C\) for some \(C>0\), let \(\varPsi \in L(H,L(Q^{-\alpha }U,H)_{U_0})\) for some \(\alpha \in (0,\infty )\), i.e., assumptions (A1) and (A2) are fulfilled. Then, it holds
for some \(C>0\) and all \(h>0\), \(t,t+h\in [0,T]\), \(D,K\in \mathbb {N}\), and \(\mathcal {J}_K \subset \mathcal {J}\) with \(|\mathcal {J}_K|=K\).
Proof
For a proof, we refer to Sect. 3. \(\square \)
For completeness, we state the following error estimate. Again, this is the estimate that we consider when incorporating the approximation of the iterated integrals into a numerical scheme; see also the notes on Corollary 1.
Corollary 2
Assume that Q is a nonnegative and symmetric trace class operator and \((W_t)_{t \ge 0}\) is a Q-Wiener process. Furthermore, let \(\varPhi \in L(U,H)_{U_0}\), \(\varPsi \in L(H,L(U,H)_{U_0})\), i.e., conditions (B1) and (B2) are fulfilled. Then, it holds
for some \(C > 0\) and all \(h>0\), \(t,t+h\in [0,T]\), \(D,K\in \mathbb {N}\), and \(\mathcal {J}_K \subset \mathcal {J}\) with \(|\mathcal {J}_K|=K\).
Proof
The proof of this corollary is detailed in the proof of Theorem 3. \(\square \)
If we assume, e.g., \(\eta _j \le Cj^{-\rho _Q}\) for \(C>0\), \(\rho _Q>1\), and all \(j \in \mathcal {J} \subset \mathbb {N}\), we can improve the result in Theorem 3 in the case \(\rho _Q <3\). Precisely, we obtain an error term that involves the factor \(K^{\frac{\rho _Q}{2}}\). The main difference is that the alternative proof works with the entries of the covariance matrices explicitly. A statement along the lines of Corollary 2 can be obtained analogously.
Theorem 4
(Convergence result for Algorithm 2) Assume that the trace class operator Q is nonnegative and symmetric and \((W_t)_{t \ge 0}\) is a Q-Wiener process. Let \(\varPhi \in L(U,H)_{U_0}\) with \(\Vert \varPhi Q^{-\alpha }\Vert _{L_{HS}(U_0,H)}<C\) and \(\varPsi \in L(H,L(Q^{-\alpha }U,H)_{U_0})\) for some \(\alpha \in (0,\infty )\), i.e., assumptions (A1) and (A2) are fulfilled. Then, it holds
for some \(C>0\) and all \(h>0\), \(t,t+h\in [0,T]\), \(D,K\in \mathbb {N}\), and \(\mathcal {J}_K \subset \mathcal {J}\) with \(|\mathcal {J}_K|=K\).
Proof
For a proof, we again refer to Sect. 3. \(\square \)
Remark 1
Note that if \((W_t)_{t\ge 0}\) is a cylindrical Wiener process, we get the same estimate (4) as in the finite dimensional case.
2.3 Comparison and concluding remarks
For some fixed interval of length \(h=\frac{T}{M}\) and a given projected Q-Wiener process \((W_t^K)_{t\ge 0}\), that is, some fixed values \(M, K\in \mathbb {N}\), we obtain convergence as \(D \rightarrow \infty \) for both algorithms that we proposed in the last sections with differing orders, respectively. However, if the approximation of the iterated integrals is incorporated in a numerical scheme to compute the solution of a given SPDE, then the interval length h has to decrease and the number K, which defines the approximation of the Q-Wiener process, has to increase in order to approximate the solution with increasing accuracy. Algorithm 1 provides convergence as \(K,M\rightarrow \infty \) if we choose \(D \ge h^{2-2\theta }\) for some \(\theta >0\) with order \(\mathcal {O}(M^{-\theta })\) where the choice of D does not depend on K. For Algorithm 2, we require \(D \ge (\min _{j\in \mathcal {J}_K}\eta _j)^{-\frac{1}{2}}h^{1-\theta }\) or, respectively, \(D \ge \sqrt{K^2(K-1)}h^{1-\theta }\) for some \(\theta >0\) to get convergence with order \(\mathcal {O}(M^{-\theta })\). In order to maintain the global rate of convergence in the mean-square sense in \(h=\frac{T}{M}\) for a given numerical scheme, we need an even more careful choice of D. The error estimates for the iterated integrals depend on the values M, D, and, for Algorithm 2, on the value K. Moreover, for the approximation of the solution of an SPDE, we have to include an approximation of the infinite dimensional solution space, see [2], for example. All these error terms, i.e., in time, in space, the truncation of the Q-Wiener process and the truncation of the Fouries series for the iterated integrals, have to be balanced such that an optimal overall order of convergence for a given numerical scheme can be observed. The full treatment of this optimization is beyond the scope of this work as it, e.g., depends on the numerical scheme and the specific SPDE. We restrict our discussion to the clarification of the relation of the value D to the number of time steps and the approximation of the Q-Wiener process which characterizes the main difference between the two algorithms that we proposed. However, this already shows that, dependent on the desired order of convergence in h and on the operator Q (which are determined by the SPDE to be solved and the numerical scheme employed for the discretization of time), either Algorithm 1 or Algorithm 2 is the optimal choice in terms of lower computational cost.
3 Proofs
3.1 Convergence analysis for Algorithm 1
Proof of Theorem 1
We determine the error resulting from the approximation of the iterated stochastic integral (7) by Algorithm 1 which also contains the projection of the Q-Wiener process in (9). Below, we employ error estimates of the following form several times, see also the proof in [2]. It holds
where we used the expression
for all \(s\in [0,T]\), \(K\in \mathbb {N}\) in the first step. We fix some arbitrary \(h>0\), \(t, t+h \in [0,T]\), and \(K\in \mathbb {N}\) throughout the proof and decompose the error into several parts
For now, we neglect the last term in (29) and estimate the other parts. By Itô’s isometry, the properties (A1) and (A2) of the operators \(\varPhi \), \(\varPsi \), and estimate (28), we get
Finally, assumption (A1) yields
Now, we concentrate on the last term in (29); this part also proves Corollary 1. We get
as \(\mathrm {E}\Big [ \big (I_{(i,j)}^Q(h)-\bar{I}_{(i,j)}^{Q,(D)}(h) \big ) \big (I_{(k,l)}^Q(h)-\bar{I}_{(k,l)}^{Q,(D)}(h) \big ) \Big ] =0\) for all \(i,j,k,l \in \mathcal {J}_K\) with \((i,j) \ne (k,l)\), \(K\in \mathbb {N}\), see [3]. By assumptions (B1) and (B2), we obtain
Due to the relations (14)–(16), it is enough to examine \(\tilde{A}^Q(h)\) and \(\tilde{A}^{Q,(D)}(h)\) which implies
By (12), (13), and the properties of \(a_r^j\), \(b_r^j\) for \(r\in \mathbb {N}_0\), \(j \in \mathcal {J}_K\), \(K\in \mathbb {N}\), we obtain
for all \(D\in \mathbb {N}\). As in [4], we finally estimate
and, in total, we obtain for this part
for all \(h>0\), \(t, t+h\in [0,T]\), \(D,K\in \mathbb {N}\). \(\square \)
3.2 Square root of the covariance matrix
Proof of Theorem 2
It holds \({\varSigma ^Q_{\infty }}= \tilde{Q}_K \varSigma _{\infty }^I\tilde{Q}_K^T\), where \(\varSigma _{\infty }^I\) is given by (23) for \(Q_K=I_K\), and \(\varDelta w_h^I = Q_K^{-1/2} \varDelta w_h^Q = \sqrt{h} V\). We assume that
holds and compute for \(a := \sqrt{1+\frac{1}{h} {\varDelta w_h^I}^T \varDelta w_h^I}\) the expression
The idea in [9] is to show that the first term, which slightly differs in [9], is zero, i.e.,
which proves that the expression for \(\sqrt{{\varSigma ^Q_{\infty }}} \) is correct. In the proof of Theorem 4.1 in [9], the author shows
arguing by the eigenvalues of the minimal polynomial of this equation. We do not repeat this ideas here but refer to [9] for further details. \(\square \)
3.3 Convergence analysis for Algorithm 2
Proof of Theorem 3
We split the error term as in the proof of Theorem 1, see Eq. (29), and obtain the same expression (30) from the approximation of the Q-Wiener process by \((W^K_{t})_{t \ge 0}\), \(K\in \mathbb {N}\). Further, we get as in Eq. (31)
for all \(h>0\), \(t, t+h\in [0,T]\), \(K\in \mathbb {N}\). The following part also proves Corollary 2. Let \(\Vert \cdot \Vert _F\) denote the Frobenius norm. With the expressions for \(\tilde{R}^{Q,(D)}(h)\) in (25), with \(\varSigma ^{Q,(D)} = \tilde{Q}_K \varSigma ^{I,(D)} \tilde{Q}_K^T\), \({\varSigma ^Q_{\infty }}= \tilde{Q}_K \varSigma _{\infty }^I\tilde{Q}_K^T\) where \(\varSigma ^{I,(D)}\), \(\varSigma _{\infty }^I\) are given by (26) and (23) for \(Q_K=I_K\), respectively, and the definition of the algorithm (27), we obtain
for all \(h>0\), \(t, t+h\in [0,T]\), \(D,K\in \mathbb {N}\). Here, we used that \({\varUpsilon ^{Q,(D)}}_{|Z^Q,\varDelta w_h^Q}\sim N(0_{L}, I_{L})\) for \(h>0\), \(D,L\in \mathbb {N}\) and that \(\tilde{Q}_K\) is a diagonal matrix. Precisely, for \(G:= \sqrt{\varSigma ^{I,(D)}}-\sqrt{\varSigma _{\infty }^I}\) with \(G:= (g_{ij})_{1\le i,j\le L}\) and \(\varUpsilon ^{Q,(D)} = (\varUpsilon ^{Q,(D)}_j)_{1\le j\le L}\), we compute
In order to relate to the proof in [9], we write
In total, we obtain
Now, we can insert the results obtained in the proofs of [9, Theorem 4.1, Theorem 4.2, Theorem 4.3]; this yields
for all \(h>0\), \(t, t+h\in [0,T]\), \(D,K\in \mathbb {N}\) where \(V = h^{-1/2} Q_K^{-1/2} \varDelta w_h^Q\). \(\square \)
Proof of Theorem 4
We split the error term as in the proof of Theorems 1 and 3, see Eq. (29), and obtain the same expression (30) from the approximation of the Q-Wiener process by \((W^K_{t})_{t \ge 0}\), \(K\in \mathbb {N}\). Moreover, as in the previous proof, we get from (33) that
for all \(h>0\), \(t, t+h\in [0,T]\), \(D,K\in \mathbb {N}\). In this alternative proof, we consider the elements of the matrices \(\varSigma ^{Q,(D)}\) and \({\varSigma ^Q_{\infty }}\) explicitly. Therefore, we define the index set of interest as \(\mathcal {I}_A = ((1,2),\ldots ,(1,K),\ldots ,(l,l+1),\ldots ,(l,K),\ldots ,(K-1,K)) = (I_1,\ldots ,I_{L})\) which selects the same entries of some matrix as the matrix transformation by \(H_K\) given in (17). The \(L \times L\)-matrix \(H_K \varSigma ^Q(V_1^Q)_{|Z_1^Q,\varDelta w_h^Q} H_K^T \) has entries of type
for some \(i,j,m,n \in \{1, \ldots , K\}\) with \(i<j\) and \(m<n\). Especially, its diagonal entries are of type
with \((i,j) \in \mathcal {I}_A\) and \(i \ne j\). The off-diagonal entries of \(H_K \varSigma ^Q(V_1^Q)_{|Z_1^Q,\varDelta w_h^Q} H_K^T\) are of the form
with \(i,j,m,n \in \{1,\ldots ,K\}\), \(i < j\) and \(m < n\). Therewith, it is easy to see that for \({\varSigma ^Q_{\infty }}= \mathrm {E}\Big [H_K \varSigma ^Q(V_1^Q)_{|Z_1^Q,\varDelta w_h^Q} H_K^T \Big |\varDelta w_h^Q\Big ]\), we get
and for the off-diagonal entries, it holds
with \(k,l\in \{1,\ldots , L\}\), \(l \ne k\), \(i,j,m,n\in \{1,\ldots ,K\}\), \(i < j\) and \(m < n\). Next, we employ the following lemma from [9] in order to rewrite (34).
Lemma 1
Let A and G be symmetric positive definite matrices and denote the smallest eigenvalue of matrix G by \(\lambda _{min}\). Then, it holds
Proof of Lemma 1
A proof can be found in [9, Lemma 4.1]. \(\square \)
For simplicity, we assume \(\eta _1\ge \eta _2\ge \ldots \ge \eta _K\) for all \(K\in \mathbb {N}\). We decompose \({\varSigma ^Q_{\infty }}\) as
to determine its smallest eigenvalue. The matrix \(\widehat{{\varSigma ^Q_{\infty }}}\) is defined as follows: For the diagonal elements, we get values
with \(k \in \{1,\ldots ,L\}\), \((i,j) \in \mathcal {I}_A\), and \(h>0\). For the off-diagonal elements, we get \(\big (\widehat{{\varSigma ^Q_{\infty }}}\big )_{(k,l)} = \big ({{\varSigma ^Q_{\infty }}}\big )_{(k,l)}\) for all \(k,l \in \{1,\ldots ,L\}\), \(k\ne l\). As the matrix \(\widehat{{\varSigma ^Q_{\infty }}}\) is symmetric and positive semi-definite, the smallest eigenvalue \(\lambda _{\min }\) of \({{\varSigma ^Q_{\infty }}}\) fulfills \(\lambda _{\min } \ge 2\eta _{K-1} \, \eta _K \ge 2\eta _K^2\).
Below, we use the notation \(c_D = \sum _{r=D+1}^{\infty }\frac{1}{r^2}\) for legibility. The matrices \(\varSigma ^{Q,(D)}\) and \({\varSigma ^Q_{\infty }}\) are symmetric positive definite. By Lemma 1 and the definitions of \(\varSigma ^{Q,(D)}\), \({\varSigma ^Q_{\infty }}\) in (26) and (23), respectively, we obtain from (34)
for \(h>0\), \(t, t+h\in [0,T]\), \(D,K \in \mathbb {N}\). Following ideas from [9], we get
Next, we compute the conditional expectation involved in this estimate. We split the sum into diagonal entries and off-diagonal elements of the matrix, that is,
Then, we insert the expressions detailed above for \(H_K \varSigma ^Q(V_r^Q)_{|Z_r^Q, \varDelta w_h^Q}H_K^T\), \(r\in \mathbb {N}\) and \({\varSigma ^Q_{\infty }}\), see (35)–(38). This yields for \(h>0\), \(t, t+h\in [0,T]\), \(D, L\in \mathbb {N}\)
We compute the terms in (39) separately and obtain
and
for all \(i,j,m,n \in \mathcal {J}_K\) with \(i< j\) and \(m<n\). For the other terms of this type, we get similar results. Moreover, we compute bounds for the following expressions
for all \(D\in \mathbb {N}\). A combination of these estimates yields
for all \(D\in \mathbb {N}\). At this point, the main difference to Algorithm 1 arises—we obtain a higher order of convergence in D. In total, we get
Finally, this implies for all \(h>0\), \(t, t+h\in [0,T]\), \(D,K\in \mathbb {N}\)
that is, more generally,
The statement of the theorem follows by combing this estimate with (30). \(\square \)
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Acknowledgements
Funding and support by the Graduate School for Computing in Medicine and Life Sciences funded by Germany’s Excellence Initiative [DFG GSC 235/2] and in addition by the Cluster of Excellence “The Future Ocean” is gratefully acknowledged. “The Future Ocean” is funded within the framework of the Excellence Initiative by the Deutsche Forschungsgemeinschaft (DFG) on behalf of the German federal and state governments.
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Leonhard, C., Rößler, A. Iterated stochastic integrals in infinite dimensions: approximation and error estimates. Stoch PDE: Anal Comp 7, 209–239 (2019). https://doi.org/10.1007/s40072-018-0126-9
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DOI: https://doi.org/10.1007/s40072-018-0126-9
Keywords
- Iterated stochastic Itô integral in infinite dimensions
- Q-Wiener process
- Lévy area simulation
- Karhunen–Loève expansion
- Fourier series expansion
- Numerical approximation
- Error bound