Adaptive step-size control for global approximation of SDEs driven by countably dimensional Wiener process

In this paper we deal with global approximation of solutions of stochastic differential equations (SDEs) driven by countably dimensional Wiener process. Under certain regularity conditions imposed on the coefficients, we show lower bounds for exact asymptotic error behaviour. For that reason, we analyse separately two classes of admissible algorithms: based on equidistant, and possibly not equidistant meshes. Our results indicate that in both cases, decrease of any method error requires significant increase of the cost term, which is illustrated by the product of cost and error diverging to infinity. This is, however, not visible in the finite dimensional case. In addition, we propose an implementable, path-independent Euler algorithm with adaptive step-size control, which is asymptotically optimal among algorithms using specified truncation levels of the underlying Wiener process. Our theoretical findings are supported by numerical simulation in Python language.


Introduction
We investigate global approximation of solutions of the following stochastic differential equations dX(t) = a(t, X(t)) dt + σ(t) dW (t), t ∈ [0, T ], where T > 0, W (t) = [W 1 (t), W 2 (t), . ..]T is a sequence of independent scalar Wiener processes on the probability space with sufficiently rich filtration (Ω, F, P, F t ) t∈[0,T ] , and x 0 ∈ R. For suitable, regular coefficients a, σ, the uniqueness of the solution X = X(t), and its finite second-order moments can be assured; see [1,4,6,23] where more general models were considered.
Recently, global approximation of solutions of SDEs driven by finite-dimensional Wiener process has been studied extensively in the literature.In particular, the algorithms with step-size were introduced in [8,9,12,19,20].Generally, the time-step adaptation linked to the equation coefficients instead of leveraging equidistant mesh can significantly decrease the asymptotic constant for the method error in the finite dimensional models.On the other hand, SDEs driven by countably dimensional Wiener noise can be found in [2,5,14], while their applications -e.g., in [3,18].Nonetheless, there are still few papers referring to the exact error behaviour and optimality issues for such SDEs in global setting.For instance, in [24] authors developed an Euler algorithm and estimated its global error for X being countably dimensional.However, the assumptions were relatively strong, and the proposed algorithm was non-implementable due to infinite dimension of the solution.
In this paper we extend some asymptotic results for global approximation from [8,12] to SDEs with countably dimensional noise structure.To that end, we utilise solution moment bounds and approximation strategy presented in [23], where a pointwise setting was investigated.In our setting, error of an algorithm A returning a process Y = (Y (t)) t∈[0,T ] is measured in the norm • 2 defined as follows Under suitable conditions imposed on the model coefficients, we analyse asymptotic exact error behaviour in two classes χ eq , χ noneq of admissible algorithms leveraging only finite dimensional evaluations of the process W ; we refer to [7,8,11,12,17,22,26] where similar approach for generic error analysis was developed.In our setting, permitted truncation levels are determined by non-decreasing sequences leveraging information about how fast σ entries vanish.In Theorem 2 we show that for any fixed, admissible truncation level sequence M = (M n ) ∞ n=1 , an exact asymptotic behaviour of the cost-error relation satisfies cost (X Mn, kn ) irrespective of the choice of admissible method (X Mn, kn ) n∈N ∈ χ M noneq based on (possibly) non-equidistant mesh with a suitable number of nodes kn + 1, and such that X Mn, kn utilises discrete information from M n , n ∈ N, first coordinates of W. When we limit ourselves to methods χ M eq based on equidistant partitions of the interval [0, T ], we get cost (X Mn, kn ) We hereinafter use the notation .
By the Hölder inequality, we have 0 ≤ C noneq ≤ C eq .We also stress that the lower bounds in (2) and (3) diverge as n tends to infinity, which illustrates significant increase of the informational cost needed to decrease the method error.Next, for fixed M , we construct truncated Euler algorithm with adaptive path-independent step-size control X step Mn,k * n which is optimal in class χ M noneq , since it attains asymptotic lower bound in (2).We also provide lower bounds which hold irrespective of the truncation level sequence, see Theorem 3.While those cannot be asymptotically achieved by any admissible algorithm, we show that the errors for the methods proposed in this paper can be arbitrary close in some sense to the obtained bound.We note that both error bounds and optimality are investigated in the spirit of IBC (Information-Based Complexity) framework.
According to our best knowledge, this is the first paper to establish lower bounds for exact asymptotic error in the global approximation setting for SDEs with countably dimensional Wiener process.Moreover, the new constructed algorithms are implementable, and their performance is verified by using the multiprocessing library in Python.
The paper is organised as follows.In Section 2 we provide basic notation, model assumptions and properties of the underlying solution X.Then, in Section 3 we investigate lower bounds for exact asymptotic error behaviour.In Section 4.1 and in Section 4.2 we introduce and show optimality of the truncated dimension Euler schemes in the classes χ M eq and χ M noneq , respectively.Next, in Section 4.3, we extend optimality investigation to the classes χ eq and χ noneq .Finally, Section 5 deals with numerical experiments in Python and alternative solver implementation utilizing Numba compiler.
For x ∈ 2 (R) we use the following notation x = (x 1 , x 2 , . ..).We introduce projection operators We also set P ∞ = Id, hence P ∞ v = v for all v ∈ 2 (R).For brevity, in this paper we write k=1 w k we understand the vector w 1 ⊕ w 2 ⊕ . . .⊕ w n .In this paper, we use the following asymptotic notation.For two real-valued sequences We also say that a n ≈ b n , n → +∞, if and only if lim n→+∞ a n /b n = 1.Furthermore, the asymptotic symbols Ω, Θ, O, o appearing in this paper are aligned with classical Landau notation for sequences.For a sequence (c(n)) ∞ n=1 of non-negative numbers converging to zero and > 0, we define the inverse c We assume that drift coefficient a : [0, T ] × R → R belongs to C 1,2 ([0, T ] × R) and satisfies the following conditions: k=1 ⊂ R be a positive, strictly decreasing sequence vanishing at infinity.For fixed δ, by G δ we denote a set of all non-decreasing sequences We assume that diffusion coefficient σ = (σ 1 , σ 2 , . ..) : [0, T ] → 2 (R) satisfies the following conditions: for C 2 > 0 and some fixed sequence δ as above.
Our idea is to first provide the approximation of truncated solution X M = X M (a, σ, W ) which depends on the first M ∈ N coordinates of the underlying Wiener process W .Then, we estimate globally the inevitable truncation error resulting from substituting the process X for X M .For convenience, we will use the notation X ∞ := X.
To this end, we consider the family of processes In particular, for M = +∞ we obtain the main problem (1).For further analysis, we need some properties of the process X M .Those are presented below, in a corollary from Lemma 1 in [23].
Lemma 1.For every M ∈ N ∪ {∞} the equation (5) admits a unique strong solution X M = (X M (t)) t∈[0,T ] .Moreover, there exists K ∈ (0, +∞), depending only on the constants C 1 , C 2 , such that for every M ∈ N ∪ {∞} we have that and for all s, t ∈ [0, T ] the following holds We also state truncation error bound for our model.This result is a corollary from Proposition 1 in [23].
We define truncated dimension time-continuous Euler algorithm X E M,n = ( X E M,n (t)) t∈[0,T ] that approximates the process (X(t)) t∈[0,T ] .Take M, n ∈ N, and let be a sequence of partitions of the interval [0, T ], with k n ∈ N. We set We stress that X E M,n is not implementable since it requires complete knowledge of the trajectories of the underlying Wiener process.Now we state the upper error bound of the truncated dimension time-continuous Euler algorithm in finite dimensional setting.Proposition 2. Under the assumptions (A1)-(A4) and (S1)-(S3), there exists a positive constant C 0 , depending only on C 1 , C 2 , such that for all M, n ∈ N the time-continuous truncated Euler process (9) based on partition (8) satisfies where ∆t j,n = t j+1,n − t j,n , j = 0, 1, . . ., k n − 1.
The proof of Proposition 2 is postponed to the Appendix.
From Proposition 1 and Proposition 2 we obtain the following result.
Theorem 1.Let the coefficients a, σ satisfy (A1)-(A4) and (S1)-(S3) with sequence δ, respectively.Then, there exists a positive constant K, depending on C 1 , C 2 , δ, such that for every M, n ∈ N and the discretisation (8), it holds At the end of this section, we present several remarks on the model, imposed assumptions, and suitability of the chosen stochastic scheme.
Remark 1. Generally, the concept of SDEs with integrals wrt countably dimensional Wiener process is used to describe the evolution driven by countably many risk factors.This modelling choice can be leveraged for a wide range of problems in, e.g., genetics, mathematical finance or physics [3,18].From a pragmatic point of view, infinite dimensional setting can be leveraged when the number of random risks is finite but still too large to be entirely captured by the electronic machines.On the other hand, the integrals appearing in this paper can be viewed as stochastic integrals with respect to cylindrical Brownian motion in Hilbert space of sequences 2 (R) and hence, our model forms a bridge between ordinary SDEs and stochastic partial differential equations (SPDEs), see [5].
Remark 2. First, the assumptions (A1) -(A4) imply the existence of a constant Second, in the presented setting, a crucial role is played by appropriate choice of corresponding truncation levels for admissible methods.Indeed, those are defined in terms of the elements of G δ as per (4).We note that for every δ the corresponding set G δ is nonempty.Furthermore, the slower rate of diffusion decay to zero, the greater values of G(n) need to be taken.We note that δ does not need to be optimal in a sense that for fixed σ there might exist δ also satisfying (S3) and δ < δ.Nevertheless, sharper bound in (S3) yields greater palette of the corresponding sequences G in (4), as δ ≤ δ implies G δ ⊂ G δ .
Remark 3. It is worth mentioning that suitable modifications of the classic Euler scheme for finite-dimensional setting have been investigated in e.g.[8,15,23].For the problem (1), the method X E M,n coincides with truncated dimension time-continuous Euler algorithm proposed in [23].However, the regularity of function a in our case enhances the rate of convergence from 1/2 to 1, see Theorem 1. Indeed, the proposed method also coincides with the Milstein scheme; see e.g., [12,13,16,[19][20][21][22] where the approximation by modified Milstein schemes for finite dimensional models was considered.
In the following sections, we investigate lower bounds for exact asymptotic error behaviour and construct optimal methods in suitable subclasses of admissible algorithms.The optimality is defined in the spirit of Information-Based Complexity (IBC) framework, see also [26] for more details.

Lower bounds for exact asymptotic error behaviour
In this section, we derive minimal global approximation error for our initial problem (1).The main results are presented in Theorem 2.
Let us fix (a, σ), and a sequence δ satisfying (S3).An arbitrary method under consideration is a sequence of the form X = (X Mn, kn ) ∞ n=1 and can be equivalently viewed as a quadruple X = ( ∆, N , M , φ), where: n=1 is a sequence of (possibly) non-expanding partitions of the interval [0, T ], i.e., ∆n : 0 where for some C 1 , C 2 > 0 it holds • N = (N Mn, kn ) ∞ n=1 is a sequence of information vectors.For fixed n ∈ N, the vector N Mn, kn consists of the points in which the method X Mn, kn evaluates the values of underlying (scalar) Wiener processes W k , k = 1, . . ., M n : n=1 ∈ G δ with M n indicating number of initial Wiener process coordinates used by the method X Mn, kn , n ∈ N.
• The method X Mn, kn is assumed to evaluate W in the time points from N Mn, kn , yielding a process X Mn, kn (a, σ, W ) being an approximation of X. Namely, we assume the existence of Borel measurable mappings φ The class of algorithms satisfying above conditions is denoted by χ noneq .In this paper, we distinguish a subclass χ eq ⊂ χ noneq of methods leveraging equidistant partitions The optimality in class of methods leveraging equidistant meshes for global approximation problem in finite dimensional model has been considered recently in e.g., [11].In this paper we also show the benefit of leveraging adaptive meshes instead of equidistant ones.
By the cost of the algorithm X Mn, kn we understand a number of evaluations of scalar Wiener processes performed by X Mn, kn .Specifically, we have cost(X Mn, kn ) = M n • kn , when σ ≡ 0, 0, when σ ≡ 0.
While in case of σ ≡ 0 we actually deal with ordinary differential equations and still some calculations need to be performed, there is no W process involved.In our setting, this is justified by zero cost.
The global approximation error is measured in a product For a fixed method X Mn, kn = ( ∆, N , M , φ) ∈ χ noneq \ χ eq we define the sequence of where k1/2 n /m n → 0 and m n / kn → 0, n → +∞, and ∆ eq mn = (t eq j ) mn j=0 is an equidistant partition, t eq j = T j/m n , j = 0, . . ., m n .In the sequel, by k n we denote the number of distinct time points t j,n in ∆ n .Consequently, we get which in turn implies that lim n→+∞ k n / kn = 1.In addition, we introduce augmented information vectors N = ( N Mn, kn (W )) ∞ n=1 , where For brevity, in the sequel we will use the notation ∆ t j,n = t j+1,n − t j,n , n ∈ N, j = 0, 1, . . ., k n − 1.
Now for fixed n ∈ N we estimate distance between truncated dimension time-continuous Euler process X E Mn, kn based on partition ∆ n and the associated time-continuous conditional Euler process X cond Mn,kn (t) = E X E Mn, kn (t) N Mn, kn (W ) , which leverages the augmented information N Mn, kn (W ).We refer to, e.g., [12] for more details on conditional Euler process in finite dimensional setting when also jumps modelled by homogeneous Poisson process are considered.
We obtain where Ŵj,k,n is a Brownian bridge on the interval [ t j,n , t j+1,n ], conditioned on W k .For more details on Brownian bridge, we refer to e.g.[19].Furthermore, Combining Proposition 1, (17), and the fact that X Mn, kn is σ( NMn, kn (W ))-measurable, we get where both D 1 , D 2 do not depend on n (explicitly or implicitly via M n or kn , k n ).Above, we also leveraged property NMn, kn (W ) ⊂ N Mn, kn (W ).Hence, by ( 4), ( 13), (18), and the definition of m n , we have By Fact 2, ( 19), the Hölder inequality, and the fact that k n ≈ kn , n → +∞, we arrive at we finally get for all X Mn, kn ∈ χ noneq that cost(X Mn, kn ) For X Mn, kn ∈ χ eq , the proof follows analogous steps, except for taking simply ∆ n = ∆n in (15).Leveraging the fact that ∆ t j,n = T kn , j = 0, . . ., kn − 1, leads to Consequently, (21) implies that for all X Mn, kn ∈ χ eq it holds cost(X Mn, kn ) Considering ( 20) and ( 22), we are ready to formulate the main result of this section.
Theorem 2. Let us denote by χ M noneq ⊂ χ noneq a set of all admissible methods with fixed truncation level sequence M .We have the following asymptotic lower bound In particular, restriction to the subclass χ eq gives sharper asymptotic lower bound Remark 4. The restriction (13) imposed on kn is not limiting, as the majority of algorithms used in practice leverage O(n) nodes.We also stress that C 1 , C 2 , as well as the related sequence kn , depend on the method X and need not be the same across all considered algorithms.Furthermore, we allow only discrete, finite-dimensional evaluations of W. Therefore, for all n ∈ N, the vector N Mn, kn is σ W 1 , W 2 , . . ., W Mn -measurable.Moreover, different partitions among various coordinates of W are not permitted in our setting.
Remark 5. We stress that while the sequences ∆ and φ might depend on a, σ, they cannot base on the trajectory of W. The resulting information about Wiener process is then called non-adaptive, while the associated algorithms are referred to as path-independent.For pathdependent version of Euler algorithm in finite dimensional setting, see e.g., [20].
Remark 6.It should be noted that lower bounds in Theorem 2 diverge to infinity when n → +∞ in both subclasses.This significantly differs from finite-dimensional case, when the truncation level sequence was bounded from above.That saying, when and the asymptotic constant appearing in ( 23) is consistent with the result from Theorem 2 in [8].
Notably, for given sequence M , in Section 4.1 and Section 4.2 we construct algorithms which asymptotically attain lower bounds appearing in Theorem 2.Then, in Section 4.3 we discuss the existence of optimal algorithms in a class of methods taking into account all permitted truncation sequences M .

Optimal algorithm in class of methods χ M
eq .First, we fix M = (M * n ) ∞ n=1 satisfying (4).Next, for any n ∈ N let us denote by X Eq M * n ,n ∈ χ eq the truncated dimension Euler method based on equidistant mesh ∆ eq n : 0 = t eq 0,n < . . .< t eq n,n = T, where t eq j,n = T j n , j = 0, 1, . . ., n.The scheme is defined as follows: The outcome of the method is a stochastic process X Eq * M * n ,n (t) t∈[0,T ] obtained by linear interpolation between two subsequent nodes t eq j,n and t eq j+1,n , i.e., , t ∈ t eq j,n , t eq j+1,n , Let us also denote by N eq M * n ,n (W ) the related information vector.In addition, by X E,eq M * n ,n we understand the truncated dimension time-continuous Euler process based on the mesh ∆ eq n , n ∈ N. Certainly, it holds Consequently, from Theorem 1 and (27) it follows Therefore, by repeating the reasoning as in (17) and the fact that Finally, by (29) we have which establishes the lower bound in Theorem 2 for fixed class χ M eq .Note that X Eq M * n ,n is an implementable algorithm, and it does not require the knowledge of the trajectories of (truncated) Wiener process.

4.2.
Optimal algorithm with adaptive path-independent step-size control in class χ M noneq .In this section, we construct an optimal method in class χ M noneq for fixed truncation level sequence M = (M * n ) ∞ n=1 ∈ G δ .To this end, we define truncated dimension Euler scheme (X step n=1 with adaptive path-independent step-size of the following form.First, let ε = (ε n ) ∞ n=1 ⊂ R + be a non-increasing sequence satisfying For fixed n ∈ N the proposed scheme utilises step-size control with t0,n := 0 and tj+1,n := tj,n + where We will denote the corresponding mesh by ∆n .Then, we set In the sequel, we will use the notation ∆ * n = ∆n \ { tk * n ,n } ∪ {T } and t * j,n = tj,n , j = 0, 1, . . ., k * n − 1, t * k * n ,n = T.The corresponding information vector will be denoted by N * M * n ,k * n (W ).Also, by ∆ tj,n , ∆t * j,n we will understand the j-th time step for the discretisation ∆n and ∆ * n , respectively.

The final process X
).We recall that in our model (1), the process X * M * n ,k * n coincides with the time-continuous conditional Euler process based on the sequence of partitions ∆ * n .Fact 1.
a) The proposed method X step M * n ,k * n with adaptive step-size control is an element of χ noneq and attains point T, irrespective of prior choice of the sequences M and ε. b) k * n is deterministic and Proof.Since for every M, n ∈ N and j = 0, . . ., k * n we have for n 0 = n(ε n + Ĉ) + 1 it holds tn 0 ,n ≥ T. Combining this, the fact that nε n = o(n), and nε n = Ω(n 1/2 ) by (31), we have that 1 This proves both assertions a) and b).Now we investigate asymptotic behaviour of the method error.Let us denote By Fact 2 we get that Consequently, from (34), (36), and the fact that for all a, b ≥ 0 Since from (34) and ( 35) Also, note that (32) results in Consequently, since for some C > 0. Consequently, (38) together with (39) yield Furthermore, by ( 39) and ( 41) we arrive at where K 6 does not depend on n on the virtue of (40 (43) By ( 42), (43), and Fact 1 we obtain Combining ( 42) and (44) results in On the other hand, by Proposition 1, Fact 1, and (33) we have for some K 10 > 0 which depends only on the constants C 1 , C 2 , T. Therefore, from ( 4), (31), and (46) it follows Rewriting (47) analogously as in ( 16) and ( 17), we conclude that (45) and (47) imply Hence, (48) yields cost(X step which establishes lower bound in (23) for class χ M noneq .
In this subsection we extend optimality results obtained for fixed truncation level sequences M to the general classes of considered methods χ eq , χ noneq .Our final conclusions are gathered in Theorem 3.
However, the lower bound in (49) cannot be asymptotically attained by any algorithm.Otherwise, the truncation level would satisfy M n ≈ δ −1 (n −1/2 ), which in turn would violate the property (4).Nevertheless, for every sequence M : and both sequences in (52) belong to G δ , with natural logarithm being composed k-times.
For fixed k, denote the sequence on the right side of (52 , and the output of an optimal method (X * ,k ) = X * ,k (t) t∈[0,T ] in corresponding class χ M k , ∈ {eq, noneq}, respectively.For every k ∈ N, X * ,k eq is a truncated dimension Euler scheme based on equidistant mesh, while X * ,k noneq is a truncated dimension Euler scheme with adaptive path-independent step-size control (32).
As a result, we have the following relation between the exact asymptotic behaviour of the method errors cost(X Mn, kn ) holding for all X Mn, kn ∈ χ M .Since the constructed sequence satisfies M k n = o(M n ), n → +∞, this concludes the proof.
Remark 7. Due to leveraging the alternative truncation sequence as per (52), the asymptotic benefit is at least of order M n / M k n 1/2 , which is unbounded and diverges to infinity, as n → +∞.In particular, this ratio can be significant also for smaller values of n, especially for sequences δ converging relatively slowly to zero.Remark 8.In class χ noneq we consider only algorithms which are non-adaptive with respect to the number of leveraged Wiener process coordinates.In particular, this results in lower bound for cost times error terms in ( 23), ( 24) diverge to infinity, as the cost rises.On the other hand, this is not the case for finite-dimensional noise structure, see e.g., [12], [20].We conjecture that this term can be significantly lowered when additional adaptation with respect to Wiener process coordinates is introduced.We plan to investigate such algorithms in our future work.

Solver implementation in Numba.
In this section we exhibit alternative implementation of X step M * n ,k * n , which leverages Numba compiler in Python.This enables us to execute specified functions, called kernels, on the GPU (Graphics Processing Unit) device which supports CUDA API developed by NVIDIA.For more details on parallel computing and kernel execution for a specified grid of blocks and threads, we refer to [10,23].Due to the fact that truncated dimension Euler scheme can be executed independently on each thread, we would expect significant decrease of the computation time when compared to the similar calculations on CPU (Central Processing Unit).This is beneficial especially when large number of the trajectories should be simulated.
The crucial part of the code responsible for the algorithm X step M * n ,k * n execution on GPU, together with relevant comments, can be found in the listing below.
1 # this decorator allows to execute the kernel on GPU device 2 @cuda .jit 3 def method _ s t e p _ s i z e ( STATES , n , M_n , x0 , timegrid , K , w_incr , out = Simulate NUM_THREADS different trajectories by method X ^{ step } _ { M_n , k_n } for fixed n and M_n .Listing 1. Truncated dimension Euler algorithm with adaptive step-sizecrucial part of the code using Numba compiler.

Results of numerical experiments in Python.
In this section, we present results of numerical experiments performed on CPU by using Multiprocessing library in Python programming language.We analyse the asymptotic error behaviour for both algorithms X step M * n ,k * n and X Eq M * n ,n by verifying if the ratio between constants C noneq and C eq , appearing in Theorem 2, is attained.
By substituting v = (2p − 1) log(x + 1), we arrive at and the integral appearing in (55) is equal to the upper incomplete gamma function Γ 1, (2p − 1) log(l + 1) .Since for all s ∈ N, x > 0 we have combining ( 55) and (56) yields Therefore, we can assume δ In our simulations, we set p = 0.9, hence M n n 5/4+ε ∈ G δ for all ε > 0. We decide to choose ε = 0.03.Consequently, it suffices to take M n = Θ(n 1.28 ), n → +∞.Now we provide the values of the constants C noneq and C eq in our model.First, for all t ∈ [0, T ] we have that σ(t) 2 = γ(e 2t + 2) with γ = 0.75638883... Finally, the constant appearing in lower bounds for χ eq is equal to T 6 0.25e 4T + 2e 2T − 9/4 + 4T The method error, err K (X alg ), is estimated by simulating K trajectories of the underlying processes.We measure the difference between each pair of trajectories by using the composite Simpson quadrature Q based on time points for which X alg is evaluated, together with the midpoints of the corresponding subintervals.To summarise, we take err K (X alg ) : where X alg l , X W ratio •M * n ,n * ,l , and W (l) are the l-th generated trajectories of the corresponding processes.Finally, we compare empirical improvement ratio err K (X The average improvement from leveraging adaptive mesh is generally visible.For n ∈ {5000, 10000} we executed smaller number of trajectories due to high complexity and time consumption.We also note that the impact of leveraging (57), Monte Carlo simulation, and rare-fine mesh comparison as an approximation of the method error is not quantified.Nevertheless, the obtained ratios are roughly aligned with the expected asymptotic error behaviour.

Conclusions
We investigated global approximation of SDEs driven by countably dimensional Wiener process, where the diffusion term depends only on the time variable.For fixed sequence δ, modelling level of decay for the diffusion term, we derived lower bounds for asymptotic error in suitable classes of algorithms leveraging specified truncation levels of the Wiener process.In particular, we quantified asymptotic benefit from leveraging step-size control instead of equidistant mesh.We also constructed two truncated dimension Euler schemes which are the (almost) optimal algorithms in the respective classes.Our results indicate that the decrease of method error requires significant increase of the cost term, which is illustrated by the product of cost and minimal error diverging to infinity.Nonetheless, we conjecture that the estimates might be beaten in case we allow for additional adaptation with respect to different Wiener process coordinates.
Therefore, we have for all t ∈ [0, T ] that

Table 1 .
noneq /C eq 0.85113723.The testing results are exhibited in Table1.Simulation results for X Eq * n.