Convergence rate of the truncated Euler-Maruyama method for highly nonlinear neutral stochastic differential equations with time-dependent delay
-
Aleksandra M. Petrović
Abstract
This article can be considered as a continuation of Petrović and Milošević [The truncated Euler-Maruyama method for highly nonlinear neutral stochastic differential equations with time-dependent delay, Filomat 35 (2021), no. 7, 2457–2484], where the authors established the
1 Introduction and preliminary results
It is well known that many complex dynamical systems depend not only on their present but also on their past states. So, neutral stochastic differential equations with time-dependent delay have become a powerful tool for modeling these systems [1–5]. The structure of these equations is specific, so it is very difficult to obtain their explicit solutions. Therefore, various numerical methods have taken on important role for generating corresponding approximate solutions of the observed equations such as those from the articles [6–16], among other publications.
Due to its convenience for computations and implementation, the Euler-Maruyama (EM) method has stood out from other methods, and as such it has often been used to obtain approximate solutions for different types of stochastic differential equations [17–21]. However, article by Hutzenthaler et al. [22] reveals a large class of ordinary stochastic differential equations with superlinearly growing coefficients for which the EM method cannot be convergent in the strong and numerically weak
On the other hand, the
Besides the previously mentioned novelty aspects of the article, it should be stressed that two more conditions, related to the increments of the coefficients and neutral term of the equations under consideration, are imposed. These conditions are essential for proving the
Since the results in this article represent a continuation of the work from [24], for convenience, we will use throughout the whole article the same notation as in the cited article. In that sense, it is assumed that all random variables and processes are defined on a filtered probability space
For a given
Let
satisfying the initial condition
where the functions
are all Borel measurable and
In order to prove main results of this article, we will use the hypotheses from [24] that are listed below.
Moreover, we suppose that
Clearly, particularly for
2 Truncated EM approximate solution
Recall the truncated EM numerical scheme defined in [24]. The function
The inverse function of
and for all
Remark 1
In [24], it was required to choose a number
and for all
Hence, we simply let
In [24], for a given step size
where we set
for
which together with (7) implies that, for all
So, we consider the following stochastic differential equation:
satisfying the initial condition (2).
Moreover, in [24], the truncated EM approximate solution
Then, the following definition is imposed:
where
are introduced, where
whenever
Then, the continuous-time truncated EM approximate solution
Since, for
On the other hand, for
Moreover, for all
In what follows, before stating the main results of this article, we give some assertions from the article [24], which will be used explicitly throughout the article.
Theorem 1
[24, Theorem 1.1] Let Assumptions
Lemma 1
[24, Lemma 2.1] Let Assumption
Moreover, for all
Lemma 2
[24, Lemma 2.2] Let Assumption
where
Lemma 3
[24, Lemma 3.1] If Assumptions
then, for any
where constants
Corollary 1
[24, Corollary 3.2] Let the conditions of
Lemma 3
hold. Then, for any integer
where
Lemma 4
[24, Lemma 3.4] Let the conditions of
Lemma 3
hold together with Assumption
where C is a positive real constant dependent on
In the sequel, we will prove that the truncated function
Lemma 5
Let Assumption
Moreover, if
Proof
Let
On the other hand, for
If
Now, on the basis of Cauchy-Schwarz inequality, we obtain
Using the condition
On the other hand, for
Now, Cauchy-Schwarz inequality yields
Based on the assumption
For
Since
The same way (32) follows for
Now, we suppose that one of the values
If
Having in mind the previous lemma, we can prove the following one under the condition on the Lipschitz constants of
Lemma 6
If Assumptions
then, for any
where constants
Proof
Fix any
When
Applying the elementary inequality
and on the basis of (32), we obtain
On the basis of (12), we see that
Now, from the proof of Lemma 3, we have that
and
Substituting (39), (40), and (41) into (38), bearing in mind that
On the basis of condition (35), there exists
Applying (36), inequality (42) yields
Thus, we find that
where
which completes the proof.□
The following lemma shows that the truncated functions
Lemma 7
Let Assumption
where
Proof
This lemma essentially represents Lemma 2, but here we show that the condition
Fix any
Furthermore, if
On the basis of
where
In view of Lemma 7, we see that the constant
3 Convergence rates
In [24], the
For further consideration, the following hypotheses are essential.
It is useful to emphasize that the truncated functions
Based on the definition of the function
Moreover, we observe from Lemma 5 that, for all
Therefore, (46) can be estimated as
On the other hand, from Assumption
In a similar way, we can estimate the expression
where
For completing this section, we establish the following two lemmas that are essential for obtaining the main results of this article. In that sense, we first impose the following remark.
Remark 2
It should be emphasized that, when applying Lemmas 3 and 4 in the proofs of the following assertions, based on Remark 1, we will consider that
In the next lemma, we will estimate the delayed terms of the continuous
Lemma 8
Let Assumptions
where
Proof
Fix any
Otherwise, for
Therefore, we obtain
On the other hand, on the basis of
For
Now, by Definition (19) of the step-process
In order to estimate
Case 1: If
Case 2: If
Now, we estimate the expression
In a similar way, using (50), we obtain
Then, on the basis of Lemma 6, we conclude that
By the Hölder and Burkholder-Davis-Gundy inequalities, as well as (12), we obtain
Substituting (56) and (57) into (53), we obtain that
In order to estimate the second summand of (52), we use Definition (17) and obtain that
whenever
Then, in a view of Lemma 6, we obtain
Therefore, since
where
Now, we are in a position to estimate the closeness between the delayed term of the continuous truncated EM solution
Corollary 2
Let the conditions of
Lemma 8
hold, together with Assumption
where
Proof
Fix any
On the other hand, by the Hölder and Chebyshev inequalities, we obtain
So, in a view of (21) and Lemma 6, we find that
Thus, substitution of (63) into (62) implies that
where
Therefore, the proof is complete.□
In what follows, we will prove the main result of this article which establishes the rate of convergence of the continuous truncated EM solution
Theorem 2
Let Assumptions
where
In particular, recalling (49), we may define, for
and let
Proof
First, we stress that Assumption
Fix
For each integer
where
Note that
Then, (69) can be estimated as
where
First, we estimate the integral
we derive that
Furthermore, in a view of Assumption
Using Assumption
Substituting the previous estimate into (74), we obtain that
So, applying Corollary 2, we can conclude that for
The function
Therefore, (75) can be estimated as
On the other hand, the integral
where
In order to estimate
where
Based on Assumption
Using the Hölder inequality, together with (21), we observe that
Applying the similar procedure as in (63), we obtain that
Therefore, based on Lemma 4, we estimate (80) as
where
Substituting (81) into (79), we obtain
Analogously to (79), we can show that
where
Applying Assumption
Now, (21), (22), and (23), together with Lemma 4, give
where
By Corollary 1, we have that for
where
Then, substitution of the estimates (82) and (85) into (78) yields
Consequently, (77) and (86), together with (70), give
Based on Assumption
Hence, we obtain that for
where
In the sequel, we will use the elementary inequality
Based on the fact that the solutions
So, for
Substituting the previous estimate into (88), we obtain
Now, the application of the Gronwall-Bellman lemma to (90) yields
where
Using the well-known Fatou lemma, we can let
Finally, if
where
Therefore, (94) is the required assertion (67). Similarly, we can show (68). The proof is therefore complete.□
The following theorem shows that the rate of
Theorem 3
Let the conditions of
Theorem 2
hold. Then, for any
Proof
Choosing
and
Let us discuss an example to illustrate our theory.
Example 1
Consider the following one-dimensional neutral stochastic differential equation with time-dependent delay:
where
Clearly, the coefficients
where
Thus, Assumption
we find that
Furthermore, it is easy to show that for any
Thus, Assumption
Thus, Assumption
So, we can have
Therefore, it follows from Theorem 3 that the truncated EM solutions of equation (99) satisfy relations (96) and (97), i.e., the order of
Now, we can choose
For such a small step size
This is almost impossible so results from [24] are much more restrictive. However, results in this article can still be applied. Also, we see that for
So, Theorem 3 is not only applicable in this situation but also shows that the truncated EM solution
We choose
On the other hand, on the basis of Theorem 3, for
where
Therefore, the problem reduces to comparing the lines
which tends to
However, in order to make a conclusion to the initial problem, the function
which tends to 0 when
Also, for the computational convenience, the constant
Moreover, for fixed parameters
Additionally, if we increase the values of
Trajectories of the truncated EM solution with
![Figure 2
Lines
a
1
2
,
Δ
a\left(\phantom{\rule[-0.75em]{}{0ex}},\frac{1}{2},\Delta \right)
(orange line) and
a
1
2
−
2
ε
,
Δ
a\left(\phantom{\rule[-0.75em]{}{0ex}},\frac{1}{2}-2\varepsilon ,\Delta \right)
(blue line) for
Δ
∈
(
0
,
1
)
,
q
¯
=
2
,
ε
=
0.1
\Delta \in \left(0,1),\bar{q}=2,\varepsilon =0.1
.](/document/doi/10.1515/math-2024-0038/asset/graphic/j_math-2024-0038_fig_002.jpg)
Lines
![Figure 3
Lines
a
1
2
,
Δ
a\left(\phantom{\rule[-0.75em]{}{0ex}},\frac{1}{2},\Delta \right)
(orange line) and
a
1
2
−
2
ε
,
Δ
a\left(\phantom{\rule[-0.75em]{}{0ex}},\frac{1}{2}-2\varepsilon ,\Delta \right)
(blue line) for
Δ
∈
(
0
,
1
)
,
p
=
13
,
q
¯
=
2
,
ε
0
=
0.001
,
\Delta \in \left(0,1),p=13,\bar{q}=2,{\varepsilon }_{0}=0.001,
(a)
ε
=
0.2
,
\varepsilon =0.2,
(b)
ε
=
0.17
,
\varepsilon =0.17,
(c)
ε
=
0.14
,
\varepsilon =0.14,
and (d)
ε
=
0.1
.
\varepsilon =0.1.](/document/doi/10.1515/math-2024-0038/asset/graphic/j_math-2024-0038_fig_003.jpg)
Lines
![Figure 4
Lines
a
1
2
,
Δ
a\left(\phantom{\rule[-0.75em]{}{0ex}},\frac{1}{2},\Delta \right)
(orange line) and
a
1
2
−
2
ε
,
Δ
a\left(\phantom{\rule[-0.75em]{}{0ex}},\frac{1}{2}-2\varepsilon ,\Delta \right)
(blue line) for
Δ
∈
(
0
,
1
)
,
p
=
20
,
q
¯
=
3
,
ε
0
=
0.001
,
\Delta \in \left(0,1),p=20,\bar{q}=3,{\varepsilon }_{0}=0.001,
(a)
ε
=
0.2
,
\varepsilon =0.2,
(b)
ε
=
0.17
,
\varepsilon =0.17,
(c)
ε
=
0.14
,
\varepsilon =0.14,
and (d)
ε
=
0.1
\varepsilon =0.1
.](/document/doi/10.1515/math-2024-0038/asset/graphic/j_math-2024-0038_fig_004.jpg)
Lines
4 Conclusions
On the basis of the previously presented results, one can conclude that the rate of the
On the other hand, when one applies this method, one should pay attention for which value of
which affects the convergence rate of the method.
Possible directions of the future research could be the extensions of the presented results to neutral stochastic differential equations with time-distributed delay, with state-dependent delay, for example, with or without some additional noise, such as Poison jump, Lévy noise, or Markovian switching. Also, the future study could be based on determining some other sufficient conditions of the
Acknowledgment
The author is grateful for the reviewer’s valuable comments that improved the manuscript.
-
Funding information: This work was supported by Ministry of Education, Science and Technological Development of the Republic of Serbia, Contract Number: 451-03-66/2024-03/200124.
-
Author contributions: The author confirms the sole responsibility for the conception of the study, presented results, and prepared manuscript.
-
Conflict of interest: The author states no conflict of interest.
-
Ethical approval: The conducted research is not related to either human or animal use.
-
Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.
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© 2024 the author(s), published by De Gruyter
This work is licensed under the Creative Commons Attribution 4.0 International License.
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- Erratum to “Infinitesimals via Cauchy sequences: Refining the classical equivalence”
- Corrigendum
- Corrigendum to “Matrix stretching”
- Corrigendum to “A comprehensive review of the recent numerical methods for solving FPDEs”
Articles in the same Issue
- Special Issue on Contemporary Developments in Graph Topological Indices
- On the maximum atom-bond sum-connectivity index of graphs
- Upper bounds for the global cyclicity index
- Zagreb connection indices on polyomino chains and random polyomino chains
- On the multiplicative sum Zagreb index of molecular graphs
- The minimum matching energy of unicyclic graphs with fixed number of vertices of degree two
- Special Issue on Convex Analysis and Applications - Part I
- Weighted Hermite-Hadamard-type inequalities without any symmetry condition on the weight function
- Scattering threshold for the focusing energy-critical generalized Hartree equation
- (p, q)-Compactness in spaces of holomorphic mappings
- Characterizations of minimal elements of upper support with applications in minimizing DC functions
- Some new Hermite-Hadamard-type inequalities for strongly h-convex functions on co-ordinates
- Global existence and extinction for a fast diffusion p-Laplace equation with logarithmic nonlinearity and special medium void
- Extension of Fejér's inequality to the class of sub-biharmonic functions
- On sup- and inf-attaining functionals
- Regularization method and a posteriori error estimates for the two membranes problem
- Rapid Communication
- Note on quasivarieties generated by finite pointed abelian groups
- Review Articles
- Amitsur's theorem, semicentral idempotents, and additively idempotent semirings
- A comprehensive review of the recent numerical methods for solving FPDEs
- On an Oberbeck-Boussinesq model relating to the motion of a viscous fluid subject to heating
- Pullback and uniform exponential attractors for non-autonomous Oregonator systems
- Regular Articles
- On certain functional equation related to derivations
- The product of a quartic and a sextic number cannot be octic
- Combined system of additive functional equations in Banach algebras
- Enhanced Young-type inequalities utilizing Kantorovich approach for semidefinite matrices
- Local and global solvability for the Boussinesq system in Besov spaces
- Construction of 4 x 4 symmetric stochastic matrices with given spectra
- A conjecture of Mallows and Sloane with the universal denominator of Hilbert series
- The uniqueness of expression for generalized quadratic matrices
- On the generalized exponential sums and their fourth power mean
- Infinitely many solutions for Schrödinger equations with Hardy potential and Berestycki-Lions conditions
- Computing the determinant of a signed graph
- Two results on the value distribution of meromorphic functions
- Zariski topology on the secondary-like spectrum of a module
- On deferred f-statistical convergence for double sequences
-
About
- Strong convergence for weighted sums of (α, β)-mixing random variables and application to simple linear EV regression model
- On the distribution of powered numbers
- Almost periodic dynamics for a delayed differential neoclassical growth model with discontinuous control strategy
- A new distributionally robust reward-risk model for portfolio optimization
- Asymptotic behavior of solutions of a viscoelastic Shear beam model with no rotary inertia: General and optimal decay results
- Silting modules over a class of Morita rings
- Non-oscillation of linear differential equations with coefficients containing powers of natural logarithm
- Mutually unbiased bases via complex projective trigonometry
- Hyers-Ulam stability of a nonlinear partial integro-differential equation of order three
- On second-order linear Stieltjes differential equations with non-constant coefficients
- Complex dynamics of a nonlinear discrete predator-prey system with Allee effect
- The fibering method approach for a Schrödinger-Poisson system with p-Laplacian in bounded domains
- On discrete inequalities for some classes of sequences
- Boundary value problems for integro-differential and singular higher-order differential equations
- Existence and properties of soliton solution for the quasilinear Schrödinger system
- Hermite-Hadamard-type inequalities for generalized trigonometrically and hyperbolic ρ-convex functions in two dimension
- Endpoint boundedness of toroidal pseudo-differential operators
- Matrix stretching
- A singular perturbation result for a class of periodic-parabolic BVPs
- On Laguerre-Sobolev matrix orthogonal polynomials
- Pullback attractors for fractional lattice systems with delays in weighted space
- Singularities of spherical surface in R4
- Variational approach to Kirchhoff-type second-order impulsive differential systems
- Convergence rate of the truncated Euler-Maruyama method for highly nonlinear neutral stochastic differential equations with time-dependent delay
- On the energy decay of a coupled nonlinear suspension bridge problem with nonlinear feedback
- The limit theorems on extreme order statistics and partial sums of i.i.d. random variables
- Hardy-type inequalities for a class of iterated operators and their application to Morrey-type spaces
- Solving multi-point problem for Volterra-Fredholm integro-differential equations using Dzhumabaev parameterization method
- Finite groups with gcd(χ(1), χc (1)) a prime
- Small values and functional laws of the iterated logarithm for operator fractional Brownian motion
- The hull-kernel topology on prime ideals in ordered semigroups
- ℐ-sn-metrizable spaces and the images of semi-metric spaces
- Strong laws for weighted sums of widely orthant dependent random variables and applications
- An extension of Schweitzer's inequality to Riemann-Liouville fractional integral
- Construction of a class of half-discrete Hilbert-type inequalities in the whole plane
- Analysis of two-grid method for second-order hyperbolic equation by expanded mixed finite element methods
- Note on stability estimation of stochastic difference equations
- Trigonometric integrals evaluated in terms of Riemann zeta and Dirichlet beta functions
- Purity and hybridness of two tensors on a real hypersurface in complex projective space
- Classification of positive solutions for a weighted integral system on the half-space
- A quasi-reversibility method for solving nonhomogeneous sideways heat equation
- Higher-order nonlocal multipoint q-integral boundary value problems for fractional q-difference equations with dual hybrid terms
- Noetherian rings of composite generalized power series
- On generalized shifts of the Mellin transform of the Riemann zeta-function
- Further results on enumeration of perfect matchings of Cartesian product graphs
- A new extended Mulholland's inequality involving one partial sum
- Power vector inequalities for operator pairs in Hilbert spaces and their applications
- On the common zeros of quasi-modular forms for Γ+0(N) of level N = 1, 2, 3
- One special kind of Kloosterman sum and its fourth-power mean
- The stability of high ring homomorphisms and derivations on fuzzy Banach algebras
- Integral mean estimates of Turán-type inequalities for the polar derivative of a polynomial with restricted zeros
- Commutators of multilinear fractional maximal operators with Lipschitz functions on Morrey spaces
- Vector optimization problems with weakened convex and weakened affine constraints in linear topological spaces
- The curvature entropy inequalities of convex bodies
- Brouwer's conjecture for the sum of the k largest Laplacian eigenvalues of some graphs
- High-order finite-difference ghost-point methods for elliptic problems in domains with curved boundaries
-
Riemannian invariants for warped product submanifolds in
- Generalized quadratic Gauss sums and their 2mth power mean
- Euler-α equations in a three-dimensional bounded domain with Dirichlet boundary conditions
- Enochs conjecture for cotorsion pairs over recollements of exact categories
- Zeros distribution and interlacing property for certain polynomial sequences
- Random attractors of Kirchhoff-type reaction–diffusion equations without uniqueness driven by nonlinear colored noise
- Study on solutions of the systems of complex product-type PDEs with more general forms in ℂ2
- Dynamics in a predator-prey model with predation-driven Allee effect and memory effect
- A note on orthogonal decomposition of 𝔰𝔩n over commutative rings
- On the δ-chromatic numbers of the Cartesian products of graphs
- Binomial convolution sum of divisor functions associated with Dirichlet character modulo 8
- Commutator of fractional integral with Lipschitz functions related to Schrödinger operator on local generalized mixed Morrey spaces
- System of degenerate parabolic p-Laplacian
- Stochastic stability and instability of rumor model
- Certain properties and characterizations of a novel family of bivariate 2D-q Hermite polynomials
- Stability of an additive-quadratic functional equation in modular spaces
- Monotonicity, convexity, and Maclaurin series expansion of Qi's normalized remainder of Maclaurin series expansion with relation to cosine
- On k-prime graphs
- On the existence of tripartite graphs and n-partite graphs
- Classifying pentavalent symmetric graphs of order 12pq
- Almost periodic functions on time scales and their properties
- Some results on uniqueness and higher order difference equations
- Coding of hypersurfaces in Euclidean spaces by a constant vector
- Cycle integrals and rational period functions for Γ0+(2) and Γ0+(3)
- Degrees of (L, M)-fuzzy bornologies
- A matrix approach to determine optimal predictors in a constrained linear mixed model
- On ideals of affine semigroups and affine semigroups with maximal embedding dimension
- Solutions of linear control systems on Lie groups
- A uniqueness result for the fractional Schrödinger-Poisson system with strong singularity
- On prime spaces of neutrosophic extended triplet groups
- On a generalized Krasnoselskii fixed point theorem
- On the relation between one-sided duoness and commutators
- Non-homogeneous BVPs for second-order symmetric Hamiltonian systems
- Erratum
- Erratum to “Infinitesimals via Cauchy sequences: Refining the classical equivalence”
- Corrigendum
- Corrigendum to “Matrix stretching”
- Corrigendum to “A comprehensive review of the recent numerical methods for solving FPDEs”
