pith. sign in

arxiv: 1907.11408 · v1 · pith:TC27AETYnew · submitted 2019-07-26 · 🧮 math.NA · cs.NA· math.PR

Semi-implicit Euler-Maruyama method for non-linear time-changed stochastic differential equations

Chang-song Deng , Wei Liu This is my paper

Pith reviewed 2026-05-24 15:44 UTC · model grok-4.3

classification 🧮 math.NA cs.NAmath.PR
keywords semi-implicit Euler-Maruyama methodtime-changed stochastic differential equationsstrong convergencemean square polynomial stabilityinverse subordinatorBernstein function
0
0 comments X

The pith

The semi-implicit Euler-Maruyama method converges strongly for time-changed SDEs whose drift grows super-linearly.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines a semi-implicit Euler-Maruyama scheme applied to stochastic differential equations whose time argument is replaced by an inverse subordinator. It establishes strong convergence of the scheme even when the drift coefficient grows faster than linearly, provided the diffusion coefficient remains globally Lipschitz. Under the additional condition that the Bernstein function of the inverse subordinator is regularly varying at zero, both the continuous equations and their numerical approximations are shown to possess mean-square polynomial stability.

Core claim

The semi-implicit Euler-Maruyama method is strongly convergent for time-changed SDEs with super-linear drift and globally Lipschitz diffusion; when the Bernstein function of the inverse subordinator is regularly varying at zero, the method also preserves the mean-square polynomial stability of the underlying equations.

What carries the argument

Semi-implicit Euler-Maruyama discretization of the time-changed SDE driven by an inverse subordinator.

If this is right

  • The scheme supplies a reliable simulation tool for the given class of equations.
  • Mean-square polynomial stability of the continuous model is inherited by the discrete trajectories.
  • The convergence rate depends on the regular-variation index of the Bernstein function.
  • The result extends the classical Euler-Maruyama theory to time-changed processes with super-linear drift.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same stability preservation argument could be tested on other implicit or semi-implicit schemes for the same equations.
  • The regular-variation condition on the Bernstein function may link the long-time behaviour of the numerical solutions to the tail properties of the subordinator.
  • If the convergence rate can be made explicit in terms of the variation index, the method becomes usable for quantitative error control in applications.

Load-bearing premise

The diffusion coefficient satisfies the global Lipschitz condition, and for the stability preservation result the Bernstein function of the inverse subordinator is regularly varying at zero.

What would settle it

A concrete counter-example in which the diffusion fails to be globally Lipschitz yet the semi-implicit scheme still converges strongly, or a time change whose Bernstein function is not regularly varying at zero yet the numerical solutions lose mean-square polynomial stability.

Figures

Figures reproduced from arXiv: 1907.11408 by Chang-song Deng, Wei Liu.

Figure 1
Figure 1. Figure 1: Numerical simulations of D(t), E(t) and X(t) step size, h0 = 10−6 , is regarded as the true solution. The step sizes of h = 10−2 , 10−3 and 10−4 are used to calculated the numerical solutions. For the given step size h, the L 1 strong error is calculated by 1 N X N i=1 |yi(Eh0 (T)) − yi(Eh(T))| . Two hundreds (N = 200) sample paths are used to draw Loglog plot of the L 1 error against the step sizes in [P… view at source ↗
Figure 2
Figure 2. Figure 2: The L 1 errors between the exact solution and the numerical solutions for step sizes ∆ = 10−2 , 10−3 , 10−4 . 0 2 4 6 8 10 0 5 10 15 20 25 t E|x(t)| 2 (a) Mean square of y(Eh(t)) 0 5 10 15 20 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 t x(t) (b) paths of y(Eh(t)) [PITH_FULL_IMAGE:figures/full_fig_p018_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Stabilities of numerical solutions 18 [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗
read the original abstract

The semi-implicit Euler-Maruyama (EM) method is investigated to approximate a class of time-changed stochastic differential equations, whose drift coefficient can grow super-linearly and diffusion coefficient obeys the global Lipschitz condition. The strong convergence of the semi-implicit EM is proved and the convergence rate is discussed. When the Bernstein function of the inverse subordinator (time-change) is regularly varying at zero, we establish the mean square polynomial stability of the underlying equations. In addition, the numerical method is proved to be able to preserve such an asymptotic property. Numerical simulations are presented to demonstrate the theoretical results.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper investigates the semi-implicit Euler-Maruyama method for approximating a class of time-changed stochastic differential equations with super-linearly growing drift and globally Lipschitz diffusion. It proves the strong convergence of the method and discusses the convergence rate. When the Bernstein function of the inverse subordinator is regularly varying at zero, the paper establishes the mean square polynomial stability of the underlying equations and proves that the numerical method preserves this asymptotic property. Numerical simulations are presented to demonstrate the theoretical results.

Significance. This work provides a valuable extension of numerical methods to time-changed SDEs with non-Lipschitz drifts, which is important for applications involving anomalous diffusion and memory effects. The proofs of convergence and the preservation of stability under the regular variation condition represent a solid contribution to the field of stochastic numerical analysis. The inclusion of numerical experiments strengthens the practical relevance of the theoretical findings.

minor comments (3)
  1. The abstract states that the convergence rate is discussed but does not indicate the specific rate obtained; adding this detail would strengthen the summary of the main results.
  2. In the numerical simulations section, additional information on the discretization of the inverse subordinator and parameter choices would improve reproducibility of the experiments.
  3. A short comparison in the introduction with prior numerical schemes for subordinated or time-changed SDEs would better highlight the novelty of the semi-implicit approach.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary and recommendation of minor revision. The assessment accurately captures the paper's contributions on strong convergence and stability preservation for the semi-implicit EM method applied to time-changed SDEs with super-linear drift. Since no specific major comments were raised, we have no points requiring detailed rebuttal or revision at this stage.

Circularity Check

0 steps flagged

No significant circularity; derivation relies on standard SDE theory

full rationale

The paper proves strong convergence of the semi-implicit Euler-Maruyama scheme for time-changed SDEs with super-linear drift and globally Lipschitz diffusion, plus mean-square polynomial stability preservation when the Bernstein function is regularly varying at zero. These results are established under explicitly stated assumptions using properties of subordinators and standard numerical SDE analysis techniques. No load-bearing step reduces by construction to a fitted parameter, self-definition, or self-citation chain; the central claims remain independent of the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract only; no free parameters, axioms, or invented entities are identifiable from the provided text.

pith-pipeline@v0.9.0 · 5625 in / 1110 out tokens · 24034 ms · 2026-05-24T15:44:25.942369+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

24 extracted references · 24 canonical work pages

  1. [1]

    N. H. Bingham, C. M. Goldie, and J. L. Teugels. Regular variation, volume 27 of Encyclopedia of Mathematics and its Applications . Cambridge University Press, Cambridge, 1987

    work page 1987

  2. [2]

    Gajda and M

    J. Gajda and M. Magdziarz. Fractional Fokker-Planck equation with tempered α- stable waiting times: Langevin picture and computer simulation. Phys. Rev. E (3) , 82(1):011117, 6, 2010

    work page 2010

  3. [3]

    D. J. Higham, X. Mao, and C. Yuan. Almost sure and moment exponential stability in the numerical simulation of stochastic differential equations. SIAM J. Numer. Anal., 45(2):592–609 (electronic), 2007

    work page 2007

  4. [4]

    Y. Hu. Semi-implicit Euler-Maruyama scheme for stiff stochastic equations. In Stochastic analysis and related topics, V (Silivri, 1994) , volume 38 of Progr. Probab., pages 183–202. Birkh¨ auser Boston, Boston, MA, 1996

    work page 1994

  5. [5]

    Hutzenthaler, A

    M. Hutzenthaler, A. Jentzen, and P. E. Kloeden. Strong and weak divergence in finite time of Euler’s method for stochastic differential equations with non-globally Lipschitz continuous coefficients. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. , 467(2130):1563–1576, 2011. 19

    work page 2011

  6. [6]

    Jin and K

    S. Jin and K. Kobayashi. Strong approximation of stochastic differential equations driven by a time-changed Brownian motion with time-space-dependent coefficients. J. Math. Anal. Appl. , 476(2):619–636, 2019

    work page 2019

  7. [7]

    E. Jum. Numerical Approximation of Stochastic Differential Equations Driven by L´ evy Motion with Infinitely Many Jumps . PhD thesis, University of Tennessee - Knoxville, 2015

    work page 2015

  8. [8]

    Jum and K

    E. Jum and K. Kobayashi. A strong and weak approximation scheme for stochastic differential equations driven by a time-changed Brownian motion. Probab. Math. Statist., 36(2):201–220, 2016

    work page 2016

  9. [9]

    Kobayashi

    K. Kobayashi. Stochastic calculus for a time-changed semimartingale and the asso- ciated stochastic differential equations. J. Theoret. Probab., 24(3):789–820, 2011

    work page 2011

  10. [10]

    X. Li, Q. Ma, H. Yang, and C. Yuan. The numerical invariant measure of stochastic differential equations with Markovian switching. SIAM J. Numer. Anal., 56(3):1435– 1455, 2018

    work page 2018

  11. [11]

    Liu and X

    W. Liu and X. Mao. Numerical stationary distribution and its convergence for non- linear stochastic differential equations. J. Comput. Appl. Math. , 276:16–29, 2015

    work page 2015

  12. [12]

    Magdziarz

    M. Magdziarz. Stochastic representation of subdiffusion processes with time- dependent drift. Stochastic Process. Appl., 119(10):3238–3252, 2009

    work page 2009

  13. [13]

    Magdziarz, S

    M. Magdziarz, S. Orzel, and A. Weron. Option pricing in subdiffusive Bachelier model. J. Stat. Phys. , 145(1):187–203, 2011

    work page 2011

  14. [14]

    X. Mao. Stochastic differential equations and applications . Horwood Publishing Limited, Chichester, second edition, 2008

    work page 2008

  15. [15]

    Mao and L

    X. Mao and L. Szpruch. Strong convergence rates for backward Euler-Maruyama method for non-linear dissipative-type stochastic differential equations with super- linear diffusion coefficients. Stochastics, 85(1):144–171, 2013

    work page 2013

  16. [16]

    M. M. Meerschaert and H.-P. Scheffler. Limit theorems for continuous-time random walks with infinite mean waiting times. J. Appl. Probab., 41(3):623–638, 2004. 20

    work page 2004

  17. [17]

    Nane and Y

    E. Nane and Y. Ni. Stability of the solution of stochastic differential equation driven by time-changed L´ evy noise.Proc. Amer. Math. Soc., 145(7):3085–3104, 2017

    work page 2017

  18. [18]

    Nane and Y

    E. Nane and Y. Ni. Path stability of stochastic differential equations driven by time-changed L´ evy noises.ALEA Lat. Am. J. Probab. Math. Stat. , 15(1):479–507, 2018

    work page 2018

  19. [19]

    Neuenkirch and L

    A. Neuenkirch and L. Szpruch. First order strong approximations of scalar SDEs defined in a domain. Numer. Math., 128(1):103–136, 2014

    work page 2014

  20. [20]

    R. L. Schilling, R. Song, and Z. Vondraˇ cek. Bernstein functions. Theory and appli- cations, volume 37 of De Gruyter Studies in Mathematics . Walter de Gruyter & Co., Berlin, second edition, 2012

    work page 2012

  21. [21]

    H. Schurz. Stability, stationarity, and boundedness of some implicit numerical meth- ods for stochastic differential equations and applications. Logos Verlag Berlin, Berlin, 1997

    work page 1997

  22. [22]

    Umarov, M

    S. Umarov, M. Hahn, and K. Kobayashi. Beyond the triangle: Brownian motion, Ito calculus, and Fokker-Planck equation—fractional generalizations . World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2018

    work page 2018

  23. [23]

    Wang and X

    L. Wang and X. Wang. Convergence of the semi-implicit Euler method for stochas- tic age-dependent population equations with Poisson jumps. Appl. Math. Model. , 34(8):2034–2043, 2010

    work page 2034

  24. [24]

    Q. Wu. Stability analysis for a class of nonlinear time-changed systems. Cogent Math., 3:Art. ID 1228273, 10, 2016. 21

    work page 2016