Skip to main content
Fachgebiete
Automobil + Motoren Bauwesen + Immobilien Business IT + Informatik Elektrotechnik + Elektronik Energie + Nachhaltigkeit Finance + Banking Management + Führung Marketing + Vertrieb Maschinenbau + Werkstoffe Versicherung + Risiko
DE
EN
Bücher
Zeitschriften
Themenseiten
Organisationspsychologie Projektmanagement Marketing Smart Manufacturing
Weitere Formate
Podcasts Webinare Technik Webinare Wirtschaft Kongresse Veranstaltungskalender Awards
Jetzt Einzelzugang starten
Zugang für Unternehmen
Referenzkunden
SLX-Digitalkonferenz: Zukunftswerkstatt 2024

Mehr Umsatz mit Top-Kontakten

Am 7. Mai erfahren Sie, wie Vertriebsteams Leadgenerierung, Leadnurturing und Leadstrategien effizient steuern können.
Erweiterte Suche
Anmelden
nach oben
BIT Numerical Mathematics
Erschienen in: BIT Numerical Mathematics 4/2020

12.05.2020

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

verfasst von: Chang-Song Deng, Wei Liu

Erschienen in: BIT Numerical Mathematics | Ausgabe 4/2020

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config
loading …

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.
Vorheriger Artikel Approximation of the matrix exponential for matrices with a skinny field of values
Nächster Artikel Symplectic dynamical low rank approximation of wave equations with random parameters

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren
Literatur
1.
Bingham, N.H., Goldie, C.M., Teugels, J.L.: Regular Variation, Volume 27 of Encyclopedia of Mathematics and Its Applications. Cambridge University Press, Cambridge (1987)
2.
Gajda, J., Magdziarz, M.: Fractional Fokker–Planck equation with tempered \(\alpha \)-stable waiting times: Langevin picture and computer simulation. Phys. Rev. E (3) 82(1), 011117 (2010)MathSciNetCrossRef
3.
Higham, D.J., Mao, X., Yuan, C.: Almost sure and moment exponential stability in the numerical simulation of stochastic differential equations. SIAM J. Numer. Anal. 45(2), 592–609 (2007). (electronic)MathSciNetCrossRef
4.
Hu, Y.: Semi-implicit Euler–Maruyama scheme for stiff stochastic equations. In: Korezlioglu, H., Oksendal, B., Ustunel, A.S. (eds.) Stochastic Analysis and Related Topics V (Silivri, 1994). Progress in Probability, pp. 183–202. Birkhäuser, Boston (1996)
5.
Hutzenthaler, M., Jentzen, A., Kloeden, P.E.: 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)MathSciNetMATH
6.
Jin, S., Kobayashi, K.: 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)MathSciNetCrossRef
7.
Jum, E.: Numerical approximation of stochastic differential equations driven by Lévy motion with infinitely many jumps. Ph.D. thesis, University of Tennessee - Knoxville (2015)
8.
Jum, E., Kobayashi, K.: A strong and weak approximation scheme for stochastic differential equations driven by a time-changed Brownian motion. Probab. Math. Stat. 36(2), 201–220 (2016)MathSciNetMATH
9.
Khasminskii, R.: Stochastic Stability of Differential Equations. With Contributions by G. N. Milstein and M. B. Nevelson. Completely Revised and Enlarged Second Edition. Stochastic Modelling and Applied Probability, vol. 66. Springer, Heidelberg (2012)MATH
10.
Kobayashi, K.: Stochastic calculus for a time-changed semimartingale and the associated stochastic differential equations. J. Theor. Probab. 24(3), 789–820 (2011)MathSciNetCrossRef
11.
Li, X., Ma, Q., Yang, H., Yuan, C.: The numerical invariant measure of stochastic differential equations with Markovian switching. SIAM J. Numer. Anal. 56(3), 1435–1455 (2018)MathSciNetCrossRef
12.
Liu, W., Mao, X.: Numerical stationary distribution and its convergence for nonlinear stochastic differential equations. J. Comput. Appl. Math. 276, 16–29 (2015)MathSciNetCrossRef
13.
Magdziarz, M.: Stochastic representation of subdiffusion processes with time-dependent drift. Stoch. Process. Appl. 119(10), 3238–3252 (2009)MathSciNetCrossRef
14.
Magdziarz, M., Orzel, S., Weron, A.: Option pricing in subdiffusive Bachelier model. J. Stat. Phys. 145(1), 187–203 (2011)MathSciNetCrossRef
15.
Mao, X.: Stochastic Differential Equations and Applications, second edn. Horwood Publishing Limited, Chichester (2008)CrossRef
16.
Mao, X.: Stability of Stochastic Differential Equations with Respect to Semimartingales. Pitman Research Notes in Mathematics Series, vol. 251. Longman Scientific & Technical, Harlow (1991). (copublished in the United States with Wiley, New York)
17.
Mao, X., Szpruch, L.: 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)MathSciNetCrossRef
18.
Meerschaert, M.M., Scheffler, H.-P.: Limit theorems for continuous-time random walks with infinite mean waiting times. J. Appl. Probab. 41(3), 623–638 (2004)MathSciNetCrossRef
19.
Nane, E., Ni, Y.: Stability of the solution of stochastic differential equation driven by time-changed Lévy noise. Proc. Am. Math. Soc. 145(7), 3085–3104 (2017)CrossRef
20.
Nane, E., Ni, Y.: Path stability of stochastic differential equations driven by time-changed Lévy noises. ALEA Lat. Am. J. Probab. Math. Stat. 15(1), 479–507 (2018)MathSciNetCrossRef
21.
Neuenkirch, A., Szpruch, L.: First order strong approximations of scalar SDEs defined in a domain. Numer. Math. 128(1), 103–136 (2014)MathSciNetCrossRef
22.
Protter, P.E.: Stochastic Integration and Differential Equations. Applications of Mathematics (New York). Stochastic Modelling and Applied Probability, vol. 21, 2nd edn. Springer, Berlin (2004)
23.
Schilling, R.L., Song, R., Vondraček, Z.: Bernstein Functions. Theory and Applications, Volume 37 of De Gruyter Studies in Mathematics, 2nd edn. Walter de Gruyter & Co., Berlin (2012)
24.
Schurz, H.: Stability, Stationarity, and Boundedness of Some Implicit Numerical Methods for Stochastic Differential Equations and applications. Logos Verlag, Berlin (1997)MATH
25.
Umarov, S., Hahn, M., Kobayashi, K.: Beyond the Triangle: Brownian Motion, Ito calculus, and Fokker–Planck Equation—Fractional Generalizations. World Scientific Publishing Co. Pte. Ltd., Hackensack (2018)CrossRef
26.
Wang, L., Wang, X.: Convergence of the semi-implicit Euler method for stochastic age-dependent population equations with Poisson jumps. Appl. Math. Model. 34(8), 2034–2043 (2010)MathSciNetCrossRef
27.
Metadaten
Titel
Semi-implicit Euler–Maruyama method for non-linear time-changed stochastic differential equations
verfasst von
Chang-Song Deng
Wei Liu
Publikationsdatum
12.05.2020
Verlag
Springer Netherlands
Erschienen in
BIT Numerical Mathematics / Ausgabe 4/2020
Print ISSN: 0006-3835
Elektronische ISSN: 1572-9125
DOI
https://doi.org/10.1007/s10543-020-00810-7

Weitere Artikel der Ausgabe 4/2020

BIT Numerical Mathematics 4/2020 Zur Ausgabe

Research Paper

Quantifying the ill-conditioning of analytic continuation

OriginalPaper

A semidiscrete Galerkin scheme for a coupled two-scale elliptic–parabolic system: well-posedness and convergence approximation rates

OriginalPaper

Lower and upper bounds for strong approximation errors for numerical approximations of stochastic heat equations

OriginalPaper

Modified averaged vector field methods preserving multiple invariants for conservative stochastic differential equations

OriginalPaper

Variational formulation for fractional inhomogeneous boundary value problems

OriginalPaper

The Least squares and line search in extracting eigenpairs in Jacobi–Davidson method

Premium Partner

    Bildnachweise