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Analytical and numerical investigation of stochastic differential equations with applications using an exponential Euler–Maruyama approach

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Abstract

This work aims to develop an explicit approximation scheme for solving stochastic differential systems based on the Euler–Maruyama scheme. We demonstrate the strong consistency and convergence properties of the proposed scheme. In addition, in a mean-square sense, we perform linear and nonlinear asymptotic stability analysis for the multiplicative and additive cases. At the end of the paper, several numerical examples arising in biology, neurology, and population dynamics are carried out to illustrate the obtained convergence and stability properties.

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References

  • Abdulle A, Cirilli S (2008) S-ROCK: Chebyshev methods for stiff stochastic differential equations. SIAM J Sci Comput 30(2):997–1014

    Article  MathSciNet  MATH  Google Scholar 

  • Abdulle A, Li T (2008) S-ROCK methods for stiff Itô SDEs. Commun Math Sci 6(4):845–868

    Article  MathSciNet  MATH  Google Scholar 

  • Abdulle A, Vilmart G, Zygalakis KC (2013) Weak second order explicit stabilized methods for stiff stochastic differential equations. SIAM J Sci Comput 35(4):1792–1814

    Article  MathSciNet  MATH  Google Scholar 

  • Abdulle A, Almuslimani I, Vilmart G (2018) Optimal explicit stabilized integrator of weak order 1 for stiff and ergodic stochastic differential equations. SIAM/ASA J Uncertain Quantif 6(2):937–964

    Article  MathSciNet  MATH  Google Scholar 

  • Arnold L (1974) Stochastic differential equations: theory and applications. John Wiley & Sons, New York

    MATH  Google Scholar 

  • Baker CTH, Buckwar E (2000) Numerical analysis of explicit one-step methods for stochastic delay differential equations. LMS J Comput Math 3:315–335

    Article  MathSciNet  MATH  Google Scholar 

  • Buckwar E, Samson A, Tamborrino M, Tubikanec I (2022) A splitting method for SDEs with locally Lipschitz drift: Illustration on the FitzHugh–Nagumo model. Appl Numer Math 179:191–220

    Article  MathSciNet  MATH  Google Scholar 

  • Calatayud J, Cortés J-C, Dorini FA, Jornet M (2020) On a stochastic logistic population model with time-varying carrying capacity. Comput Appl Math 39(4):Paper No. 288, 16

    Article  MathSciNet  MATH  Google Scholar 

  • Cortés J-C, Navarro-Quiles A, Romero J-V, Roselló M-D (2022) Solving fully randomized first-order linear control systems: application to study the dynamics of a damped oscillator with parametric noise under stochastic control. J Comput Appl Math 404:Paper No. 113389, 18

    Article  MathSciNet  MATH  Google Scholar 

  • Dalal N, Greenhalgh D, Mao X (2008) A stochastic model for internal HIV dynamics. J Math Anal Appl 341(2):1084–1101

    Article  MathSciNet  MATH  Google Scholar 

  • David W (1991) Probability with martingales. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  • Díaz-Infante S, Jerez S (2016) Convergence and asymptotic stability of the explicit Steklov method for stochastic differential equations. J Comput Appl Math 291:36–47

    Article  MathSciNet  MATH  Google Scholar 

  • Díaz-Infante S, Jerez S (2017) The linear Steklov method for SDEs with non-globally Lipschitz coefficients: strong convergence and simulation. J Comput Appl Math 309:408–423

    Article  MathSciNet  MATH  Google Scholar 

  • Erdoğan U, Lord GJ (2019) A new class of exponential integrators for SDEs with multiplicative noise. IMA J Numer Anal 39(2):820–846

    Article  MathSciNet  MATH  Google Scholar 

  • Erdoğan U, Lord GJ (2022) Strong convergence of a GBM based tamed integrator for SDEs and an adaptive implementation. J Comput Appl Math 399:Paper No. 113704, 21

    Article  MathSciNet  MATH  Google Scholar 

  • Hernández DB, Spigler R (1992) \(A\)-stability of Runge–Kutta methods for systems with additive noise. BIT 32(4):620–633

    Article  MathSciNet  MATH  Google Scholar 

  • Higham DJ (2000) Mean-square and asymptotic stability of the stochastic theta method. SIAM J Numer Anal 38(3):753–769

    Article  MathSciNet  MATH  Google Scholar 

  • Hu L, Chan A, Bao X (2021) Numerical analysis of the balanced methods for stochastic Volterra integro-differential equations. Comput Appl Math 40(6):Paper No. 203, 18

    Article  MathSciNet  MATH  Google Scholar 

  • Hutzenthaler M, Jentzen A, Kloeden PE (2012) Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients. Ann Appl Probab 22(4):1611–1641

    Article  MathSciNet  MATH  Google Scholar 

  • Jimenez JC (2002) A simple algebraic expression to evaluate the local linearization schemes for stochastic differential equations. Appl Math Lett 15(6):775–780

    Article  MathSciNet  MATH  Google Scholar 

  • Jimenez JC, Shoji I, Ozaki T (1999) Simulation of stochastic differential equations through the local linearization method. A comparative study. J Stat Phys 94(3–4):587–602

    Article  MathSciNet  MATH  Google Scholar 

  • Jornet M, Calatayud J, Le Maître OP, Cortés J-C (2020) Second order linear differential equations with analytic uncertainties: stochastic analysis via the computation of the probability density function. J Comput Appl Math 374:112770, 20

    Article  MathSciNet  MATH  Google Scholar 

  • Khasminskii R (2012) Stochastic stability of differential equations, vol. 66 of stochastic modelling and applied probability, 2nd edn. Springer, Heidelberg

    Book  Google Scholar 

  • Kloeden PE, Platen E (1992) Numerical solution of stochastic differential equations. Springer, Berlin

    Book  MATH  Google Scholar 

  • Komori Y, Burrage K (2012) Weak second order S-ROCK methods for Stratonovich stochastic differential equations. J Comput Appl Math 236(11):2895–2908

    Article  MathSciNet  MATH  Google Scholar 

  • Komori Y, Burrage K (2013) Strong first order S-ROCK methods for stochastic differential equations. J Comput Appl Math 242:261–274

    Article  MathSciNet  MATH  Google Scholar 

  • Komori Y, Burrage K (2014) A stochastic exponential Euler scheme for simulation of stiff biochemical reaction systems. BIT 54(4):1067–1085

    Article  MathSciNet  MATH  Google Scholar 

  • Komori Y, Cohen D, Burrage K (2017) Weak second order explicit exponential Runge–Kutta methods for stochastic differential equations. SIAM J Sci Comput 39(6):2857–2878

    Article  MathSciNet  MATH  Google Scholar 

  • Mao X (2007) Stochastic differential equations and applications, 2nd edn. Horwood, Chichester

    MATH  Google Scholar 

  • Milstein GN, Tretyakov MV (2004) Stochastic numerics for mathematical physics. Springer-Verlag, Berlin

    Book  MATH  Google Scholar 

  • Nouri K (2022) Improving split-step forward methods by ODE solver for stiff stochastic differential equations. Math Sci 16(1):51–57

    Article  MathSciNet  MATH  Google Scholar 

  • Nouri K, Ranjbar H, Cortés J-C (2020a) Modifying the split-step \(\theta \)-method with harmonic-mean term for stochastic differential equations. Int J Numer Anal Model 17(5):662–678

  • Nouri K, Ranjbar H, Torkzadeh L (2020b) The explicit approximation approach to solve stiff chemical Langevin equations. Eur Phys J Plus 135:758

  • Nouri K, Ranjbar H, Torkzadeh L (2020c) Study on split-step Rosenbrock type method for stiff stochastic differential systems. Int J Comput Math 97(4):816–836

  • Nouri K, Ranjbar H, Baleanu D, Torkzadeh L (2021) Investigation on Ginzburg–Landau equation via a tested approach to benchmark stochastic Davis–Skodje system. Alex Eng J 60(6):5521–5526

    Article  Google Scholar 

  • Saito Y (2008) Stability analysis of numerical methods for stochastic systems with additive noise. http://www.shotoku.ac.jp/data/facilities/library/publication/education--keizai08_2_06.pdf

  • Saito Y, Mitsui T (1996) Stability analysis of numerical schemes for stochastic differential equations. SIAM J Numer Anal 33(6):2254–2267

    Article  MathSciNet  MATH  Google Scholar 

  • Schurz H (2007) Modeling, analysis and discretization of stochastic logistic equations. Int J Numer Anal Model 4(2):178–197

    MathSciNet  MATH  Google Scholar 

  • Senosiain MJ, Tocino A (2019) On the numerical integration of the undamped harmonic oscillator driven by independent additive gaussian white noises. Appl Numer Math 137:49–61

    Article  MathSciNet  MATH  Google Scholar 

  • Shahmoradi M, Ahmadian D, Ranjbar M (2021) Mean-square stability of 1.5 strong convergence orders of diagonally drift Runge–Kutta methods for a class of stochastic differential equations. Comput Appl Math 40(4):Paper No. 108, 17

    Article  MathSciNet  MATH  Google Scholar 

  • Wang P (2015) A-stable Runge–Kutta methods for stiff stochastic differential equations with multiplicative noise. Comput Appl Math 34(2):773–792

    Article  MathSciNet  MATH  Google Scholar 

  • Yin Z, Gan S (2015) An error corrected Euler–Maruyama method for stiff stochastic differential equations. Appl Math Comput 256:630–641

    MathSciNet  MATH  Google Scholar 

  • Zhang Z, Ma H (2017) Order-preserving strong schemes for SDEs with locally Lipschitz coefficients. Appl Numer Math 112:1–16

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Acknowledgements

The authors would like to thank the reviewers and editor for their conscientious reading and comments which were extremely helpful and useful for improving the article.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Mathematics, Statistics and Computer Sciences, Semnan University, P. O. Box 35195-363, Semnan, Iran

    H. Ranjbar, L. Torkzadeh & K. Nouri

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  1. H. Ranjbar
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  2. L. Torkzadeh
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  3. K. Nouri
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Correspondence to L. Torkzadeh.

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Communicated by Juan Carlos Cortes.

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Ranjbar, H., Torkzadeh, L. & Nouri, K. Analytical and numerical investigation of stochastic differential equations with applications using an exponential Euler–Maruyama approach. Comp. Appl. Math. 42, 23 (2023). https://doi.org/10.1007/s40314-022-02164-y

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  • DOI: https://doi.org/10.1007/s40314-022-02164-y

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