In this paper, we investigate Euler–Maruyama approximate solutions of stochastic differential equations (SDEs) with multiple delay functions. Stochastic differential delay equations (SDDEs) are generalizations of SDEs. Solutions of SDDEs are influenced by both the present and past states. Because these solutions may include past information, they are not necessarily Markov processes. This makes representations of solutions complicated; therefore, approximate solutions are practical. We estimate the rate of convergence of approximate solutions of SDDEs to the exact solutions in the $ L^p $-mean for $ p \geq 2 $ and apply the result to obtain confidence interval estimations for the approximate solutions.
Citation: Masataka Hashimoto, Hiroshi Takahashi. On the rate of convergence of Euler–Maruyama approximate solutions of stochastic differential equations with multiple delays and their confidence interval estimations[J]. AIMS Mathematics, 2023, 8(6): 13747-13763. doi: 10.3934/math.2023698
In this paper, we investigate Euler–Maruyama approximate solutions of stochastic differential equations (SDEs) with multiple delay functions. Stochastic differential delay equations (SDDEs) are generalizations of SDEs. Solutions of SDDEs are influenced by both the present and past states. Because these solutions may include past information, they are not necessarily Markov processes. This makes representations of solutions complicated; therefore, approximate solutions are practical. We estimate the rate of convergence of approximate solutions of SDDEs to the exact solutions in the $ L^p $-mean for $ p \geq 2 $ and apply the result to obtain confidence interval estimations for the approximate solutions.
| [1] | D. Llemit, J. M. Escaner Ⅳ, Value functions in a regime switching jump diffusion with delay market model, AIMS Math., 6 (2021), 11595–11609. http://www.aimspress.com/article/doi/10.3934/math.2021673 |
| [2] |
K. Ito, M. Nisio, On stationary solution of a stochastic differential equation, J. Math. Kyoto Univ., 4 (1964), 1–75. https://doi.org/10.1215/kjm/1250524705 doi: 10.1215/kjm/1250524705
|
| [3] | A. F. Ivanov, Y. I. Kazmerchuk, A. V. Swishchuk, Theory, stochastic stability and applications of stochastic delay differential equations: a survey of recent results, Differ. Equ. Dyn. Syst., 11 (2003), 55–115. |
| [4] | U. Küchler, E. Platen, Strong discrete time approximation of stochastic differential equations with time delay, Math. Comput. Simul., 54 (2000), 189–205. |
| [5] |
X. Mao, S. Sabanis, Numerical solutions of stochastic differential delay equations under local Lipschitz condition, J. Comput. Appl. Math., 151 (2003), 215–227. https://doi.org/10.1016/S0377-0427(02)00750-1 doi: 10.1016/S0377-0427(02)00750-1
|
| [6] |
G. Song, J. Hu, S. Gao, X. Li, The strong convergence and stability of explicit approximations for nonlinear stochastic delay differential equations, Numer. Algorithms, 89 (2022), 855–883. https://doi.org/10.1007/s11075-021-01137-2 doi: 10.1007/s11075-021-01137-2
|
| [7] |
S. You, L. Hu, J. Lu, X. Mao, Stabilization in distribution by delay feedback control for hybrid stochastic differential equations, IEEE Trans. Automat. Contr., 67 (2022), 971–977. https://doi.org/10.1109/TAC.2021.3075177 doi: 10.1109/TAC.2021.3075177
|
| [8] |
W. Zhang, M. H. Song, M. Z. Liu, Almost Sure Exponential Stability of Stochastic Differential Delay Equations, Filomat, 33 (2019), 789–814. https://doi.org/10.2298/FIL1903789Z doi: 10.2298/FIL1903789Z
|
| [9] | S. Kanagawa, S. Ogawa, Numerical solutions of stochastic differential equations and their applications, Sugaku Expositions, 18 (2005), 75–99. |
| [10] |
S. Kanagawa, The rate of convergence for approximate solutions of stochastic differential equation, Tokyo J. Math., 12 (1989), 33–48. https://doi.org/10.3836/tjm/1270133546 doi: 10.3836/tjm/1270133546
|
| [11] | S. Kanagawa, On the rate of convergence for Maruyama's approximate solutions of stochastic differential equations, Yokohama Math. J., 36 (1988), 79–86. http://doi.org/10131/5546 |
| [12] |
H. Takahashi, Confidence intervals for Euler–Maruyama approximate solutions of stochastic delay equations, Theor. Appl. Mech. Japan, 63 (2015), 127–132. https://doi.org/10.11345/nctam.63.127 doi: 10.11345/nctam.63.127
|
| [13] |
M. Hashimoto, H. Takahashi, Interval estimations for Euler–Maruyama approximate solutions of stochastic differential equations with multiple delays, J. Jpn. Soc. Civ. Eng., Ser. A2, 75 (2019), 105–111. https://doi.org/10.2208/jscejam.75.2_I_105 doi: 10.2208/jscejam.75.2_I_105
|
| [14] |
S. Kanagawa, Confidence intervals of discretized Euler–Maruyama approximate solutions of SDE's, Nonlinear Anal. Theory Methods Appl., 30 (1997), 4101–4104. https://doi.org/10.1016/S0362-546X(97)00159-4 doi: 10.1016/S0362-546X(97)00159-4
|
| [15] |
S. Kanagawa, S. Hasegawa, A confidence interval for the Euler–Maruyama approximate solution of SDE with unbounded coefficients, Theor. Appl. Mech. Japan, 62 (2014), 91–98. https://doi.org/10.11345/nctam.62.91 doi: 10.11345/nctam.62.91
|
| [16] | E. Platen, N. Bruti-Liberati, Numerical solution of stochastic differential equations with jumps in finance, Stochastic Modelling and Applied Probability, volume 64, Berlin: Springer-Verlag, 2010. https://doi.org/10.1007/978-3-642-13694-8 |
| [17] |
H. Takahashi, K. Yoshihara, Approximation of solutions of multi-dimensional linear stochastic differential equations defined by weakly dependent random variables, AIMS Math., 2 (2017), 377–384. https://doi.org/10.3934/Math.2017.3.377 doi: 10.3934/Math.2017.3.377
|
| [18] |
Y. A. Alnafisah, A New Approach to Compare the Strong Convergence of the Milstein Scheme with the Approximate Coupling Method, Fractal Fract., 6 (2022), 339. https://doi.org/10.3390/fractalfract6060339 doi: 10.3390/fractalfract6060339
|
| [19] |
M. Milošević, The Euler–Maruyama approximation of solutions to stochastic differential equations with piecewise constant arguments, J. Comput. Appl. Math., 298 (2016), 1–12. https://doi.org/10.1016/j.cam.2015.11.019 doi: 10.1016/j.cam.2015.11.019
|
Masataka Hashimoto, Hiroshi Takahashi. On the rate of convergence of Euler–Maruyama approximate solutions of stochastic differential equations with multiple delays and their confidence interval estimations[J]. AIMS Mathematics, 2023, 8(6): 13747-13763. doi: 10.3934/math.2023698
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