Convergence rates for Euler schemes of Lévy-driven SDE using dynamic cutting
Pub. online: 20 August 2026
Type: Research Article
Open Access
Open Access
Received
21 April 2025
21 April 2025
Revised
27 July 2026
27 July 2026
Accepted
27 July 2026
27 July 2026
Published
20 August 2026
20 August 2026
Abstract
We introduce a dynamic cutting approach for the numerical approximation of Lévy-driven stochastic differential equations. The key idea is to remove small jumps according to a time-dependent threshold, so that the retained jumps form a time-inhomogeneous compound Poisson process. We derive ${L^{p}}$-strong convergence rates for the adjusted Euler scheme. Numerical experiments at matched computational cost compare the dynamic cutting scheme with the classical Asmussen–Rosiński truncation and demonstrate consistently smaller strong errors when the time-dependent jump coefficient has a singularity at $t=0$.
References
Asmussen, S.r., Rosiński, J.: Approximations of small jumps of Lévy processes with a view towards simulation. J. Appl. Probab. 38(2), 482–493 (2001). https://doi.org/https://doi.org/10.1017/s0021900200019987. MR1834755
Bingham, N.H., Goldie, C.M., Teugels, J.L.: Regular Variation. Encyclopedia of Mathematics and Its Applications, vol. 27, p. 491. Cambridge University Press, Cambridge (1987). https://doi.org/https://doi.org/10.1017/CBO9780511721434. https://doi.org/10.1017/CBO9780511721434. MR898871
Bossy, M., Maurer, P.: On the ε-Euler-Maruyama scheme for time inhomogeneous jump-driven SDEs. Stoch. Process. Appl. 190, Paper No. 104747, 34 pp. (2025). https://doi.org/https://doi.org/10.1016/j.spa.2025.104747. MR4943239
Breton, J.-C., Privault, N.: Integrability and regularity of the flow of stochastic differential equations with jumps. Teor. Veroâtn. Primen. 65(1), 103–125 (2020). https://doi.org/https://doi.org/10.4213/tvp5291. MR4061867
Butkovsky, O., Dareiotis, K., Gerencsér, M.: Strong Rate of Convergence of the Euler Scheme for SDEs with Irregular Drift Driven by Lévy Noise (2024). arXiv:2204.12926. MR4982003. https://doi.org/10.1214/24-aihp1506
Çinlar, E.: Introduction to Stochastic Processes, p. 402. Prentice-Hall, Inc., Englewood Cliffs, NJ (1975). MR380912
Fournier, N.: Simulation and approximation of Lévy-driven stochastic differential equations. ESAIM, Probab. Stat. 15, 233–248 (2011). https://doi.org/https://doi.org/10.1051/ps/2009017. MR2870514
Ikeda, N., Watanabe, S.: Stochastic Differential Equations and Diffusion Processes, 2nd edn. North-Holland Mathematical Library, vol. 24, p. 555. North-Holland Publishing Co., Amsterdam; Kodansha, Ltd., Tokyo (1989). MR1011252
Ivanenko, D., Knopova, V., Platonov, D.: On approximation of some Lévy processes. Austrian J. Stat. 54(1), 177–199 (2025). https://doi.org/https://doi.org/10.17713/ajs.v54i1.1991
Jacod, J.: The Euler scheme for Lévy driven stochastic differential equations: limit theorems. Ann. Probab. 32(3A), 1830–1872 (2004). https://doi.org/https://doi.org/10.1214/009117904000000667. MR2073179
Knopova, V.: Compound kernel estimates for the transition probability density of a Lévy process in ${\mathbb{R}^{n}}$. Teor. Imovir. Mat. Stat. 89, 51–63 (2013). https://doi.org/https://doi.org/10.1090/s0094-9000-2015-00935-2. MR3235175
Knopova, V., Kulik, A.: Intrinsic small time estimates for distribution densities of Lévy processes. Random Oper. Stoch. Equ. 21(4), 321–344 (2013). https://doi.org/https://doi.org/10.1515/rose-2013-0015. MR3139314
Knopova, V., Kulik, A.: Parametrix construction of the transition probability density of the solution to an SDE driven by α-stable noise. Ann. Inst. Henri Poincaré Probab. Stat. 54(1), 100–140 (2018). https://doi.org/https://doi.org/10.1214/16-AIHP796. MR3765882
Knopova, V., Kulik, A., Schilling, R.L.: Construction and heat kernel estimates of general stable-like Markov processes. Diss. Math. 569, 86 (2021). https://doi.org/https://doi.org/10.4064/dm824-8-2021. MR4361582
Kohatsu-Higa, A., Tankov, P.: Jump-adapted discretization schemes for Lévy-driven SDEs. Stoch. Process. Appl. 120(11), 2258–2285 (2010). https://doi.org/https://doi.org/10.1016/j.spa.2010.07.001. MR2684745
Kühn, F.: Existence and estimates of moments for Lévy–type processes. Stoch. Process. Appl. 127(3), 1018–1041 (2017). https://doi.org/https://doi.org/10.1016/j.spa.2016.07.008. MR3605719
Kühn, F., Schilling, R.L.: Strong convergence of the Euler-Maruyama approximation for a class of Lévy-driven SDEs. Stoch. Process. Appl. 129(8), 2654–2680 (2019). https://doi.org/https://doi.org/10.1016/j.spa.2018.07.018. MR3980140
Kühn, F., Schilling, R.L.: Maximal inequalities and some applications. Probab. Surv. 20, 382–485 (2023). https://doi.org/https://doi.org/10.1214/23-ps17. MR4568704
Lewis, P.A.W., Shedler, G.S.: Simulation of nonhomogeneous Poisson processes by thinning. Nav. Res. Logist. Q. 26(3), 403–413 (1979). https://doi.org/https://doi.org/10.1002/nav.3800260304. MR546120
Li, Y., Zhao, G.: Euler-Maruyama scheme for SDE driven by Lévy process with Hölder drift. Stat. Probab. Lett. 215, Paper No. 110220, 6 pp. (2024). https://doi.org/https://doi.org/10.1016/j.spl.2024.110220. MR4782995
Menoukeu Pamen, O., Taguchi, D.: Strong rate of convergence for the Euler-Maruyama approximation of SDEs with Hölder continuous drift coefficient. Stoch. Process. Appl. 127(8), 2542–2559 (2017). https://doi.org/https://doi.org/10.1016/j.spa.2016.11.008. MR3660882
Mikulevičius, R., Xu, F.: On the rate of convergence of strong Euler approximation for SDEs driven by Levy processes. Stochastics 90(4), 569–604 (2018). https://doi.org/https://doi.org/10.1080/17442508.2017.1381095. MR3784978
Orsingher, E., Ricciuti, C., Toaldo, B.: Time-inhomogeneous jump processes and variable order operators. Potential Anal. 45(3), 435–461 (2016). https://doi.org/https://doi.org/10.1007/s11118-016-9551-4. MR3554398
Platen, E.: An approximation method for a class of Itô processes with jump component. Liet. Mat. Rink. 22(2), 124–136 (1982). MR659025
Platen, E., Bruti-Liberati, N.: Numerical Solution of Stochastic Differential Equations with Jumps in Finance. Stochastic Modelling and Applied Probability, vol. 64, p. 856. Springer (2010). https://doi.org/https://doi.org/10.1007/978-3-642-13694-8. https://doi.org/10.1007/978-3-642-13694-8. MR2723480
Protter, P., Talay, D.: The Euler scheme for Lévy driven stochastic differential equations. Ann. Probab. 25(1), 393–423 (1997). https://doi.org/https://doi.org/10.1214/aop/1024404293. MR1428514
Pruitt, W.E.: General one-sided laws of the iterated logarithm. Ann. Probab. 9(1), 1–48 (1981). https://doi.org/https://doi.org/10.1214/aop/1176994508. MR606797
