# American Institute of Mathematical Sciences

March  2021, 26(3): 1405-1446. doi: 10.3934/dcdsb.2020167

## On Milstein-type scheme for SDE driven by Lévy noise with super-linear coefficients

 Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, Uttarakhand, India, 247 667

* Corresponding author: C. Kumar (chaman.kumar@ma.iitr.ac.in)

Received  July 2019 Revised  December 2019 Published  March 2021 Early access  May 2020

Fund Project: The author is supported by Professional Development Allowance (PDA) and Faculty Initiation Grant provided by the Indian Institute of Technology Roorkee

A new explicit Milstein-type scheme for SDE driven by Lévy noise is proposed where both drift and diffusion coefficients are allowed to grow super-linearly. The strong rate of convergence (in $\mathcal{L}^2$-sense) is shown to be arbitrarily close to one which is consistent with the corresponding result on the classical Milstein scheme obtained for coefficients satisfying global Lipschitz conditions.

Citation: Chaman Kumar. On Milstein-type scheme for SDE driven by Lévy noise with super-linear coefficients. Discrete and Continuous Dynamical Systems - B, 2021, 26 (3) : 1405-1446. doi: 10.3934/dcdsb.2020167
##### References:
 [1] W.-J. Beyn, E. Isaak and R. Kruse, Stochastic C-stability and B-consistency of explicit and implicit Milstein-type schemes, Journal of Scientific Computing, 70 (2017), 1042-1077.  doi: 10.1007/s10915-016-0290-x. [2] K. Dareiotis, C. Kumar and S. Sabanis, On tamed Euler approximations of SDEs driven by Lévy noise with application to delay equations, SIAM Journal on Numerical Analysis, 54 (2016), 1840-1872.  doi: 10.1137/151004872. [3] D. Fudenberg and C. Harris, Evolutionary dynamics with aggregate shocks, Journal of Economic Theory, 57 (1992), 420-441.  doi: 10.1016/0022-0531(92)90044-I. [4] F. B. Hanson, Applied Stochastic Processes and Control for Jump-Diffusions: Modeling, Analysis and Computation, Advances in Design and Control, 13. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2007. doi: 10.1137/1.9780898718638. [5] F. B. Hanson and H. C. Tuckwell, Population growth with randomly distributed jumps, Journal of Mathematical Biology, 36 (1997), 169-187.  doi: 10.1007/s002850050096. [6] M. Hutzenthaler and A. Jentzen, Numerical approximations of stochastic differential equations with non-globally Lipschitz continuous coefficients, Memoirs of the American Mathematical Society, 236 (2015). doi: 10.1090/memo/1112. [7] 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 (2011), 1563-1576.  doi: 10.1098/rspa.2010.0348. [8] M. Hutzenthaler, A. Jentzen and P. E. Kloeden, Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients, Annals of Applied Probability, 22 (2012), 1611-1641.  doi: 10.1214/11-AAP803. [9] C. Kumar and S. Sabanis, On explicit approximations for Lévy driven SDEs with super-linear diffusion coefficients, Electronic Journal of Probability, 22 (2017), Paper No. 73, 19 pp. doi: 10.1214/17-EJP89. [10] C. Kumar and S. Sabanis, On tamed Milstein scheme of SDEs driven by Lévy noise, Discrete and Continuous Dynamical Systems-Series B, 22 (2017), 421-463.  doi: 10.3934/dcdsb.2017020. [11] C. Kumar and S. Sabanis, On Milstein approximations with varying coefficients: The case of super-linear diffusion coefficients, BIT Numerical Mathematics, 59 (2019), 929-968.  doi: 10.1007/s10543-019-00756-5. [12] E. Platen and N. Bruti-Liberati, Numerical Solution of Stochastic Differential Equations with Jumps in Finance, Stochastic Modelling and Applied Probability, 64. Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-13694-8. [13] S. Sabanis, A note on tamed Euler approximations, Electronic Communications in Probability, 18 (2013), 10 pp. doi: 10.1214/ECP.v18-2824. [14] S. Sabanis, Euler approximations with varying coefficients: The case of superlinearly growing diffusion coefficients, Annals of Applied Probability, 26 (2016), 2083-2105.  doi: 10.1214/15-AAP1140. [15] S. Sabanis and Y. Zhang, On explicit order 1.5 approximations with varying coefficients: The case of super-linear diffusion coefficients, Journal of Complexity, 50 (2019), 84-115.  doi: 10.1016/j.jco.2018.09.004. [16] R. Situ, Theory of Stochastic Differential Equations with Jumps and Applications: Mathematical and Analytic Techniques with Applications to Engineering, Springer, New York, 2005. [17] M. V. Tretyakov and Z. Zhang, A fundamental mean-square convergence theorem for SDEs with locally Lipschitz coefficients and its applications, SIAM Journal on Numerical Analysis, 51 (2013), 3135-3162.  doi: 10.1137/120902318. [18] X. J. Wang and S. Q. Gan, The tamed Milstein method for commutative stochastic differential equations with non-globally Lipschitz continuous coefficients, Journal of Difference Equations and Applications, 19 (2013), 466-490.  doi: 10.1080/10236198.2012.656617. [19] Z. Q. Zhang and H. P. Ma, Order-preserving strong schemes for SDEs with locally Lipschitz coefficients, Applied Numerical Mathematics, 112 (2017), 1-16.  doi: 10.1016/j.apnum.2016.09.013.

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##### References:
 [1] W.-J. Beyn, E. Isaak and R. Kruse, Stochastic C-stability and B-consistency of explicit and implicit Milstein-type schemes, Journal of Scientific Computing, 70 (2017), 1042-1077.  doi: 10.1007/s10915-016-0290-x. [2] K. Dareiotis, C. Kumar and S. Sabanis, On tamed Euler approximations of SDEs driven by Lévy noise with application to delay equations, SIAM Journal on Numerical Analysis, 54 (2016), 1840-1872.  doi: 10.1137/151004872. [3] D. Fudenberg and C. Harris, Evolutionary dynamics with aggregate shocks, Journal of Economic Theory, 57 (1992), 420-441.  doi: 10.1016/0022-0531(92)90044-I. [4] F. B. Hanson, Applied Stochastic Processes and Control for Jump-Diffusions: Modeling, Analysis and Computation, Advances in Design and Control, 13. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2007. doi: 10.1137/1.9780898718638. [5] F. B. Hanson and H. C. Tuckwell, Population growth with randomly distributed jumps, Journal of Mathematical Biology, 36 (1997), 169-187.  doi: 10.1007/s002850050096. [6] M. Hutzenthaler and A. Jentzen, Numerical approximations of stochastic differential equations with non-globally Lipschitz continuous coefficients, Memoirs of the American Mathematical Society, 236 (2015). doi: 10.1090/memo/1112. [7] 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 (2011), 1563-1576.  doi: 10.1098/rspa.2010.0348. [8] M. Hutzenthaler, A. Jentzen and P. E. Kloeden, Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients, Annals of Applied Probability, 22 (2012), 1611-1641.  doi: 10.1214/11-AAP803. [9] C. Kumar and S. Sabanis, On explicit approximations for Lévy driven SDEs with super-linear diffusion coefficients, Electronic Journal of Probability, 22 (2017), Paper No. 73, 19 pp. doi: 10.1214/17-EJP89. [10] C. Kumar and S. Sabanis, On tamed Milstein scheme of SDEs driven by Lévy noise, Discrete and Continuous Dynamical Systems-Series B, 22 (2017), 421-463.  doi: 10.3934/dcdsb.2017020. [11] C. Kumar and S. Sabanis, On Milstein approximations with varying coefficients: The case of super-linear diffusion coefficients, BIT Numerical Mathematics, 59 (2019), 929-968.  doi: 10.1007/s10543-019-00756-5. [12] E. Platen and N. Bruti-Liberati, Numerical Solution of Stochastic Differential Equations with Jumps in Finance, Stochastic Modelling and Applied Probability, 64. Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-13694-8. [13] S. Sabanis, A note on tamed Euler approximations, Electronic Communications in Probability, 18 (2013), 10 pp. doi: 10.1214/ECP.v18-2824. [14] S. Sabanis, Euler approximations with varying coefficients: The case of superlinearly growing diffusion coefficients, Annals of Applied Probability, 26 (2016), 2083-2105.  doi: 10.1214/15-AAP1140. [15] S. Sabanis and Y. Zhang, On explicit order 1.5 approximations with varying coefficients: The case of super-linear diffusion coefficients, Journal of Complexity, 50 (2019), 84-115.  doi: 10.1016/j.jco.2018.09.004. [16] R. Situ, Theory of Stochastic Differential Equations with Jumps and Applications: Mathematical and Analytic Techniques with Applications to Engineering, Springer, New York, 2005. [17] M. V. Tretyakov and Z. Zhang, A fundamental mean-square convergence theorem for SDEs with locally Lipschitz coefficients and its applications, SIAM Journal on Numerical Analysis, 51 (2013), 3135-3162.  doi: 10.1137/120902318. [18] X. J. Wang and S. Q. Gan, The tamed Milstein method for commutative stochastic differential equations with non-globally Lipschitz continuous coefficients, Journal of Difference Equations and Applications, 19 (2013), 466-490.  doi: 10.1080/10236198.2012.656617. [19] Z. Q. Zhang and H. P. Ma, Order-preserving strong schemes for SDEs with locally Lipschitz coefficients, Applied Numerical Mathematics, 112 (2017), 1-16.  doi: 10.1016/j.apnum.2016.09.013.
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