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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 |
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.
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. |
show all references
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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