Electronic Journal of Probability

On explicit approximations for Lévy driven SDEs with super-linear diffusion coefficients

Abstract

Motivated by the results of [21], we propose explicit Euler-type schemes for SDEs with random coefficients driven by Lévy noise when the drift and diffusion coefficients can grow super-linearly. As an application of our results, one can construct explicit Euler-type schemes for SDEs with delays (SDDEs) which are driven by Lévy noise and have super-linear coefficients. Strong convergence results are established and their rate of convergence is shown to be equal to that of the classical Euler scheme. It is proved that the optimal rate of convergence is achieved for $\mathcal{L} ^2$-convergence which is consistent with the corresponding results available in the literature.

Article information

Source
Electron. J. Probab., Volume 22 (2017), paper no. 73, 19 pp.

Dates
Received: 10 January 2017
Accepted: 4 August 2017
First available in Project Euclid: 13 September 2017

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1505268104

Digital Object Identifier
doi:10.1214/17-EJP89

Mathematical Reviews number (MathSciNet)
MR3698742

Zentralblatt MATH identifier
1379.60076

Citation

Kumar, Chaman; Sabanis, Sotirios. On explicit approximations for Lévy driven SDEs with super-linear diffusion coefficients. Electron. J. Probab. 22 (2017), paper no. 73, 19 pp. doi:10.1214/17-EJP89. https://projecteuclid.org/euclid.ejp/1505268104

References

• [1] Aït-Sahalia, Y. and Jacod, J.: Estimating the Degree of Activity of Jumps in High Frequency Data. Annals of Statistics 37(5A), (2009), 2202-44.
• [2] Beyn, W.-J., Isaak, E. and Kruse, R.: Stochastic C-Stability and B-Consistency of Explicit and Implicit Euler-Type Schemes. Journal of Scientific Computing 67(3), (2016), 955-987.
• [3] Dareiotis, K., Kumar, C. and Sabanis, S.: On Tamed Euler Approximations of SDEs Driven by Lévy Noise with Applications to Delay Equations. SIAM J. Numer. Anal. 54(3), (2016), 1840-1872.
• [4] Duffie, D., Pan, J. and Singleton, K.: Transform analysis and asset pricing for affine jump-diffusions. Econometrica 68, (2000), 1343-1376.
• [5] Eraker, B.: Do Stock Prices and Volatility Jump? Reconciling Evidence from Spot and Option Prices. The Journal of Finance LIX(3), (2004), 1367-1403.
• [6] Fudenberg, D. and Harris, C.: Evolutionary dynamics with aggregate shocks. Journal of Economic Theory 57(2), (1992), 420-441.
• [7] Hanson, F.B.: Applied Stochastic Processes and Control for Jump-Diffusions: Modeling, Analysis and Computation. SIAM Books, Philadelphia, PA, (2006).
• [8] Hanson, F. B. and Tuckwell, H. C.: Population growth with randomly distributed jumps. Journal of Mathematical Biology 36(2), (1997), 169-187.
• [9] Gyöngy, I. and Krylov, N. V.: On Stochastic Equations with Respect to Semimartingales I. Stochastics 4, (1980), 1-21.
• [10] Gyöngy, I. and Sabanis, S.: A note on Euler approximation for stochastic differential equations with delay. Applied Mathematics and Optimization 68, (2013), 391-412.
• [11] Hutzethaler, M., Jentzen, A. and Kloeden, P. E.: Strong and weak divergence in finite time of Euler’s method for stochastic differential equations with non-globally Lipschitz continuous coefficients. Proceedings of the Royal Society A 467, (2010), 1563-1576.
• [12] Hutzenthaler, M., Jentzen, A. and Kloeden, P. E.: Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients. The Annals of Applied Probability 22, (2012), 1611-1641.
• [13] Hutzenthaler, M. and Jentzen, A.: On a perturbation theory and on strong convergence rates for stochastic ordinary and partial differential equations with non-globally monotone coefficients. arXiv:1401.0295 [math.PR], (2014).
• [14] Hutzenthaler, M. and Jentzen, A.: Numerical approximations of stochastic differential equations with non-globally Lipschitz continuous coefficients. Memoirs of the American Mathematical Society, 236, (2015), no. 1112.
• [15] Jentzen, A., Müller-Gronbach, T. and Yaroslavtseva, L.: On stochastic differential equations with arbitrary slow convergence rates for strong approximation. Communications in Mathematical Sciences 14(6), (2016), 1477-1500.
• [16] Kumar, C. and Sabanis, S.: Strong Convergence of Euler Approximations of Stochastic Differential Equations with Delay under Local Lipschitz Condition. Stochastic Analysis and Applications 32, (2014), 207-228.
• [17] Kumar, C. and Sabanis, S.: On Tamed Milstein Schemes of SDEs Driven by Lévy Noise. Discrete and Continuous Dynamical Systems-Series B 22(2), (2017), 421-463.
• [18] Kumar, C. and Sabanis, S.: On Milstein approximations with varying coefficients: the case of super-linear diffusion coefficients. arXiv:1601.02695 [math.PR], (2016).
• [19] Platen, E. and Bruti-Liberati, N.: Numerical Solution of Stochastic Differential Equations with Jumps in Finance. Springer-Verlag, Berlin, (2010).
• [20] Sabanis, S.: A note on tamed Euler approximations. Electron. Commun. in Probab. 18, (2013), 1-10.
• [21] Sabanis, S.: Euler approximations with varying coefficients: the case of superlinearly growing diffusion coefficients. The Annals of Applied Probability 26(4), (2016), 2083-2105.
• [22] Sepp, A.: Pricing Options on Realized Variance in Heston Model with Jumps in Returns and Volatility. Journal of Computational Finance 11(4), (2008), 33-70.
• [23] Tretyakov, M. V. and Zhang, Z.: A fundamental mean-square convergence theorem for SDEs with locally Lipschitz coefficients and its applications. SIAM J. Numer. Anal. 51(6), (2013), 3135-3162.
• [24] Yaroslavtseva, L.: On non-polynomial lower error bounds for adaptive strong approximation of SDEs. Journal of Complexity 42, (2017), 1-18.