# American Institute of Mathematical Sciences

July  2011, 16(1): 1-14. doi: 10.3934/dcdsb.2011.16.1

## The Euler-Maruyama approximations for the CEV model

 1 School of Mathematical Sciences, Monash University, Clayton Campus, Building 28, Wellington road, Victoria, 3800, Australia 2 School of Mathematical Sciences, Monash University, Clayton Campus, Building 28,, Wellington road, Victoria, 3800, Australia 3 Department of Engineering Systems, Tel Aviv University, Tel Aviv, Ramat Aviv, 69978, Israel

Received  April 2010 Revised  August 2010 Published  April 2011

The CEV model is given by the stochastic differential equation $X_t=X_0+\int_0^t\mu X_s ds+\int_0^t\sigma (X^+_s)^p dW_s$, $\frac{1}{2}\le p<1$. It features a non-Lipschitz diffusion coefficient and gets absorbed at zero with a positive probability. We show the weak convergence of Euler-Maruyama approximations $X_t^n$ to the process $X_t$, $0 \le t \le T$, in the Skorokhod metric, by giving a new approximation by continuous processes. We calculate ruin probabilities as an example of such approximation. The ruin probability evaluated by simulations is not guaranteed to converge to the theoretical one, because the limiting distribution is discontinuous at zero. To approximate the size of the jump at zero we use the Levy metric, and also confirm the convergence numerically.
Citation: Vyacheslav M. Abramov, Fima C. Klebaner, Robert Sh. Lipster. The Euler-Maruyama approximations for the CEV model. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 1-14. doi: 10.3934/dcdsb.2011.16.1
##### References:
 [1] A. Alfonsi, Higher order discretization schemes for the CIR process: application to affine term structure and Heston models,, Mathematics of Computation, 79 (2010), 209. doi: 10.1090/S0025-5718-09-02252-2. Google Scholar [2] V. Bally and D. Talay, The law of the Euler scheme for stochastic differential equations I. Convergence rate of the distribution function,, Probability Theory and Related Fields, 104 (1996), 43. doi: 10.1007/BF01303802. Google Scholar [3] P. Billingsley, "Converges of Probability Measures,", John Wiley & Sons, (1968). Google Scholar [4] M. Bossy and A. Diop, An efficient discretisation scheme for one dimensional SDEs with a diffusion coefficient function of the form $|x|^\alpha, \alpha\in [1/2, 1)$,, preprint, (2007). Google Scholar [5] J. C. Cox, The constant elasticity of variance option. Pricing model,, The Journal of Portfolio Management, 23 (1997), 15. Google Scholar [6] D. Dawson, http://www.math.ubc.ca/ db5d/SummerSchool09/lectures-dd/lecture4.pdf,, (Accessed on August 3, (2010). Google Scholar [7] G. Deelstra and F. Delbaen, Convergence of discretized stochastic (interest rate) processes with stochastic drift term,, Applied Stochastic Models and Data Analysis, 14 (1998), 77. doi: 10.1002/(SICI)1099-0747(199803)14:1<77::AID-ASM338>3.0.CO;2-2. Google Scholar [8] F. Delbaen and H. A. Shirakawa, Note of option pricing for constant elasticity of variance model,, Asia-Pacific Financial Markets, 9 (2002), 159. doi: 10.1023/A:1024173029378. Google Scholar [9] W. Feller, Two singular diffusion problems,, Annals of Mathematics, 54 (1951), 173. doi: 10.2307/1969318. Google Scholar [10] I. Gy, A note on Eulers approximations,, Potential Analysis, 8 (1998), 205. doi: 10.1023/A:1008605221617. Google Scholar [11] I. Gy，I. ongy and N. Krylov, Existence of strong solutions for Itó's stochastic equations via approximations,, Probability Theory and Related Fields, 105 (1996), 143. doi: 10.1007/BF01203833. Google Scholar [12] N. Halidias and P. Kloeden, A note on the Euler-Maruyama scheme for stochastic differential equations with a discontinuous monotone drift coefficient,, BIT Numerical Mathematics, 48 (2008), 51. doi: 10.1007/s10543-008-0164-1. Google Scholar [13] P. L. Hennequin and A. Tortrat, "Théorie des Probabilités et Quelques Applications,", Masson et Cie, (1965). Google Scholar [14] D. J. Higham and X. Mao, Convergence of Monte Carlo simulations involving the mean-reverting square root process,, Computational Finance, 8 (2005), 35. Google Scholar [15] D. J. Higham, X. Mao and A. M. Stuart, Strong convergence of Euler-type methods for nonlinear stochastic differential equations,, SIAM Journal of Numerical Analysis, 40 (2002), 1041. doi: 10.1137/S0036142901389530. Google Scholar [16] M. Hutzenthaler and A. Jentzen, Non-globally Lipschitz counterexamples for the stochastic Euler scheme,, preprint, (). Google Scholar [17] A. Jentzen, P. E. Kloeden and A. Neuenkirch, Pathwise approximation of stochastic differential equations on domains: Higher order convergence rates without global Lipschitz coeffcients,, Numerische Mathematik, 112 (2009), 41. doi: 10.1007/s00211-008-0200-8. Google Scholar [18] Yu. Kabanov, R. Liptser and A. N. Shiryaev, Estimates of closeness in variation of probability measures (Russian),, Dokl. Akad. Nauk SSSR, 278 (1984), 265. Google Scholar [19] F. C. Klebaner, "Introduction to Stochastic Calculus with Applications,", 2$^{nd}$ edition, (2005). Google Scholar [20] P. E. Kloeden and K. Platten, "Numerical Solution of Stochastic Differential Equations,", Springer-Verlag, (1992). Google Scholar [21] R. Sh. Liptser and A. N. Shiryayev, "Theory of Martingales" [Translated from the Russian by K. Dzjaparidze], Mathematics and its Applications (Soviet Series), 49 (1989). Google Scholar [22] G. N. Milstein and M. V. Tretyakov, "Stochastic Numerics for Mathematical Physics,", Springer-Verlag, (2004). Google Scholar [23] L. C. G. Rogers and D. Williams, "Diffusions, Markov Processes, and Martingales. Vol. 2. Itô Calculus,", Reprint of the 2$^{nd}$ edition, (2000). Google Scholar [24] T. Shiga and S. Watanabe, Bessel diffusions as a one parameter family of diffusion processes,, Zeitschrift f\, 27 (1973), 37. Google Scholar [25] D. Talay, Simulation and numerical analysis of stochastic differential systems: A review,, in, 451 (1995), 63. Google Scholar [26] L. Yan, The Euler scheme with irregular coefficients,, Annals of Probability, 30 (2002), 1172. doi: 10.1214/aop/1029867124. Google Scholar [27] H. Zähle, Weak approximation of SDEs by discrete time processes,, Journal of Applied Mathematics and Stochastic Analysis, 2008 (2008). Google Scholar

show all references

##### References:
 [1] A. Alfonsi, Higher order discretization schemes for the CIR process: application to affine term structure and Heston models,, Mathematics of Computation, 79 (2010), 209. doi: 10.1090/S0025-5718-09-02252-2. Google Scholar [2] V. Bally and D. Talay, The law of the Euler scheme for stochastic differential equations I. Convergence rate of the distribution function,, Probability Theory and Related Fields, 104 (1996), 43. doi: 10.1007/BF01303802. Google Scholar [3] P. Billingsley, "Converges of Probability Measures,", John Wiley & Sons, (1968). Google Scholar [4] M. Bossy and A. Diop, An efficient discretisation scheme for one dimensional SDEs with a diffusion coefficient function of the form $|x|^\alpha, \alpha\in [1/2, 1)$,, preprint, (2007). Google Scholar [5] J. C. Cox, The constant elasticity of variance option. Pricing model,, The Journal of Portfolio Management, 23 (1997), 15. Google Scholar [6] D. Dawson, http://www.math.ubc.ca/ db5d/SummerSchool09/lectures-dd/lecture4.pdf,, (Accessed on August 3, (2010). Google Scholar [7] G. Deelstra and F. Delbaen, Convergence of discretized stochastic (interest rate) processes with stochastic drift term,, Applied Stochastic Models and Data Analysis, 14 (1998), 77. doi: 10.1002/(SICI)1099-0747(199803)14:1<77::AID-ASM338>3.0.CO;2-2. Google Scholar [8] F. Delbaen and H. A. Shirakawa, Note of option pricing for constant elasticity of variance model,, Asia-Pacific Financial Markets, 9 (2002), 159. doi: 10.1023/A:1024173029378. Google Scholar [9] W. Feller, Two singular diffusion problems,, Annals of Mathematics, 54 (1951), 173. doi: 10.2307/1969318. Google Scholar [10] I. Gy, A note on Eulers approximations,, Potential Analysis, 8 (1998), 205. doi: 10.1023/A:1008605221617. Google Scholar [11] I. Gy，I. ongy and N. Krylov, Existence of strong solutions for Itó's stochastic equations via approximations,, Probability Theory and Related Fields, 105 (1996), 143. doi: 10.1007/BF01203833. Google Scholar [12] N. Halidias and P. Kloeden, A note on the Euler-Maruyama scheme for stochastic differential equations with a discontinuous monotone drift coefficient,, BIT Numerical Mathematics, 48 (2008), 51. doi: 10.1007/s10543-008-0164-1. Google Scholar [13] P. L. Hennequin and A. Tortrat, "Théorie des Probabilités et Quelques Applications,", Masson et Cie, (1965). Google Scholar [14] D. J. Higham and X. Mao, Convergence of Monte Carlo simulations involving the mean-reverting square root process,, Computational Finance, 8 (2005), 35. Google Scholar [15] D. J. Higham, X. Mao and A. M. Stuart, Strong convergence of Euler-type methods for nonlinear stochastic differential equations,, SIAM Journal of Numerical Analysis, 40 (2002), 1041. doi: 10.1137/S0036142901389530. Google Scholar [16] M. Hutzenthaler and A. Jentzen, Non-globally Lipschitz counterexamples for the stochastic Euler scheme,, preprint, (). Google Scholar [17] A. Jentzen, P. E. Kloeden and A. Neuenkirch, Pathwise approximation of stochastic differential equations on domains: Higher order convergence rates without global Lipschitz coeffcients,, Numerische Mathematik, 112 (2009), 41. doi: 10.1007/s00211-008-0200-8. Google Scholar [18] Yu. Kabanov, R. Liptser and A. N. Shiryaev, Estimates of closeness in variation of probability measures (Russian),, Dokl. Akad. Nauk SSSR, 278 (1984), 265. Google Scholar [19] F. C. Klebaner, "Introduction to Stochastic Calculus with Applications,", 2$^{nd}$ edition, (2005). Google Scholar [20] P. E. Kloeden and K. Platten, "Numerical Solution of Stochastic Differential Equations,", Springer-Verlag, (1992). Google Scholar [21] R. Sh. Liptser and A. N. Shiryayev, "Theory of Martingales" [Translated from the Russian by K. Dzjaparidze], Mathematics and its Applications (Soviet Series), 49 (1989). Google Scholar [22] G. N. Milstein and M. V. Tretyakov, "Stochastic Numerics for Mathematical Physics,", Springer-Verlag, (2004). Google Scholar [23] L. C. G. Rogers and D. Williams, "Diffusions, Markov Processes, and Martingales. Vol. 2. Itô Calculus,", Reprint of the 2$^{nd}$ edition, (2000). Google Scholar [24] T. Shiga and S. Watanabe, Bessel diffusions as a one parameter family of diffusion processes,, Zeitschrift f\, 27 (1973), 37. Google Scholar [25] D. Talay, Simulation and numerical analysis of stochastic differential systems: A review,, in, 451 (1995), 63. Google Scholar [26] L. Yan, The Euler scheme with irregular coefficients,, Annals of Probability, 30 (2002), 1172. doi: 10.1214/aop/1029867124. Google Scholar [27] H. Zähle, Weak approximation of SDEs by discrete time processes,, Journal of Applied Mathematics and Stochastic Analysis, 2008 (2008). Google Scholar
 [1] Pavel Chigansky, Fima C. Klebaner. The Euler-Maruyama approximation for the absorption time of the CEV diffusion. Discrete & Continuous Dynamical Systems - B, 2012, 17 (5) : 1455-1471. doi: 10.3934/dcdsb.2012.17.1455 [2] Mikhail Krastanov, Michael Malisoff, Peter Wolenski. On the strong invariance property for non-Lipschitz dynamics. Communications on Pure & Applied Analysis, 2006, 5 (1) : 107-124. doi: 10.3934/cpaa.2006.5.107 [3] Boris Hasselblatt and Amie Wilkinson. Prevalence of non-Lipschitz Anosov foliations. Electronic Research Announcements, 1997, 3: 93-98. [4] Wei Mao, Liangjian Hu, Xuerong Mao. Asymptotic boundedness and stability of solutions to hybrid stochastic differential equations with jumps and the Euler-Maruyama approximation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 587-613. doi: 10.3934/dcdsb.2018198 [5] Yavdat Il'yasov. On critical exponent for an elliptic equation with non-Lipschitz nonlinearity. Conference Publications, 2011, 2011 (Special) : 698-706. doi: 10.3934/proc.2011.2011.698 [6] Chunhong Li, Jiaowan Luo. Stochastic invariance for neutral functional differential equation with non-lipschitz coefficients. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3299-3318. doi: 10.3934/dcdsb.2018321 [7] Wei Mao, Liangjian Hu, Surong You, Xuerong Mao. The averaging method for multivalued SDEs with jumps and non-Lipschitz coefficients. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 4937-4954. doi: 10.3934/dcdsb.2019039 [8] Tomás Caraballo, David Cheban. On the structure of the global attractor for non-autonomous dynamical systems with weak convergence. Communications on Pure & Applied Analysis, 2012, 11 (2) : 809-828. doi: 10.3934/cpaa.2012.11.809 [9] Zheng-Hai Huang, Shang-Wen Xu. Convergence properties of a non-interior-point smoothing algorithm for the P*NCP. Journal of Industrial & Management Optimization, 2007, 3 (3) : 569-584. doi: 10.3934/jimo.2007.3.569 [10] Fuke Wu, Xuerong Mao, Peter E. Kloeden. Discrete Razumikhin-type technique and stability of the Euler--Maruyama method to stochastic functional differential equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (2) : 885-903. doi: 10.3934/dcds.2013.33.885 [11] Tomás Caraballo, David Cheban. On the structure of the global attractor for infinite-dimensional non-autonomous dynamical systems with weak convergence. Communications on Pure & Applied Analysis, 2013, 12 (1) : 281-302. doi: 10.3934/cpaa.2013.12.281 [12] Qingguang Guan, Max Gunzburger. Stability and convergence of time-stepping methods for a nonlocal model for diffusion. Discrete & Continuous Dynamical Systems - B, 2015, 20 (5) : 1315-1335. doi: 10.3934/dcdsb.2015.20.1315 [13] Yeping Li, Jie Liao. Stability and $L^{p}$ convergence rates of planar diffusion waves for three-dimensional bipolar Euler-Poisson systems. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1281-1302. doi: 10.3934/cpaa.2019062 [14] Wolf-Jüergen Beyn, Janosch Rieger. The implicit Euler scheme for one-sided Lipschitz differential inclusions. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 409-428. doi: 10.3934/dcdsb.2010.14.409 [15] Wen Tan, Chunyou Sun. Dynamics for a non-autonomous reaction diffusion model with the fractional diffusion. Discrete & Continuous Dynamical Systems - A, 2017, 37 (12) : 6035-6067. doi: 10.3934/dcds.2017260 [16] Lukáš Poul. Existence of weak solutions to the Navier-Stokes-Fourier system on Lipschitz domains. Conference Publications, 2007, 2007 (Special) : 834-843. doi: 10.3934/proc.2007.2007.834 [17] Corrado Lattanzio, Pierangelo Marcati. The relaxation to the drift-diffusion system for the 3-$D$ isentropic Euler-Poisson model for semiconductors. Discrete & Continuous Dynamical Systems - A, 1999, 5 (2) : 449-455. doi: 10.3934/dcds.1999.5.449 [18] Benoît Merlet, Morgan Pierre. Convergence to equilibrium for the backward Euler scheme and applications. Communications on Pure & Applied Analysis, 2010, 9 (3) : 685-702. doi: 10.3934/cpaa.2010.9.685 [19] Y. S. Choi, Roger Lui, Yoshio Yamada. Existence of global solutions for the Shigesada-Kawasaki-Teramoto model with weak cross-diffusion. Discrete & Continuous Dynamical Systems - A, 2003, 9 (5) : 1193-1200. doi: 10.3934/dcds.2003.9.1193 [20] Alexander Mielke. Weak-convergence methods for Hamiltonian multiscale problems. Discrete & Continuous Dynamical Systems - A, 2008, 20 (1) : 53-79. doi: 10.3934/dcds.2008.20.53

2018 Impact Factor: 1.008