Markov chain approximations for transition densities of Lévy processes

Aleksandar Mijatovic (Imperial College London)
Matija Vidmar (University of Warwick)
Saul Jacka (University of Warwick)


We consider the convergence of a continuous-time Markov chain approximation $X^h$, $h>0$, to an $\mathbb{R}^d$-valued Lévy process $X$. The state space of $X^h$ is an equidistant lattice and its $Q$-matrix is chosen to approximate the generator of $X$. In dimension one ($d=1$), and then under a general sufficient condition for the existence of transition densities of $X$, we establish sharp convergence rates of the normalised probability mass function of $X^h$ to the probability density function of $X$. In higher dimensions ($d>1$), rates of convergence are obtained under a technical condition, which is satisfied when the diffusion matrix is non-degenerate.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 1-37

Publication Date: January 13, 2014

DOI: 10.1214/EJP.v19-2208


  • Al-Mohy, Awad H.; Higham, Nicholas J. Computing the action of the matrix exponential, with an application to exponential integrators. SIAM J. Sci. Comput. 33 (2011), no. 2, 488--511. MR2785959
  • Bally, Vlad; Talay, Denis. The law of the Euler scheme for stochastic differential equations. II. Convergence rate of the density. Monte Carlo Methods Appl. 2 (1996), no. 2, 93--128. MR1401964
  • Bertoin, Jean. Lévy processes. Cambridge Tracts in Mathematics, 121. Cambridge University Press, Cambridge, 1996. x+265 pp. ISBN: 0-521-56243-0 MR1406564
  • Böttcher, Björn; Schilling, René L. Approximation of Feller processes by Markov chains with Lévy increments. Stoch. Dyn. 9 (2009), no. 1, 71--80. MR2502474
  • Carr, Peter P.; Geman, Hélyette; Madan, Dilip B.; Yor, Marc. The fine structure of asset returns: an empirical investigation, Journal of Business 75 (2002), no. 2, 305-332.
  • Cohen, Serge; Rosiński, Jan. Gaussian approximation of multivariate Lévy processes with applications to simulation of tempered stable processes. Bernoulli 13 (2007), no. 1, 195--210. MR2307403
  • Cont, Rama; Tankov, Peter. Financial modelling with jump processes. Chapman & Hall/CRC Financial Mathematics Series. Chapman & Hall/CRC, Boca Raton, FL, 2004. xvi+535 pp. ISBN: 1-5848-8413-4 MR2042661
  • Cont, Rama; Voltchkova, Ekaterina. A finite difference scheme for option pricing in jump diffusion and exponential Lévy models. SIAM J. Numer. Anal. 43 (2005), no. 4, 1596--1626 (electronic). MR2182141
  • Crosby, John; Le Saux, Nolwenn; Mijatović, Aleksandar. Approximating Lévy processes with a view to option pricing. Int. J. Theor. Appl. Finance 13 (2010), no. 1, 63--91. MR2646974
  • Dudley, R. M. Real analysis and probability. Revised reprint of the 1989 original. Cambridge Studies in Advanced Mathematics, 74. Cambridge University Press, Cambridge, 2002. x+555 pp. ISBN: 0-521-00754-2 MR1932358
  • Evard, J.-Cl.; Jafari, F. A complex Rolle's theorem. Amer. Math. Monthly 99 (1992), no. 9, 858--861. MR1191706
  • Figueroa-López, José E. Approximations for the distributions of bounded variation Lévy processes. Statist. Probab. Lett. 80 (2010), no. 23-24, 1744--1757. MR2734238
  • Filipović, Damir; Mayerhofer, Eberhard; Schneider, Paul. Density approximations for multivariate affine jump-diffusion processes. J. Econometrics 176 (2013), no. 2, 93--111. MR3084047
  • Garroni, M. G.; Menaldi, J.-L. Green functions for second order parabolic integro-differential problems. Pitman Research Notes in Mathematics Series, 275. Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1992. xvi+417 pp. ISBN: 0-582-02156-1 MR1202037
  • Glasserman, Paul. Monte Carlo methods in financial engineering. Applications of Mathematics (New York), 53. Stochastic Modelling and Applied Probability. Springer-Verlag, New York, 2004. xiv+596 pp. ISBN: 0-387-00451-3 MR1999614
  • Higham, Nicholas J. The scaling and squaring method for the matrix exponential revisited. SIAM J. Matrix Anal. Appl. 26 (2005), no. 4, 1179--1193 (electronic). MR2178217
  • Jacod, Jean; Shiryaev, Albert N. Limit theorems for stochastic processes. Second edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 288. Springer-Verlag, Berlin, 2003. xx+661 pp. ISBN: 3-540-43932-3 MR1943877
  • Kiessling, Jonas; Tempone, Raúl. Diffusion approximation of Lévy processes with a view towards finance. Monte Carlo Methods Appl. 17 (2011), no. 1, 11--45. MR2784742
  • Kloeden, Peter E.; Platen, Eckhard. Numerical solution of stochastic differential equations. Applications of Mathematics (New York), 23. Springer-Verlag, Berlin, 1992. xxxvi+632 pp. ISBN: 3-540-54062-8 MR1214374
  • Knopova, Viktorya; Schilling, René L. Transition density estimates for a class of Lévy and Lévy-type processes. J. Theoret. Probab. 25 (2012), no. 1, 144--170. MR2886383
  • Kohatsu-Higa, Arturo; Ortiz-Latorre, Salvador; Tankov, Peter. Optimal simulation schemes for Lévy driven stochastic differential equations, Math. Comp. (to appear).
  • Kyprianou, Andreas E. Introductory lectures on fluctuations of Lévy processes with applications. Universitext. Springer-Verlag, Berlin, 2006. xiv+373 pp. ISBN: 978-3-540-31342-7; 3-540-31342-7 MR2250061
  • Madan, Dilip B.; Carr, Peter; Chang, Eric C. The Variance Gamma Process and Option Pricing, European Finance Review 2 (1998), 79-105.
  • Mijatović, Aleksandar. Spectral properties of trinomial trees. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 463 (2007), no. 2083, 1681--1696. MR2331531
  • Moler, Cleve; Van Loan, Charles. Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later. SIAM Rev. 45 (2003), no. 1, 3--49 (electronic). MR1981253
  • Norris, J. R. Markov chains. Reprint of 1997 original. Cambridge Series in Statistical and Probabilistic Mathematics, 2. Cambridge University Press, Cambridge, 1998. xvi+237 pp. ISBN: 0-521-48181-3 MR1600720
  • Orey, Steven. On continuity properties of infinitely divisible distribution functions. Ann. Math. Statist. 39 1968 936--937. MR0226701
  • Poirot, Jérémy; Tankov, Peter. Monte Carlo Option Pricing for Tempered Stable (CGMY) Processes, Asia-Pacific Financial Markets 13 (2006), no. 4, 327-344.
  • Rosinski, Jan. Simulation of Lévy processes, Encyclopedia of Statistics in Quality and Reliability: Computationally Intensive Methods and Simulation, Wiley, 2008.
  • Sato, Ken-iti. Lévy processes and infinitely divisible distributions. Translated from the 1990 Japanese original. Revised by the author. Cambridge Studies in Advanced Mathematics, 68. Cambridge University Press, Cambridge, 1999. xii+486 pp. ISBN: 0-521-55302-4 MR1739520
  • Sidje, Roger B. Expokit: a software package for computing matrix exponentials, ACM Trans. Math. Softw. 24 (1998), no. 1, 130-156, MATLAB codes at
  • Stewart, Ian; Tall, David. Complex analysis. The hitchhiker's guide to the plane. Cambridge University Press, Cambridge-New York, 1983. ix+290 pp. ISBN: 0-521-24513-3; 0-521-28763-4 MR0698076
  • Szimayer, Alex; Maller, Ross A. Finite approximation schemes for Lévy processes, and their application to optimal stopping problems. Stochastic Process. Appl. 117 (2007), no. 10, 1422--1447. MR2353034
  • Sztonyk, Paweł. Transition density estimates for jump Lévy processes. Stochastic Process. Appl. 121 (2011), no. 6, 1245--1265. MR2794975
  • Tanaka, Hideyuki; Kohatsu-Higa, Arturo. An operator approach for Markov chain weak approximations with an application to infinite activity Lévy driven SDEs. Ann. Appl. Probab. 19 (2009), no. 3, 1026--1062. MR2537198

Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.