Abstract
The estimation of Lévy process has received a lot of attention in recent years. Evidence of this is the extensive amount of literature concerning this problem which can be classified in two categories: the nonparametric approach, and the parametric approach. In this paper, we shall concentrate on the latter, and in particular the parameters will be estimated within a stochastic programming framework. To be more specific, the first derivative of the characteristic function and its empirical version shall be used in objective function. Furthermore, the parameter estimates are recursively estimated by making use of a modified extended Kalman filter (MEKF). Some properties of the parameter estimates are studied. Finally, a number of simulations will be carried out and the results are presented and discussed.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alessandri A, Cuneo M, Pagnan S, Sanguinetti M (2007) A recursive algorithm for nonlinear least-squares problems. Comput Optim Appl 38:195–216
Basawa IV, Brockwell PJ (1982) Non-Parametric Estimation for Non-Decreasing Lévy Processes. J R Stat Soc Ser B Methodol 44(2):262–269
Belomestny D, Reiss M (2006) Spectral calibration of exponential Lévy models. Finance Stoch 10:449–474
Belomestny D, Comte F, Genon-Catalot V, Masuda H, Reiss M (2015) Lévy Matters IV: Estimation for Discreetly observed Lévy processes. Springer, Cham, Heidelberg, New York, Dordrecht, London
Bertsekas DP (1996) Incremental least squares methods and the extended Kalman filter. J Soc Ind Appl Math 6(3):807–822
Comte F, Genon-Catalot V (2009) Nonparametric estimation for pure jump Lévy processes based on high frequency data. Stochastic Processes and their Applications 119(12):4088–4123
Comte F, Genon-Catalot V (2010) Nonparametric adaptive estimation for pure jump Lvy processes. Annales de lOIstitut Henri Poincare Proabilites et Statistique 46(3):595–617
Comte F, Genon-Catalot V (2011) Estimation for Lévy processes from high frequency data within a long time interval. Ann Stat 39(2):803–837
Figueroa-Lopez JE (2006) Estimation methods for Lévy based model of asset prices, Financial Mathematics Seminar
Gegler G, Stadmuller U (2010) Estimation of the characteristics of a Lévy process. J Stat Planning and Inference 140:1481–1496
Gugushvili S (2009) Nonparametric estimation of the characteristic triplet of a discretely observed Lévy Process. J Non Stat 21(3):321–343
Herman H, Tucker HG (1959) Estimating the Parameters of a Differential Process. Ann Math Statist 30:641–658
Jain NC, Marcus MB (1975) Central limit theorems for C(S)-valued random variables. J Funct Anal 19(3):216–231
Kappus J, Reiss M (2007) Estimation of the Characteristics of a Lévy process observed at arbitrary frequency. Economic Risk 27
Karmista N, Tanaka M, Herskovits J (2009) Nonconvex nonsmooth minimisation via cutting planes and feasible direction interior point method, Turku Centre for Computer Science
Levin A, Khramtsov V (2012) Effective Empirical Characteristic Function Methods for Estimation of Affine Diffusions Using the Realized Variance, Available at SSRN. http://ssrn.com/abstract=1993301 or doi:10.2139/ssrn.1993301
Morijama H, Yamashita N, Fukushima M (2003) The incremental Gauss-Newton algorithm with adaptive stepsize rule. Comput Optim Appl 26:107–141
Neumann M H, Reiss M (2009) Nonparametric estimation for Lévy processes from low-frequency observations. Bernoulli 15(1):223–248
Shapiro A, Dentcheva D, Ruszczynski A (2009) Lectures on Stochastic Programming Modelling and Theory, Siam
Sueishi N, Nishiyama Y (2005) Estimation of Lévy processes in mathematical Finance: A comparative Study. International Congress on Modelling and Simulation: Advances and Applications for Management and Decision Making, 953–959
Ueltzhofer FAJ, Kluppelberg C (2011) An oracle inequality for penalised projection estmation of Lévy densities from high-frequency observation. Journal of Nonparametric Statistics 23(4):967–989
Sant L, Caruana MA (2014) Estimation of Lévy Processes through Stochastic Programming. Stochastic Modelling Techniques and Data Analysis Conference Proceedings 4:43–50
van der Vaart W (2000) Asymptotic Statistics. Cambridge University Press, Cambridge
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Caruana, M.A. Estimation of Lévy Processes via Stochastic Programming and Kalman Filtering. Methodol Comput Appl Probab 19, 1211–1225 (2017). https://doi.org/10.1007/s11009-017-9552-9
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11009-017-9552-9