Abstract
This paper deals with the problem of parameter estimation in a class of stochastic differential equations driven by a fractional Brownian motion with \(H \ge 1/2\) and a discontinuous coefficient in the diffusion. Two Bayesian type estimators are proposed for the diffusion parameters based on Markov Chain Monte Carlo and Approximate Bayesian Computation methods.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Andrade P, Rifo L (2017) Long-range dependence and approximate Bayesian computation. Commun Stat Simul Comput 46(2):1219–1237
Beaumont MA, Zhang W, Balding DJ (2002) Approximate Bayesian computation in population genetics. Genetics 162:2025–2035
Bertin K, Torres S, Tudor CA (2011a) Drift parameter estimation in fractional diffusions, martingales and random walks. Stat Probab Lett 81(2):243–249
Bertin K, Torres S, Tudor CA (2011b) Maximum likelihood estimators and random walks in long memory models. Statistics 45(4):361–374
Bokil VA, Gibson NL, Nguyen SL, Thomann EA, Waymire EC (2020) An Euler–Maruyama method for diffusion equations with discontinuous coefficients and a family of interface conditions. J Comput Appl Math 368:112545
Calvet LE, Czellar V (2014) Accurate methods for approximate Bayesian computation filtering. J Financ Econom 13:798–838
De la Cruz R, Meza C, Arribas-Gil A, Carroll RJ (2016) Bayesian regression analysis of data with random effects covariates from nonlinear longitudinal measurements. J Multivar Anal 143:94–106
Dieker A, Mandjes M (2003) On spectral simulation of fractional Brownian motion. Probab Eng Inf Sci 17(3):417–434
Fay D, Moore AW, Brown K, Filosi M, Jurman G (2015) Graph metrics as summary statistics for approximate Bayesian computation with application to network model parameter estimation. J Complex Netw 3:52–83
Garzón J, León JA, Torres S (2017) Fractional stochastic differential equation with discontinuous diffusion. Stoch Anal Appl 35(6):1113–1123
Hu Y, Nualart D (2010) Parameter estimation for fractional Ornstein–Uhlenbeck processes. Stat Probab Lett 80:1030–1038
Ilmonen P, Torres S, Viitasaari L (2020) Oscillating Gaussian processes. Stat Inference Stoch Process. https://doi.org/10.1007/s11203-020-09212-6
Jasra A (2015) Approximate Bayesian computation for a class of time series models. Int Stat Rev 83:405–435
Jasra A, Kantas N, Ehrlich E (2014) Approximate inference for observation-driven time series models with intractable likelihoods. ACM Trans Model Comput Simul 24(3):1–13
Johnston ST, Simpson MJ, McElwain DLS, Binder BJ, Ross JV (2014) Interpreting scratch assays using pair density dynamics and approximate Bayesian computation. Open Biol 4:140097
Karabatsos G, Leisen F (2018) Approximate likelihood perspective on ABC methods. Stat Surv 12:66–104
Keilson J, Wellner JA (1978) Oscillating Brownian motion. J Appl Probab 15(2):300–310
Kubilius K, Mishura Y, Ralchenko K (2017) Estimation in fractional diffusion models. Bocconi and Springer Series, English Edition
Lejay A, Pigato P (2018) Statistical estimation of the oscillating Brownian motion. Bernoulli 24:3568–3602
Lejay A, Pigato P (2019) A threshold model for local volatility: evidence of leverage and mean reversion effects on historical data. Int J Theor Appl Finance 22(04):1950017
Marin JM, Pudlo P, Robert C, Ryder R (2012) Approximate Bayesian computational methods. Stat Comput 22:1167–1180
Nualart D (2006) Malliavin calculus and related topics, 2nd edn. Springer, Berlin
Peters GW, Fan Y, Sisson SA (2012) On sequential Monte Carlo, partial rejection control and approximate Bayesian computation. Stat Comput 22:1209–1222
Peters GW, Panayi E, Septier F (2018) Sequential Monte Carlo-ABC methods for estimation of stochastic simulation models of the limit order book. In: Sisson SA, Fan Y, Beaumont MA (eds) Handbook of approximate Bayesian computation. Chapman and Hall/CRC Press, Boca Raton, pp 437–480
Picchini U (2014) Inference for SDE models via approximate Bayesian computation. J Comput Graph Stat 23:1080–1100
Prakasa Rao BLS (2010) Statistical inference for fractional diffusion processes
Pritchard JK, Seielstad MT, Perez-Lezaun A, Feldman MW (1999) Population growth of human Y chromosomes: a study of Y chromosome microsatellites. Mol Biol Evol 16:1791–1798
Shirota S, Gelfand A (2017) Approximate Bayesian computation and model validation for repulsive spatial point processes. J Comput Graph Stat 26(3):646–657
Sottinen T (2001) Fractional Brownian motion, random walks and binary market models. Finance Stoch 5:343–355
Tavare S, Balding DJ, Griffiths RC, Donnelly P (1997) Inferring coalescence times from DNA sequence data. Genetics 145:505–518
Toni T, Welch D, Strelkowa N, Ipsen A, Stumpf MPH (2009) Approximate Bayesian computation scheme for parameter inference and model selection in dynamical systems. J R Soc Interface 6:187–202
Torres S, Viitasaari L (2019) Stochastic Differential Equations with Discontinuous Diffusion. https://arxiv.org/abs/1908.03183
Acknowledgements
This research was partially supported by Project ECOS - CONICYT C15E05, REDES 150038 and MATHAMSUD 19-MATH-06 SARC. Héctor Araya was partially supported by Proyecto FONDECYT Post-Doctorado 3190465, Soledad Torres was partially supported by FONDECYT Grant 1171335.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Araya, H., Slaoui, M. & Torres, S. Bayesian inference for fractional Oscillating Brownian motion. Comput Stat 37, 887–907 (2022). https://doi.org/10.1007/s00180-021-01146-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00180-021-01146-8