On the Efficiency and Robustness of Commingle Wiener and Lévy Driven Processes for Vasciek Model

The driven processes of Wiener and Lévy are known
self-standing Gaussian-Markov processes for fitting non-linear
dynamical Vasciek model. In this paper, a coincidental Gaussian
density stationarity condition and autocorrelation function of the
two driven processes were established. This led to the conflation
of Wiener and Lévy processes so as to investigate the efficiency
of estimates incorporated into the one-dimensional Vasciek model
that was estimated via the Maximum Likelihood (ML) technique.
The conditional laws of drift, diffusion and stationarity process
was ascertained for the individual Wiener and Lévy processes as
well as the commingle of the two processes for a fixed effect
and Autoregressive like Vasciek model when subjected to financial
series; exchange rate of Naira-CFA Franc. In addition, the model
performance error of the sub-merged driven process was miniature
compared to the self-standing driven process of Wiener and Lévy.




References:
[1] M.S. Ldeo, An Introduction to Dynamical Systems and Chaos, 1997.
[2] J. Gleick, C. Penguin, Q172.5.C45G54, 1987.
[3] S. Roweis, Z. Ghahramani, An EM algorithm for identification
of non-linear dynamical systems. Gatsby Computational Neuroscience
Unit, University College London, London WCIN 3AR, U.K, 2010.
http://gatsby.ucl.ac.uk.s
[4] D. Grahova, N.N. Leonenko, A. Sikorskii, M.S. Taqqu, “The unusual
properties of aggregated superpositions of Ornstein–Uhlenbeck type
Processes,” Bernoulli, 2017, pp. 1-34.
[5] C. Lee, J.P.N. Bishwal, M.H. Lee, “Sequential maximum likelihood
estimation for reflected Ornstein - Uhlenbeck processes,” Journal of
Statistics Planning Inference, vol.142, 2012, pp. 1234-1242.
[6] R. Kleeman, “Information theory and dynamical system predictability,”
Entropy, vol. 13, 2011, pp. 612-649. doi:10.3390/e13030612.
[7] E. Bibbona, G. Panfilo, P. Tavella, “The Ornstein–Uhlenbeck process as
a model for filtered white noise,” dell’Universit di Torino, vol. 45, 2008,
pp. 117-126. doi:10.1088/0026-1394/45/6/S17.
[8] G.E. Uhlenbeck, L.S. Ornstein, Physical Review, vol. 36, 1930, pp.
823-84.
[9] S. Kullback, R.A. Leibler, “On information and sufficiency,” Annals of
Mathematical Statistics,vol. 22 (1), 1951, pp.7986.
[10] C. Archambeau, D. Cornford, M. Opper, J. Shawe-Taylor, “Gaussian
process approximations of stochastic dierential equations,” JMLR:
Workshop and Conference Proceedings 1:,2007,1-16.
[11] L. Eyinck, O. Gregory, R.R. Juan, “Most probable histories for nonlinear
dynamics: tracking climate transitions,” Journal of Statistical Physics,
vol.101, 2004, 459472.
[12] A. Apte, M. Hairer, A. Stuart, J. Voss, “Sampling
the posterior: An approach to non–Gaussian data
assimilation,” Physica D, 2006, Submitted, available from
http://www.maths.warwick.ac.uk/ stuart/sample.html.
[13] O¨ . O¨ nalan, “Financial Modelling with OrnsteinUhlenbeck processes
driven by L´evy process”, Proceedings of the World Congress on
Engineering, Vol. II, WCE 2009, London, U.K.
[14] L. Valdivieso, W. Schoutens, F. Tuerlinckx, “Maximum likelihood
estimation in processes of Ornstein-Uhlenbeck type,” Stat. Infer. Stoch.
Process, vol.12, 2009, 119. doi: 10.1007/s11203-008-9021-8.
[15] A. Kyprianou, Introductory lectures on fluctuations of L´evy processes
with applications, Springer, 2006. [16] O.I. Shittu, O.O. Otekunrin, C.G. Udomboso, K. Adepoju, Introduction
to probability and stochastic processes with applications, 2014, ISBN
978-978-2890-0-8.