Modeling the Processes with Deterministic and Stochastic Trends

Олександр Миколайович Трофимчук, Тарас Юрійович Кутовий


Most of actual financial and economic processes exhibit nonstationary behavior. Their mathematical expectation and/or variance are functions of time what requires constructing of adequate forecasting models for the processes of this class. The purpose of the work is review of deterministic trends models and their applications, the study of possibilities for application of stochastic processes combinations for describing stochastic trends, as well as development of recommendations for modeling stochastic trends. As the examples of deterministic trends models we used time polynomials, exponents, splines and combinations of harmonic functions. To describe stochastic trends the following combinations of random processes were used: random walk, random walk with noise and shift, and the model of linear local trend. The procedure proposed for mathematical description of stochastic trends provides a possibility for constructing adequate candidate models for estimating of short and medium period forecasts. The study of the modeling procedure proposed with the use of heteroskedastic processes models for forecasting nonstationary process variance provide a possibility for reaching acceptable forecasting quality for stochastic trends. The mean absolute percentage errors for the generated forecasts was in the following limits: 7 – 20%.


Nonstationary random processes; Deterministic and stochastic trends; Mathematical modeling; Model constructing methodology


R.S. Tsay, Analysis of financial time series. New York: John Wiley & Sons, Ltd, 2002, 455 p.

J.D. Hamilton, Time series analysis. Princeton, New Jer­sey: Princeton University Press, 1994, 813 p.

Бідюк П.І., Романенко В.Д., Тимощук О.Л. Аналіз часових рядів. – К.: Політехніка, 2013. – 600 с.

M. Small, Applied nonlinear time series analysis. Dan­vers, MA: World Scientific Publishing, 2005, 261 p.

N. Shepard, Stochastic volatility. New York: Oxford Uni­versity Press Inc., 2005, 534 p.

S.-H. Poon, A practical guide to forecasting financial mar­ket volatility. New York: John Wiley & Sons, Ltd, 2005, 238 p.

J.L. Devore and K.N. Berk, Mathematical statistics with ap­plications. New York: Springer, 2012, 858 p.

B.B. Rao, “Deterministic and stochastic trends in the ti­me series models” MPRA, Rep. 3580, 2007, 19 p.

P. Glasserman, Monte Carlo methods in financial engi­ne­ering. New York: Springer, 2004, 614 p.

GOST Style Citations





  • There are currently no refbacks.