Abstract
Exact analytic solutions of some stochastic differential equations are given along with characteristic futures of these models as the Mean and Variance. The procedure is based on the Ito calculus and a brief description is given. Classical stochastic models and also new models are provided along with a related bibliography. Stochastic models included are the Gompertz, Linear models with multiplicative noise term, the Revised Exponential and the Generalized Logistic. Emphasis is given in the presentation of stochastic models with a sigmoid form for the mean value. These models are of particular interest when dealing with the innovation diffusion into a specific population, including the spread of epidemics, diffusion of information and new product adoption.
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References
Gardiner CW (1990) Handbook of stochastic methods for physics, chemistry and natural science, 2nd edn. Springer, Berlin
Gompertz B (1825) On the nature of the function expressive of the law of human mortality, and on the mode of determining the value of life contingencies. Philos Trans R Soc 115:513–585
Gihman II, Skorokhod AV (1972) Stochastic differential equations. Springer, Berlin
Giovanis AN, Skiadas CH (1995) Forecasting the electricity consumption by applying stochastic modeling techniques: the case of Greece. In: Janssen J, Skiadas CH, Zopounidis C (eds) Advances in applying stochastic modeling and data analysis. Kluwer, Dordrecht
Ito K (1944) Stochastic integral. Proc Imp Acad Tokyo 20:519–524
Ito K (1951) On stochastic differential equations. Mem Am Math Soc 4:1–51
Kloeden PE, Platen E (1999) Numerical solution of stochastic differential equations. Springer, Berlin
Kloeden PE, Platen E, Schurz H (2003) Numerical solution of SDE through computer experiments. Springer, Berlin
Kloeden PE, Schurz H, Platten E, Sorensen M (1992) On effects of discretization on estimators of drift parameters for diffusion processes. Research Report no. 249, department of theoretical statistics. Institute of Mathematics, University of Aarhus
Richards FJ (1959) A flexible growth function for empirical use. J Exp Bot 10:290–300
Skiadas CH (1985) Two generalized rational models for forecasting innovation diffusion. Technol Forecast Soc Change 27:39–61
Skiadas CH (1986) Innovation diffusion models expressing asymmetry and/or positively or negatively influencing forces. Technol Forecast Soc Change 30:313–330
Skiadas CH (1987) Two simple models for the early and middle stage prediction of innovation diffusion. IEEE Trans Eng Manage 34:79–84
Skiadas CH, Giovanis AN (1997) A stochastic bass innovation diffusion model studying the growth of electricity consumption in Greece. Appl Stoch Models Data Anal 13:85–101
Skiadas CH, Giovanis AN, Dimoticalis J (1993) A sigmoid stochastic growth model derived from the revised exponential. In: Janssen J, Skiadas CH (eds) Applied stochastic models and data analysis. World Scientific, Singapore, pp 864–870
Skiadas CH, Giovanis AN, Dimoticalis J (1994) Investigation of stochastic differential models: the gompertzian case. In: Gutierez R, Valderama Bonnet MJ (eds) Selected topics on stochastic modeling. World Scientific, Singapore
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Skiadas, C.H. Exact Solutions of Stochastic Differential Equations: Gompertz, Generalized Logistic and Revised Exponential. Methodol Comput Appl Probab 12, 261–270 (2010). https://doi.org/10.1007/s11009-009-9145-3
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DOI: https://doi.org/10.1007/s11009-009-9145-3
Keywords
- Stochastic simulation
- Analytic solutions
- Stochastic modeling
- Stochastic Gompertz model
- Stochastic generalized Logistic model
- Revised exponential
- Stochastic simulation