Exact Solutions of Stochastic Differential Equations: Gompertz, Generalized Logistic and Revised Exponential

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.

[1]  Christos H. Skiadas.  Two simple models for the early and middle stage prediction of innovation diffusion , 1987, IEEE Transactions on Engineering Management.

[2]  F. J. Richards.  A Flexible Growth Function for Empirical Use , 1959 .

[3]  S. Swain.  Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences , 1984 .

[4]  Kiyosi Itô.  109. Stochastic Integral , 1944 .

[5]  M. Ogborn.  On the Nature of the Function Expressive of the Law of Human Mortality , 1953 .

[6]  Benjamin Gompertz,et al.  XXIV. On the nature of the function expressive of the law of human mortality, and on a new mode of determining the value of life contingencies. In a letter to Francis Baily, Esq. F. R. S. &c , 1825, Philosophical Transactions of the Royal Society of London.

[7]  Christos H. Skiadas,et al.  A stochastic Bass innovation diffusion model for studying the growth of electricity consumption in Greece , 1997 .

[8]  Kiyosi Itô.  Stochastic Differential Equations , 2018, The Control Systems Handbook.

[9]  E. Renshaw,et al.  STOCHASTIC DIFFERENTIAL EQUATIONS , 1974 .

[10]  B. Gompertz,et al.  On the Nature of the Function Expressive of the Law of Human Mortality , 1825 .

[11]  Christos H. Skiadas,et al.  Two generalized rational models for forecasting innovation diffusion , 1985 .

[12]  C. Skiadas,et al.  Forecasting the Electricity Consumption by Applying Stochastic Modelling Techniques: The Case of Greece , 1995 .

[13]  P. Kloeden,et al.  Numerical Solution of Stochastic Differential Equations , 1992 .

[14]  Michael Pecht,et al.  Investigation of Stochastic Differential Models and a Recursive Nonlinear Filtering Approach for Fusion-Prognostics , 2011 .

[15]  C. Skiadas.  Innovation diffusion models expressing asymmetry and/or positively or negatively influencing forces , 1986 .

[16]  P. Kloeden,et al.  Numerical Solution of Sde Through Computer Experiments , 1993 .

[17]  Peter E. Kloeden,et al.  On effects of discretization on estimators of drift parameters for diffusion processes , 1996, Journal of Applied Probability.