For citation:
Parphenova A. Y., Saraev L. A. Stochastic model of innovation diffusion that takes into account the changes in the total market volume. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2022, vol. 22, iss. 2, pp. 152-158. DOI: 10.18500/1816-9791-2022-22-2-152-158, EDN: ISFCVE
Stochastic model of innovation diffusion that takes into account the changes in the total market volume
The article proposes a stochastic mathematical model of the diffusion of consumer innovations, which takes into account changes over time in the total number of potential buyers of an innovative product. A stochastic differential equation is constructed for a random value of the number of consumers of an innovative product. The interaction of random changes in the number of consumers with changes in the total market volume of the product under consideration is investigated. Following the Euler – Maruyama method, an algorithm for the numerical solution of the stochastic differential equation for the diffusion of innovations is constructed. For each implementation of this algorithm, the corresponding stochastic trajectories are constructed for a random function of the number of consumers of an innovative product. A variant of the method for calculating the mathematical expectation of a random function of the number of consumers of an innovative product is developed and the corresponding differential equation is obtained. It is shown that the numerical solution of this equation and the average value of the function of the number of consumers calculated for all the implemented implementations of stochastic trajectories give practically the same results. Numerical analysis of the developed model showed that taking into account an external random disturbing factor in the stochastic model leads to significant deviations from the classical deterministic model of smooth market development with innovative goods.
- Bass F. M. A new product growth model for consumer durables. Management Science, 1969, vol. 15, no. 5, pp. 215–227. https://doi.org/10.1287/mnsc.1040.0264
- Bass F. M. The future of research in marketing: Marketing science. Journal of Marketing Research, 1993, vol. 30, pp. 1–6. https://doi.org/10.1177/002224379303000101
- Bass F. M. Empirical generalizations and marketing science: A personal view. Marketing Science, 1995, vol. 14, pp. 6–19. https://doi.org/10.1287/mksc.14.3.G6
- Rogers E. M. Diffusion of Innovations. 5th ed. New York, Free Press, 2003. 576 p.
- Srinivasan V., Mason C. H. Nonlinear least squares estimation of new product diffusion model. Marketing Science, 1986, vol. 5, iss. 2, pp. 169–178.
- Schmittlein D. C., Mahajan V. Maximum likelihood estimation for an innovational diffusion model of new-product acceptance. Management Science, 1982, vol. 1, no. 1, pp. 57–78. https://doi.org/10.1287/MKSC.1.1.57
- Mahajan V., Sharma S. A simple algebraic estimation procedure for innovation diffusion models of new product acceptance. Technological Forecasting and Social Change, 1986, vol. 30, iss. 4, pp. 331–345. https://doi.org/10.1016/0040-1625(86)90031-4
- Mahajan V., Mason C. H., Srinivasan V. An evaluation of estimation procedures for new product diffusion models. In: V. Mahajan, Y. Wind, eds. Innovation Diffusion Models of New Product Acceptance. Cambridge, Ballinger Publishing Company, 1986, pp. 203–232.
- Ito K., McKean H. P. Diffusion Processes and their Sample Paths. Berlin, Heidelberg, Springer, 1996. 323 p. https://doi.org/10.1007/978-3-642-62025-6 (Russ. ed.: Moscow, Mir, 1986. 329 p.).
- Allen E. Modeling with Ito Stochastic Differential Equations. Mathematical Modelling: Theory and Applications, vol. 22. Springer, 2007. 230 p. https://doi.org/10.1007/978-1- 4020-5953-7
- Kanellos N., Katsianis D., Varoutas D. On the Introduction of Diffusion Uncertainty in Telecommunications’ Market Forecasting. Engineering Proceedings, 2021, vol. 5, no. 1, Art. 13. https://doi.org/10.3390/engproc2021005013
- Wang H., Sun B. Diffusion Mechanism of Leading Technology in the New Energy Industry Based on the Bass Model. Frontiers in Energy Research, 2021, vol. 9. https://doi.org/10.3389/fenrg.2021.586787
- Grasman J., Kornelis M. Forecasting product sales with a stochastic Bass model. Journal of Mathematics in Industry, 2019, vol. 9, Art. 2. https://doi.org/10.1186/s13362-019-0059-6
- Ilyina E. A., Parphenova A. Yu., Saraev L. A. Influence of changes to the total volume of the market on the kinetics of the process of diffusion of innovations. Vestnik Altaiskoi akademii ekonomiki i prava [Bulletin of the Altai Academy of Economics and Law], 2019, no. 12, pp. 61–67 (in Russian). https://doi.org/10.17513/vaael.848
- Saraev A. L., Saraev L. A. Stochastic calculation of curves dynamics of enterprise. Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, 2020, vol. 24, no. 2, pp. 343–364 (in Russian). https://doi.org/10.14498/vsgtu1700
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