Sethi model

小结：

1、 销量 广告

微分方程 动态系统 市场份额

https://en.wikipedia.org/wiki/Sethi_model

The Sethi model was developed by Suresh P. Sethi and describes the process of how sales evolve over time in response to advertising.^[1][2] The rate of change in sales depend on three effects: response to advertising that acts positively on the unsold portion of the market, the loss due to forgetting or possibly due to competitive factors that act negatively on the sold portion of the market, and a random effect that can go either way.

Suresh Sethi published his paper "Deterministic and Stochastic Optimization of a Dynamic Advertising Model" in 1983.^[1] The Sethi model is a modification as well as a stochastic extension of the Vidale-Wolfe advertising model.^[3] The model and its competitive extensions have been used extensively in the literature.^{[4][5][6][7][8][9][10][11][12][13][14][15][16][17]} Moreover, some of these extensions have been also tested empirically.^{[5][6][9][12]}

The Sethi advertising model or simply the Sethi model provides a sales-advertising dynamics in the form of the following stochastic differential equation:

{\displaystyle dX_{t}=\left(rU_{t}{\sqrt {1-X_{t}}}-\delta X_{t}\right)\,dt+\sigma (X_{t})\,dz_{t},\qquad X_{0}=x}.

Where:

• {\displaystyle X_{t}} is the market share at time {\displaystyle t}
• {\displaystyle U_{t}} is the rate of advertising at time {\displaystyle t}
• {\displaystyle r} is the coefficient of the effectiveness of advertising
• {\displaystyle \delta } is the decay constant
• {\displaystyle \sigma (X_{t})} is the diffusion coefficient
• {\displaystyle z_{t}} is the Wiener process (Standard Brownian motion); {\displaystyle dz_{t}} is known as White noise.