Introduction to Stochastic Differential Equations with Applications to Modelling in Biology and Finance

8Study of geometric Brownian motion (the stochastic Malthusian model or Black–Scholes model)

8.1 Study using Itô calculus

The Black–Scholes model or stochastic Malthusian model, which was introduced in Chapter , can, among other applications, be used to model the price of a stock in the stock market or the growth of a population of living beings with abundant resources under environmental stochasticity. By environmental stochasticity it is meant that the population lives in an environment subject to random fluctuations that affect the per capita growth rate. You should be aware that this model is not appropriate to deal with demographic stochasticity, i.e. the random sampling variations in the number of births and deaths that occur even when the environment is steady and therefore birth and death rates are not influenced by environmental conditions. The model is described by the SDE

(8.1)

or by the corresponding integral form

(8.2)

Here, represents the ‘average’ return or growth rate. It is always a per unit (per capita in populations, per unit of capital in stocks) rate of growth, but we will simply call it average return rate in case of stocks and average growth rate in case of ...