Stochastic Differential Equations
Many mathematical modelling scenarios involve an inherent level of uncertainty. For example, rate constants in a chemical reaction model might be obtained experimentally, in which case they are subject to measurement errors. Or the simulation of an epidemic might require an educated guess for the initial number of infected individuals. More fundamentally, there may be microscopic effects that (a) we are not able or willing to account for directly, but (b) can be approximated stochastically. For example, the dynamics of a coin toss could, in principle, be simulated to high precision if we were prepared to measure initial conditions sufficiently accurately and take account of environmental effects, such as wind speed and air pressure. However, for most practical purposes it is perfectly adequate, and much more straightforward, to model the outcome of the coin toss using a random variable that is equally likely to take the value heads or tails. Stochastic models may also be used in an attempt to deal with ignorance. For example, in mathematical finance, there appears to be no universal “law of motion” for the movement of stock prices, but random models seem to fit well to real data.
KeywordsProbability Density Function Lower Picture Straight Line Approximation Weak Error Positive Initial Condition
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