In the previous section we saw that difference equations can be used to model quite a diverse phenomena but their applicability is limited by the fact that the system should not change between subsequent time steps. These steps can vary from fractions of a second to years or centuries but they must stay fixed in the model. On the other hand, there are numerous situations when changes can occur at all times. These include the growth of populations in which breeding is not restricted to specific seasons, motion of objects, where the velocity and acceleration may change at every instant, spread of an epidemic with no restriction on infection times, and many others. In such cases it is not feasible to model the process by relating the state of the system at a particular instant to a finite number of earlier states (although this part remains as an intermediate stage of the modelling process). Instead, we have to find relations between the rates of change of quantities relevant to the process. The rates of change typically are expressed as derivatives and thus continuous time modelling leads to differential equations which involve the derivatives of the function describing the state of the system.

In what follows we shall derive several differential equation models trying to provide continuous counterparts of some discrete systems described above.

Equations related to financial mathematics

In this section we shall provide continuous counterparts of equations (2.2) and (2.5) and compare the results.

In this chapter we first introduce discrete mathematical models of phenomena happening in the real world. We begin with some explanatory words. Apart from the simplest cases such as the compound interest equation, where the equation is a mathematical expression of rules created by ourselves, the mathematical model attempts to find equations describing events happening according to their own rules, our understanding of which is far from complete. At best, the model can be an approximation of the real world. This understanding guides the way in which we construct the model: we use the principle of economy (similar to the Ockham razor principle) to find the simplest equation which incorporates all relevant features of the modelled events. Such a model is then tested against experiment and only adjusted if we find that its description of salient properties of the real phenomenon we try to model is unsatisfactory.

This explains why we often begin modelling by fitting a linear function to the data and why such linear, or only slightly more complicated, models are commonly used, although everybody agrees that they do not properly describe the real world. The reason is that often they supply sufficient, if not exact, answers at a minimal cost. One must remember, however, that using such models is justified only if we understand their limitations and that, if necessary, are ready to move in with more fine-tuned ones.

Engineers, natural scientists and, increasingly, researchers and practitioners working in economics and other social sciences, use mathematical modelling to solve problems arising in their disciplines. There are at least two identifiable kinds of mathematical modelling. One involves translating the rules of nature or society into mathematical formulae, applying mathematical methods to analyse them and then trying to understand the implications of the obtained results for the original disciplines. The other kind is to use mathematical reasoning to solve practical industrial or engineering problems without necessarily building a mathematical theory for them.

This book is predominantly concerned with the first kind of modelling: that is, with the analysis and interpretation of models of phenomena and processes occurring in the real world. It is important to understand, however, that models only give simplified descriptions of real-life problems but, nevertheless, they can be expressed in terms of mathematical equations and thus can be solved in one way or another.