1.5. MATHEMATICAL MODELING 45
automatically or naturally. The craft of creating, solving, using, and in-
terpreting a mathematical model must be practiced and developed. The
structured approach to modeling helps distinguish the distinct steps, each
requiring separate intellectual skills. It also provides a framework for devel-
oping and explaining a mathematical model.
An example from physical chemistry
This section illustrates the cycle of mathematical modeling with a simple
example from physical chemistry. This simple example provides us with a
powerful analogy about the role of mathematical modeling in mathematical
finance. I have slightly modified the historical order of discovery to illustrate
the idealized modeling cycle. Scientific progress rarely proceeds in such a
direct line.
Scientists observed that diverse gases such as air, water vapor, hydrogen,
and carbon dioxide all behave predictably and similarly. After many obser-
vations, scientists derived empirical relations such as Boyle’s law, and the
law of Charles and Gay-Lussac about the gas. These laws express relations
among the volumeV, the pressureP, the amountn, and the temperature.
Tof the gas.
In classical theoretical physics, we can define anideal gasby making the
following assumptions [19]:
- A gas consists of particles called molecules which have mass, but es-
sentially have no volume, so the molecules occupy a negligibly small
fraction of the volume occupied by the gas. - The molecules can move in any direction with any speed.
- The number of molecules is large.
- No appreciable forces act on the molecules except during a collision.
- The collisions between molecules and the container are elastic, and
of negligible duration so both kinetic energy and momentum are con-
served. - All molecules in the gas are identical.