The continuous time limit

The simplest models in finance come from continuous time models. In these models, the time variable, t, may take any value. The asset price, S(t) is defined for any value of t. In the simplest models, S(t) is a continuous and random function of t. Without randomness, we would describe the dynamics of S(t) using an ordinary differential equation. With randomness, we use instead a stochastic differential equation. Stochastic differential equations are more subtle than ordinary differential equations, but they build on many of the same ideas. Stochastic differential equation models have the same relation to discrete tree models that ordinary calculus has to algebra. Tree models and algebra have simpler foundations, but differential equations have simpler formulas. For example, the sum