behaves for large values of time; the answer depends on a relationship between the drift and diffusion parameters:
In general it is not possible to find an exact solution, and in these cases we must resort to numerical approximation techniques.
Equation (3.17) is a one-factor equation because there is only one dependent variable (namely
where:
In general the drift and diffusion terms in (3.20) are non-linear:
This is a generalisation of Equation (3.17). Thus, instead of scalars this system employs vectors for the solution, drift and random number terms while the diffusion term is a rectangular matrix.
Existence and uniqueness theorems for the solution of the SDE system (3.20) are similar to those in the one-factor case. For example, theorem 5.2.1 in Øksendal (1998) addresses these issues. We discuss SDEs in more detail in Chapter 13.
3.5 NUMERICAL METHODS FOR ODES
In this section we introduce a class of one-step methods to approximate the solution of ODE system (3.1).
The first step is to replace continuous time by discrete time. To this end, we divide the interval [0, T] into a number of subintervals. We define
In this case we define a set of subintervals
In general, we speak of a non-uniform mesh when the sizes of the subintervals are not necessarily the same. However, in this book we consider in the main a class of finite difference schemes where the N subintervals have the same length (we then speak of a uniform mesh), namely
In general, we define
where
The increment function represents the increment of the approximate solution. In general, the goal is to produce a formula for
