The Riccati ODE is a non-linear ODE of the form:
This ODE has many applications, for example to interest-rate models (Duffie and Kan (1996)). In some cases a closed-form solution to Equation (3.10) is possible, but in this book our focus is on approximating it using the finite difference method.
We now discuss the relationship between the Riccati equation and the pricing of a zero-coupon bond P(t, T), which is a contract that offers one dollar at maturity T. By definition, an affine term structure model assumes that P(t, T) has the form:
Let us assume that the short-term interest rate is described by the following stochastic differential equation (SDE):
where
Duffie and Kan proved that P(t, T) is exponential-affine if and only if the drift
where
The coefficients A(t, T) and B(t, T) in this case are determined by the following ordinary differential equations:
and:
The first Equation (3.11) for B(t, T) is the Riccati equation and the second one (3.12) is solved easily from the first one by integration.
3.3.3 Predator-Prey Models
ODEs can be used as simple models of population growth, for example, by assuming that the rate of reproduction of a population of size P is proportional to the existing population and to the amount of available resources. The ODE is:
where r is the growth rate and K is the carrying capacity. The initial population is
Transformation of this equation leads to the logistic ODE:
(3.13)
where n is the population in units of carrying capacity
For systems, we can consider the predator-prey model in an environment consisting of foxes and rabbits:
where:
