Introduction to Stochastic Differential Equations with Applications to Modelling in Biology and Finance

12Some biological and financial applications

12.1 The Vasicek model and some applications

The Vasicek model (11.3) is studied both in terms of its applications in finance (mainly on exchange rates or on interest rates) and its application in growth of individual living organisms (using as state variable an appropriate function of the size of an individual organism). We have studied the transient distributions in Section 11.1 and obtained the stationary density as a limit when . Since both boundaries and are non‐attractive and the speed measure is finite, we have seen in Section 11.3 that the stationary density exists and is proportional to the speed density, so we can also obtain it directly using the speed density. Furthermore, the process is ergodic.

We have seen in Chapter  that, for the Vasicek model, the stationary distribution is Gaussian with mean equal to the reference rate and variance equal to .

Exercise 12.1

Suppose that , the exchange rate British pound (£) – euro (€), ...