Models of Instantaneous Forward Rates

The Heath, Jarrow and Morton approach to term structure modelling is based on an exogenous specification of the dynamics of instantaneous, continuously compounded forward rates f(t,T). For any fixed maturity T≤T^*, the dynamics of the forward rate f(t,T) are (cf. Heath et al. (1990a, 1992b))

$$df(t,T)=\alpha (t,T)\,dt+\sigma (t,T)\cdot dW_{t},$$

(1)

where α and σ are adapted stochastic processes with values in ℝ and ℝ^d respectively, and W is a d-dimensional standard Brownian motion with respect to the underlying probability measure ℙ (to be interpreted as the actual probability).