= Diagonal short-rate dynamics in the Heath-Jarrow-Morton model
{title2=$dr_t=[\partial_Tf(t,t)+\alpha(t,t)]dt+\sigma(t,t)dW_t$}
For a maturity-differentiable <Heath-Jarrow-Morton model> forward field with the integrability needed for stochastic differentiation, the <short rate> $r_t=f(t,t)$ has the displayed dynamics. Indeed changing observation time uses the forward-rate <stochastic differential equation>, while changing maturity adds $\partial_Tf(t,t)dt$. Under a <risk-neutral measure>, $\alpha(t,T)=\sigma(t,T)\int_t^T\sigma(t,u)du$, so $\alpha(t,t)=0$. The remaining drift depends on the diagonal slope of the whole forward curve, which need not be determined by $r_t$; a <Markov process> for the <short rate> is an additional modelling restriction. Without sufficient maturity regularity, the diagonal need not admit this classical <Itô process> formula.
Back to article page
