# Markovian short rate in HJM framework

In Bjork it is proven in proposition 20.5 that a forward rate dynamics: $$\begin{equation} f(t,T) = f(0,T) + \int_0^t\alpha(s,T)ds + \int_0^t\sigma(s,T)dW(s) \end{equation}$$ imply a dynamics for the short rate $$\begin{equation} dr(t)=a(t)dt + b(t)dW(t) \end{equation}$$ where: $$\begin{equation} \begin{cases} a(t)=\frac{\partial f}{\partial T}(t,t) + \alpha(t,t) \\ b(t)=\sigma(t,t) \end{cases} \end{equation}$$ Written in this way it seems like this is a Markovian dynamics. At the same time we say that in an HJM model a no arbitrage dynamics for the forward rate is

$$\begin{equation} \begin{split} df(t,T) =\sigma(t,T)\sigma^*(t,T)dt + \sigma(t,T)d\overline{W}(t) \end{split} \end{equation}$$ where $$\overline{W}$$ is a brownian motion under the risk neutral measure and

$$\begin{equation} \sigma^*(t,T)=\int_t^T\sigma(t,v)dv \end{equation}$$ In general we say that this forward rate doesn't imply a markovian short rate. To me these formulas seem similar so I don't see why one imply a markovian rate and the other doesn't. Can someone explain this to me or am I missing something?

• It is not clear why you can conclude that "Written in this way it seems like this is a Markovian dynamics"? – Gordon Sep 24 '19 at 18:17
• Actually the question is written badly and I have undertstood my error. I would remove the question but I don't know how to do it. – ab94 Sep 24 '19 at 21:36