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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?

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  • $\begingroup$ It is not clear why you can conclude that "Written in this way it seems like this is a Markovian dynamics"? $\endgroup$ – Gordon Sep 24 '19 at 18:17
  • $\begingroup$ 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. $\endgroup$ – ab94 Sep 24 '19 at 21:36

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