# Hull White and HJM model not Markov

In HJM model we have instaneous forward rate $f(t,T):$

$$d f(t,T) = v(t,T)v_T(t,T)d t - v_T(t,T)d W_t,$$

is Markov. And the spot rate $r(t)$

$$d r(t) = \left\{f_t(0,t) + \int^t_0 [v(\tau,t)v_{tt}(\tau,t)+v_{t}(\tau,t)^2]d\tau - \int^t_0 v_{tt}(\tau,t)d W_\tau\right\}d t$$ $$+ v_{tt}(\tau,t)|_{\tau = t}d W_\tau.$$

Term $$\int^t_0 v_{tt}(\tau,t)d W_\tau$$ makes $r(t)$ non-Markovgenerally(Ho-Lee is not).

But one thing confused me is that, Hull-White (One-Factor) Model: $$d r = [\theta(t) - ar]d t + \sigma d W_t.$$ is HJM model, and $r$ is obviously Markov. But we have $v(t,T)$ in Hull-White: $$v(t,T) = \sigma\dfrac{1 - e^{-a(T-t)}}{a}$$ should make $r(t)$ non-Markov.

So what's wrong here?