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Here is a solution without using the PDE technique, which is preferred as we do not need to assume the affine form of a zero-coupon price from the start. we assume that, under the risk-neutral measure, \begin{align*} dr_t = (\theta(t)-a r_t) dt + \sigma dW_t, \end{align*} where $a$ and $\sigma$ are constants, $a(t)$ is a deterministic function, and $W_t$ is ...

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