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Note that under Risk Neutral measure, Hull White SDE for short rate is $r_{t}=X_{t}+\varphi(t)+f^{M}(0, t)$ where $X_t$ is pure mean reverting process under Risk Neutral measure: $ \mathrm{d} \mathrm{X}_{\mathrm{t}}=-\mathrm{a} \mathrm{X}_{\mathrm{t}} \mathrm{dt}+\sigma(\mathrm{t}) \mathrm{d} W_{\mathrm{t}}$ Other terms in $r_t$ are just used to match ...


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