# HJM drift condition problem: Show that the HJM drift condition implies $b(t) \equiv b, \rho^{2}(t) \equiv a$

I need your help with understanding and solving the HJM framework. I am hoping I can get some help as I feel so lost with HJM and learning online because of the pandemic is adding more stress. Anyway this is the problem:

Problem

An HJM forward curve evolution by parallel shifts is then of the form $$f(t, T)=h(T-t)+Z(t)$$ \begin{aligned} &\text { for some deterministic initial curve } f(0, T)=h(T) \text { and some Itô process } d Z(t)=\\ &b(t) d t+\rho(t) d W^*(t) \text { with } Z(0)=0 \end{aligned}

Show that the HJM drift condition implies $$b(t) \equiv b, \rho^{2}(t) \equiv a$$, and $$h(x)=-\frac{a}{2} x^{2}+b x+c$$ for some constants $$a \geq 0$$, and $$b, c \in \mathbb{R}$$.

My attempt:

We take the derivative of $$f(t,T)$$

$$d f((t, T))=\left(-h^{\prime}(T-t)+b(t)\right) d t+\rho(t) d w^{*}(t)$$

The Q dynamics of the forward rates of the HJM framework is in the form of:

$$f(t, T)=f(0, T)+\int_{0}^{t}\left(\sigma(s, T) \int_{S}^{T} \sigma(s, u) d u\right) d s+\int_{0}^{t} \sigma(s, T) d u_{t}^{*}$$

$$d f(t, T)=\sigma(t, T) \int_{t}^{T} \sigma(t, u) d u+\sigma\left(s, T\right) d \omega_{t}^{*}$$

Hence the HJM drift equals:

$$d f(t, T)=\rho(t) \int_{t}^{T} \rho(t) d u$$

$$d f(t, T)=\rho^{2}(t)(T-t)$$

Setting $$x = T- t$$ we get:

$$\rho^{2}(t) x=-h(x)+b(t)$$ <- not sure about this part.

Taking the derivative with respect to $$x$$ on both sides we get:

$$\rho^{2}(t)=-h^{\prime \prime}(x)$$

Setting $$x = 0$$ we have:

$$\rho^{2}(t)=-h^{\prime \prime}(0)=a$$