One interpretation can be by letting $X=\frac{dT}{dt}$ and so the Itô formulation is $$dX=(t-A)(t-B)dt+(t-A)(t-B)dB_{t},$$ which is a linear SDE.

So we have

$$X_{t}=X_{0}+\int^{t}(s-A)(s-B)ds+\int^{t} (s-A)(s-B)dB_{s}$$

and thus

$$T_{t}=X_{0}t+\int^{t}(s-A)(s-B)ds+\int\int^{t} (s-A)(s-B)dB_{s}dt.$$

Here we are assuming that by white noise $W(r)$ the OP meant

$$W(r)=\frac{dB_{r}}{dr}.$$

If meant something different, please comment below.

Then one interpretation can be done by letting $X=\frac{dT}{dt}$ and so the Itô formulation is $$dX=(t-A)(t-B)dt+(t-A)(t-B)dB_{t},$$ which is a linear SDE.

So we have

$$X_{t}=X_{0}+\int^{t}(s-A)(s-B)ds+\int^{t} (s-A)(s-B)dB_{s}$$

and thus

$$T_{x}=X_{0}t+\int^{x}\int^{t}(s-A)(s-B)dsdt+\int^{x}\int^{t} (s-A)(s-B)dB_{s}dt.$$