Giving meaning to and solving a second-order stochastic differential equation with white noise

Question

I have encountered a second-order stochastic differential equation (SDE) of the form: $$ \frac{d^2 T}{dr^2} = (1 + W(r)) (r - A)(r - B)$$ where $r \in (A, B)$ and $W(r)$ is, for example, white noise. I am aware that second-order SDEs with white noise can pose challenges both in terms of interpretation and solution methods, due to the irregularity of the white noise term.

1. Interpretation: How can I give a rigorous mathematical meaning to this equation, considering the presence of the white noise term? Is there a suitable framework or space where this equation can be properly defined? I've come across the concept of generalized functions (distributions), but I'm unsure how they might apply here.

2. Solution: Assuming we manage to give a proper meaning to the equation, what would be the possible methods or approaches to solve or at least approximate the solution? Are there any known results or references that deal with similar types of equations?

Any guidance, references, or insights would be greatly appreciated!

Answer 1

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

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