I am trying to solve an backwards integrated Ito Stochastic Differential equation of the form $$ dx = a dt + bdW \tag{1}$$ where dW is the random variable associated with a Wiener process. However I am confused about the corrosponding PDE equation it represents. Different sources show differences by a minus sign in the time derivative. It comes down to
$$\partial f/\partial t = v \partial f/\partial x + D \partial^2f / \partial x ^2 \tag{2}$$
versus
$$-\partial f/\partial s = v \partial f/\partial x + D \partial^2f / \partial x ^2 \tag{3}$$
Which one here is the correct one? I noticed that one is labled as "s" but I assume that can be simply relabeled to "t" or am I wrong?
In summary: I am wondering if a=v or a=-v, when trying to represent the diffusion equation as a backwards integrated SDE.
I don't have very thorough understanding of the mathmatics behind this concept so please keep the answers simple. I would very much appreicate it thanks!
I found the answer to my question in https://doi.org/10.1007/s11214-017-0351-y
I was wrong on my first assumption where I state "t" and "s" are interchangable. They are not! They are related by $$s = T - t$$ where T is the final time (or "starting time" for integrating the SDE backwards in time).
As a result by change of variables, equations 2 and 3 are identical, which the answer to my question is: $$a = v$$ I hope this helps.