I am interested in solving diffusion equations but where the drift field is treated as unknown and to be inferred instead of fixed. For example, suppose one is interested in modeling a stock where they initially believe the drift is 0. If the stock increases 20% in the first month, this would give evidence that there was a positive drift while if the stock decreased 20% this would imply a negative drift. Formally, let’s define this problem as follows and for simplicity let’s start with one dimension:
$dX(t)=Μ(t)dt+σ_x dW_1(t)$
$dΜ(t)=σ_μ dW_2(t)$
In general, I am interested in the distribution of {$X(t),Μ(t)$} but since the future dynamics depend only on the average of Μ(t), we can instead define $ϕ(t,x)≡⟨Μ(t)│X(t)=x⟩$. With this in mind, we are interested in the evolution of two fields: $p(t,x)≡P(X(t)=x)$ and $ϕ(t,x)$. Note that the evolution is coupled – in each time step the change in $p(t,x)$ depends on the drift and the change in $ϕ(t,x)$ depends on the distribution. The evolution of $p(t,x)$ is simply given by the continuity equation $∂_t[p(t,x)]=∇∙(p(t,x)ϕ(t,x))$.
What I need help with is putting together the differential form for the evolution of $ϕ(t,x)$. The short step evolution is as follows: $ϕ(t+ϵ,x)=∫_Rϕ ̃(t,ξ)p(t,ξ)P(X(t+ϵ)=x│X(t)=ξ)dξ$ Here, $ϕ ̃(t,ξ)$ is the updated drift given that $X(t)=ξ→ X(t+ϵ)=x$. I had solved for the exact formulas here but can’t seem to find them, suffice it to say that it’s a linear combination of the prior $ϕ(t,ξ)$ and $x-ξ$ where the mixing coefficient is dependent on $σ_x$ and $σ_μ$. So, to recap, I am looking for a system of two partial differential equations that describe the joint evolution of the distribution $p(t,x)$ and that of the expected inferred drift given the location $ϕ(t,x)≡⟨Μ(t)│X(t)=x⟩$. Is any part of my logic to this point incorrect and if not any ideas on how to get the PDE for the evolution of $ϕ(t,x)$?