I consider following situation described by 2-dimensional Langevin equation with Gaussian noise.(I deal with 2 dimensional case, but even if it is 1 dimensional or higher dimensional case, it will be good for me.)
$dx_{i} = A_i(x,t)dt + B_i(x,t)dW_i (i=1,2)$
As initial condition, random variable $x_1,x_2$ at $t=t_{ini}$ is given, and these random variables are stochastic until the threshold $t=t_{th}$, then finally gets the values $x_{1,th},x_{2,th}$.
Here, I want to compute a joint probability $P_{joint}(t_{th},x_{1,th},x_{2,th})$, but I am confusing and stuck. I thought $P_{joint}$ is the product of $P_{FPT}(t_{th})$ and $P(x_{1,th},x_{2,th};t)$ apparently. $P_{FPT}(t_{th})$ is the probability density function(PDF) of the Frist Passage Time (which is the solution of Adjoint Fokker Planck equation), and $P(x_{1,th},x_{2,th};t)$ is the PDF taking $x_{1,th},x_{2,th}$ at $t$, solution of Fokker Planck equation. However, these two PDF are seemed not to be independent.
Generally speaking, How can we solve this kind of problem for joint probability. Or is there any information or reference about this problem? Please let me know.
In Proposition 5.4.3.1 in Mathematical Methods for Financial Markets, they study the joint density of 1d-Brownian motion and a hitting time using a Fokker-Plank pde with boundary conditions.
In the article First passage times of two-dimensional correlated processes: Analytical results for the Wiener process and a numerical method for diffusion processes, they even study the 2d-version and contain various references for that case.