Suppose I have an Ito diffusion SDE: $$ dX_t = b(X_t) \,dt + \sigma(X_t)\,dB_t $$ in $\mathbb{R}^n$. I only know the values of $b$ and $\sigma$ at specific (incidentally random) points $R=\{r_i\;|\;i=1\text{ to }n\}$, but I can interpolate the values to simulate the SDE if needed.
Then, $$ \mathcal{A} = \sum_i b^i \frac{\partial}{\partial x_i} + \frac{1}{2}\sum_i\sum_j [\sigma \sigma^T]_{ij} \frac{\partial^2}{\partial x_i \partial x_j} $$ is the infinitesimal generator of the process.
I want to estimate the transition density function of $X_t$. That is, estimate the heat kernel solution of $u_t = \mathcal{A} u$. (Please point out any inaccuracies in the above!)
To simplify, I'll try to estimate $p(t,x|s,x)$ only. Specifically, I want the values at $R$; i.e. for each $i$, I'd like $p_i(t) := p(t,r_i|0,r_i)$.
My first thought is to use the Fokker-Planck equation: $$ \frac{\partial}{\partial t} p = \mathcal{A}^*p, $$ where $\mathcal{A^*}$ is the adjoint infinitesimal generator given by: $$ \mathcal{A^*}f(t,y) = -\frac{\partial}{\partial y} \left[ b(y)f(t,y) \right] + \frac{1}{2}\frac{\partial^2}{\partial y^2} \left[ \sigma^2(y) f(t,y) \right]. $$ So I can numerically integrate this starting from $p_0(0,x)=1_{x=r_i}$, for each $i$.
Questions:
- Does this make sense?
- Is there a better way to do this by simulating $X_t$ and doing statistics on it (that would be more robust or faster)?