I'm playing around with a simulation of a particle that drifts around in space as time passes and looks like this $$ x_{t+1} = x_{t} + \sqrt{dt}\cdot f(x_{t})\cdot Y + dt \cdot g(x_{t}) $$ Where the $dt$ is the time discretization of this process that we use for simulation processes and $f$ and $g$ are non-linear differentiable functions of $X$ and $Y\sim \mathcal{N}(0,1)$. $X\in\mathbf{R}$.
I think I can write out the conditional density then as the following $$ X_{t+1}\mid X_{t} \sim \mathcal{N}(X_{t} + dt\,g(X_{t}),dt\,\left(f(x)\right)^{2}) $$
My questions are as follows:
- Can we think of this as a continuous time process if we take the limit $dt\to 0$?
- Can we think of this as a brownian process? I think the standard deviation being a function on the current position means we don't have stationary increments, so the answer (I think) is no
- Is there a way to find a (nice?) closed form for the density $P(X_{t}\mid X_{0},t)$?
What you wrote there is the Euler-Maruyama discretization of the Ito SDE $dX_t=g(X_t) dt + f(X_t) dW_t$. It typically converges as $dt \to 0$. (Specifically, it is consistent but not necessarily stable, and a breakdown of stability can cause it to fail to converge). You can describe the time evolution of the density exactly using the forward Kolmogorov equation, also called the Fokker-Planck equation in the physics literature. This reads
$$\frac{\partial p}{\partial t} = -\frac{\partial}{\partial x}[g p] + \frac{1}{2} \frac{\partial^2}{\partial x^2}[f^2 p].$$
The update of the conditional density on a short time interval converges to what you said as the length of the interval goes to zero, but unless $f$ and $g$ are actually constants then this is not exact for finite $dt$.