diffusion process

References:

• http://www0.cs.ucl.ac.uk/staff/C.Archambeau/SDE_web/figs_files/ca07_RgIto_text.pdf
• https://www.ma.imperial.ac.uk/~pavl/lec_diff_proc.pdf

A diffusion process is a Markov process with continuous sample paths. Under some regularity conditions, diffusion processes are fully characterized by their first and second moments.TODO: I don't fully understand this yet.

Formally, a Markov process $x_t$ with joint density function $p(x_t, x_s)$ is a diffusion process if

1. There are no instantaneous jumps, i.e., sample paths are continuous almost surely:
   $$\lim_{t\downarrow s} \frac{1}{|t - s|}\int_{|x_t - x_s| \ge \epsilon} p(x_s, x_t ) dx_t = 0$$
2. The mean has instantaneous rate of change
   $$\lim_{t\downarrow s} \frac{1}{|t - s|}\int_{|x_t - x_s| < \epsilon} (x_t - x_s) p(x_s, x_t ) dx_t = \alpha_s(x_s)$$
3. The squared fluctuations have instantaneous rate of change
   $$\lim_{t\downarrow s} \frac{1}{|t - s|}\int_{|x_t - x_s| < \epsilon} (x_t - x_s)^2 p(x_s, x_t ) dx_t = \beta^2_s(x_s)$$
   Such a process can be constructed as a solution to the stochastic differential equation
   $$dx_t = \alpha_t(x_t) dt + \beta_t(x_t) dW_t.$$
   As a Markov process it is characterized by a transition density $p_{t|s}(x_t | x_s)$. This evolves according to the Fokker-Planck equation aka Kolmogorov forward equation
   $$\frac{\partial p}{\partial t} + \frac{\partial}{\partial x_t} \{\alpha_t(x_t) p\} - \frac{1}{2}\frac{\partial^2}{\partial x_t^2}\{\beta^2_t(x_t)p\}=0$$