# Writing a given process as a diffusion

When is a stochastic process a diffusion process?

Wikipedia says "A diffusion process is a Markov process with continuous sample paths for which the Kolmogorov forward equation is the Fokker–Planck equation." I don't know how to interpret this; doesn't this just mean it has the typical diffusion form $$$$dX_t = \mu(X_t) dt + \sigma(X_t) dW_t, \;\; X_0 = x_0$$$$ and how would one check this for an arbitrary process?

I understand this means there's an average drift and a variance around that drift. But I don't know if this form is valid for an arbitrary process $$X_t$$, since we can always define the "drift" as $$\mu(X_t) = \mathbb{E}[X_t]$$ and variance $$\sigma(X_t) = \text{Var}(X_t)$$.

For example, define a process $$X_t$$ that increases by $$dX$$ with probability $$p \in [0,1]$$ in each time interval $$dt$$. This resembles a Poisson process with rate $$p$$ and infinitesimal jump size. Since the mean and variance of Poi$$(p)$$ are both $$p$$, then I guess $$\mu(X_t)dt = p dt$$ and $$\sigma^2 = \sqrt{p}$$. But I still don't know if the Kolmogorov forward equation is the Fokker-Planck.

• I don't have an answer, but on Oksendal's book on SDEs, chapter 7 or 8 if I remember correctly there's a section entitled "when is an Ito process a diffusion?", You should check it out Commented Mar 3, 2021 at 21:44
• For others interested: I looked up this chapter and it asks the question "if $X_t$ is an Ito diffusion, for what $C^2$ functions $\phi$ are $\phi(X_t)$ also an Ito diffusion?" I'm not sure whether/how this can answer my question. Does this mean I would have to look for a $\phi^{-1}$ that maps my given process back to a Brownian motion? Commented Mar 8, 2021 at 0:53

Definition: Diffusion process (see here)

A Markov process $$X_t$$ with transition probability $$P(\Gamma, t|x,s)$$ is a diffusion process if

1. for every $$x$$ and every $$\epsilon>0$$, $$\int_{|x-y|>\epsilon} P(dy,t|x,s) = o(t-s).$$ I.e. the probability of moving more than $$\epsilon$$ in a small time frame is small.

2. the drift is given by $$a(x,s)$$ such that for every $$x$$, and every $$\epsilon >0$$, $$\int_{|y-x|\leq \epsilon} (y-x) P(dy,t|x,s) = a(x,s)(s-t) + o(s-t)$$

3. Diffusion coefficient is given by $$b(x,s)$$ such that $$\int_{|y-x|\leq \epsilon} (y-x)^2 P(dy,t|x,s) = b(x,s)(s-t) + p(s-t).$$

I.e. the drift and diffusion coefficients specify the distribution of the movement, giving a necessary concentration of measure.

Characterization by Fokker-Planck equation In my question I said that a diffusion process is a Markov process whose forward equation is the Fokker-Planck equation. How can this be reconciled with the above definition?

This says that if we define a pdf for the transition density, $$P(\Gamma,t|x,s) = \int_\Gamma p(s,x,t,y)dy$$ and that assume $$a, b$$ are differentiable, and assume 1., 2., 3. above, then the transition probability density satisfies the forward Kolmogorov equation.

So in addition note that the equation $$dX_t = a(X,t) dt + b(X_t, t) dB_t, \;\; X_0 = x_0$$, does not specify a diffusion process unless the coefficients are continuous.

Something I am still confused about: I saw somewhere else that the drift is given by $$\mathbb{E}[X_{t+dt}-X_t| X_t = x] = a(x,t)dt + O(dt^2)$$ I'm not sure how this is equivalent to the integral in 2) above, and why there is a $$O(dt^2)$$ term (since the expectation should be fixed).