Need help to interpret the definition of a diffusion process

Question

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In these lecture notes at page 15 and 16 I am looking at the definition of diffusion process and the three coniditions which are stated at the top of page 16. These can be difficult to read mathematically.

How would you explain what those conditions are and what are their implications? For instance, let's look at a simple stock price prosses and deterministic volatilty function: $$dS_t/S_t=a(t)dt+b(t)dW_t$$ What does $a$ and $b$ need to satisfy in order for the stock process to be a diffusion process?

Answer 1

Regarding the conditions on page 16, each one of them points to a different property of the SDE solution.

1. Continuity of the process. Notice that the integral represents the probability of ending at a distance larger than $\epsilon$ after $t-s$ units of time have passed.
2. Drift of increments. In this case, the integral represents the expected movement from the starting point. In particular, since the term is normalized by $t-s$, it accounts for the ratio of movement per unit of time.
3. Variance of the increments. As mark leeds pointed out, this integral is computing the variance of the movement.

Answer 2

I would suggest keeping two ideas separate:

(1) Among all Markov Processes, the Diffusion Processes have certain smoothness properties as described on Page 16. The Brownian motion is a classic example. There are also MP's whose statistical properties are not smooth, i.e. do not satisfy the Page 16 properties; a major category are the Jump Processes, of which the classic example is the Poisson Counting Process.

(2) Stochastic Differential Equations SDE's are widely used to generate and study specific examples of diffusion. Which allows us to look at many other types of diffusion beyond BM. If the SDE has a solution, then the solution is always a diffusion. In more advanced books like Oksendal Page 66 there are specific conditions on $a(X,t)$ and $b(X,t)$ that are required for the solution to exist and be unique; roughly speaking these require that A and B do not increase too fast as $X$ increases, or else the stochastic process is going to diverge to $\pm \infty$. But to repeat: if the solution of the SDE exists, it is a diffusion.

Answer 3

The conditions just define a diffusion process.

You know a Markov process has jumps, drift, and a random process. Diffusion process is a Markov process that has continuous paths, drift and diffusion (no jumps), and is completely specified by its first two moments. So the first condition just states continuity, and the other two conditions specify its first two moments (drift and diffusion coefficients).