# diffusion processes and Ito diffusion processes

If I am correct, a diffusion process is defined as a Markov process with a.s. continuous sample paths.

A Ito diffusion process is defined via a SDE. From Wikipedia:

A (time-homogeneous) Itō diffusion in n-dimensional Euclidean space $ℝ^n$ is a process $X : [0, +∞) × Ω → ℝ^n$ defined on a probability space $(Ω, Σ, ℙ)$ and satisfying a stochastic differential equation of the form $$\mathrm{d} X_{t} = b(X_{t}) \, \mathrm{d} t + \sigma (X_{t}) \, \mathrm{d} B_{t},$$ where $B$ is an m-dimensional Brownian motion and $b : ℝ^n → ℝ^n$ and $σ : ℝ^n → ℝ^{n×m}$ satisfy the usual Lipschitz continuity condition $$| b(x) - b(y) | + | \sigma (x) - \sigma (y) | \leq C | x - y |$$ for some constant $C$ and all $x$ and $y$ in $ℝ^n$; this condition ensures the existence of a unique strong solution X to the stochastic differential equation given above.

I was wondering if an Ito diffusion is a diffusion process, but a diffusion process may not be an Ito diffusion. For example, a diffusion process can be defined by a SDE similar to the one above for an Ito diffusion, except that $b$ and $\sigma$ can explicitly depend on $t$ as $b(X_t,t)$ and $\sigma(X_t,t)$?

Thanks and regards!

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As far as I know, there is not widely accepted precise definition of the diffusion process. I would say it is a Markov process with cont. paths without fancy boundary behavior like killing, reflection,.. –  Julian Wergieluk Mar 11 '13 at 7:41
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