Definition 3.1.5 (Diffusion Type SDE).label Let $\sigma: \real^{d} \to L(\real^{d}; \real^{n})$ and $b: \real^{n} \to \real^{n}$ be measurable functions, then a SDE of diffusion type is the equation
\[X_{t} = \xi + \int_{0}^{t} \sigma(X_{s}) dB_{s} + \int_{0}^{t} b(X_{s})ds\]
under the constraint
\begin{equation}\int_{0}^{t} \norm{\sigma(X_s)}_{\real^n}^{2} + \norm{b(X_s)}_{\real^n}ds < \infty \tag{3.1}\end{equation}
for all $t > 0$, where
$B$ is a standard Brownian motion on a filtered probability space $(\Omega, \bracs{\cf_t}, \bp)$.
$\xi$ is a $\cf_{0}$-measurable random variable.