Stochastic differential equation

Definition

Let $(\Omega, \mathcal{F}, (\mathcal{F}_t)_{t \ge 0}, P)$ be a probability space with a filtration that satisfies the usual conditions. Let $W = (W_t)_{t \ge 0}$ be an $m$-dimensional Wiener process with respect to this filtration. Let $T > 0$, and let $b \colon [0, T] \times \R^n \to \R^n$ and $\sigma \colon [0, T] \times \R^n \to \R^{n \times m}$ be Borel measurable. Let $\xi$ be an $\R^n$-valued $\mathcal{F}_0$-measurable random variable. The stochastic differential equation with drift $b$, diffusion coefficient $\sigma$ and initial value $\xi$ is $$dX_t = b(t, X_t) \, dt + \sigma(t, X_t) \, dW_t, \qquad X_0 = \xi.$$ A strong solution is a continuous adapted process $X$ with values in $\R^n$ with two properties. Almost surely, $\int_0^T \left( \abs{b(s, X_s)} + \abs{\sigma(s, X_s)}^2 \right) ds < \infty$. Almost surely, $$X_t = \xi + \int_0^t b(s, X_s) \, ds + \int_0^t \sigma(s, X_s) \, dW_s \quad \text{for all } t \in [0, T],$$ where the last integral is the Itô integral of Itô calculus.