Let $f \in C^2_b(\mathbb{R}^d)$ and $V \in C_b(\mathbb{R}^d)$. Suppose $u : \mathbb{R}_+ \times \mathbb{R}^d \to \mathbb{R}$ satisfies the equation \begin{align*} \partial_t u &= Lu + V(x) u \quad \text{on } \mathbb{R}_+ \times \mathbb{R}^d, \\ u(0, \cdot) &= f \quad \text{on } \mathbb{R}^d. \end{align*} Let $X$ be a solution to $\mathcal{E}_x(\sigma, b)$. Then for all $t > 0$ and $x \in \mathbb{R}^d$, \begin{align*} u(t, x) = \mathbb{E}_x\!\left[f(X_t) \exp\!\left(\int_0^t V(X_s) \, ds\right)\right]. \end{align*}

Assume $U$ has smooth boundary (or satisfies the exterior cone condition), $a$ and $b$ are Hölder continuous on $\bar{U}$, and $a$ is uniformly elliptic. Then for every Hölder continuous $f : \bar{U} \to \mathbb{R}$ and every continuous $g : \partial U \to \mathbb{R}$, the Dirichlet–Poisson problem has a solution.

Assume that weak existence holds for $\mathcal{E}(\sigma, b)$ and that pathwise uniqueness holds. Then: 1. Uniqueness in law holds. 2. For every filtered probability space $(\Omega, \mathcal{F}, (\mathcal{F}_t), \mathbb{P})$ satisfying the usual conditions, every $(\mathcal{F}_t)$-Brownian motion $W$, and every $x \in \mathbb{R}^d$, there exists a unique strong solution to $\mathcal{E}_x(\sigma, b)$.

Let $W$ be a standard Brownian motion on $(\Omega, \mathcal{F}, (\mathcal{F}_t)_{t \geq 0}, \mathbb{P})$, where $(\mathcal{F}_t)$ is the natural filtration of $W$. Then every $(\mathcal{F}_t)$-local martingale $M$ can be written as \begin{align*} M_t = M_0 + \int_0^t H_s \, dW_s \end{align*} for some $(\mathcal{F}_t)$-previsible process $H$ with $\int_0^t H_s^2 \, ds < \infty$ a.s. for all $t$.

Let $M$ be a continuous local martingale with $M_0 = 0$ and $\langle M \rangle_\infty = \infty$ a.s. Define the right-continuous inverse \begin{align*} T_s = \inf\{t \geq 0 : \langle M \rangle_t > s\}, \end{align*} the process $B_s = M_{T_s}$, and the filtration $\mathcal{G}_s = \mathcal{F}_{T_s}$. Then: - Each $T_s$ is an $(\mathcal{F}_t)$-stopping time and $T_s < \infty$ for all $s \geq 0$. - $\langle M \rangle_{T_s} = s$ for all $s \geq 0$. - $B$ is a $(\mathcal{G}_s)$-Brownian motion. - $M_t = B_{\langle M \rangle_t}$ for all $t \geq 0$.

Let $M$ be a continuous local martingale with $M_0 = 0$ and $\langle M \rangle_\infty = \infty$ almost surely. Define the time change $\tau_s = \inf\{t \ge 0 : \langle M \rangle_t > s\}$ for $s \ge 0$. Then the process $W_s = M_{\tau_s}$ is a standard Brownian motion (with respect to the filtration $(\mathcal{F}_{\tau_s})_{s \ge 0}$), and \begin{align*} M_t = W_{\langle M \rangle_t} \quad \text{almost surely for all } t \ge 0. \end{align*}

Let $M$ be a continuous local martingale with $M_0 = 0$, and let $H \in L^2_{\mathrm{loc}}(M)$. Then there exists a unique continuous local martingale $H \cdot M$ with $(H \cdot M)_0 = 0$ satisfying the following: (i) For every continuous local martingale $N$, \begin{align*} \langle H \cdot M, N \rangle = H \cdot \langle M, N \rangle. \end{align*} (ii) For every stopping time $T$, \begin{align*} (\mathbb{1}_{[0,T]} H) \cdot M = (H \cdot M)^T = H \cdot M^T. \end{align*} (iii) If $K$ is previsible, then $K \in L^2_{\mathrm{loc}}(H \cdot M)$ if and only if $HK \in L^2_{\mathrm{loc}}(M)$, and in that case \begin{align*} K \cdot (H \cdot M) = (KH) \cdot M. \end{align*} (iv) If $M \in \mathcal{M}^2_c$ and $H \in L^2(M)$, the definition coincides with the one from Section 3.2.

Let $M$ be a continuous local martingale with $M_0 = 0$. Then there exists a unique (up to indistinguishability) continuous adapted increasing process $(\langle M \rangle_t)_{t \ge 0}$ such that $\langle M \rangle_0 = 0$ and $M^2_t - \langle M \rangle_t$ is a continuous local martingale. Moreover, $\langle M \rangle$ is the u.c.p. limit of the Riemann-sum approximants \begin{align*} \langle M \rangle^{(n)}_t = \sum_{i=1}^{\lfloor 2^n t \rfloor} (M_{i \cdot 2^{-n}} - M_{(i-1) \cdot 2^{-n}})^2. \end{align*}

Let $X \in \mathcal{M}^2$. Then \begin{align*} \mathbb{E}\!\left[\sup_{t \geq 0} X_t^2\right] \leq 4\,\mathbb{E}[X_\infty^2]. \end{align*}

A sequence $(X_n)$ converges to $X$ in $L^1$ if and only if $(X_n)$ is uniformly integrable and $X_n \xrightarrow{\mathbb{P}} X$.