$\require{physics}\newcommand{\cbrt}[1]{\sqrt[3]{#1}}\newcommand{\sgn}{\text{sgn}}\newcommand{\ii}[1]{\textit{#1}}\newcommand{\eps}{\varepsilon}\newcommand{\EE}{\mathbb E}\newcommand{\PP}{\mathbb P}\newcommand{\Var}{\mathrm{Var}}\newcommand{\Cov}{\mathrm{Cov}}\newcommand{\pperp}{\perp\kern-6pt\perp}\newcommand{\LL}{\mathcal{L}}\newcommand{\pa}{\partial}\newcommand{\AAA}{\mathscr{A}}\newcommand{\BBB}{\mathscr{B}}\newcommand{\CCC}{\mathscr{C}}\newcommand{\DDD}{\mathscr{D}}\newcommand{\EEE}{\mathscr{E}}\newcommand{\FFF}{\mathscr{F}}\newcommand{\WFF}{\widetilde{\FFF}}\newcommand{\GGG}{\mathscr{G}}\newcommand{\HHH}{\mathscr{H}}\newcommand{\PPP}{\mathscr{P}}\newcommand{\Ff}{\mathcal{F}}\newcommand{\Gg}{\mathcal{G}}\newcommand{\Hh}{\mathbb{H}}\DeclareMathOperator{\ess}{ess}\newcommand{\CC}{\mathbb C}\newcommand{\FF}{\mathbb F}\newcommand{\NN}{\mathbb N}\newcommand{\QQ}{\mathbb Q}\newcommand{\RR}{\mathbb R}\newcommand{\ZZ}{\mathbb Z}\newcommand{\KK}{\mathbb K}\newcommand{\SSS}{\mathbb S}\newcommand{\II}{\mathbb I}\newcommand{\conj}[1]{\overline{#1}}\DeclareMathOperator{\cis}{cis}\newcommand{\abs}[1]{\left\lvert #1 \right\rvert}\newcommand{\norm}[1]{\left\lVert #1 \right\rVert}\newcommand{\floor}[1]{\left\lfloor #1 \right\rfloor}\newcommand{\ceil}[1]{\left\lceil #1 \right\rceil}\DeclareMathOperator*{\range}{range}\DeclareMathOperator*{\nul}{null}\DeclareMathOperator*{\Tr}{Tr}\DeclareMathOperator*{\tr}{Tr}\newcommand{\id}{1\!\!1}\newcommand{\Id}{1\!\!1}\newcommand{\der}{\ \mathrm {d}}\newcommand{\Zc}[1]{\ZZ / #1 \ZZ}\newcommand{\Zm}[1]{\left(\ZZ / #1 \ZZ\right)^\times}\DeclareMathOperator{\Hom}{Hom}\DeclareMathOperator{\End}{End}\newcommand{\GL}{\mathbb{GL}}\newcommand{\SL}{\mathbb{SL}}\newcommand{\SO}{\mathbb{SO}}\newcommand{\OO}{\mathbb{O}}\newcommand{\SU}{\mathbb{SU}}\newcommand{\U}{\mathbb{U}}\newcommand{\Spin}{\mathrm{Spin}}\newcommand{\Cl}{\mathrm{Cl}}\newcommand{\gr}{\mathrm{gr}}\newcommand{\gl}{\mathfrak{gl}}\newcommand{\sl}{\mathfrak{sl}}\newcommand{\so}{\mathfrak{so}}\newcommand{\su}{\mathfrak{su}}\newcommand{\sp}{\mathfrak{sp}}\newcommand{\uu}{\mathfrak{u}}\newcommand{\fg}{\mathfrak{g}}\newcommand{\hh}{\mathfrak{h}}\DeclareMathOperator{\Ad}{Ad}\DeclareMathOperator{\ad}{ad}\DeclareMathOperator{\Rad}{Rad}\DeclareMathOperator{\im}{im}\renewcommand{\BB}{\mathcal{B}}\newcommand{\HH}{\mathcal{H}}\DeclareMathOperator{\Lie}{Lie}\DeclareMathOperator{\Mat}{Mat}\DeclareMathOperator{\span}{span}\DeclareMathOperator{\proj}{proj}$ Here are an important class of [[Martingales in Continuous Time|CTMs]]. We say that a process $(Z_t)_{t\geq 0}$ on a vector space has ==**independent increments**== if $Z$ is adapted and, for every $0\leq s < t$, $Z_t - Z_s$ is independent of $\FFF_s$. >[!example] Martingales seeded by independent increments > If $Z$ is real-valued, then > 1. If $Z\in L^1$, then $Z_t - \EE[Z_t]$ is a martingale. > 2. If $Z\in L^2$, then $Z_t^2 - \EE[Z_t^2]$ is a martingale. > 3. If, for some $\theta\in \RR$, $\EE[e^{\theta Z_t}] < \infty$ for all $t\geq 0$, then $e^{\theta Z_t} / \EE[e^{\theta Z_t}]$ is a martingale. >[!example] Doob >Let $B_t$ be a brownian motion. Then $B_t^2 - t$ is a continuous martingale >[!example] Exponential Martingales of Brownian Motion >$\exp\left(\theta B_t - \theta^2 t/2\right)$ for $\theta > 0$ is a continuous martingale. >[!example] GWN Martingales > More generally, for any $f\in L^2(\RR_+, \BBB, dt)$, centered Gaussians $Z_t = \int_0^t f(s)dB_s$ have independent increments, thus$\int_0^t f(s)dB_s,\qquad \left(\int_0^t f(s)dB_s\right) - \int_0^t f(s)^2ds,\qquad \exp\left(\theta\int_0^t f(s)dB_s - \frac{\theta^2}{2}\int_0^t f(s)^2ds\right)$ > are all martingales. > [!idea]- After you've read about the stochastic integral > These are continuous (admit continuous modifications)! Indeed, $f(s)\in L^2(\RR_+, \BBB, dt)$ is the (deterministic) progressive process $f_s = f(s)$. This satisfies $\EE\left[\int_0^\infty f(s)^2 d\braket{B,B}_s\right] = \int_0^\infty f(s)^2ds < \infty$, thus $f\in L^2(B)\subset L^2_{\text{loc}}(B)$. Then, the stochastic integral with respect to the continuous martingale $B$ yields a continuous random process, which is a priori a continuous local martingale, but of we just showed $\EE[\braket{f\cdot B, f\cdot B}_\infty] < \infty$, thus $f\cdot B$ is a true martingale bounded in $L^2$. > > Notably, $\EE[(f\cdot B)_t] = 0$, and $\EE[(f\cdot B)_t^2] = \EE[\braket{f\cdot B, f\cdot B}_t] = \EE\left[\left(f^2\cdot \braket{B,B}\right)_t\right] = \int_0^t f(s)^2 ds$. (You already knew that though, because the theory of [[Gaussian Spaces]] told you that $\int_0^t \bullet dB_s\equiv G(\bullet)$ was an isometry from $L^2(\RR_+,\BBB,dt)$ to the Gaussian space $\subset L^2(\Omega,\FFF,\PP)$). > > Ito's formula doesn't really help you get the $\EE[e^{\theta (f\cdot B)_t}]$; instead use your brain cells and observe this is a Laplace transform of a Gaussian. The [[Poisson Process]] is another example of a process, this time discrete-valued, which has independent increments. This creates its own class of martingales.