$\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}$ >[!definition] SDE > Let $d, m\in \ZZ_{\geq 1}$. Let $\sigma$ be a locally bounded $\Hom(\RR^d, \RR^m)$-valued function and $b$ be a locally bounded $\RR^d$-valued function. A ==**stochastic differential equation**== is the formal equation > $ > E(\sigma, b): \qquad dX_t = \sigma(t, X_t)dB_t + b(t, X_t)dt. > $ > Such a SDE $E(\sigma, b)$ is solved by: > - A filtered complete probability space $(\Omega, \FFF, (\FFF_t)_{t\in [0,\infty]}, \PP)$. > - An $m$-dimensional BM $B$. > - An $(\FFF_t)$-adapted path-continuous process $X$ on $\RR^d$ such that$X_t = X_0 + \int_0^t \sigma(s, X_s)dB_s + \int_0^t b(s, X_s)ds.$ > > If $X_0 = x$ AS, then we say $X$ is a solution of $E_x(\sigma, b)$. >[!idea] >It is very important to remember that an SDE does not specify the probability space nor the noise variable. This is analogous to physics modelling; we've been given a process and we can construct all this mathematical stuff however we want, so long as we describe what we see. > >But anyways, we'll often say that "$X$ is a solution to $Equot; and leave implicit that there is also a suitable space and BM admitting this solution. >[!definition] Some nice properties: >Adjectives regarding $E(\sigma,b)$: >- ==**Weak existence**== means that for every $x$, $E_x(\sigma, b)$ has a solution. >- ==**Weak existence and weak uniqueness**== if additionally, all solutions to $E_x(\sigma, b)$ have the same law. >- ==**Pathwise uniqueness**== if, upon fixing $(\Omega, \FFF, (\FFF_t), \PP)$ and $B$, any two solutions $X, X'$ such that $X_0 = X_0'$ AS are indistinguishable. > >Adjectives regarding a solution $X$ to $E_x(\sigma, b)$: >- ==**Strong solution**== if $X$ is adapted wrt the completed canonical filtration of $B$. Here is the general result, unproved and unused: >[!theorem] Yamada-Watanabe >Weak existence and pathwise uniqueness imply weak uniqueness. >Under these conditions, for any choice of filtered space and BM $B$, there exists for every $x$ a unique strong solution of $E_x(\sigma, b)$.