Forward Itô SDE Well-Posedness — Examples

Given: The data above, the coefficient maps $b(t,x)=-\theta x$ and $\sigma (t,x)={\sigma }_0$, and the Itô SDE conventions of the A page.

The global well-posedness contract requires jointly Borel coefficients, uniform global state Lipschitz and linear-growth bounds, a filtration-relative Brownian motion, and an independent ${\mathcal{F}}_0$ initial value in $L^2$. Itô SDE data and strong solution

Under those hypotheses the SDE has an adapted continuous strong solution, unique up to indistinguishability, with finite $E{\sup }_{t\le T}\mathrm{\mid }{X}_t{\mathrm{\mid }}^2$. Global strong well-posedness for Lipschitz Itô SDEs

Given: the data above, the Itô SDE convention of Itô SDE data and strong solution, and Full AC.

The SDE definition requires jointly Borel coefficients, uniform global state Lipschitz and linear-growth bounds, and an ${\mathcal{F}}_0$-measurable $L^2$ initial value independent of future Brownian increments. Itô SDE data and strong solution

Under those hypotheses, there is a unique adapted continuous strong solution on the given space. Global strong well-posedness for Lipschitz Itô SDEs

The law of a random element $X_0$ assigns a measurable target set $A$ the probability $P(X_0\in A)$. Law or distribution of a random element

Lebesgue outer measure is the infimum of the sums of ${\mu }_0$ over countable elementary covers. Every half-open box is elementary and has ${\mu }_0(B)=\mathrm{vol}(B)$; in dimension one, $(-1\mathrm{/}n,1\mathrm{/}n]$ is a half-open interval of volume $2\mathrm{/}n$. Lebesgue outer measure on ${\mathbb{R}}^n$ Elementary sets: the finite unions of half-open boxes in ${\mathbb{R}}^n$ The sum of the volumes of a disjoint box decomposition of an elementary set does not depend on the decomposition Half-open boxes in ${\mathbb{R}}^n$ and their volume

Full AC supplies countable choice. Under countable choice, a set of Lebesgue outer measure zero is Lebesgue measurable and has Lebesgue measure zero. The Axiom of Choice AC supplies countable selections and prescribed serial paths The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$) Assuming countable choice, $\mathcal{L}({\mathbb{R}}^n)$ is a sigma-algebra containing every elementary set and ${\lambda }_n$ is a complete measure extending elementary volume Lebesgue measurable sets, the family $\mathcal{L}({\mathbb{R}}^n)$, and the restricted set function ${\lambda }_n$

The integral over a measurable set is the nonnegative Lebesgue integral of the function times its indicator; that integral is the supremum of nonnegative simple minorants, and a simple integral is the sum of its values times the measures of their level sets. Integral over a measurable subset The nonnegative Lebesgue integral The integral of a nonnegative simple function

The singleton $\{0\}$ is Borel: for each $x\mathrm{\ne }0$, $\mathrm{\mid }x\mathrm{\mid }\mathrm{/}2>0$, and if $y\in N_{\mathrm{\mid }x\mathrm{\mid }\mathrm{/}2}(x)$ then $\mathrm{\mid }x\mathrm{\mid }\le \mathrm{\mid }x-y\mathrm{\mid }+\mathrm{\mid }y\mathrm{\mid }<\mathrm{\mid }x\mathrm{\mid }\mathrm{/}2+\mathrm{\mid }y\mathrm{\mid }$, so $y\mathrm{\ne }0$. Thus $\mathbb{R}\setminus \{0\}$ is open and $\{0\}$ is closed; closed sets are Borel. The $\epsilon$-neighbourhood and the punctured $\epsilon$-neighbourhood of a point of $\mathbb{R}$ Basic properties of the absolute value The triangle inequality Open subset of $\mathbb{R}$ (every point has a neighbourhood inside it), closed subset (complement open), and clopen The Borel sigma-algebra of a topological space

For any real $a\ge 0$ with $a\le 2\mathrm{/}n$ for every $n\ge 1$, one has $a=0$: if $a>0$, the Archimedean property supplies $n>2\mathrm{/}a$, so $2\mathrm{/}n<a$, a contradiction. Every complete ordered field is Archimedean