Itô SDE data and strong solution

Assume the Axiom of Choice as required by the stochastic-integral interfaces below. Fix a finite horizon $T\ge 0$ and finite integers $d,m\ge 1$. Work on a complete filtered probability space $(\mathrm{\Omega },\mathcal{F},({\mathcal{F}}_t{)}_{0\le t\le T},P)$ whose filtration is right-continuous and contains every $P$-null subset of $\mathcal{F}$ in ${\mathcal{F}}_0$ (the usual conditions).

Let $W=(W_t{)}_{0\le t\le T}$ be a standard $m$-dimensional Brownian motion $d$-dimensional Brownian motion adapted to this filtration, with $W_t-W_s$ independent of ${\mathcal{F}}_s$ and distributed as $N_m(0,(t-s)I_m)$ for $0\le s<t\le T$. Let $X_0:\mathrm{\Omega }\to {\mathbb{R}}^d$ be ${\mathcal{F}}_0$-measurable, satisfy $E\mathrm{\mid }{X}_0{\mathrm{\mid }}^2<\mathrm{\infty }$, and be independent of the future Brownian increments. Explicitly, $X_0$ is independent of the sigma-algebra $\sigma (W_t-W_0:0\le t\le T)$.

Continuity of a process means continuity on one measurable event of probability one. We use an everywhere-continuous adapted representative: if the chosen process is continuous only on such an event $A$, replace it off $A$ by its initial value. Since $A^c$ is a null set and the filtration is complete, $A^c\in {\mathcal{F}}_0$; the replacement preserves adaptedness and changes the SDE identity only on a null event, hence preserves it up to indistinguishability.

The drift and diffusion coefficients are jointly Borel maps $b:[0,T]\times {\mathbb{R}}^d\to {\mathbb{R}}^d,\sigma :[0,T]\times {\mathbb{R}}^d\to {\mathbb{R}}^{d\times m}\mathrm{.}$ For nonnegative finite constants $L,K$, assume for every $t\in [0,T]$ and $x,y\in {\mathbb{R}}^d$ that $\mathrm{\mid }b(t,x)-b(t,y)\mathrm{\mid }+\mathrm{\parallel }\sigma (t,x)-\sigma (t,y){\mathrm{\parallel }}_F\le L\mathrm{\mid }x-y\mathrm{\mid },$ $\mathrm{\mid }b(t,x)\mathrm{\mid }+\mathrm{\parallel }\sigma (t,x){\mathrm{\parallel }}_F\le K(1+\mathrm{\mid }x\mathrm{\mid })\mathrm{.}$ Here $\mathrm{\mid }\cdot \mathrm{\mid }$ is the Euclidean norm and $\mathrm{\parallel }A{\mathrm{\parallel }}_F^2={\sum }_{i=1}^d{\sum }_{k=1}^m\mathrm{\mid }{A}^{ik}{\mathrm{\mid }}^2$ is the Frobenius norm.

For an adapted continuous candidate $X:[0,T]\times \mathrm{\Omega }\to {\mathbb{R}}^d$, the vector Itô equation $X_t=X_0+{\int }_0^{t}b(s,X_s)ds+{({\sum }_{k=1}^m{\int }_0^t{\sigma }^{ik}(s,X_s)d{W}_s^k)}_{i=1}^d,0\le t\le T\mathrm{.}$ uses a pathwise Lebesgue integral for the drift and, in each coordinate, the localized Itô integral from Continuous Brownian Ito processes, clause 2. Require ${\int }_0^T\mathrm{\mid }b(s,X_s)\mathrm{\mid }ds<\mathrm{\infty }$ and ${\int }_0^T\mathrm{\parallel }\sigma (s,X_s){\mathrm{\parallel }}_F^{2}ds<\mathrm{\infty }$ almost surely; these are the local coefficient-integrability conditions in that definition. When a coordinate integrand has finite expected energy, its integral is the square-integrable Itô integral of Ito integral for square-integrable predictable processes; the Picard estimates below establish that stronger condition for every input in ${\mathcal{H}}_T$. Equality of processes means equality for every $t$ on one measurable event of probability one, that is, indistinguishability Process law, modification, and indistinguishability. A strong solution on $[0,T]$ is an adapted continuous process satisfying this integral equation indistinguishably. No moment bound is included in the definition; the existence theorem proves its $L^2$ path-sup bound. The usual-filtration, continuity, and finite-coordinate conventions agree with the multidimensional class of continuous Brownian Itô processes Continuous Brownian Ito processes. This definition makes no ellipticity, transition-density, forward-equation, or time-reversal assertion.

Brownian motion here means the finite-horizon version of the cited definition: $W_0=0$ almost surely, with continuous paths and the stated increments on $[0,T]$. All cited stochastic-calculus interfaces are used on this interval. Their constructions require no extension of $W$ beyond $T$: elementary integrals use only increments between partition points in $[0,T]$, and their $L^2$ limits and continuous versions are constructed on that same interval. For a predictable scalar integrand $H$ with finite pathwise energy, localize at ${\rho }_n=\inf \{t\in [0,T]:{\int }_0^t{H}_s^{2}ds\ge n\}\wedge T$. Here $\inf \mathrm{\varnothing }=+\mathrm{\infty }$. The stopped integrands have expected energy at most $n$; their continuous integrals agree up to the smaller stop by the finite-energy stopping identity. On the full event of finite energy, ${\rho }_n=T$ for all sufficiently large $n$, so these integrals paste uniquely on the whole closed interval $[0,T]$, exactly as in the localized construction of Continuous Brownian Ito processes. At $T=0$ every integral is zero.

Van Handel's Chapter 5.1 specifies the integral-equation interpretation and finite-horizon filtered setup. Van der Vaart's Section 7.1 treats strong solutions on a given filtered space and indistinguishability. The local definition fixes the vector/matrix coordinate convention and the exact coefficient and initial-value assumptions used by the following proofs.