Global strong well-posedness for Lipschitz Itô SDEs

If $T=0$, set $X_0$ as the sole value of the process; both integrals vanish, the path-sup bound follows from $(1+3E\mathrm{\mid }{X}_0{\mathrm{\mid }}^2)\ge E\mathrm{\mid }{X}_0{\mathrm{\mid }}^2$, and any two solutions with that initial value agree. For $T>0$, let $\mathrm{\Phi }$ be the map of [F1], set $X_t^0=X_0$, and recursively set $X^{n+1}=\mathrm{\Phi }(X^n)$.

Put $a_n(t)=E{\sup }_{u\le t}\mathrm{\mid }{X}_u^n-X_u^{n-1}{\mathrm{\mid }}^2$ for $n\ge 1$. By [F1], $a_1(T)\le A_T:=(4K^2{T}^2+16m{K}^{2}T)(1+E\mathrm{\mid }{X}_0{\mathrm{\mid }}^2)$ and $a_{n+1}(t)\le C_d{\int }_0^t{a}_n(s)ds$. Induction on $n$ gives $a_n(t)\le A_T{C}_d^{n-1}{t}^{n-1}\mathrm{/}(n-1)!$.

The majorant series ${\sum }_{n\ge 1}\sqrt{A_T{C}_d^{n-1}{T}^{n-1}\mathrm{/}(n-1)!}$ converges. If $A_T=0$, it is identically zero; if $C_{d}T=0$, every term after the first is zero; otherwise the ratio of consecutive terms is $\sqrt{C_{d}T\mathrm{/}n}\to 0$. By [F4], $E\sup \mathrm{\mid }{X}^n-X^{n-1}\mathrm{\mid }\le \sqrt{a_n(T)}$ and monotone convergence gives $E{\sum }_n\sup \mathrm{\mid }{X}^n-X^{n-1}\mathrm{\mid }={\sum }_{n}E\sup \mathrm{\mid }{X}^n-X^{n-1}\mathrm{\mid }<\mathrm{\infty }$; hence that nonnegative series is finite almost surely. On this event, for every $t$ the vector sequence $(X_t^n{)}_n$ is Cauchy, since ${\sup }_{u\le T}\mathrm{\mid }{X}_u^m-X_u^n\mathrm{\mid }\le {\sum }_{j=n+1}^m{\sup }_{u\le T}\mathrm{\mid }{X}_u^j-X_u^{j-1}\mathrm{\mid }$; [F7] gives the coordinate limits and therefore a vector limit $X_t$. Letting $m\to \mathrm{\infty }$ in the same inequality bounds ${\sup }_{t\le T}\mathrm{\mid }{X}_t^n-X_t\mathrm{\mid }$ by the tail of the summable series, which tends to zero. Thus the convergence is almost surely uniform. For finite tails the $L^2$ triangle inequality follows by expanding the square and applying Cauchy–Schwarz, so $\mathrm{\parallel }{X}^m-X^n{\mathrm{\parallel }}_{L^2(\sup )}\le {\sum }_{j=n+1}^m\sqrt{a_j(T)}$. Fatou's lemma [F4] then gives $E{\sup }_{t\le T}\mathrm{\mid }{X}_t^n-X_t{\mathrm{\mid }}^2\le ({\sum }_{j=n+1}^{\mathrm{\infty }}\sqrt{a_j(T)}{)}^2\to 0$.

Let $A$ be the measurable probability-one event on which that series of path suprema is finite; measurability follows because each supremum is a supremum over the countable dense set in [F5]. Uniform convergence makes $X$ continuous on $A$; completeness gives $A^c\in {\mathcal{F}}_0$, so set $X_t=X_0$ on $A^c$. For each fixed $t$, the limit on $A$ of the ${\mathcal{F}}_t$-measurable $X_t^n$, together with the ${\mathcal{F}}_0$-measurable value on $A^c$, is ${\mathcal{F}}_t$-measurable. Thus $X$ is adapted and continuous, and its $L^2$ path-sup convergence places it in the class of [F2].

For every $n$, $X^{n+1}=\mathrm{\Phi }(X^n)$. The difference estimate of [F1] gives $E{\sup }_{u\le T}\mathrm{\mid }\mathrm{\Phi }(X^n{)}_u-\mathrm{\Phi }(X{)}_u{\mathrm{\mid }}^2\le C_{d}TE{\sup }_{u\le T}\mathrm{\mid }{X}_u^n-X_u{\mathrm{\mid }}^2\to 0$. Since $X^{n+1}\to X$ in the same norm by step 3.1, uniqueness of an $L^2$ limit implies $\mathrm{\Phi }(X)=X$ indistinguishably. Hence $X$ satisfies the SDE.

Write $f(t)=E{\sup }_{u\le t}\mathrm{\mid }{X}_u{\mathrm{\mid }}^2$. The growth estimate of [F1] and $X=\mathrm{\Phi }(X)$ imply $f(t)\le 3E\mathrm{\mid }{X}_0{\mathrm{\mid }}^2+C_T{\int }_0^t(1+f(s))ds$. Put $g=1+f$ and $H(t)=1+3E\mathrm{\mid }{X}_0{\mathrm{\mid }}^2+C_T{\int }_0^{t}g(s)ds$. Then $g\le H$, $H(0)=1+3E\mathrm{\mid }{X}_0{\mathrm{\mid }}^2$, and $H^{\mathrm{\prime }}(t)=C_{T}g(t)\le C_{T}H(t)$ almost everywhere. Consequently $(e^{-C_{T}t}H(t){)}^{\mathrm{\prime }}\le 0$ almost everywhere, so $f(T)\le g(T)\le (1+3E\mathrm{\mid }{X}_0{\mathrm{\mid }}^2)e^{C_{T}T}$.

If $Y$ is any other strong solution with the same data, then $Y=\mathrm{\Phi }(Y)$ by [F2] and $Y\in {\mathcal{H}}_T$ by [F6]. The difference estimate in [F1] gives $q(t):=E{\sup }_{u\le t}\mathrm{\mid }{X}_u-Y_u{\mathrm{\mid }}^2\le C_d{\int }_0^{t}q(s)ds$. With $Q(t)=C_d{\int }_0^{t}q(s)ds$, one has $Q(0)=0$ and $Q^{\mathrm{\prime }}\le C_{d}Q$ almost everywhere, hence $Q=q=0$. Thus ${\sup }_{t\le T}\mathrm{\mid }{X}_t-Y_t\mathrm{\mid }=0$ almost surely, which is indistinguishability. AC is exactly the inherited assumption [F3].