Well-definedness and estimates for the Lipschitz SDE map

For $Z\in {\mathcal{H}}_T$, the path supremum is measurable by [F5], [F3] makes each coefficient composition predictable, and [F1] and Tonelli's theorem [F2] give $E{\int }_0^T(\mathrm{\mid }b(s,Z_s){\mathrm{\mid }}^2+\mathrm{\parallel }\sigma (s,Z_s){\mathrm{\parallel }}_F^2)ds\le 2K^2{\int }_0^T(1+E\mathrm{\mid }{Z}_s{\mathrm{\mid }}^2)ds\le 2K^{2}T(1+\mathrm{\parallel }Z{\mathrm{\parallel }}_T^2)<\mathrm{\infty }\mathrm{.}$ This nonnegative energy is finite almost surely; on each such path, applying $0\le {\int }_0^T(g-\bar{g}{)}^{2}ds$ to $g(s)=\mathrm{\mid }b(s,Z_s)\mathrm{\mid }$ and its average gives ${\int }_0^T\mathrm{\mid }b(s,Z_s)\mathrm{\mid }ds\le \sqrt{T}({\int }_0^T\mathrm{\mid }b(s,Z_s){\mathrm{\mid }}^{2}ds{)}^{1\mathrm{/}2}<\mathrm{\infty }$. Every scalar diffusion coordinate has finite energy. By [F4] each scalar integral has a continuous adapted version; their finite sum and the drift integral define $\mathrm{\Phi }(Z)$ continuously and adaptedly.

Write $S_t=({\sum }_k{\int }_0^t{\sigma }^{ik}(s,Z_s)d{W}_s^k{)}_{i=1}^d$. For vectors $z_1,\dots ,z_m$, expansion gives $m{\sum }_k\mathrm{\mid }{z}_k{\mathrm{\mid }}^2-\mathrm{\mid }{\sum }_k{z}_k{\mathrm{\mid }}^2={\sum }_{k<\mathrm{\ell }}\mathrm{\mid }{z}_k-z_{\mathrm{\ell }}{\mathrm{\mid }}^2\ge 0$. Hence ${\sup }_{u\le t}\mathrm{\mid }{S}_u{\mathrm{\mid }}^2\le m{\sum }_k{\sum }_i{\sup }_{u\le t}\mathrm{\mid }{M}_u^{ik}{\mathrm{\mid }}^2$ for the scalar coordinate integrals $M^{ik}$. Applying [F4] to each gives $E{\sup }_{u\le t}\mathrm{\mid }{S}_u{\mathrm{\mid }}^2\le 4mE{\int }_0^t\mathrm{\parallel }\sigma (s,Z_s){\mathrm{\parallel }}_F^{2}ds$.

For a square-integrable vector path $h$ on $[0,u]$, put $\bar{h}=u^{-1}{\int }_0^{u}h(s)ds$ when $u>0$. The identity $0\le {\int }_0^u\mathrm{\mid }h-\bar{h}{\mathrm{\mid }}^{2}ds={\int }_0^u\mathrm{\mid }h{\mathrm{\mid }}^{2}ds-u^{-1}\mathrm{\mid }{\int }_0^{u}h(s)ds{\mathrm{\mid }}^2$ gives $\mathrm{\mid }{\int }_0^{u}h(s)ds{\mathrm{\mid }}^2\le u{\int }_0^u\mathrm{\mid }h(s){\mathrm{\mid }}^{2}ds$; for $u=0$ both sides vanish. Applying this pathwise to the drift and using $\mathrm{\mid }{X}_0+D+S{\mathrm{\mid }}^2\le 3(\mathrm{\mid }{X}_0{\mathrm{\mid }}^2+\mathrm{\mid }D{\mathrm{\mid }}^2+\mathrm{\mid }S{\mathrm{\mid }}^2)$ give $E{\sup }_{u\le t}\mathrm{\mid }\mathrm{\Phi }(Z{)}_u{\mathrm{\mid }}^2\le 3E\mathrm{\mid }{X}_0{\mathrm{\mid }}^2+3tE{\int }_0^t\mathrm{\mid }b(s,Z_s){\mathrm{\mid }}^{2}ds+12mE{\int }_0^t\mathrm{\parallel }\sigma (s,Z_s){\mathrm{\parallel }}_F^{2}ds\mathrm{.}$ By [F1], each squared coefficient is at most $2K^2(1+\mathrm{\mid }{Z}_s{\mathrm{\mid }}^2)$. Since $t\le T$, the last display is at most $3E\mathrm{\mid }{X}_0{\mathrm{\mid }}^2+(6K^{2}T+24m{K}^2){\int }_0^t(1+\mathrm{\parallel }Z{\mathrm{\parallel }}_s^2)ds$. Hence $\mathrm{\Phi }(Z)\in {\mathcal{H}}_T$ and the first bound holds.

For $Z,V\in {\mathcal{H}}_T$, the initial terms cancel. The two-term square inequality, the drift Cauchy–Schwarz bound, the coordinate estimate of step 1.2, and the Lipschitz part of [F1] give $E{\sup }_{u\le t}\mathrm{\mid }\mathrm{\Phi }(Z{)}_u-\mathrm{\Phi }(V{)}_u{\mathrm{\mid }}^2\le 2t{L}^2{\int }_0^{t}E\mathrm{\mid }{Z}_s-V_s{\mathrm{\mid }}^{2}ds+8m{L}^2{\int }_0^{t}E\mathrm{\mid }{Z}_s-V_s{\mathrm{\mid }}^{2}ds\le (2T{L}^2+8m{L}^2){\int }_0^{t}E{\sup }_{r\le s}\mathrm{\mid }{Z}_r-V_r{\mathrm{\mid }}^{2}ds\mathrm{.}$ This is the second bound.

For the constant input $Z_t=X_0$, apply $\mathrm{\mid }D+S{\mathrm{\mid }}^2\le 2\mathrm{\mid }D{\mathrm{\mid }}^2+2\mathrm{\mid }S{\mathrm{\mid }}^2$ to $\mathrm{\Phi }(Z)-X_0$. Cauchy–Schwarz bounds the drift term by $2T{\int }_0^{T}E\mathrm{\mid }b(s,X_0){\mathrm{\mid }}^{2}ds\le 4K^2{T}^2(1+E\mathrm{\mid }{X}_0{\mathrm{\mid }}^2)$; step 1.2 and [F1] bound the diffusion term by $8m{\int }_0^{T}E\mathrm{\parallel }\sigma (s,X_0){\mathrm{\parallel }}_F^{2}ds\le 16m{K}^{2}T(1+E\mathrm{\mid }{X}_0{\mathrm{\mid }}^2)$. Their sum is the stated base-iterate bound.

If $T=0$, both integrals vanish, $\mathrm{\Phi }(Z{)}_0=X_0$, and all three estimates hold with zero integral terms; for $T>0$, the preceding steps establish every assertion. The coordinate sums are finite and no nondegeneracy of $\sigma$ is used. The Axiom of Choice enters only through the interfaces recorded in [F6].