Euler–Maruyama one-step moment and increment estimates

The law in [F1] has the representation $\sqrt{\delta }Z$ with $Z$ realized as in [F3]. By [F4] each coordinate has density $e^{-x^2\mathrm{/}2}\mathrm{/}\sqrt{2\pi }$. Its $p$th absolute moment is finite: the integral on $[-1,1]$ is finite, and for $\mathrm{\mid }x\mathrm{\mid }\ge 1$ the function $\mathrm{\mid }x{\mathrm{\mid }}^p{e}^{-x^2\mathrm{/}4}$ is bounded, so $\mathrm{\mid }x{\mathrm{\mid }}^p{e}^{-x^2\mathrm{/}2}\le C_p{e}^{-x^2\mathrm{/}4}$. The finite-dimensional inequality $\mathrm{\mid }Z{\mathrm{\mid }}^p\le m^{p\mathrm{/}2-1}{\sum }_{j=1}^m\mathrm{\mid }{Z}_j{\mathrm{\mid }}^p$ then gives $\mathbb{E}\mathrm{\mid }Z{\mathrm{\mid }}^p<\mathrm{\infty }$ and $\mathbb{E}(\mathrm{\mid }{\sigma }_s(U)(W_{s+\delta }-W_s){\mathrm{\mid }}^p\mid {\mathcal{F}}_s)\le c_{p,m}{\delta }^{p\mathrm{/}2}\mathrm{\parallel }{\sigma }_s(U){\mathrm{\parallel }}_F^p$. For $\delta =0$ the increment is zero.

For $\varphi (x)=\mathrm{\mid }x{\mathrm{\mid }}^p$, direct differentiation gives $\mathrm{\parallel }{D}^2\varphi (x)\mathrm{\parallel }\le C_p\mathrm{\mid }x{\mathrm{\mid }}^{p-2}$ for $p\ge 2$. Apply the one-dimensional Taylor formula with integral remainder to $r\mapsto \varphi (U+rR)$, where $R=\delta b_s(U)+{\sigma }_s(U)(W_{s+\delta }-W_s)$. The triangle inequality and $(a+b{)}^q\le C_q(a^q+b^q)$ for $q\ge 0$ yield $\varphi (U+R)\le \varphi (U)+D\varphi (U)R+C_p(\mathrm{\mid }U{\mathrm{\mid }}^{p-2}\mathrm{\mid }R{\mathrm{\mid }}^2+\mathrm{\mid }R{\mathrm{\mid }}^p)$.

The drift term in $V-U$ is ${\mathcal{F}}_s$-measurable and the Brownian increment is conditionally centered with covariance $\delta I_m$. Therefore the cross term has conditional expectation zero and $\mathbb{E}(\mathrm{\mid }V-U{\mathrm{\mid }}^2\mid {\mathcal{F}}_s)={\delta }^2\mathrm{\mid }{b}_s(U){\mathrm{\mid }}^2+\delta \mathrm{\parallel }{\sigma }_s(U){\mathrm{\parallel }}_F^2\le C_{2,m,K}\delta (1+\mathrm{\mid }U{\mathrm{\mid }}^2)$. This also holds for $\delta =0$.

The coefficient growth assumption [F2] and $\delta \le 1$ give $\mathrm{\mid }{b}_s(U){\mathrm{\mid }}^p+\mathrm{\parallel }{\sigma }_s(U){\mathrm{\parallel }}_F^p\le C_{p,K}(1+\mathrm{\mid }U{\mathrm{\mid }}^p)$. Conditional centering gives $\mathbb{E}(R\mid {\mathcal{F}}_s)=\delta b_s(U)$. By the growth bound and $\mathrm{\mid }a+b{\mathrm{\mid }}^2\le 2\mathrm{\mid }a{\mathrm{\mid }}^2+2\mathrm{\mid }b{\mathrm{\mid }}^2$, $\mathbb{E}(\mathrm{\mid }R{\mathrm{\mid }}^2\mid {\mathcal{F}}_s)\le 2K^2({\delta }^2+\delta )(1+\mathrm{\mid }U{\mathrm{\mid }}^2)\le 4K^2\delta (1+\mathrm{\mid }U{\mathrm{\mid }}^2)$. For the $p$th moment, $\mathrm{\mid }a+b{\mathrm{\mid }}^p\le 2^{p-1}(\mathrm{\mid }a{\mathrm{\mid }}^p+\mathrm{\mid }b{\mathrm{\mid }}^p)$ and step 1.1 give $\mathbb{E}(\mathrm{\mid }R{\mathrm{\mid }}^p\mid {\mathcal{F}}_s)\le C_{p,m,K}({\delta }^p+{\delta }^{p\mathrm{/}2})(1+\mathrm{\mid }U{\mathrm{\mid }}^p)\le C_{p,m,K}\delta (1+\mathrm{\mid }U{\mathrm{\mid }}^p)$, since $p\ge 2$ and $\delta \le 1$. Also $\mathrm{\mid }D\varphi (U)\cdot \delta b_s(U)\mathrm{\mid }\le C_{p,K}\delta (1+\mathrm{\mid }U{\mathrm{\mid }}^p)$. Substituting these bounds into step 1.2 and using $\mathrm{\mid }U{\mathrm{\mid }}^{p-2}(1+\mathrm{\mid }U{\mathrm{\mid }}^2)\le 2(1+\mathrm{\mid }U{\mathrm{\mid }}^p)$ proves $\mathbb{E}(\mathrm{\mid }V{\mathrm{\mid }}^p\mid {\mathcal{F}}_s)\le (1+C_{p,m,K}\delta )\mathrm{\mid }U{\mathrm{\mid }}^p+C_{p,m,K}\delta$. The AC assumption is inherited from the cited Brownian and normal-law interfaces; these finite-dimensional estimates make no choice.