Euler–Maruyama step process and continuous interpolation

Use the data and grid of Euler–Maruyama recursion, and let $Y_0,\dots ,Y_N$ be its recursion. The left-endpoint map is ${\eta }_h(t)=\max \{t_k:t_k\le t\},0\le t\le T\mathrm{.}$ The right-continuous step process is ${\bar{Y}}_t=Y_k(t_k\le t<t_{k+1}),{\bar{Y}}_T=Y_N\mathrm{.}$ The continuous interpolation is the pathwise process $Y_t=Y_k+b(t_k,Y_k)(t-t_k)+\sigma (t_k,Y_k)(W_t-W_{t_k})(t_k\le t\le t_{k+1})\mathrm{.}$ At a shared grid endpoint the formulas agree because the recursion gives $Y_{k+1}=Y_k+b(t_k,Y_k)h+\sigma (t_k,Y_k)(W_{t_{k+1}}-W_{t_k})$. In particular $Y_{t_k}=Y_k$ for every $k$, including $Y_T=Y_N$.

For stochastic integration, define the predictable left-endpoint version by ${\bar{Y}}_0^{-}=Y_0$ and ${\bar{Y}}_s^{-}=Y_k$ on $(t_k,t_{k+1}]$, and set ${\eta }_h^{-}(0)=0$ and ${\eta }_h^{-}(s)=t_k$ on $(t_k,t_{k+1}]$. Its coefficient integrands use the same fixed left endpoints. When the recursion is adapted, each coordinate of ${\bar{Y}}^{-}$ and each coefficient composition is a predictable finite-step process: on $(t_k,t_{k+1}]$ its value is ${\mathcal{F}}_{t_k}$-measurable. These coefficients need not be bounded, so the integrands need not be elementary in the bounded-coefficient sense of Elementary predictable Brownian integrands. Values at the finitely many grid points do not change either integral. Once square integrability is established, A square-integrable predictable step Itô integral is its finite sum identifies each finite-coordinate Itô integral of Ito integral for square-integrable predictable processes with its finite-sum version, giving the equivalent formula $Y_t=X_0+{\int }_0^{t}b({\eta }_h^{-}(s),{\bar{Y}}_s^{-})ds+{\sum }_{j=1}^m{\int }_0^t{\sigma }_{\cdot j}({\eta }_h^{-}(s),{\bar{Y}}_s^{-})d{W}_s^j\mathrm{.}$ where ${\eta }_h^{-}(s)=t_k$ on $(t_k,t_{k+1}]$ and ${\sigma }_{\cdot j}$ denotes column $j$ of the matrix.

The justified-by theorem Euler–Maruyama uniform moments and interpolation increments ↗ establishes the required integrability and well-definedness under the stated coefficient and initial-moment conditions.

For well-definedness on any allowed grid, including $h>1$, put $u=t\mathrm{/}T$, ${\tilde{\mathcal{F}}}_u={\mathcal{F}}_{Tu}$, ${\tilde{W}}_u=T^{-1\mathrm{/}2}{W}_{Tu}$, $\tilde{b}(u,x)=Tb(Tu,x)$, and $\tilde{\sigma }(u,x)=\sqrt{T}\sigma (Tu,x)$ on $[0,1]$. The rescaled Brownian increments have covariance $(v-u)I_m$ and are independent of ${\tilde{\mathcal{F}}}_u$ for $u<v$. The rescaled coefficients inherit the Borel, state-Lipschitz, linear-growth and half-Hölder time-modulus assumptions with finite constants depending on $T$. The normalized mesh is $1\mathrm{/}N\le 1$, and substitution in the recursion and interpolation gives exactly $Y_k$ and $Y_{Tu}$, respectively. Thus Euler–Maruyama uniform moments and interpolation increments ↗, applied to these rescaled data with $p=2$, gives adapted square-integrable grid values and an adapted continuous interpolation for every original $h>0$.

In particular, the original coefficient integrands satisfy $\mathbb{E}{\int }_0^T(\mathrm{\mid }b({\eta }_h^{-}(s),{\bar{Y}}_s^{-}){\mathrm{\mid }}^2+\mathrm{\parallel }\sigma ({\eta }_h^{-}(s),{\bar{Y}}_s^{-}){\mathrm{\parallel }}_F^2)ds\le K^2{\sum }_{k=0}^{N-1}h(1+\mathbb{E}\mathrm{\mid }{Y}_k{\mathrm{\mid }}^2)<\mathrm{\infty }\mathrm{.}$ Their predictability follows from left-endpoint measurability as above. The drift integral is its finite sum, and A square-integrable predictable step Itô integral is its finite sum identifies every coordinate Itô integral with its finite sum on one common measurable probability-one event. This gives the displayed integral representation without any mesh restriction. The quantitative estimates in the justified-by theorem retain their stated $0<h\le 1$ restriction when applied directly to the original data.

The distinction between $\bar{Y}$ and ${\bar{Y}}^{-}$ is only at deterministic grid times. The first records the value used at each time by the left-endpoint convention; the second is the predictable representative used by the stochastic integral.