Euler–Maruyama Scheme and Coefficient Conditions

Assume the Axiom of Choice and use the finite-dimensional Brownian-motion convention of $d$-dimensional Brownian motion. Let the state dimension $d$ and noise dimension $m$ be positive finite integers, and fix $T>0$ and $N\in \mathbb{N}$ with $N\ge 1$. Set $h=T\mathrm{/}N>0$ and $t_k=kh$ for $0\le k\le N$. Let $b:[0,T]\times {\mathbb{R}}^d\to {\mathbb{R}}^d$ and $\sigma :[0,T]\times {\mathbb{R}}^d\to {\mathbb{R}}^{d\times m}$ be jointly Borel. Write $\mathrm{\mid }\cdot \mathrm{\mid }$ for the Euclidean norm and $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_F$ for the Frobenius norm. Suppose finite constants $L,K$ satisfy, for every $s,t\in [0,T]$ and $x,y\in {\mathbb{R}}^d$, $\mathrm{\mid }b(t,x)-b(t,y)\mathrm{\mid }+\mathrm{\parallel }\sigma (t,x)-\sigma (t,y){\mathrm{\parallel }}_F\le L\mathrm{\mid }x-y\mathrm{\mid },$ $\mathrm{\mid }b(t,x){\mathrm{\mid }}^2+\mathrm{\parallel }\sigma (t,x){\mathrm{\parallel }}_F^2\le K^2(1+\mathrm{\mid }x{\mathrm{\mid }}^2),$ and $\mathrm{\mid }b(t,x)-b(s,x)\mathrm{\mid }+\mathrm{\parallel }\sigma (t,x)-\sigma (s,x){\mathrm{\parallel }}_F\le K(1+\mathrm{\mid }x\mathrm{\mid })\mathrm{\mid }t-s{\mathrm{\mid }}^{1\mathrm{/}2}\mathrm{.}$ Let $W$ be an adapted standard $m$-dimensional Brownian motion on a filtered probability space, with $W_t-W_s$ independent of ${\mathcal{F}}_s$ and law $N_m(0,(t-s)I_m)$ whenever $0\le s<t\le T$. Let $X_0$ be ${\mathcal{F}}_0$-measurable, independent of future Brownian increments, and $\mathbb{E}\mathrm{\mid }{X}_0{\mathrm{\mid }}^2<\mathrm{\infty }$.

The Euler–Maruyama algorithm takes these data as input and returns $Y_0,\dots ,Y_N$:

1. Set $Y_0=X_0$.
2. For $k=0,\dots ,N-1$, form $\mathrm{\Delta }{W}_k=W_{t_{k+1}}-W_{t_k}$ and set $Y_{k+1}=Y_k+b(t_k,Y_k)h+\sigma (t_k,Y_k)\mathrm{\Delta }{W}_k\mathrm{.}$

The output is the grid approximation for the coefficient pair $(b,\sigma )$. The companion step-process and interpolation definition uses this same grid and endpoint convention.

This algorithm defines finite-dimensional recursions. Its definition alone does not assert a convergence rate, density regularity, or a path-supremum moment bound.

Assume AC. Fix $T>0$ and a finite partition $0=t_0<t_1<\cdots <t_N=T$. Let $W$ be a scalar Brownian motion adapted to $({\mathcal{F}}_t)$, with $W_t-W_s$ independent of ${\mathcal{F}}_s$ and distributed as $N(0,t-s)$ for every $0\le s<t\le T$. If each ${\xi }_k$ is real-valued, ${\mathcal{F}}_{t_k}$-measurable and square-integrable, then the predictable step process $H_s={\sum }_{k=0}^{N-1}{\xi }_k{1}_{(t_k,t_{k+1}]}(s)$ is square-integrable. Every continuous version of its Itô integral agrees, at every time on one measurable event of probability one, with $S_t={\sum }_{k=0}^{N-1}{\xi }_k(W_{t\wedge t_{k+1}}-W_{t\wedge t_k}),0\le t\le T\mathrm{.}$ If the probability space is complete, this agreement is indistinguishability. The coefficients need not be bounded.

Van der Vaart, §5.3, Definitions 5.20 and 5.25 and Lemmas 5.21–5.23, constructs the square-integrable integral by continuous extension from bounded simple predictable integrands. The proof above supplies the finite-partition truncation and common continuous-version argument for the exact step processes used by Euler–Maruyama.

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.

Let $p\ge 2$, $0\le \delta \le 1$, and $s\ge 0$ with $s+\delta \le T$. Let $U$ be an ${\mathcal{F}}_s$-measurable ${\mathbb{R}}^d$-valued random variable with $\mathbb{E}\mathrm{\mid }U{\mathrm{\mid }}^p<\mathrm{\infty }$. Assume $W_{s+\delta }-W_s$ is independent of ${\mathcal{F}}_s$ with law $N_m(0,\delta I_m)$. Let $b_s:{\mathbb{R}}^d\to {\mathbb{R}}^d$ and ${\sigma }_s:{\mathbb{R}}^d\to {\mathbb{R}}^{d\times m}$ be measurable and satisfy $\mathrm{\mid }{b}_s(x){\mathrm{\mid }}^2+\mathrm{\parallel }{\sigma }_s(x){\mathrm{\parallel }}_F^2\le K^2(1+\mathrm{\mid }x{\mathrm{\mid }}^2)\mathrm{.}$ Set $V=U+\delta b_s(U)+{\sigma }_s(U)(W_{s+\delta }-W_s)$. There is a constant $C_{p,m,K}$, independent of $s,\delta ,U$, such that $\mathbb{E}(\mathrm{\mid }V{\mathrm{\mid }}^p\mid {\mathcal{F}}_s)\le (1+C_{p,m,K}\delta )\mathrm{\mid }U{\mathrm{\mid }}^p+C_{p,m,K}\delta$ and $\mathbb{E}(\mathrm{\mid }V-U{\mathrm{\mid }}^2\mid {\mathcal{F}}_s)\le C_{2,m,K}\delta (1+\mathrm{\mid }U{\mathrm{\mid }}^2)\mathrm{.}$ The first estimate controls a grid step and, with $\delta =t-t_k$, the interpolated value on one grid interval. The second estimate is the interpolation increment bound. The global state-Lipschitz and time-Hölder assumptions of Euler–Maruyama recursion are not needed for these one-step bounds; the uniform linear-growth bound is enough.

Assume the coefficient, filtered Brownian-motion, and initial-value hypotheses of Euler–Maruyama recursion on a uniform grid with $0<h\le 1$ and $N=T\mathrm{/}h\in \mathbb{N}$, with $\mathbb{E}\mathrm{\mid }{X}_0{\mathrm{\mid }}^p<\mathrm{\infty }$ for a fixed finite $p\ge 2$. Let $Y_k$, $\bar{Y}$, and $Y_t$ be the recursion, step process, and continuous interpolation of Euler–Maruyama step process and continuous interpolation. Then the step process has a predictable representative, the interpolation formula is well-defined, and $Y$ is adapted and continuous. There is a constant $C_{p,T,L,K,d,m}$ independent of $h$ such that ${\max }_{0\le k\le N}\mathbb{E}\mathrm{\mid }{Y}_k{\mathrm{\mid }}^p+{\sup }_{0\le t\le T}\mathbb{E}\mathrm{\mid }{\bar{Y}}_t{\mathrm{\mid }}^p+{\sup }_{0\le t\le T}\mathbb{E}\mathrm{\mid }{Y}_t{\mathrm{\mid }}^p\le C_{p,T,L,K,d,m}(1+\mathbb{E}\mathrm{\mid }{X}_0{\mathrm{\mid }}^p)\mathrm{.}$ For every $t\in [0,T]$, $\mathbb{E}\mathrm{\mid }{Y}_t-{\bar{Y}}_t{\mathrm{\mid }}^2\le C_{T,K,d,m}h(1+\mathbb{E}\mathrm{\mid }{X}_0{\mathrm{\mid }}^2)\mathrm{.}$ These are pointwise-in-time moment bounds; no claim is made about $\mathbb{E}{\sup }_{t\le T}\mathrm{\mid }{Y}_t{\mathrm{\mid }}^p$. The estimates hold under the shared half-Hölder time-modulus contract, although this proof uses only the uniform linear-growth bound and the Brownian increment assumptions.

The half-Hölder time modulus is retained because this pair shares its coefficient contract with later Euler error results. The present moment and increment estimates do not use that modulus, and they do not imply a strong or weak convergence rate.