Euler–Maruyama recursion

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.