Have anyone read book "Paul Glasserman Monte Carlo MIFE", it's good, but i'm stuck in chapter 6 page 341
let $$ dX_t=a(X_t)dt+b(X_t)\,dW_t $$ they said that $$ \int_{t}^{t+\Delta t}a(X_{u}) \, du \approx (\Delta t)a(X_{j-1})=\Delta t.a(X_{t}) $$ and the error of this approximation is $O(\Delta t)$ whereas $$ \begin{align} \int_{t}^{t+\Delta t}b(X_{u})dW_{u} &\approx b(X_{t})\cdot(W_{t+\Delta t}-W_{t})\\ &=b(X_{t})\cdot\sqrt{\Delta t}\cdot G \end{align} $$ with $G$ is Gaussian $N(0,1)$ and the error of this approximation is $O(\sqrt{\Delta t})$ i have two questions : First question: how they know that in (6.6) from page 341: $$ \begin{align} b(X_{u})&\approx b(X_{t})+b'(X_{t})b(X_{t})(W_{u}-W_{t}), \forall u \in[t,t+\Delta]\\ &=b(X_{t})+b'(X_{t})b(X_{t})\cdot \sqrt{\Delta t}\cdot G \end{align} $$ that the error of this approximation is $O(\Delta t)$ and not $O(\sqrt{\Delta t})$ Second question: in general how can i determinate the error of this kind of approximation
As a general remark to these process transformations, one should take care what is the same, what is almost the same and what is a distributionally equivalent process. That is, is the $dW$ in the original equation still the same as that used in the solution? This question only makes practical sense if there are coupled equations using the same Wiener process.
As to your question, the terms of magnitude $\sqrt{Δt}$ are included in the approximation, so the remaining terms are either of order $Δt$ or of order $(ΔW(t))^2\sim Δt$ and $ΔW(t)\sqrt{Δt}$, which is also in all relevant error expectations of order $Δt$, and after that come the terms of size $\sqrt{Δt}^3$ and higher powers of that.