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\begin{document}

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\begin{center}
\textbf{\Large Homework assignment, week 5: numerical SDEs} \\[0pt]
% \vspace{4mm} Fei Lu\\ feilu@math.jhu.edu \\
%Last updated: \today  % Started: 2018-4-10
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% \title{Reduction using local average and rescaling}
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% \section{Problem 1}
\noindent 1. Consider the SDE with smooth bounded $a, b$: 
\begin{equation}\label{eq:sde}
dX_t=a(X_t)dt + b(X_t)dW_t
\end{equation}
Derive the strong order 1.5
Ito-Taylor scheme (see note or Kloeden+Platen Chapter 10). \\
(Hint: you will need to use the multiple integrals and to pay attention to their relations: 
\begin{align*}
I_{1}&=\int_{0}^{h}dW_{s},\ I_{10}=\int_{0}^{h}\int_{0}^{s}dW_{s_{1}}ds,\
I_{11}=\int_{0}^{h}\int_{0}^{s}dW_{s_{1}}dWs, \\
I_{111} & =\int_{0}^{h}\int_{0}^{s}\int_0^{r}dW_u dW_rdWs
\end{align*}%
Optional: Write a code to test the order of convergence for your IT1.5 scheme. 

\bigskip
\noindent 2. Consider the Ornstein-Uhlenbeck equation with $\lambda <0$:
\[
dX_t = \lambda X_t dt + \sigma dW_t,   
\]
(a) Find the range of the time step size $\delta$ such that the Euler-Maruyama scheme  
 \begin{equation*}
Y_{n+1}= Y_{n} +  \lambda Y_{n}\delta +  \sigma \sqrt{\delta} \xi_n; \quad Y_0= 0;  \text{ where }  \xi_n\sim \mathcal{N}(0,1)
\end{equation*}
is stable in the sense that $\E{Y_n^2}<\infty$ for all $n$ and compute $\lim_{n\to \infty}\E{Y_n^2}$. \\
\noindent  (b) Find  the range of  the time step size $\delta$ so that the implicit Euler scheme  
 \begin{equation*}
Y_{n+1}= Y_{n} +  \lambda Y_{n+1}\delta +  \sigma \sqrt{\delta} \xi_n; \quad Y_0= 0;  \text{ where }  \xi_n\sim \mathcal{N}(0,1) 
\end{equation*}
is stable in the sense that $\E{Y_n^2}<\infty$ for all $n$ and compute $\lim_{n\to \infty}\E{Y_n^2}$. \\



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