Euler Maruyama Method

• $X: [0, T] \times \Omega \rightarrow \mathbb{R}$,
• $X_{0}$,
  • initial value
• $d \in \mathbb{N}$,
• $W_{i}(t)$,
  • $d$-factor Brownian motion.
• $n \in \mathbb{N}$,
  • dimension of $X$,
• $V_{i}: \mathbb{R} \rightarrow \mathbb{R}$,

\[\begin{eqnarray} X(t) & = & X_{0} + \int_{0}^{t} V_{0}(X(s)) \ ds + \sum_{i=1}^{d} \int_{0}^{t} V_{i}(X(s)) \ dW(s) \nonumber \end{eqnarray}\]

Method

\(\{Z_{i}^{n}\}_{i=1, \ldots, d}^{n=1, \ldots, N}\) is standard normal random variables. \(\{Z_{i}^{n}\}_{i=1, \ldots, d}^{n=1, \ldots, N}\) are independent.

\[\begin{eqnarray} Y_{0} & := & X_{0} \\ Y_{n+1} & = & Y_{n} + V_{0}(Y_{n}) h + Z_{i}^{n} + \sum_{i=1}^{d} V_{i}(Y_{n}) Z_{i}^{n} \end{eqnarray}\] \[\begin{eqnarray} \end{eqnarray}\]

Reference

• Euler–Maruyama method - Wikipedia