Euler–Maruyama method¶
In Itô calculus, the Euler–Maruyama method (also called the Euler method) is a method for the approximate numerical solution of a stochastic differential equation (SDE). It is an extension of the Euler method for ordinary differential equations to stochastic differential equations. It is named after Leonhard Euler and Gisiro Maruyama. Unfortunately, the same generalization cannot be done for any arbitrary deterministic method.[1]
Itô 미적분학에서 오일러-마루야마 방법(오일러 방법이라고도 함)은 확률적 미분 방정식(SDE)의 근사 수치 해법입니다. 이는 상미분 방정식에 대한 오일러 방법을 확률적 미분 방정식으로 확장한 것입니다. Leonhard Euler와 Gisiro Maruyama의 이름을 따서 명명되었습니다. 불행히도 임의의 결정론적 방법에 대해 동일한 일반화를 수행할 수 없습니다.[1]
Consider the stochastic differential equation (see Itô calculus)
$$
d X_{t}=a(X_{t},t) dt+b(X_{t},t) dW_{t}
$$
with initial condition $X_0 = x_0$, where $W_t$ stands for the Wiener process, and suppose that we wish to solve this SDE on some interval of time $[0, T]$. Then the Euler–Maruyama approximation to the true solution $X$ is the Markov chain $Y$ defined as follows:
- partition the interval $[0, T]$ into $N$ equal subintervals of width $\Delta t>0$:
$$
0=\tau _{{0}}<\tau _{{1}}<\cdots <\tau _{{N}}=T{\text{ and }}\Delta t=T/N;
$$
- set $Y_0 = x_0$
- recursively define $Y_n$ for $0 \leq n \leq N-1 $by
$$
Y_{n+1}=Y_{n}+a(Y_{n},\tau_{n}) \Delta t+b(Y_{n},\tau_{n}) \Delta W_{n},
$$
where
$$
\Delta W_{n}=W_{\tau_{n+1}}-W_{\tau_{n}}.
$$
The random variables ΔWn are independent and identically distributed normal random variables with expected value zero and variance $\Delta t$.
