---
tag: 22T0NVC56P
kind: concept
name: Euler–Maruyama method
msc: [65C30, 60H35, 60H10]
sources:
  - url: https://en.wikipedia.org/wiki/Euler%E2%80%93Maruyama_method
same_as:
  wikidata: Q761604
  wikipedia: Euler–Maruyama method
---

## Summary

The Euler–Maruyama method is the simplest numerical scheme for stochastic differential equations.
It extends the [[Euler method|0JARTXX4XP]] for ordinary differential equations by adding, at each step, the diffusion coefficient times a Brownian increment.
Under Lipschitz and growth conditions on the coefficients, it converges with strong order $1/2$.
Under further smoothness conditions, it converges with weak order $1$.

## Explanations

### L1 human

To follow a random path driven by noise, one walks forward in small time steps.
At each step, one moves by the drift times the step length, as in the ordinary Euler method.
Then one adds a random kick, a normal random number with variance equal to the step length, scaled by the noise coefficient.
For example, for $dX = -X \, dt + dB$ with step $0.1$, the update is $Y_{n+1} = 0.9 \, Y_n + \sqrt{0.1} \, Z_n$ with $Z_n$ standard normal.