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  "body": "\n## Summary\n\nThe Euler–Maruyama method is the simplest numerical scheme for stochastic differential equations.\nIt extends the [[Euler method|0JARTXX4XP]] for ordinary differential equations by adding, at each step, the diffusion coefficient times a Brownian increment.\nUnder Lipschitz and growth conditions on the coefficients, it converges with strong order $1/2$.\nUnder further smoothness conditions, it converges with weak order $1$.\n\n## Explanations\n\n### L1 human\n\nTo follow a random path driven by noise, one walks forward in small time steps.\nAt each step, one moves by the drift times the step length, as in the ordinary Euler method.\nThen one adds a random kick, a normal random number with variance equal to the step length, scaled by the noise coefficient.\nFor 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.\n",
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