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add missing term in OU noise notebook
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clinssen authored Aug 9, 2023
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"It turns out that the infinitesimal step in Brownian motion is white noise, that is, an independent and identically distributed sequence of Gaussian $\\mathcal{N}(0, 1)$ random variables. The noise $dB(t)/dt$ can be sampled at time $t$ by drawing a sample from that Gaussian distribution, so if the process is sampled at discrete intervals of length $h$, we can write (equation 2.47 from [\\[1\\]](#References)):\n",
"\n",
"\\begin{align}\n",
"U(t + h) = (U(t) - \\mu)\\exp(-h/\\tau) + \\sigma\\sqrt{(1 - \\exp(-2h / \\tau ))} \\cdot\\mathcal{N}(0, 1)\n",
"U(t + h) = \\mu + (U(t) - \\mu)\\exp(-h/\\tau) + \\sigma\\sqrt{(1 - \\exp(-2h / \\tau ))} \\cdot\\mathcal{N}(0, 1)\n",
"\\end{align}"
]
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