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  "body": "\n## Summary\n\nA stochastic Volterra equation is a [[Volterra integral equation|DND8CBEVXT]] driven by noise.\nIt has the form $X(t) = X_0(t) + \\int_0^t a(t, s, X(s)) \\, ds + \\int_0^t b(t, s, X(s)) \\, dW(s)$, where the last integral is an Itô integral against a [[Wiener process|22735FGJV6]].\nWhen the coefficients $a$ and $b$ do not depend on $t$ and $X_0$ is constant, it reduces to a [[stochastic differential equation|B219G9SHJA]].\nThe dependence on $t$ lets the noise and the drift have memory.\nSuch memory is used in models of rough volatility in finance and of viscoelastic materials under random forcing.\n\n## Explanations\n\n### L1 human\n\nA stochastic differential equation moves a quantity by small random kicks whose effect lasts forever and unchanged.\nIn a stochastic Volterra equation the effect of a kick received at time $s$ may change with the elapsed time.\nFor example, it may fade away.\nSo the system remembers its past noise in a weighted way.\nThe process $\\int_0^t (t - s)^{H - 1/2} \\, dW(s)$ with $H \u003e 0$ is a simple example: recent kicks are weighted differently from old ones.\n",
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