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Mon, 20 Apr 2026 00:00:00 +0000

Stochastic Volterra Integral Equations: Asymptotic Stationarity!

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Emmanuel G., Gilles Pagès (2026). On Path-dependent Volterra Integral Equations: Strong Well-posedness and Stochastic Numerics..

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Emmanuel G.

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, Gilles Pagès

Equal contribution

(2025). On a Stationarity Theory for Stochastic Volterra Integral Equations. SPA.

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$$\begin{align*} &\sum_{z \in \mathbb{C} \setminus \{-1\}: z^\alpha=-1} \text{Res}(J_{\alpha}(t, \cdot), z) = \frac{1}{2\pi i} \oint_{\Gamma_{\gamma, \delta, R}} J_{\alpha}(t, z) \, dz = \frac{1}{2\pi i}\int_{\textit{Br}(\gamma, R)} J_{\alpha}(t, z) \, dz + \frac{1}{2\pi i}\int_{C^+} J_{\alpha}(t, z) \, dz \\ &\quad \hspace{1.5cm} + \frac{1}{2\pi i}\int_{C_R^+} J_{\alpha}(t, z) \, dz - \frac{1}{2\pi i}\int_{\textit{H}(\delta, \frac{1}{R})} J_{\alpha}(t, z) \, dz + \frac{1}{2\pi i}\int_{C_R^-} J_{\alpha}(t, z) \, dz + \frac{1}{2\pi i}\int_{C^-} J_{\alpha}(t, z) \, dz.\end{align*} $$

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\begin{align*}
 &\sum_{z \in \mathbb{C} \setminus \{-1\}: z^\alpha=-1} \text{Res}(J_{\alpha}(t, \cdot), z)
 = \frac{1}{2\pi i} \oint_{\Gamma_{\gamma, \delta, R}} J_{\alpha}(t, z) \, dz
 = \frac{1}{2\pi i}\int_{\textit{Br}(\gamma, R)} J_{\alpha}(t, z) \, dz + \frac{1}{2\pi i}\int_{C^+} J_{\alpha}(t, z) \, dz \\
 &\quad \hspace{1.5cm} + \frac{1}{2\pi i}\int_{C_R^+} J_{\alpha}(t, z) \, dz - \frac{1}{2\pi i}\int_{\textit{H}(\delta, \frac{1}{R})} J_{\alpha}(t, z) \, dz + \frac{1}{2\pi i}\int_{C_R^-} J_{\alpha}(t, z) \, dz + \frac{1}{2\pi i}\int_{C^-} J_{\alpha}(t, z) \, dz.
\end{align*}

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👩🏼‍🏫 Outreach in mathematics

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