Weak existence and uniqueness for affine stochastic Volterra equations with L1-kernels

Eduardo Abi Jaber (1)

1 CES - Centre d'économie de la Sorbonne

We provide existence, uniqueness and stability results for affine stochastic Volterra equations with L1-kernels. Such equations arise as scaling limits of branching processes in population genetics and self-exciting Hawkes processes in mathematical finance. The strategy we adopt for the existence part is based on approximations using stochastic Volterra equations with L 2-kernels. Most importantly, we establish weak uniqueness using a duality argument on the Fourier-Laplace transform via a deterministic Riccati-Volterra integral equation. We illustrate the applicability of our results on a class of hyper-rough Volterra Heston models with a Hurst index H ∈ (−1/2, 1/2].

Mots clés: Stochastic Volterra equations; Affine Volterra processes; Riccati-Volterra equations; superprocesses; rough volatility

Domaines: Probabilités [math.PR]; Finance quantitative [q-fin.CP]

Eduardo Abi Jaber. Weak existence and uniqueness for affine stochastic Volterra equations with L1-kernels. 2019. ⟨hal-02412741v1⟩