Large deviations for stochastic Volterra equations in the plane
The authors establish a large deviation principle for the family of two-parameter \(d\)-dimensional stochastic processes \(\{X^{\varepsilon}(z), z\in [0,1]^2\}\), depending on a small parameter \(\varepsilon\in (0,1]\), which satisfy the following stochastic Volterra equation in the plane \[ X^{\varepsilon}(z)=H(z)+\int_{[0,z]}\varepsilon f(z,\eta,X^{\varepsilon}(\eta)) W(d\eta)+\int_{[0,z]}b(z,\eta,X^{\varepsilon}(\eta))d\eta, \] where \(W\) is a \(k\)-dimensional Wiener sheet. The function \(H\) is a deterministic Lipschitz function. The coefficients \(f(z,\eta,x)\) and \(b(z,\eta,x)\) and the partial derivatives \({\partial f\over\partial z_1}\), \({\partial f\over\partial z_2}\) and \({\partial^2 f\over\partial z_1 \partial z_2}\) are supposed to be Lipschitz functions in all their variables. The proof of this result is based on the \textit{R. Azencott}'s method for diffusion process [in: École d'été de probabilités de Saint-Flour VIII-1978. Lect. Notes Math. 774, 1-176 (1980; Zbl 0435.60028)]. The required exponential inequality for the stochastic integral \(\int_{[0,z]} f(z,\eta,X^{\varepsilon}(\eta)) W(d\eta)\) is obtained by expressing this integral as a representable two-parameter semimartingale in the sense of \textit{E. Wong} and \textit{M. Zakai} [Ann. Probab. 4, 570-586 (1976; Zbl 0359.60053)], and applying exponential inequalities for different classes of stochastic integrals in the plane.
stochastic partial differential equations
scientific article; zbMATH DE number 1507483