Large deviations for a stochastic Volterra-type equation in the Besov-Orlicz space (Q1593625)

Let \(W_{z}\), \(z\in [0,1]^{2}\), be a Brownian sheet, \(b,\sigma : [0,1]^{2}\times [0,1]^{2}\times \mathbb R\to \mathbb R\) bounded continuous functions globally Lipschitz in the last variable, and \(\varepsilon >0\). Let us consider a two-parameter stochastic integral equation \[ X^{\varepsilon}_{z} = x_{z} + \int _{[0,z]} b(z,\eta , X^{\varepsilon}_{\eta}) d\eta + \sqrt \varepsilon \int _{[0,z]} \sigma (z,\eta ,X^{\varepsilon}_{\eta}) dW(\eta), \quad z\in [0,1]^{2}, \tag{1} \] of the Volterra type. \textit{C. Rovira} and \textit{M. Sanz-Solé} [Potential Anal. 12, No. 4, 359-383 (2000)] showed that solutions to (1) obey a large deviations principle in the uniform norm as \(\varepsilon \downarrow 0\). In the paper under review, their results are further strengthened. Under suitable regularity assumptions on the coefficients \(b\), \(\sigma \) and the initial condition \(x\) it is proved that the paths of the solution \(X^\varepsilon \) belong almost surely to a Besov-Orlicz space \(B^{0}_{\tau ,\omega}\) defined by the Young function \(\tau (t) = \exp (t^{2}) - 1\) and the modulus of continuity \(\omega (t) = t^{1/2}(1+\log t^{-1})^{1/2}\). Moreover, a large deviations principle on \(B^0_{\tau ,\omega}\) is established for the family \(\{X^{\varepsilon}; \varepsilon >0\}\).