Large deviations for stochastic Volterra equations. (English) Zbl 0959.60050

Nualart, David; Rovira, Carles

Bernoulli 6, No. 2, 339-355 (2000).

The authors establish a large deviations principle for the family of processes \(\{X^{\varepsilon}:\varepsilon>0\}\) of solutions of the stochastic Volterra equation \[ X_{t}^{\varepsilon}=x_{0}+\sum_{j=1}^{k}\int_{0}^{t} \varepsilon\sigma_{j}(t,s,X_{s}^{\varepsilon}) dW_{s}^{j} + \int_{0}^{t}b(t,s,X_{s}^{\varepsilon}) ds, \] where the coefficients satisfy Lipschitz condition in \(x\) and Hölder condition in \(t\) uniformly with respect to the other variables. Such problems were also studied under more restrictive conditions by C. Rovira and M. Sanz-Solé. The large deviations principle is proved using a refinement [see also R. Sowers, Probab. Theory Relat. Fields 92, No. 3, 393-421 (1992; Zbl 0767.60025), C. Rovira and M. Sanz-Solé, J. Theor. Probab. 9, No. 4, 863-901 (1996; Zbl 0878.60040), M. Ledoux, in: Séminaire de probabilités XXIV. Lect. Notes Math. 1426, 1-14 (1990; Zbl 0701.60020) and E. Mayer-Wolf, D. Nualart and V. Pérez-Abreu, in: Séminaire de probabilités XXVI. Lect. Notes Math. 1526, 11-31 (1992; Zbl 0782.60026)] of Azencott’s method [see R. Azencott, in: Ecole d’été de probabilités de Saint-Flour VIII-1978. Lect. Notes Math. 774, 1-176 (1980; Zbl 0435.60028) and also H. Doss and P. Priouret, in: Séminaire de probabilités XVII. Lect. Notes Math. 986, 353-370 (1983; Zbl 0529.60061)]. As applications, a stochastic differential equation driven by the fractional Brownian motion and a hyperbolic stochastic partial differential equation are discussed.

Reviewer: Mihai Gradinaru (Nancy)

MSC:

60H20	Stochastic integral equations
60F10	Large deviations
60H15	Stochastic partial differential equations (aspects of stochastic analysis)

Keywords:

large deviations; stochastic Volterra equations

Citations:

Zbl 0767.60025; Zbl 0878.60040; Zbl 0701.60020; Zbl 0782.60026; Zbl 0435.60028; Zbl 0529.60061