Large Deviations for Stochastic Volterra Equations

Abstract

This paper is devoted to prove a large-deviation principle for solutions to multidimensional stochastic Volterra equations.

Stochastic Volterra equations in Banach spaces and stochastic partial differential equation

“…Using the Skorohod integral, Pardoux and Protter [47] also investigated stochastic Volterra equations with anticipating coefficients. The study of stochastic Volterra equations with singular kernels can be found in [14,16,65,36,44], etc. Recently, the present author [68] studied the approximation of Euler's type and the LDP of Freidlin-Wentzell's type for stochastic Volterra equations with singular kernels.…”

Euler schemes and large deviations for stochastic Volterra equations with singular kernels

“…Then X ε · (·, ω) ∈ C for almost all ω ∈ Ω by Theorem 1.1. Another main result in this paper is to prove the following Freidlin-Wentzell's large deviation estimate that extends Nualart-Rovira's result [14].…”

“…(1) can be proved directly by successive approximation and Gronwall's inequality. A Freidlin-Wentzell's type large deviation principle in continuous functions space for stochastic Volterra equation has been established by Nualart and Rovira in [14]. This result was later improved to the BesovOrlicz space by Lakhel in [11].…”

Strassen's law of the iterated logarithm for stochastic Volterra equations and applications

“…The distance on C 0 induced by the supremum norm will be denoted by d. Denote by H the Cameron -Martin space associated with Brownian motion, i.e. the set of the absolutely continuous functions h [ C 0 such that khk Nualart and Rovira [10] have proved that the solution of (1.1) satisfies the large deviation principle (LDP). In their paper it is supposed that s 1 ¼ s and b 1 ¼ b; the same proofs, however, still work in our case and we have the following: Theorem 1.…”