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

Abstract

We prove a version of Strassen's law of iterated logarithm (LIL) for a family of stochastic Volterra equation. Our approach is similar to that of Baldi [2], Large deviations and functional iterated logarithm law for diffusion processes for diffusion process. As applications of our results, we firstly prove Strassen's LIL for hyperbolic stochastic partial differential equation and secondly we give a version of Strassen's LIL for fractional Brownian motion (fBm) and its local time for Hurst parameter 1=2 # H , 1:

The law of the iterated logarithm for a class of SPDEs

“…Using the Freidlin-Wentzell inequality derived in the previous section, we prove the Strassen's compact LIL for the family {Z ε } ε∈(0,ε 0 ) . We first provide a general definition of this type of LIL and for more background and similar results we recommend [1,10,20,34,43].…”

“…The connection between large or moderate deviations and the Strassen's compact LIL was first observed by J. Deuschel and D. Stroock in Lemma 1.4.1 of [17]. Afterwards, following the use of a family of contractions introduced in [5], authors such as those in [1,10,20] have proved the compact Strassen's LIL for their SPDEs as an application of their large deviation principle (LDP) or MDP result. The family of contractions were used to prove that the process formed for LIL is relatively compact.…”

“…The rate function from the LDP or MDP result is used to define this limit set forming the main connection to Strassen's compact LIL. For more results applying this form of connection between the two theorems we refer the reader to [1,21,23,46]. The MDP is a special case of the LDP, where for MDP the considered process converges to its deterministic counterpart at a slower rate than that used for the LDP.…”