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

Abstract

In this paper, an Euler type approximation is constructed for stochastic Volterra equation with singular kernels, which provides an algorithm for numerical calculation. Then, the large deviation estimates of small perturbation to equations of this type are obtained. We finally apply them to SDEs with the kernel of fractional Brownian motion with Hurst parameter H ∈ (0, 1).

Stochastic Volterra equations in Banach spaces and stochastic partial differential equation

“…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. In particular, the kernels in [68] can be used to deal with fractional Brownian motion kernels as well as fractional order integral kernels.…”

“…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. In particular, the kernels in [68] can be used to deal with fractional Brownian motion kernels as well as fractional order integral kernels. The study of LDP for stochastic Volterra equations is also referred to [44,36].…”

“…This variational representation has already been proved to be very effective for various finite and infinite-dimensional stochastic dynamical systems even with irregular coefficients (cf. [54,55,11,68,56], etc.). One of the main advantages of this argument is that one only needs to make some simple moment estimates (see Section 4 below).…”

An efficient computational method for solving nonlinear stochastic Itô integral equations: Application for stochastic problems in physics

“…Also, nowadays, there is an increasing demand to investigate the behavior of even more sophisticated dynamical systems in physical, medical, engineering and financial applications [6][7][8][9][10][11][12][13]. These systems often depend on a noise source, like a Gaussian white noise, governed by certain probability laws, so that modeling such phenomena naturally involves the use of various stochastic differential equations (SDEs) [4,[14][15][16][17][18][19][20], or in more complicated cases, stochastic Volterra integral equations and stochastic integro-differential equations [21][22][23][24][25][28][29][30]. In most cases it is difficult to solve such problems explicitly.…”

Large Deviations for Stochastic Tamed 3D Navier-Stokes Equations

“…This method has been proved to be very effective for various types of stochastic equations (cf. [4,18,19,26,14,27]). The main advantage of this method is that one avoids the use of the usual complicated time discretization and the proof of exponential tightness.…”