On stochastic differential equations driven by the renormalized square of the Gaussian white noise

TL;DR

This paper develops a new method to analyze stochastic differential equations driven by the renormalized square of Gaussian white noise, establishing existence, uniqueness, and positivity of solutions under standard conditions.

Contribution

It introduces a novel approach for nonlinear operations on Hida distributions, bridging Wick product and classical definitions, and applies it to solve SDEs driven by the renormalized noise.

Findings

Proves an Ito-type formula for the Wick square of Gaussian white noise.

Establishes existence and uniqueness of solutions to the SDEs under Lipschitz and linear growth conditions.

Shows positivity of solutions in the linear case.

Abstract

We investigate the properties of the Wick square of Gaussian white noises through a new method to perform non linear operations on Hida distributions. This method lays in between the Wick product interpretation and the usual definition of nonlinear functions. We prove on Ito-type formula and solve stochastic differential equations driven by the renormalized square of the Gaussian white noise. Our approach works with standard assumptions on the coefficients of the equations, Lipschitz continuity and linear growth condition, and produces existence and uniqueness results in the space where the noise lives. The linear case is studied in details and positivity of the solution is proved.