Stochastic Bernoulli equation on the algebra of generalized functions

Hafedh Rguigui

doi:10.3842/umzh.v75i8.7223

Hafedh Rguigui Department of Mathematics, AL-Qunfudhah University College, Umm Al-Qura University, KSA, High School of Sciences and Technology of Hammam Sousse, University of Sousse, Hammam Sousse, Tunisia

DOI: https://doi.org/10.3842/umzh.v75i8.7223

Keywords: Wick product; Generalized stochastic Bernoulli Wick differential equation; Space of entire functions with $\theta$-exponential growth condition of minimal type.

Abstract

UDC 519.21

Based on the topological dual space $\mathcal{F}_\theta^*(\mathcal{S'}_{\mathbb{C}})$ of the space of entire functions with $\theta$-exponential growth of finite type, we introduce the generalized stochastic Bernoulli–Wick differential equation (or the stochastic Bernoulli equation on the algebra of generalized functions) by using the Wick product of elements in $\mathcal{F}_\theta^*(\mathcal{S'}_{\mathbb{C}})$. This equation is an infinite-dimensional stochastic distributions analog of the classical Bernoulli differential equation. This stochastic differential equation is solved and exemplified by several examples.

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