Local P-th Mean Stability of Stochastic Differential Equations with Infinite Memory by Contraction Mapping Principle on Strong Banach Spaces

Source abstract

Abstract The aim of this work is to provide details how the well-known Banach’s contraction mapping principle (CMP) can be used to derive criteria for local asymptotic p -th mean stability of the trivial equilibrium of stochastic differential equations (SDEs) with infinite memory, driven by standard Wiener processes on the strong Banach space $$\mathbb {M}_p$$ M p of cádlág or cáglád stochastic processes with bounded p -th moments $$\mathbb {E}[\sup _{-\infty<t\le T} |X(t)|^p]<\infty$$ E [ sup - ∞ &lt; t ≤ T | X ( t ) | p ] &lt; ∞ . First, we prove existence, uniqueness and continuity of $$L^p$$ L p -solutions to those SDEs by using CMP. Thereafter, for the investigation of local asymptotic p -th mean stability, we carry over some techniques of Burton [9] for nonrandom differential equations to the case of SDEs with infinite memory. The derived criteria strongly depend on efficient estimates of the upper Burkholder constant of Itô-type stochastic integrals. Three examples illustrate the practical applicability of our approach. The work is supplemented by a $$L^p$$ L p -error estimate of successive iterations on strong Banach spaces $$\mathbb {M}_p$$ M p . The derived criteria help to control and optimize the long-term stability behavior of stochastic integro-differential systems with memory in the uniform moment-sense (i.e., for uncertainty quantification).