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

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 Mₚ M p of cádlág or cáglád stochastic processes with bounded p -th moments E[ _-∞ E [ sup - ∞ < t ≤ T | X ( t ) | p ] < ∞ . First, we prove existence, uniqueness and continuity ofL^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 Lᵖ L p -error estimate of successive iterations on strong Banach spaces Mₚ 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).