Stochastic Stability of Differential Equations in Abstract Spaces

By Kai Liu. 2019

211 citations recorded by the source corpus.

We begin by recalling some basic definitions and results, especially those from functional analysis, partial differential equations (PDEs), probability, and stability theories. We review basic notions and notations from deterministic systems and recall important results from stochastic differential equations. We introduce two notions of solutions, mild and strong, for infinite-dimensional stochastic differential equations and consider the existence and uniqueness of solutions under suitable assumptions. We introduce and clarify various definitions of stochastic stability in Hilbert spaces, which are a natural generalization of deterministic stability concepts. To present the proofs of all the results here would require preparatory background material that would significantly increase both the size and scope of this book. Therefore, we adopt the approach of omitting those proofs, which are treated in detail in wellknown standard textbooks such as Da Prato and Zabczyk [53], Pazy [187], and Yosida [224]. However, those proofs will be presented that are not available in the existing books and are to be found scattered in the literature, or that discuss ideas specially relevant to our purpose.

Subfields: Computational Theory and Mathematics, Control and Systems Engineering, Finance

View Paper: https://www.semanticscholar.org/paper/f0740a1f4a40392be66e07e012255c3f7949391e

Citations in the corpus

• The partially truncated Euler–Maruyama method and its stability and boundedness (2017, unknown): “, [13, 16, 17]) that the scalar stochastic differential equation (SDE) dx(t) = − ( x(t) + x(5)(t) ) dt+ x(2)(t)dB(t), t ≥ 0, (1.”