Adiabatic theorem for a class of stochastic differential equations on a Hilbert space

2017 ◽  
pp. 223-243
Author(s):  
Martin Fraas
Keyword(s):  
Hilbert Space ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Adiabatic Theorem

Existence of almost periodic solutions for fractional impulsive neutral stochastic differential equations with infinite delay

2019 ◽  
Vol 20 (01) ◽  
pp. 2050003
Author(s):  
Xiao Ma ◽  
Xiao-Bao Shu ◽  
Jianzhong Mao
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Fractional Calculus ◽  
Stochastic Differential Equations ◽  
Periodic Solutions ◽  
Infinite Delay ◽  
Almost Periodic Solutions ◽  
Almost Periodic

In this paper, we investigate the existence of almost periodic solutions for fractional impulsive neutral stochastic differential equations with infinite delay in Hilbert space. The main conclusion is obtained by using fractional calculus, operator semigroup and fixed point theorem. In the end, we give an example to illustrate our main results.

scholarly journals Stochastic differential equations in infinite dimensional Hilbert space and its optimal control problem with Lévy processes

2021 ◽  
Vol 7 (2) ◽  
pp. 2427-2455
Author(s):  
Meijiao Wang ◽  
◽  
Qiuhong Shi ◽  
Maoning Tang ◽  
Qingxin Meng ◽  
...  
Keyword(s):  
Optimal Control ◽  
Hilbert Space ◽  
Differential Equations ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Stochastic Differential Equations ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space ◽  
Dimensional Hilbert Space ◽  
Teugels Martingales

<abstract><p>The paper is concerned with a class of stochastic differential equations in infinite dimensional Hilbert space with random coefficients driven by Teugels martingales which are more general processes and the corresponding optimal control problems. Here Teugels martingales are a family of pairwise strongly orthonormal martingales associated with Lévy processes (see Nualart and Schoutens <sup>[<xref ref-type="bibr" rid="b21">21</xref>]</sup>). There are three major ingredients. The first is to prove the existence and uniqueness of the solutions by continuous dependence theorem of solutions combining with the parameter extension method. The second is to establish the stochastic maximum principle and verification theorem for our optimal control problem by the classic convex variation method and dual techniques. The third is to represent an example of a Cauchy problem for a controlled stochastic partial differential equation driven by Teugels martingales which our theoretical results can solve.</p></abstract>

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