Stochastically bounded solutions and stationary solutions of stochastic differential equations in Hilbert spaces

2009 ◽  
Vol 79 (21) ◽  
pp. 2260-2265 ◽  
Author(s):  
Jiaowan Luo ◽  
Guolie Lan
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Hilbert Spaces ◽  
Stationary Solutions ◽  
Bounded Solutions ◽  
Stochastically Bounded

DISSIPATIVE BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS IN INFINITE DIMENSIONS

2006 ◽  
Vol 09 (01) ◽  
pp. 155-168 ◽  
Author(s):  
FULVIA CONFORTOLA
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Hilbert Spaces ◽  
Stochastic Partial Differential Equations ◽  
Existence And Uniqueness ◽  
Backward Stochastic Differential Equations ◽  
Uniqueness Result ◽  
Infinite Dimensions ◽  
Partial Differential

We prove an existence and uniqueness result for a class of backward stochastic differential equations (BSDE) with dissipative drift in Hilbert spaces. We also give examples of stochastic partial differential equations which can be solved with our result.

STATIONARY SOLUTIONS OF STOCHASTIC DIFFERENTIAL EQUATIONS WITH MEMORY AND STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS

2005 ◽  
Vol 07 (05) ◽  
pp. 553-582 ◽  
Author(s):  
YURI BAKHTIN ◽  
JONATHAN C. MATTINGLY
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Stationary Solution ◽  
Landau Equation ◽  
Stationary Solutions ◽  
Navier Stokes ◽  
Partial Differential ◽  
Navier Stokes Equation ◽  
With Memory

We explore Itô stochastic differential equations where the drift term possibly depends on the infinite past. Assuming the existence of a Lyapunov function, we prove the existence of a stationary solution assuming only minimal continuity of the coefficients. Uniqueness of the stationary solution is proven if the dependence on the past decays sufficiently fast. The results of this paper are then applied to stochastically forced dissipative partial differential equations such as the stochastic Navier–Stokes equation and stochastic Ginsburg–Landau equation.

scholarly journals Optimization of functionals under uncertainties for Ito-Skorokhod stochastic differential equations in Hilbert spaces

2018 ◽  
pp. 65-70
Author(s):  
A. Nikitin ◽  
O. Baliasnikova
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Hilbert Spaces ◽  
Liquid Fuel ◽  
Degrees Of Freedom ◽  
Applied Mathematics ◽  
Optimal Controls ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Carry Over

In the article for the stochastic differential equations of Ito-Skorokhod, problems of optimization of functionals under conditions of uncertainty in Hilbert spaces are investigated. Purpose of the article is to investigate some properties of stochastic differential equations in Hilbert spaces. These objects arise in diverse areas of applied mathematics as models for various natural phenomena, in particular, the evolution of complex systems with infinitely many degrees of freedom. For instance, one may think of the liquid fuel motion in the tank of a spacecraft. Spacecraft constructors should take into account this motion, for it influences heavily the path of a spacecraft. Also, optimization of the motion is an issue of principal importance. It is not trivial to carry over the results concerning stochastic differential equations in finite-dimensional spaces to the infinite dimensional case. We give some statements, in which the existence, uniqueness is proved and the explicit form μ-optimal controls for such equations is constructed, in particular, μ-optimal control is found as a linear inverse relationship.

Sign in / Sign up

Export Citation Format

Share Document