An Inverse Problem for a Class of Linear Stochastic Evolution Equations

Yuhuan Zhao

doi:10.1155/2012/921038

Abstract

An inverse problem for a linear stochastic evolution equation is researched. The stochastic evolution equation contains a parameter with values in a Hilbert space. The solution of the evolution equation depends continuously on the parameter and is Fréchet differentiable with respect to the parameter. An optimization method is provided to estimate the parameter. A sufficient condition to ensure the existence of an optimal parameter is presented, and a necessary condition that the optimal parameter, if it exists, should satisfy is also presented. Finally, two examples are given to show the applications of the above results.

Highlights

The purpose of this paper is to study an inverse problem for the following linear stochastic evolution equation: dy A t; p B t; p ydt f t; p dt σ t; p dw t, t ∈ t0, tf ≡ T, 1.1 y t0 φ ξ, where tf < ∞, p ∈ Pad ⊂ P is a parameter to be determined, and Pad is a convex domain in P
There are many papers dealing with parameter estimation problems for stochastic partial differential equations, for instance, see 1–7, but only a few papers to estimate directly parameters involved in stochastic evolution equations in infinite dimensional spaces, for example, 8, 9
In this paper we consider solving of the parameter contained in a linear stochastic evolution equation LSEE by means of smooth optimization methods

Summary

Introduction

The purpose of this paper is to study an inverse problem for the following linear stochastic evolution equation: dy A t; p B t; p ydt f t; p dt σ t; p dw t , t ∈ t0, tf ≡ T, 1.1 y t0 φ ξ, where tf < ∞, p ∈ Pad ⊂ P is a parameter to be determined, and Pad is a convex domain in P. Determine the parameter p in the system 1.1 and 1.2 It is transformed into an optimization problem as most researchers expect. 1.10 where Eg is a conditional expectation, that is, Egf E f | Gt , 1.11 and Gt is the sub-σ-algebra induced by the stochastic process g s , 0 ≤ s ≤ t, which is adapted to Ft. 1.12 Po is called a relative optimal parameter.

Continuity and Differentiability with Respect to a Parameter

Existence and Necessary Conditions for Optimality

Applications

The System Governed by a Stochastic Parabolic Partial Differential Equations

Findings

Conclusion