Implementing Galerkin Finite Element Methods for Semilinear Elliptic Differential Inclusions

2013 ◽  
Vol 13 (1) ◽  
pp. 95-118 ◽  
Author(s):  
Janosch Rieger
Keyword(s):  
Finite Element ◽  
Differential Inclusions ◽  
Dimensional Subspace ◽  
Finite Dimensional Subspace ◽  
Basic Algorithm ◽  
Galerkin Finite Element ◽  
Finite Dimensional ◽  
Partial Differential Inclusions ◽  
Galerkin Finite Element Methods ◽  
Element Approach

Abstract. This paper presents the first feasible method for the approximation of solution sets of semi-linear elliptic partial differential inclusions. It is based on a new Galerkin Finite Element approach that projects the original differential inclusion to a finite-dimensional subspace of . The problem that remains is to discretize the unknown solution set of the resulting finite-dimensional algebraic inclusion in such a way that efficient algorithms for its computation can be designed and error estimates can be proved. One such discretization and the corresponding basic algorithm are presented along with several enhancements, and the algorithm is applied to two model problems.

scholarly journals Functional ARCH and GARCH Models: A Yule-Walker Approach

2020 ◽  
Author(s):  
Sebastian Kühnert
Keyword(s):  
Financial Time Series ◽  
Sufficient Conditions ◽  
Dimensional Subspace ◽  
Stationary Solutions ◽  
Finite Dimensional Subspace ◽  
Estimation Errors ◽  
Linear Processes ◽  
Finite Dimensional ◽  
Garch Processes ◽  
Asymptotic Upper Bound

Conditional heteroskedastic financial time series are commonly modelled by ARCH and GARCH. ARCH(1) and GARCH processes were recently extended to the function spaces C[0,1] and L2[0,1], their probabilistic features were studied and their parameters were estimated. The projections of the operators on finite-dimensional subspace were estimated, as were the complete operators in GARCH(1,1). An explicit asymptotic upper bound of the estimation errors was stated in ARCH(1). This article provides sufficient conditions for the existence of strictly stationary solutions, weak dependence and finite moments of ARCH and GARCH processes in various Lp[0,1] spaces, C[0,1] and other spaces. In L2[0,1] we deduce explicit asymptotic upper bounds of the estimation errors for the shift term and the complete operators in ARCH and GARCH and for the projections of the operators on a finite-dimensional subspace in ARCH. The operator estimaton is based on Yule-Walker equations. The estimation of the GARCH operators also involves a result concerning the estimation of the operators in invertible, linear processes which is valid beyond the scope of ARCH and GARCH. Through minor modifications, all results in this article regarding functional ARCH and GARCH can be transferred to functional ARMA.

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