tailieunhanh - Báo cáo hoa học: " Discretisation of abstract linear evolution equations of parabolic type"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Discretisation of abstract linear evolution equations of parabolic type | Discretisation of abstract linear evolution equations of parabolic type Fernando Ferreira Goncalves 1 2 Maria do Rosario Grossinho1 2 and Eva Morais1 3 1CEMAPRE ISEG Technical University of Lisbon Rua do Quelhas 6 1200-781 Lisboa Portugal 2Department of Mathematics ISEG Technical University of Lisbon Rua do Quelhas 6 1200-781 Lisboa Portugal 3Department of Mathematics University of Tras-os-Montes e Alto Douro Apartado 1013 5001-801 Vila Real Portugal Corresponding author fgoncalves@ Email addresses MRG mrg@ EM emorais@ Abstract We investigate the discretisation of the linear parabolic equation du dt A t u f t in abstract spaces making use of both the implicit and the explicit finite-difference schemes. The stability of the explicit scheme is obtained 1 and the schemes rates of convergence are estimated. Additionally we study the special cases where A and f are approximated by integral averages and also by weighted arithmetic averages. MSC 2010 65J10. Keywords parabolic evolution equations finite-difference methods financial mathematics. 1 Introduction In this article we study the discretisation with finite-difference methods of the evolution equation problem - A t u f t in 0 T u 0 g 1 where for every t 2 0 T with T 2 0 1 A t is a linear operator from a reflexive separable Banach space V to its dual V u 0 T V is an unknown function f 0 T V g belongs to a Hilbert space H with f and g given and V is continuously and densely embedded into H. We assume that operator A t is continuous and impose a coercivity condition. Our motivation lies in the numerical approximation of multidimensional PDE problems arising in European financial option pricing. Let us consider the stochastic modeling of a multi-asset financial option of European type under the framework of a general version of Black-Scholes model where the vector of asset appreciation rates and the volatility matrix are taken time and space-dependent. Owing to a Feynman-Kac type formula .