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 .
