Abstract
An alternative approach is proposed for constructing a strongly continuous semigroup based on the classical method of successive approximations, or Picard iterations, together with generating functions. An application to a Black–Scholes integro-differential operator which arises in the pricing of European options under jump-diffusion dynamics is provided. The semigroup is expressed as the Mellin convolution of time-inhomogeneous jump and Black–Scholes kernel functions. Other applications to the heat and transport equations are also given. The connection of the proposed approach to the Adomian decomposition method is explored.
Highlights
Let pX, kkq be a Banach space and A : DpAq Ñ X a linear operator with domain DpAq Ă X
In this article we propose an alternative approach to constructing a strongly continuous semigroup based on the classical method of successive approximations together with generating functions
In this article we proposed an alternative construction method for a strongly continuous group using the classical method of successive approximations together with the employment of generating functions
Summary
Let pX, kkq be a Banach space and A : DpAq Ñ X a linear operator with domain DpAq Ă X. Fix f P DpAq and consider the abstract Cauchy problem. Functions: An Application to a du “ Au, dt. Received: 3 February 2021 uptq “ Tptq f. Published: 10 March 2021 with regard to jurisdictional claims in published maps and institutional affiliations. It is well known [1,2,3] that if A is the generator of a C0 -semigroup { Tptq : t ě 0}, the unique solution of (1) is given by Accepted: 28 February 2021. The abstract Cauchy problem (1) is uniformly well posed
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