Constructing <i>C</i>₀-Semigroups via Picard Iterations and Generating Functions: An Application to a Black–Scholes Integro-Differential Operator

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.

Keywords

• strongly continuous semigroup
• Picard iterations
• generating functions
• Black–Scholes theory
• jump-diffusion process
• partial integro-differential equation