Characteristic functions of affine processes via calculus of their operator symbols

TL;DR

This paper presents a method to explicitly compute characteristic functions of multivariate affine processes using power series expansions of their operator symbols, enabling efficient numerical calculations without solving Riccati equations.

Contribution

It introduces a convergent power series representation of characteristic functions based on the symbol functions and derivatives, simplifying numerical computation for affine processes.

Findings

Power series representations are globally convergent in time and bounded domains.

Efficient computation of characteristic functions avoids solving Riccati equations.

Generalized symbol functions lead to broader applicability of the method.

Abstract

The characteristic functions of multivariate Feller processes with generator of affine type, and with smooth symbol functions have an explicit representation in terms of power series with rational number coefficients and with monmoms consisting of powers of the the symbol functions and formal derivatives of the symbol functions. The power series repesentations are convergent globally in time and on bounded domains of arbitrary size. Generalized symbol functions can be derived leading to power series expansions which are convergent on arbitrary domains in special cases. The rational number coefficients can be efficiently computed by an integer recursion. As a numerical consequence characteristic functions of multivariate affine processes can be efficiently computed from the symbol function avoiding computation of the generalized Riccati equations (an observation first made recently in a more general context).