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Holomorphic jump-diffusions

Christa Cuchiero, Francesca Primavera, Sara Svaluto-Ferro

Abstract

We introduce a class of jump-diffusions, called holomorphic, of which the well-known classes of affine and polynomial processes are particular instances. The defining property concerns the extended generator, which is required to map a (subset of) holomorphic functions to themselves. This leads to a representation of the expectation of power series of the process' marginals via a potentially infinite dimensional linear ODE. We apply the same procedure by considering exponentials of holomorphic functions, leading to a class of processes named affine-holomorphic for which a representation for quantities as the characteristic function of power series is provided. Relying on powerful results from complex analysis, we obtain sufficient conditions on the process' characteristics which guarantee the holomorphic and affine-holomorphic properties and provide applications to several classes of jump-diffusions.

Holomorphic jump-diffusions

Abstract

We introduce a class of jump-diffusions, called holomorphic, of which the well-known classes of affine and polynomial processes are particular instances. The defining property concerns the extended generator, which is required to map a (subset of) holomorphic functions to themselves. This leads to a representation of the expectation of power series of the process' marginals via a potentially infinite dimensional linear ODE. We apply the same procedure by considering exponentials of holomorphic functions, leading to a class of processes named affine-holomorphic for which a representation for quantities as the characteristic function of power series is provided. Relying on powerful results from complex analysis, we obtain sufficient conditions on the process' characteristics which guarantee the holomorphic and affine-holomorphic properties and provide applications to several classes of jump-diffusions.

Paper Structure

This paper contains 39 sections, 29 theorems, 185 equations.

Table of Contents

  1. Introduction
  2. Relation to the literature
  3. Terminology
  4. Preliminaries
  5. Notation
  6. Multi-index notation
  7. The space of sequences
  8. The set of holomorphic functions on polydiscs
  9. Operations on holomorphic functions and sequences
  10. Jump-diffusion processes
  11. Convergent power series on a given set
  12. Kernels extension
  13. Holomorphic jump-diffusions
  14. Characteristics of holomorphic jump-diffusions
  15. Examples of holomorphic jump-diffusions
  16. ...and 24 more sections

Key Result

Lemma 3.3

Consider a subset $\mathcal{V}$ such that and let $X=(X_t)_{t\in [0,T]}$ be an $S$-valued $\mathcal{V}$-holomorphic process. Then $\int_{\mathbb R^d}\|\xi\|^k \ K(x,d\xi)<\infty$ for all for all $k\geq 2$ and $x\in S$, and the characteristics $(b,a,K)$ of $X$ satisfy

Theorems & Definitions (70)

  • Definition 3.1
  • Remark 3.2
  • Lemma 3.3
  • proof
  • Remark 3.4
  • Theorem 3.6
  • Remark 3.7
  • Corollary 3.8
  • Definition 3.9
  • Corollary 3.11
  • ...and 60 more