Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Stochastic equations for two-type continuous-state branching processes in varying environments

Zenghu Li, Junyan Zhang

TL;DR

This work constructs a two-type continuous-state branching process in varying environments (TCBVE) as the pathwise unique solution to a system of stochastic integral equations driven by time-space noises, and it characterizes the evolution via a cumulant semigroup $\mathbf v_{r,t}$ that defines the transition semigroup $Q_{r,t}$ through a Laplace transform. A key contribution is the comparison property that yields pathwise uniqueness, enabling a full weak-solution and then strong-solution framework for the SIE. The paper proves that a TCVE-process with the cumulant semigroup is a weak solution to the stochastic equation system and develops a truncation-based argument to handle general coefficients, yielding a robust SIE representation. It further provides Laplace-transform formulas for positive integral functionals of the process in terms of backward equations, enabling explicit computation of a broad class of functionals.

Abstract

A two-type continuous-state branching process in varying environments is constructed as the pathwise unique solution of a system of stochastic equations driven by time-space noises, where the pathwise uniqueness is derived from a comparison property of solutions. As an application of the main result, we give characterizations of some positive integral functionals of the process in terms of Laplace transforms.

Stochastic equations for two-type continuous-state branching processes in varying environments

TL;DR

This work constructs a two-type continuous-state branching process in varying environments (TCBVE) as the pathwise unique solution to a system of stochastic integral equations driven by time-space noises, and it characterizes the evolution via a cumulant semigroup that defines the transition semigroup through a Laplace transform. A key contribution is the comparison property that yields pathwise uniqueness, enabling a full weak-solution and then strong-solution framework for the SIE. The paper proves that a TCVE-process with the cumulant semigroup is a weak solution to the stochastic equation system and develops a truncation-based argument to handle general coefficients, yielding a robust SIE representation. It further provides Laplace-transform formulas for positive integral functionals of the process in terms of backward equations, enabling explicit computation of a broad class of functionals.

Abstract

A two-type continuous-state branching process in varying environments is constructed as the pathwise unique solution of a system of stochastic equations driven by time-space noises, where the pathwise uniqueness is derived from a comparison property of solutions. As an application of the main result, we give characterizations of some positive integral functionals of the process in terms of Laplace transforms.

Paper Structure

This paper contains 5 sections, 10 theorems, 88 equations.

Table of Contents

  1. Introduction
  2. Comparison property
  3. Weak solution
  4. General stochastic equations
  5. Positive integral functionals

Key Result

Theorem 1.1

There is a pathwise unique solution $\{\mathbf X(t):t\geq 0\}$ to SIE and the solution is a TCBVE-process with transition semigroup $(Q_{r,t})_{t\geq r}$ defined by Laplace transform and backward eq.

Theorems & Definitions (15)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 3.1
  • Proposition 3.2
  • ...and 5 more