[Paper Review] Existence of densities for multi-type CBI processes

Martin Friesen, Peng Jin+1 more|arXiv (Cornell University)|Sep 30, 2018

Stochastic processes and statistical mechanics11 references4 citations

TL;DR

This paper establishes sufficient conditions for the existence of transition probability densities for multi-type CBI processes on $\mathbb{R}_{+}^{d}$, using anisotropic Besov space regularity. It avoids Malliavin calculus and Laplace transform analysis, instead employing moment estimates for stochastic integrals and a novel anisotropic framework to prove that the finite-dimensional distributions possess densities with controlled smoothness across components.

ABSTRACT

Let X be a multi-type continuous-state branching process with immigration (CBI process) on state space $\mathbb{R}^d$. Denote by $g_t$, $t \geq 0$, the law of $X_{t}$. We provide sufficient conditions under which $g_t$ has, for each $t > 0$, a density with respect to the Lebesgue measure. Such density has, by construction, some anisotropic Besov regularity. Our approach neither relies on the use of Malliavin calculus nor on the study of corresponding Laplace transform.

Motivation & Objective

• To establish the existence of transition probability densities for multi-type CBI processes on $\mathbb{R}_{+}^{d}$, which are otherwise difficult to analyze due to degenerate diffusion and complex jump mechanisms.
• To overcome the limitations of existing methods that rely on Malliavin calculus or Laplace transform analysis, particularly in the absence of non-degenerate diffusion ($c_i = 0$).
• To develop a framework that accounts for the heterogeneous behavior of different components in multi-type CBI processes by introducing anisotropic Besov spaces.
• To provide a general condition under which the transition law $g_t$ of a CBI process admits a density with respect to Lebesgue measure for all $t > 0$.
• To extend analytical tools to affine processes with general Lévy-driven jump mechanisms, especially in cases where the diffusion coefficient vanishes.

Proposed method

• Introduces anisotropic Besov spaces $B_{1, ext{infty}}^{\lambda,a}(\mathbb{R}^d)$ with parameter $a = (a_1,\dots,a_d)$ satisfying $a_i > 0$ and $\sum a_i = d$, to measure component-specific regularity of the density.
• Uses moment estimates for stochastic integrals with respect to Poisson random measures (Lemma 19), particularly for $\eta \leq \gamma \leq 2$, to control the regularity of the solution path.
• Applies a technique from [6, 13, 20] that establishes density existence via regularity estimates in anisotropic Besov spaces without relying on Malliavin calculus.
• Analyzes the generator of the CBI process in terms of drift, diffusion, and jump components, with the jump mechanism decomposed into branching ($\mu_i$) and immigration ($\nu$) parts.
• Imposes integrability conditions on $\nu$ and $\mu_i$, including $\nu(dz) \sim |z|^{-d-\alpha}$ for $\alpha \in (0,1)$, to ensure sufficient smoothing from jumps.
• Derives sufficient conditions on the parameters $(c,\beta,B,\nu,\mu)$ such that the transition law $g_t$ has a density in anisotropic Besov space $B_{1,\infty}^{\lambda,a}$ for $t > 0$.

Experimental results

Research questions

• RQ1 Under what conditions does the finite-dimensional distribution of a multi-type CBI process on $\mathbb{R}_{+}^{d}$ admit a Lebesgue density?
• RQ2 Can the existence of a density be established without using Malliavin calculus or Laplace transform analysis?
• RQ3 How can the anisotropic behavior of different components in a multi-type CBI process be captured in the regularity of the density?
• RQ4 What role do jump mechanisms—particularly those with $\alpha$-stable-like decay—play in ensuring the smoothing effect necessary for density existence?
• RQ5 What is the precise regularity class (in terms of anisotropic Besov spaces) that the density belongs to, and how does it depend on the jump and diffusion parameters?

Key findings

• For any $t > 0$, the transition law $g_t$ of a multi-type CBI process has a density with respect to Lebesgue measure under mild integrability and non-degeneracy conditions on the Lévy components.
• The density belongs to anisotropic Besov space $B_{1,\infty}^{\lambda,a}(\mathbb{R}^d)$ for some $\lambda > 0$, reflecting component-specific regularity determined by the anisotropy $a$.
• The method applies even when $c_i = 0$ for all $i$, i.e., in the pure jump case, overcoming a key limitation of prior approaches relying on non-degenerate diffusion.
• For the case where the immigration mechanism satisfies $\nu(dz) = \mathbf{1}_{\{|z| \leq 1\}}(z) \frac{dz}{|z|^{d+\alpha}} + \nu'(dz)$ with $\alpha \in (0,1)$, the density exists if $\alpha > \frac{\gamma_0}{1+\gamma_0}$, where $\gamma_0$ is a parameter related to the jump intensity.
• The approach avoids the use of Malliavin calculus and Laplace transform techniques, relying instead on moment estimates for Poisson stochastic integrals and Besov regularity theory.
• The result generalizes to affine processes on $\mathbb{R}^n \times \mathbb{R}_+^d$, with the key insight that jump smoothing suffices for density existence even without diffusion.