[Paper Review] Affine processes on positive semidefinite matrices

Christa Cuchiero, Damir Filipovi 'c+2 more|RePEc: Research Papers in Economics|Oct 1, 2009

Stochastic processes and financial applications4 citations

TL;DR

This paper establishes the mathematical foundation for stochastically continuous affine processes on the cone of positive semidefinite matrices, proving that such processes are necessarily Feller and have affine generators with admissible parameters. The key contribution is a complete characterization of all such processes, showing existence and uniqueness for any admissible parameter set, and establishing a one-to-one correspondence between parameters and solutions to generalized Riccati equations governing the characteristic function.

ABSTRACT

This article provides the mathematical foundation for stochastically continuous affine processes on the cone of positive semidefinite symmetric matrices. This analysis has been motivated by a large and growing use of matrix-valued affine processes in finance, including multi-asset option pricing with stochastic volatility and correlation structures, and fixed-income models with stochastically correlated risk factors and default intensities.

Motivation & Objective

• To provide a rigorous mathematical foundation for affine processes on the cone of positive semidefinite symmetric matrices.
• To address the need for tractable models in multi-asset option pricing with stochastic volatility and correlation.
• To generalize existing models like the Wishart process by allowing full generality in drift, diffusion, and jump components.
• To establish necessary and sufficient conditions for the existence and uniqueness of such processes.
• To prove that all stochastically continuous, infinitely decomposable Markov processes on the positive semidefinite cone are affine with zero diffusion, and vice versa.

Proposed method

• The authors define affine processes on the cone $ S_d^+ $ as time-homogeneous, stochastically continuous Markov processes with affine generators.
• They derive necessary and sufficient conditions on the generator's coefficients (drift, diffusion, killing, jump components) for the process to remain in $ S_d^+ $.
• The method relies on solving generalized Riccati differential equations for the characteristic function, derived from the affine transform formula.
• The existence proof uses the martingale problem approach and an alternative construction via approximation of jump-diffusions.
• A key technical tool is the use of a space of rapidly decreasing functions $ /mathcal{S} $ on $ S_d^+ $, and a density result for Laplace transforms of measures supported on $ S_d^+ $.
• The proof of the main result uses the Bros–Epstein–Glaser theorem to show that vanishing Laplace transforms imply zero measures, ensuring uniqueness of the solution.

Experimental results

Research questions

• RQ1 What are the necessary and sufficient conditions on the generator coefficients for a time-homogeneous Markov process on $ S_d^+ $ to be affine and stochastically continuous?
• RQ2 How can the affine transform formula be characterized for matrix-valued processes, and what ODE system governs its solution?
• RQ3 Under what parameter conditions does a unique affine process on $ S_d^+ $ exist, and how is this related to the Feller property?
• RQ4 Can all stochastically continuous, infinitely decomposable Markov processes on $ S_d^+ $ be characterized as affine with zero diffusion?
• RQ5 What is the precise relationship between the parameters of the generator and the solutions of the generalized Riccati equations?

Key findings

• All stochastically continuous affine processes on $ S_d^+ $ are necessarily Feller processes with affine generators, ensuring strong Markov and Feller continuity properties.
• For any admissible parameter set (including general drift, diffusion, and jump components), there exists a unique strong solution to the corresponding SDE on $ S_d^+ $, ensuring model well-posedness.
• The characteristic function of the process satisfies an affine transform formula, with the cumulant generating function solving a system of generalized Riccati ODEs.
• The parameter space is fully characterized by admissibility conditions, including the requirement that $ b - (d-1) ilde{ heta} ilde{ heta}^T ∈ S_d^+ $ for the drift matrix.
• The class of stochastically continuous, infinitely decomposable Markov processes on $ S_d^+ $ coincides exactly with the class of affine processes with zero diffusion.
• The Laplace transform of any measure on $ S_d^+ $ is uniquely determined by its values on the set $ \{ \exp(-\langle u, \cdot \rangle) \}_{u \in S_d^{++}} $, which ensures the uniqueness of the law of the process.