[Paper Review] Stochastic invariance of closed sets with non-Lipschitz coefficients

Eduardo Abi Jaber, Bruno Bouchard+2 more|arXiv (Cornell University)|Jul 29, 2016

Markov Chains and Monte Carlo Methods13 references4 citations

TL;DR

This paper establishes a new characterization of stochastic invariance for closed sets in diffusion processes with non-Lipschitz diffusion coefficients. By reformulating the classical Stratonovich drift condition using the Jacobian of the covariance matrix $ C = \sigma\sigma^\top $, it extends the invariance criterion to cases where $ \sigma $ is non-differentiable, enabling the construction of affine and polynomial diffusions on arbitrary closed sets, including those with boundary singularities like the square-root process.

ABSTRACT

This paper provides a new characterization of the stochastic invariance of a closed subset of R^d with respect to a diffusion. We extend the well-known inward pointing Stratonovich drift condition to the case where the diffusion matrix can fail to be differentiable: we only assume that the covariance matrix is. In particular, our result can be directly applied to construct affine diffusions and polynomial preserving diffusions on any arbitrary closed set.

Motivation & Objective

• To extend the classical Stratonovich drift condition for stochastic invariance to cases where the diffusion coefficient $ \sigma $ is non-differentiable.
• To provide a practical characterization of stochastic invariance that avoids the complex second-order normal cone used in prior work.
• To enable the construction of affine and polynomial diffusions on arbitrary closed sets, including those with non-smooth boundaries.
• To unify and generalize existing invariance criteria by focusing on the differentiability of $ C = \sigma\sigma^\top $ rather than $ \sigma $.

Proposed method

• Reformulate the Stratonovich drift condition using the Jacobian of the covariance matrix $ C = \sigma\sigma^\top $, which may be differentiable even when $ \sigma $ is not.
• Use the first-order normal cone $ \mathcal{N}^1_{\mathcal{D}}(x) $ for geometric analysis, which is computationally simpler than the second-order cone.
• Establish a necessary and sufficient condition for stochastic invariance in terms of $ \langle u, b(x) - \frac{1}{2} \sum_{j=1}^d D\Phi^j(x) \rangle \leq 0 $, where $ \Phi^j $ are the columns of $ C $.
• Apply the Moore-Penrose pseudoinverse and Kronecker product identities to handle matrix derivatives and vectorize the Jacobian computations.
• Prove that the condition remains valid under linear growth and continuity assumptions on $ b $ and $ \sigma $, with $ C \in C^2 $ in the interior of the domain.
• Use localization and pathwise uniqueness arguments to establish existence and uniqueness of strong solutions when the boundary is almost surely not hit.

Experimental results

Research questions

• RQ1 Can the classical Stratonovich drift condition for stochastic invariance be extended to cases where the diffusion coefficient $ \sigma $ is non-differentiable?
• RQ2 Is it possible to characterize stochastic invariance using the Jacobian of the covariance matrix $ C = \sigma\sigma^\top $ instead of $ \sigma $ itself?
• RQ3 How can affine and polynomial diffusions be constructed on arbitrary closed sets, including those with non-smooth boundaries?
• RQ4 What conditions ensure that a diffusion process remains almost surely within a closed set $ \mathcal{D} \subset \mathbb{R}^d $, even when $ \sigma $ fails to be Lipschitz?

Key findings

• The paper establishes a necessary and sufficient condition for stochastic invariance of a closed set $ \mathcal{D} \subset \mathbb{R}^d $ in terms of the Jacobian of the covariance matrix $ C = \sigma\sigma^\top $, replacing the need for $ \sigma $ to be differentiable.
• The condition reduces to $ \langle u, b(x) - \frac{1}{2} \sum_{j=1}^d D\Phi^j(x) \rangle \leq 0 $ for all $ x \in \mathcal{D} $ and $ u \in \mathcal{N}^1_{\mathcal{D}}(x) $, where $ \Phi^j $ are the columns of $ C $, enabling application to non-Lipschitz $ \sigma $.
• The result allows the construction of affine and polynomial diffusions on any closed set, such as the square-root process on $ \mathbb{R}_+ $, even when $ \sigma $ is non-differentiable at the boundary.
• For the Jacobi diffusion on $ (0,1] $, the condition yields the dynamics $ dX_t = \kappa(\theta - X_t)dt + \eta\sqrt{X_t(1-X_t)}dW_t $ with $ \kappa\theta \geq \frac{\eta^2}{2} $ ensuring stochastic invariance.
• The method avoids the use of second-order normal cones, which are hard to compute, while retaining the computational simplicity of first-order normal cones.
• Under mild regularity on $ C $ and local Lipschitzness of $ b $, the paper ensures existence and pathwise uniqueness of strong solutions when the boundary is almost surely not attained.