Stochastic invariance of closed sets with non-Lipschitz coefficients

Eduardo Abi Jaber (1) , Bruno Bouchard (1) , Camille Illand (2) , Eduardo Abi Jaber

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 AXA Investment Managers, Multi Asset Client Solutions, Quantitative Research

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.

Mots clés: Stochastic differential equation; stochastic invariance; polynomial preserving diffusions; affine diffusions; polynomial diffusions

Domaines: Probabilités [math.PR]; Finance quantitative [q-fin.CP]

Eduardo Abi Jaber, Bruno Bouchard, Camille Illand, Eduardo Abi Jaber. Stochastic invariance of closed sets with non-Lipschitz coefficients. Stochastic Processes and their Applications, 2018, 129 (5), pp.1726-1748. ⟨10.1016/j.spa.2018.06.003⟩. ⟨hal-01349639v3⟩