Invariance of closed convex cones for stochastic partial differential equations

Abstract

The goal of this paper is to clarify when a closed convex cone is invariant for a stochastic partial differential equation (SPDE) driven by a Wiener process and a Poisson random measure, and to provide conditions on the parameters of the SPDE, which are necessary and sufficient.2010 Mathematics Subject Classification. 60H15, 60G17.

On the Viability of Solutions to Conformable Stochastic Differential Equations

“…Since then, the viability of the classical stochastic differential equation has been studied extensively. One can refer to the results in [18][19][20][21][22][23][24][25][26][27][28][29][30][31][32], etc. Up to now, to the best of the author's knowledge, the viability of the conformable stochastic differential equations has not been studied in the literature.…”

Distance between closed sets and the solutions to stochastic partial differential equations

“…In several papers stochastic invariance of K has been studied; that is, necessary and sufficient conditions have been derived such that for every starting point x ∈ K the mild solution X( • ; x) to the SPDE (1.1) stays in the set K. This has been done in [22] and, based on the support theorem presented in [26], in [27] for the case where K is an arbitrary closed subset. In the particular case where K is a closed convex cone, we refer to the papers [24,14,35]; the latter two articles consider the situation where the SPDE (1.1) is additionally driven by a Poisson random measure. If K is a finite dimensional submanifold of H, the invariance problem has been studied in [12] and [27] (see also the review article [32]), and in [15] even in the situation where the SPDE (1.1) has an additional Poisson random measure as driving noise.…”

Translation Invariant Diffusions and Stochastic Partial Differential Equations in ${\cal S}^{\prime}

“…We refer to Example 6 in Section 6 below, for some connections between distribution dependent SDE's and our results. Our work also relates to the problem of identifying ' invariant submanifolds' of solutions of SPDEs that arise in finance (see [8], [9], [14], [44]). In effect, the set of translates {τ x y : x ∈ R d } serves as an invariant manifold for the above SPDE with initial distribution y ∈ S p , under some smoothness assumptions on y.…”