Stochastic viability and a comparison theorem

Milian, Anna

doi:10.4064/cm-68-2-297-316

Cited by 22 publications

(28 citation statements)

References 5 publications

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“…The first condition is equivalent to the invariance of K for (B, S, W), and the second condition means that K ⊂ K B,±S (see Milian [25], Theorem 2 (proof)). Therefore, these conditions are equivalent by Theorem 3.4.…”

Section: A Comparison Theorem For Rough Differential Equationsmentioning

confidence: 99%

Invariance for rough differential equations

Coutin

Marie

2017

Stochastic Processes and their Applications

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Abstract. In 1990, in Itô's stochastic calculus framework, Aubin and Da Prato established a necessary and sufficient condition of invariance of a nonempty compact or convex subset C of R d (d ∈ N * ) for stochastic differential equations (SDE) driven by a Brownian motion. In Lyons rough paths framework, this paper deals with an extension of Aubin and Da Prato's results to rough differential equations. A comparison theorem is provided, and the special case of differential equations driven by a fractional Brownian motion is detailed. Contents

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Section: A Comparison Theorem For Rough Differential Equationsmentioning

confidence: 99%

Invariance for rough differential equations

Coutin

Marie

2017

Stochastic Processes and their Applications

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“…under the flow of a stochastic differential equation in the Itô or Stratonovich sense. This result is by itself not new and many general results are known in particular a stochastic Naguno-Brezis Theorem as proved by Aubin-Da Prato in [8] or A. Milian in [44]. However, most of these results are difficult to read for a non-specialist in the field of stochastic calculus.…”

Section: Appendix a Reminder About Stochastic Invariance Of Submanifoldsmentioning

confidence: 93%

Selection of a stochastic Landau-Lifshitz equation and the stochastic persistence problem

Cresson

Kheloufi²,

Nachi³

et al. 2019

Journal of Mathematical Physics

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In this article, we study the persistence of properties of a given classical deterministic differential equation under a stochastic perturbation of two distinct forms: external and internal. The first case corresponds to add a noise term to a given equation using the framework of Itô or Stratonovich stochastic differential equations. The second case corresponds to consider a parameters dependent differential equations and to add a stochastic dynamics on the parameters using the framework of random ordinary differential equations. Our main concerns for the preservation of properties is stability/instability of equilibrium points and symplectic/Poisson Hamiltonian structures. We formulate persistence theorem in these two cases and prove that the cases of external and internal stochastic perturbations are drastically different. We then apply our results to develop a stochastic version of the Landau-Lifshitz equation. We discuss in particular previous results obtain by Etore and al. in [28] and we finally propose a new family of stochastic Landau-Lifshitz equations.

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“…The following theorem (see (Milian, 1995, Theorem 1)) characterizes the class of functions f and g such that the stochastic system (1) preserves the domain invariance of solutions.…”

Section: Summary About Continuous/discrete Stochastic Dynamical Systemsmentioning

confidence: 99%

A non-standard-Euler–Maruyama scheme

Pierret

2015

Journal of Difference Equations and Applications

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Abstract. We construct a nonstandard finite difference numerical scheme to approximate stochastic differential equations (SDEs) using the idea of weighed step introduced by R.E. Mickens. We prove the strong convergence of our scheme under locally Lipschitz conditions of a SDE and linear growth condition. We prove the preservation of domain invariance by our scheme under a minimal condition depending on a discretization parameter and unconditionally for the expectation of the approximate solution. The results are illustrated through the geometric Brownian motion. The new scheme shows a greater behavior compared to the Euler-Maruyama scheme and balanced implicit methods which are widely used in the literature and applications.

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Stochastic viability and a comparison theorem

Abstract: We give explicit necessary and sufficient conditions for the viability of polyhedrons with respect to Itô equations. Using the viability criterion we obtain a comparison theorem for multi-dimensional Itô processes.

Cited by 22 publications

References 5 publications

Invariance for rough differential equations

Invariance for rough differential equations

Selection of a stochastic Landau-Lifshitz equation and the stochastic persistence problem

A non-standard-Euler–Maruyama scheme

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