On Stability of Measure Driven Differential Equations

International audienceWe consider the problem of stability in a class of differential equations which are driven by a differential measure associated with the inputs of locally bounded variation. After discussing some existing notions of solution for such systems, we derive conditions on the system's vector fields for asymptotic stability under a specific class of inputs. These conditions present a trade-off between the Lebesgue-integrable and the measure-driven components of the system. In case the system is not asymptotically stable, we derive weaker conditions such that the norm of the resulting trajectory is bounded by some function of the total variation of the input, which generalizes the notion of integral input-to-state stability in measure-driven systems

Dissipative Linear Stochastic Hamiltonian Systems

This paper is concerned with stochastic Hamiltonian systems which model a class of open dynamical systems subject to random external forces. Their dynamics are governed by Ito stochastic differential equations whose structure is specified by a Hamiltonian, viscous damping parameters and system-environment coupling functions. We consider energy balance relations for such systems with an emphasis on linear stochastic Hamiltonian (LSH) systems with quadratic Hamiltonians and linear coupling. For LSH systems, we also discuss stability conditions, the structure of the invariant measure and its relation with stochastic versions of the virial theorem. Using Lyapunov functions, organised as deformed Hamiltonians, dissipation relations are also considered for LSH systems driven by statistically uncertain external forces. An application of these results to feedback connections of LSH systems is outlined.Comment: 10 pages, 1 figure, submitted to ANZCC 201

On the monotone stability approach to BSDEs with jumps: Extensions, concrete criteria and examples

We show a concise extension of the monotone stability approach to backward stochastic differential equations (BSDEs) that are jointly driven by a Brownian motion and a random measure for jumps, which could be of infinite activity with a non-deterministic and time inhomogeneous compensator. The BSDE generator function can be non convex and needs not to satisfy global Lipschitz conditions in the jump integrand. We contribute concrete criteria, that are easy to verify, for results on existence and uniqueness of bounded solutions to BSDEs with jumps, and on comparison and a-priori $L^{\infty}$-bounds. Several examples and counter examples are discussed to shed light on the scope and applicability of different assumptions, and we provide an overview of major applications in finance and optimal control.Comment: 28 pages. Added DOI https://link.springer.com/chapter/10.1007%2F978-3-030-22285-7_1 for final publication, corrected typo (missing gamma) in example 4.1

Schr\"odinger equations with smooth measure potential and general measure data

We study equations driven by Schr\"odinger operators consisting of a self-adjoint Dirichlet operator and a singular potential, which belongs to a class of positive Borel measures absolutely continuous with respect to a capacity generated by the operator. In particular, we cover positive potentials exploding on a set of capacity zero. The right-hand side of equations is allowed to be a general bounded Borel measure. The class of self-adjoint Dirichlet operators is quite large. Examples include integro-differential operators with the local part of divergence form. We give a necessary and sufficient condition for the existence of a solution, and prove some regularity and stability results

Jump-Diffusions in Hilbert Spaces: Existence, Stability and Numerics

By means of an original approach, called "method of the moving frame", we establish existence, uniqueness and stability results for mild and weak solutions of stochastic partial differential equations (SPDEs) with path dependent coefficients driven by an infinite dimensional Wiener process and a compensated Poisson random measure. Our approach is based on a time-dependent coordinate transform, which reduces a wide class of SPDEs to a class of simpler SDE problems. We try to present the most general results, which we can obtain in our setting, within a self-contained framework to demonstrate our approach in all details. Also several numerical approaches to SPDEs in the spirit of this setting are presented.Comment: fully revised and extended versio

Exponential stability of stochastic evolution equations driven by small fractional Brownian motion with Hurst parameter in (1/2,1)(1/2,1)

This paper addresses the exponential stability of the trivial solution of some types of evolution equations driven by H\"older continuous functions with H\"older index greater than $1/2$. The results can be applied to the case of equations whose noisy inputs are given by a fractional Brownian motion $B^H$ with covariance operator $Q$, provided that $H\in (1/2,1)$ and ${\rm tr}(Q)$ is sufficiently small.Comment: 19 page

Asymptotic stability of stochastic differential equations driven by Lévy noise

Using key tools such as Ito's formula for general semimartingales, Kunita's moment estimates for Levy-type stochastic integrals, and the exponential martingale inequality, we find conditions under which the solutions to the stochastic differential equations (SDEs) driven by Levy noise are stable in probability, almost surely and moment exponentially stable