Existence and Uniqueness of Solution of Stochastic Dynamic Systems with Markov Switching and Concentration Points (original) (raw)
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arXiv (Cornell University), 2023
This article aims to investigate sufficient conditions for the stability of stochastic differential equations with a random structure, particularly in contexts involving the presence of concentration points. The proof of asymptotic stability leverages the use of Lyapunov functions, supplemented by additional constraints on the magnitudes of jumps and jump times, as well as the Markov property of the system solutions. The findings are elucidated with an example, demonstrating both stable and unstable conditions of the system.
Journal of Differential Equations, 2010
This work is concerned with several properties of solutions of stochastic differential equations arising from hybrid switching diffusions. The word "hybrid" highlights the coexistence of continuous dynamics and discrete events. The underlying process has two components. One component describes the continuous dynamics, whereas the other is a switching process representing discrete events. One of the main features is the switching component depending on the continuous dynamics. In this paper, weak continuity is proved first. Then continuous and smooth dependence on initial data are demonstrated. In addition, it is shown that certain functions of the solutions verify a system of Kolmogorov's backward differential equations. Moreover, rates of convergence of numerical approximation algorithms are dealt with.
Springer Proceedings in Mathematics & Statistics, 2016
We consider one-dimensional stochastic differential equations with jumps in the general case. We introduce new technics based on local time and we prove new results on pathwise uniqueness and comparison theorems. Our approach are very easy to handled and don't need any approximation approach. Similar equations without jumps were studied in the same context by [15], [20] and others authors. As an application we get a new condition on the pathwise uniqueness for the solutions to stochastic differential equations driven by a symmetric stable Lévy processes.
Stability in distribution and stabilization of switching jump diffusions
ESAIM: Control, Optimisation and Calculus of Variations
This paper aims to study stability in distribution of Markovian switching jump diffusions. The main motivation stems from stability and stabilizing hybrid systems in which there is no trivial solution. An explicit criterion for stability in distribution is derived. The stabilizing effects of Markov chains, Brownian motions, and Poisson jumps are revealed. Based on these criteria, stabilization problems of stochastic differential equations with Markovian switching and Poisson jumps are developed.
Mean reflected stochastic differential equations with jumps
Advances in Applied Probability, 2020
In this paper, a reflected stochastic differential equation (SDE) with jumps is studied for the case where the constraint acts on the law of the solution rather than on its paths. These reflected SDEs have been approximated by Briand et al. (2016) using a numerical scheme based on particles systems, when no jumps occur. The main contribution of this paper is to prove the existence and the uniqueness of the solutions to this kind of reflected SDE with jumps and to generalize the results obtained by Briand et al. (2016) to this context.
Journal of Computational and Applied Mathematics, 2005
In principle, once the existence of the stationary distribution of a stochastic di erential equation with Markovian switching is assured, we may compute it by solving the associated system of the coupled Kolmogorov-Fokker-Planck equations. However, this is nontrivial in practice. As a viable alternative, we use the Euler-Maruyama scheme to obtain the stationary distribution in this paper.