888 research outputs found

    Stability of Numerical Methods for Jump Diffusions and Markovian Switching Jump Diffusions

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    This work focuses on stability analysis of numerical solutions to jump diffusions and jump diffusions with Markovian switching. Due to the use of Poisson processes, using asymptotic expansions as in the usual approach of treating diffusion processes does not work. Different from the existing treatments of Euler-Maurayama methods for solutions of stochastic differential equations, we use techniques from stochastic approximation. We analyze the almost sure exponential stability and exponential pp-stability. The benchmark test model in numerical solutions, namely, one-dimensional linear scalar jump diffusion is examined first and easily verifiable conditions are presented. Then Markovian regime-switching jump diffusions are dealt with. Moreover, analysis on stability of numerical methods for linearizable and multi-dimensional jump diffusions is carried out.Comment: This paper has been withdrawn by the author due to a private reaso

    Stability of Nonlinear Regime-switching Jump Diffusions

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    Motivated by networked systems, stochastic control, optimization, and a wide variety of applications, this work is devoted to systems of switching jump diffusions. Treating such nonlinear systems, we focus on stability issues. First asymptotic stability in the large is obtained. Then the study on exponential p-stability is carried out. Connection between almost surely exponential stability and exponential p-stability is exploited. Also presented are smooth-dependence on the initial data. Using the smooth-dependence, necessary conditions for exponential p-stability are derived. Then criteria for asymptotic stability in distribution are provided. A couple of examples are given to illustrate our results

    Stability of Regime-Switching Diffusion Systems with Discrete States Belonging to a Countable Set

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    This work focuses on stability of regime-switching diffusions consisting of continuous and discrete components, in which the discrete component switches in a countably infinite set and its switching rates at current time depend on the continuous component. In contrast to the existing approach, this work provides more practically viable approach with more feasible conditions for stability. A classical approach for asymptotic stabilityusing Lyapunov function techniques shows the Lyapunov function evaluated at the solution process goes to 0 as time t→∞t\to \infty. A distinctive feature of this paper is to obtain estimates of path-wise rates of convergence, which pinpoints how fast the aforementioned convergence to 0 taking place. Finally, some examples are given to illustrate our findings.Comment: 25page

    Recurrence and Ergodicity for A Class of Regime-Switching Jump Diffusions

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    This work develops asymptotic properties of a class of switching jump diffusion processes. The processes under consideration may be viewed as a number of jump diffusion processes modulated by a random switching mechanism. The underlying processes feature in the switching process depends on the jump diffusions. In this paper, conditions for recurrence and positive recurrence are derived. Ergodicity is examined in detail. Existence of invariant probability measures is proved.Comment: To appear in Applied Mathematics and Optimizatio

    Stabilization of regime-switching processes by feedback control based on discrete time observations II: state-dependent case

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    This work investigates the almost sure stabilization of a class of regime-switching systems based on discrete-time observations of both continuous and discrete components. It develops Shao's work [SIAM J. Control Optim., 55(2017), pp. 724--740] in two aspects: first, to provide sufficient conditions for almost sure stability in lieu of moment stability; second, to investigate a class of state-dependent regime-switching processes instead of state-independent ones. To realize these developments, we establish an estimation of the exponential functional of Markov chains based on the spectral theory of linear operator. Moreover, through constructing order-preserving coupling processes based on Skorokhod's representation of jumping process, we realize the control from up and below of the evolution of state-dependent switching process by state-independent Markov chains.Comment: 33 page

    Stability and recurrence of regime-switching diffusion processes

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    We provide some criteria on the stability of regime-switching diffusion processes. Both the state-independent and state-dependent regime-switching diffusion processes with switching in a finite state space and an infinite countable state space are studied in this work. We provide two methods to deal with switching processes in an infinite countable state space. One is a finite partition method based on the nonsingular M-matrix theory. Another is an application of principal eigenvalue of a bilinear form. Our methods can deal with both linear and nonlinear regime-switching diffusion processes. Moreover, the method of principal eigenvalue is also used to study the recurrence of regime-switching diffusion processes

    Strong solutions and strong Feller properties for regime-switching diffusion processes in an infinite state space

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    We establish the existence and pathwise uniqueness of regime-switching diffusion processes in an infinite state space, which could be time-inhomogeneous and state-dependent. Then the strong Feller properties of these processes are investigated by using the theory of parabolic differential equations and dimensional-free Harnack inequalities

    Regime-Switching Jump Diffusions with Non-Lipschitz Coefficients and Countably Many Switching States: Existence and Uniqueness, Feller, and Strong Feller Properties

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    This work focuses on a class of regime-switching jump diffusion processes, which is a two component Markov processes (X(t),Λ(t))(X(t),\Lambda(t)), where Λ(t)\Lambda(t) is a component representing discrete events taking values in a countably infinite set. Considering the corresponding stochastic differential equations, our main focus is on treating those with non-Lipschitz coefficients. We first show that there exists a unique strong solution to the corresponding stochastic differential equation. Then Feller and strong Feller properties are investigated

    Jump-diffusion processes in random environments

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    In this paper we investigate jump-diffusion processes in random environments which are given as the weak solutions to SDE's. We formulate conditions ensuring existence and uniqueness in law of solutions. We investigate Markov property. To prove uniqueness we solve a general martingale problem for \cadlag processes. This result is of independent interest. In the last section we present application of our results considering generalized exponential Levy model

    Regime-switching diffusion processes: strong solutions and strong Feller property

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    We investigate the existence and uniqueness of strong solutions up to an explosion time for regime-switching diffusion processes in an infinite state space. Instead of concrete conditions on coefficients, our existence and uniqueness result is established under the general assumption that the diffusion in every fixed environment has a unique non-explosive strong solution. Moreover, non-explosion conditions for regime-switching diffusion processes are given. The strong Feller property is proved by further assuming that the diffusion in every fixed environment generates a strong Feller semigroup
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