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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–Maruyama methods for solutions of stochastic differential equations, we use techniques from stochastic approximation. We analyze the almost sure exponential stability and exponential p -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 of stability of numerical methods for linearizable and multi-dimensional jump diffusions is carried out. [ABSTRACT FROM AUTHOR]

Electrolytes with a high ionic conductivity are the prerequisite for the success of solid state rechargeable ion batteries. In this paper, density functional theory (DFT) calculations are applied in combination with a climbing-image nudged elastic band (CI-NEB) method to obtain the diffusion barriers of the lithium (Li) and sodium (Na) in stoichiometric AM2(PO4)3 (A = Li, Na, M = Ti, Sn and Zr) compounds. In the AM2(PO4)3, Li and Na ions occupy the interstitial sites, M1, which is coordinated by a trigonal antiprism of oxygen, and M2, which has a distorted eight-fold coordination. Results show that the diffusion barriers are closely related with the average bond length of A–O bond when the atom A occupies the M2 site in the AM2(PO4)3 compounds, and the diffusion barriers decrease with increasing the average bond length. Among the various types of the AM2(PO4)3 compounds, the AM2(PO4)3 has a good potential for solid electrolytes due to its low diffusion barriers for the Li and Na. [ABSTRACT FROM AUTHOR]

In this paper, sodium and nickel ion dynamics in layered cathode materials of NaNi1/2Mn1/2O2 and NaNi1/3Mn1/3Co1/3O2 for Na-ion battery applications were investigated using first principles based on density functional theory. The diffusion pathways for sodium and nickel migration inside the layered cathode materials were optimized and their energy barriers were calculated based on the nudged elastic band method. The energy barrier of Na diffusion in NaNi1/2Mn1/2O2 and NaNi1/3Mn1/3Co1/3O2 is about 1.0 eV. However, when the Ni is located at the Na layer, i.e. occupying the Na position, the energy barrier for Ni diffusion inside the layered cathode materials is about 0.36 eV, which indicates that Ni will diffuse to the surface of electrode materials, thus will block or slow down the sodium diffusion, limiting the battery performance. [ABSTRACT FROM AUTHOR]

This work develops asymptotic properties for a parabolic system with two-time scales associated with a transient switching diffusion. Although the problem is motivated by stochastic systems, the techniques that we are using are purely analytic. Asymptotic expansions are constructed; their validity is justified. [ABSTRACT FROM AUTHOR]

Abstract: 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. One of the distinct features considered here is that the switching process depends on the jump diffusions. First asymptotic stability in the large is obtained. Then the study on exponential -stability is carried out. Connection between almost surely exponential stability and exponential -stability is exploited. Also presented are smooth-dependence on the initial data. Using the smooth-dependence, necessary conditions for exponential -stability are derived. Then criteria for asymptotic stability in distribution are provided. A couple of examples are given to illustrate our results. [Copyright &y& Elsevier]

This work aims to developing asymptotic expansions of solutions of a system of coupled differential equations with applications to option price under regime-switching diffusions. The main motivation stems from using switching diffusions to model stochastic volatility so as to obtain uniform asymptotic expansions of European-type options. By focusing on fast mean reversion, our effort is placed on finding the “effective volatility”. Under simple conditions, asymptotic expansions are developed with uniform asymptotic error bounds. The leading term in the asymptotic expansions satisfies a Black–Scholes equation in which the mean return rate and volatility are averaged out with respect to the stationary measure of the switching process. In addition, the full asymptotic series is developed, which will help us to gain insight on the behavior of the option price when the time approaches maturity. The asymptotic expansions obtained in this paper are interesting in their own right and can be used for other problems in control optimization of systems involving fast varying switching processes. [ABSTRACT FROM AUTHOR]

In response to the increasing needs for control and optimization of hybrid systems, this work is concerned with such asymptotic properties as recurrence (also known as weak stochastic stability in the literature) and ergodicity of regime-switching diffusions. Using Liapunov functions, necessary and sufficient conditions for positive recurrence are developed. Then, ergodicity of positive recurrent regime-switching diffusions is obtained by constructing cycles using the associated discretetime Markov chains. [ABSTRACT FROM AUTHOR]

Abstract: This work is concerned with numerical methods for controlled regime-switching diffusions, and regime-switching jump diffusions. Numerical procedures based on Markov chain approximation techniques are developed. Convergence of the algorithms is derived by means of weak convergence methods. In addition, examples are also provided for demonstration purpose. [Copyright &y& Elsevier]

This work is concerned with diffusion processes having fast and slow components. It was known that under suitable assumptions the slow component can be approximated by the Markov process with averaged characteristics. In this work, asymptotic expansions for the solutions of the Kolmogorov backward equations are constructed and justified. Certain probabilistic conclusions and examples are also provided. [ABSTRACT FROM AUTHOR]

Abstract: This work is devoted to stability of regime-switching diffusion processes. After presenting the formulation of regime-switching diffusions, the notion of stability is recalled, and necessary conditions for -stability are obtained. Then main results on stability and instability for systems arising in approximation are presented. Easily verifiable conditions are established. An example is examined as a demonstration. A remark on linear systems is also provided. [Copyright &y& Elsevier]