Numerical solutions of neutral stochastic functional differential equations

This paper examines the numerical solutions of neutral stochastic functional differential equations (NSFDEs) $d[x(t)-u(x_t)]=f(x_t)dt+g(x_t)dw(t)$, $t\geq 0$. The key contribution is to establish the strong mean square convergence theory of the Euler-Maruyama approximate solution under the local Lipschitz condition, the linear growth condition, and contractive mapping. These conditions are generally imposed to guarantee the existence and uniqueness of the true solution, so the numerical results given here are obtained under quite general conditions. Although the way of analysis borrows from [X. Mao, LMS J. Comput. Math., 6 (2003), pp. 141-161], to cope with $u(x_t)$, several new techniques have been developed

Managing the Transition from PSTN to IP Networks

This study tries to evaluate problems surrounding the transition from public switched telephone networks (PSTNs) to IP networks that affect both the current users of PSTN and competitive carriers. For this purpose, this study surveys discussions on the transition based on four perspectives: (1) consumer-oriented, (2) forward-looking, (3) competitive neutrality and (4) proportionality of regulation. It is organised in the following manner. First, it clarifies the background of the transition, especially the impact of the diffusion of broadband Internet and mobile technologies. Second, it surveys a series of the Nippon Telegraph and Telephone Corporation’s (NTT) migration proposals published since 2010. Third, the Ministry of Internal Affairs and Communications’ (MIC) investigation responding to NTT’s proposals is summarised. Fourth, problems regarding the transition are evaluated by contrasting NTT’s proposals and MIC’s reports. Finally, the study infers five important remarks based on the above analysis. It must be acknowledged that the evaluations and remarks are premature because discussions on the transition are still in progress and the details of the transition have not yet been officially published

Uncertainty and economic growth in a stochastic R&D model

The paper examines an R&D model with uncertainty from the population growth, which is a stochastic cooperative Lotka-Volterra system, and obtains a suciently condition for the existence of the globally positive solution. The long-run growth rate of the economic system is ultimately bounded in mean and fluctuation of its growth will not be faster than the polynomial growth. When uncertainty of the population growth, in comparison with its expectation, is suciently large, the growth rate of the technological progress andthe capital accumulation will converge to zero. Inversely, when uncertainty of the population growth is suciently small or its expected growth rate is suciently high, the economic growth rate will not decay faster than the polyno-mial speed. The paper explicitly computes the sample average of the growth rates of both the technology and the capital accumulation in time and compares them with their counterparts in the corresponding deterministic model

Almost sure exponential stability of the Euler–Maruyama approximations for stochastic functional differential equations

By the continuous and discrete nonnegative semimartingale convergence theorems, this paper investigates conditions under which the Euler–Maruyama (EM) approximations of stochastic functional differential equations (SFDEs) can share the almost sure exponential stability of the exact solution. Moreover, for sufficiently small stepsize, the decay rate as measured by the Lyapunov exponent can be reproduced arbitrarily accurately

Multi-Agent Consensus With Relative-State-Dependent Measurement Noises

In this note, the distributed consensus corrupted by relative-state-dependent measurement noises is considered. Each agent can measure or receive its neighbors' state information with random noises, whose intensity is a vector function of agents' relative states. By investigating the structure of this interaction and the tools of stochastic differential equations, we develop several small consensus gain theorems to give sufficient conditions in terms of the control gain, the number of agents and the noise intensity function to ensure mean square (m. s.) and almost sure (a. s.) consensus and quantify the convergence rate and the steady-state error. Especially, for the case with homogeneous communication and control channels, a necessary and sufficient condition to ensure m. s. consensus on the control gain is given and it is shown that the control gain is independent of the specific network topology, but only depends on the number of nodes and the noise coefficient constant. For symmetric measurement models, the almost sure convergence rate is estimated by the Iterated Logarithm Law of Brownian motions