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

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

Some New Results on the Lotka-Volterra System with Variable Delay

This paper discusses the stochastic Lotka-Volterra system with time-varying delay. The nonexplosion, the boundedness, and the polynomial pathwise growth of the solution are determined once and for all by the same criterion. Moreover, this criterion is constructed by the parameters of the system itself, without any uncertain one. A two-dimensional stochastic delay Lotka-Volterra model is taken as an example to illustrate the effectiveness of our result

The Boundedness and Exponential Stability Criterions for Nonlinear Hybrid Neutral Stochastic Functional Differential Equations

Neutral differential equations have been used to describe the systems that not only depend on the present and past states but also involve derivatives with delays. This paper considers hybrid nonlinear neutral stochastic functional differential equations (HNSFDEs) without the linear growth condition and examines the boundedness and exponential stability. Two illustrative examples are given to show the effectiveness of our theoretical results

A highly sensitive mean-reverting process in finance and the Euler-Maruyama approximations

Empirical studies show that the most successful continuous-time models of the short term rate in capturing the dynamics are those that allow the volatility of interestchanges to be highly sensitive to the level of the rate. However, from the mathematics, the high sensitivity to the level implies that the coeffcients do not satisfy the lineargrowth condition, so we can not examine its properties by traditional techniques. This paper overcomes the mathematical difculties due to the nonlinear growth and examines its analytical properties and the convergence of numerical solutions in probability. The convergence result can be used to justify the method within Monte-Carlo simulations that compute the expected payoff of financial products. For illustration, we apply our results compute the value of a bond with interest rate given by the highly sensitive mean-reverting process as well as the value of a single barrier call option with the asset price governed by this process

Stability in distribution of stochastic functional differential equations

In this paper we investigate the stability in distribution for a class of stochastic functional differential equations (SFDEs), which include stochastic differential delay equations (SDDEs). Although stability in distribution has been studied by several authors recently, there is so far no stability-in-distribution criterion on SFDEs where the terms involved the delay components are highly nonlinear (not bounded by linear functions). In this paper we will establish the sufficient criteria on the stability in distribution for a class of highly nonlinear SFDEs. Two examples will be given to illustrate our new results. We also explain the reason why the existing stability-in-distribution criteria are not applicable by these two examples