Simulation of a nonlinear stochastic ecology model

A nonlinear ecological system model is formulated as a stochastic process. The formulation traces a chemical nutrient as it undergoes random exchanges between the phytoplankton, the zooplankton, and the euphotic zone of an aquatic ecosystem. The Chapman-Kolmogorov equations are derived, and simulation techniques are used to study both single realizations and the statistical parameters which govern the system.     

Exponential moment bounds and strong convergence rates for tamed-truncated numerical approximations of stochastic convolutions

Numerical Algorithms - In this article we establish exponential moment bounds, moment bounds in fractional order smoothness spaces, a uniform Hölder continuity in time, and strong convergence...     

Parameter estimation in nonlinear stochastic differential equations

We discuss the problem of parameter estimation in nonlinear stochastic differential equations (SDEs) based on sampled time series. A central message from the theory of integrating SDEs is that there exist in general two time scales, i.e. that of integrating these equations and that of sampling. We argue that therefore, maximum likelihood estimation is computationally extremely expensive. We discuss the relation between maximum likelihood and quasi maximum likelihood estimation. In a simulation study, we compare the quasi maximum likelihood method with an approach for parameter estimation in nonlinear SDEs that disregards the existence of the two time scales.     

Parameter estimation for the stochastic SIS epidemic model

In this paper we estimate the parameters in the stochastic SIS epidemic model by using pseudo-maximum likelihood estimation (pseudo-MLE) and least squares estimation. We obtain the point estimators and $100 (1-\alpha )\%$ confidence intervals as well as $100 (1-\alpha )\%$ joint confidence regions by applying least squares techniques. The pseudo-MLEs have almost the same form as the least squares case. We also obtain the exact as well as the asymptotic $100 (1-\alpha )\%$ joint confidence regions for the pseudo-MLEs. Computer simulations are performed to illustrate our theory.     

Correlations and bounds for stochastic volatility models

We investigate here, systematically and rigorously, various stochastic volatility models used in Mathematical Finance. Mathematically, such models involve coupled stochastic differential equations with coefficients that do not obey the natural and classical conditions required to make these models “well-posed”. And we obtain necessary and sufficient conditions on the parameters, such as correlation, of these models in order to have integrable or Lp$L^p$ solutions (for 1<p<∞$1<p<\infty$).     

Upper bounds for Gaussian stochastic programs

We present a construction which gives deterministic upper bounds for stochastic programs in which the randomness appears on the right–hand–side and has a multivariate Gaussian distribution. Computation of these bounds requires the solution of only as many linear programs as the problem has variables. Received December 2, 1997 / Revised version received January 5, 1999? Published online May 12, 1999     

The moment Lyapunov exponent for a three-dimensional stochastic system

This paper presents a method, through which the pth moment stability of a linear multiplicative stochastic system, that is a linear part of a co-dimension two-bifurcation system upon a three-dimensional center manifold and is subjected to a parametric excitation by an ergodic real noise, is obtained. The excitation included is assumed to be an integrable function of an n-dimensional Ornstein–Uhlenbeck vector process that is the output of a linear filter system and both the strong mixing condition, which is the sufficient condition for the stochastic averaging method, and the delicate balance condition are removed in the present study. By using a perturbation method and the spectrum representations of both the Fokker Planck operator and its adjoint one of the linear filter system, the asymptotic expressions of the moment Lyapunov exponent are obtained, which match the numerical results well.     

Parameter analysis based on stochastic model for differential evolution algorithm

In this paper, a stochastic model is used to describe and analyze the evolution process of differential evolution (DE) for numerical optimization. With the model, it illustrates how the probability distribution of the whole population is changed by mutation, selection and crossover operations. Based on the theoretical analysis, some guidelines about the parameter setting for DE are provided. In addition, numerical simulations are carried out to verify the conclusions drawn from model analysis.     

Monotonicity and bounds for convex stochastic control models

We consider a general convex stochastic control model. Our main interest concerns monotonicity results and bounds for the value functions and for optimal policies. In particular, we show how the value functions depend on the transition kernels and we present conditions for a lower bound of an optimal policy. Our approach is based on convex stochastic orderings of probability measures. We derive several interesting sufficient conditions of these ordering concepts, where we make also use of the Blackwell ordering. The structural results are illustrated by partially observed control models and Bayesian information models.     

Almost sure and moment Lyapunov exponents for a stochastic beam equation

The transverse vibrations of an Euler-Bernoulli beam with axial tension P and axial white noise forcing are given by

     

Research on nonlinear stochastic dynamical price model

In consideration of many uncertain factors existing in economic system, nonlinear stochastic dynamical price model which is subjected to Gaussian white noise excitation is proposed based on deterministic model. One-dimensional averaged Itô stochastic differential equation for the model is derived by using the stochastic averaging method, and applied to investigate the stability of the trivial solution and the first-passage failure of the stochastic price model. The stochastic price model and the methods presented in this paper are verified by numerical studies.     

Error bounds for stochastic shortest path problems

For stochastic shortest path problems, error bounds for value iteration due to Bertsekas elegantly generalize the classic MacQueen–Porteus error bounds for discounted infinite-horizon Markov decision problems, but incur prohibitive computational overhead. We derive bounds on these error bounds that can be computed with little or no overhead, making them useful in practice—especially so, since easily-computed error bounds have not previously been available for this class of problems.     

Simulation-based confidence bounds for two-stage stochastic programs

This paper provides a rigorous asymptotic analysis and justification of upper and lower confidence bounds proposed by Dantzig and Infanger (A probabilistic lower bound for two-stage stochastic programs, Stanford University, CA, 1995) for an iterative sampling-based decomposition algorithm, introduced by Dantzig and Glynn (Ann. Oper. Res. 22:1–21, 1990) and Infanger (Ann. Oper. Res. 39:41–67, 1992), for solving two-stage stochastic programs. The paper provides confidence bounds in the presence of both independent sampling across iterations, and when common samples are used across different iterations. Confidence bounds for sample-average approximation then follow as a special case. Extensions of the theory to cover use of variance reduction and the dropping of cuts are also presented. An extensive empirical investigation of the performance of these bounds establishes that the bounds perform reasonably on realistic problems.

     

Hopf bifurcation of the stochastic model on HAB nonlinear stochastic dynamics

A nonlinear stochastic dynamical model on a typical HAB algae diatom and dianoflagellate densities was created and presented in this paper. Simplifying the model through a stochastic averaging method, we obtained a two-dimensional diffusion process of averaged amplitude and phase. The singular boundary theory of diffusion process and the invariant measure theory were applied in analyzing the bifurcation of stability and the Hopf bifurcation of the stochastic system. The critical value of the stochastic Hopf bifurcation parameter was obtained and the conclusion that the position of Hopf bifurcation drifting with the parameter increase is presented as a result.     

Some moment inequalities for stochastic integrals and for solutions of stochastic differential equations

Some inequalities concerning the Itô stochastic integral and solutions of stochastic different equations are obtained.     

Computable error bounds for nonlinear programming

This paper presents a method for obtaining computable bounds for the error in an approximate Kuhn—Tucker point of a nonlinear program. Techniques of interval analysis are employed to compute the error bounds.Sponsored by the United States Army under Contract No. DA-31-124-ARO-D-462.     

Convergence bounds for nonlinear programming algorithms

Bounds on convergence are given for a general class of nonlinear programming algorithms. Methods in this class generate at each interation both constraint multipliers and approximate solutions such that, under certain specified assumptions, accumulation points of the multiplier and solution sequences satisfy the Fritz John or the Kuhn—Tucker optimality conditions. Under stronger assumptions, convergence bounds are derived for the sequences of approximate solution, multiplier and objective function values. The theory is applied to an interior—exterior penalty function algorithm modified to allow for inexact subproblem solutions. An entirely new convergence bound in terms of the square root of the penalty controlling parameter is given for this algorithm.