Alexandra Rodkina
University of the West Indies
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Featured researches published by Alexandra Rodkina.
Systems & Control Letters | 1998
Xuerong Mao; Natalia Koroleva; Alexandra Rodkina
In this paper we first discuss the robust stability of uncertain linear stochastic differential delay equations. We then extend the theory to cope with the robust stability of uncertain semi-linear stochastic differential delay equations. We shall also give several examples to illustrate our theory.
Mathematics of Control, Signals, and Systems | 2006
Alexandra Rodkina; Michael V. Basin
Global asymptotic stability conditions for nonlinear stochastic systems with state delay are obtained based on the convergence theorem for semimartingale inequalities, without assuming the Lipschitz conditions for nonlinear drift functions. The Lyapunov–Krasovskii and degenerate functionals techniques are used. The derived stability conditions are directly expressed in terms of the system coefficients. Nontrivial examples of nonlinear systems satisfying the obtained stability conditions are given.
Systems & Control Letters | 2007
Alexandra Rodkina; Michael V. Basin
Global asymptotic stability conditions for discrete vector nonlinear stochastic systems with state delay and Volterra diffusion term are obtained based on the convergence theorem for semimartingale inequalities, without assuming the Lipschitz conditions for nonlinear drift functions. The derived stability conditions are directly expressed in terms of the system coefficients. A number of nontrivial examples of nonlinear systems satisfying the obtained stability conditions are given. The obtained results are compared to some previously known asymptotic stability conditions for discrete nonlinear stochastic systems.
Stochastics An International Journal of Probability and Stochastic Processes | 2009
John A. D. Appleby; Gregory Berkolaiko; Alexandra Rodkina
We consider the stochastic difference equation where f and g are nonlinear, bounded functions, is a sequence of independent random variables, and h>0 is a nonrandom parameter. We establish results on asymptotic stability and instability of the trivial solution . We also show that, for some natural choices of f and g, the rate of decay of is approximately polynomial: there exists such that decays faster than but slower than , for any . It turns out that, if decays faster than as , the polynomial rate of decay can be established precisely: tends to a constant limit. On the other hand, if g does not decay quickly enough, the approximate decay rate is the best possible result.
Stochastic Analysis and Applications | 2000
Alexandra Rodkina; Xuerong Mao; Vladimir Kolmanovskii
The stochastic difference equations with Volterra type linear and nonlinear main term are considered in the paper.Conditions on a.s. boundedness of the solutions,asymptotic stability,exponential stability,and stability with a speed which differs from the exponential,are obtained.A numerous examples are presented.
Journal of Difference Equations and Applications | 2008
John A. D. Appleby; Gregory Berkolaiko; Alexandra Rodkina
We consider the nonlinear stochastic difference equation Here, (ξ n ) n ∈ ℕ is a sequence of independent random variables with zero mean and unit variance and with distribution functions F n . The function f : ℝ → ℝ is continuous, f(0) = 0, xf(x)>0 for x ≠ 0. We establish a condition on the noise intensity σ and the rate of decay of the tails of the distribution functions F n , under which the convergence of solutions to zero occurs with probability zero. If this condition does not hold, and f is bounded by a linear function with slope 2 − γ, for γ ∈ (0, 2), all solutions tend to zero a.s. On the other hand, if f grows more quickly than linear function with slope 2 + γ, for γ>0, the solutions tend to infinity in modulus with arbitrarily high probability, once the initial condition is chosen sufficiently large. Such equations can still demonstrate local stability; for a wide class of highly nonlinear f, it is shown that solutions tend to zero with arbitrarily high probability, once the initial condition is chosen appropriately. Results which elucidate the relationship between the rate of decay of the noise intensities and the rate of decay of the tails, and the necessary condition for stability, are presented. The connection with the asymptotic dynamical consistency of the system, when viewed as a discretisation of an Itô stochastic differential equation, is also explored.
Journal of Difference Equations and Applications | 2001
Alexandra Rodkina; Xuerong Mao
Consider a stochastic difference equation with the Volterra type nonlinear main term G and Volterra type noise Functions Gf, σ supposed to be random, x;i is a martingale—difference. In general so Eq, (1) cannot be considered as a stochastic equation with respect to the discrete semimartingale because the term is not a martingale—difference. The main aim of this paper is to establish the sufficient conditions on the almost sure boundedness, asymptotic and exponential stability of the solutions. Equation (1) can be interpreted as a generalization of the equation describing the gain incurred by the insurance company in a year i+1. We therefore explore the possible application of our theory in this area
Lms Journal of Computation and Mathematics | 2012
Gregory Berkolaiko; Evelyn Buckwar; Cónall Kelly; Alexandra Rodkina
We perform an almost sure linear stability analysis of the θ-Maruyama method, selecting as our test equation a two-dimensional system of Itô differential equations with diagonal drift coefficient and two independent stochastic perturbations which capture the stabilising and destabilising roles of feedback geometry in the almost sure asymptotic stability of the equilibrium solution. For small values of the constant step-size parameter, we derive close-to-sharp conditions for the almost sure asymptotic stability and instability of the equilibrium solution of the discretisation that match those of the original test system. Our investigation demonstrates the use of a discrete form of the Itô formula in the context of an almost sure linear stability analysis.
Journal of Difference Equations and Applications | 2006
Gregory Berkolaiko; Alexandra Rodkina
We consider a non-homogeneous non-linear stochastic difference equation and its linear counterpart both with initial value , non-random decaying free coefficient S n and independent random variables . We establish results on a.s. convergence of solutions X n to zero. Obtained necessary conditions tie together certain moments of the noise and the rate of decay of S n . To ascertain sharpness of our conditions we discuss some situations when X n diverges. We also establish a result concerning the rate of decay of X n to zero. Several examples are given to illustrate the ideas of the paper.
Applied Mathematics and Computation | 2010
John A. D. Appleby; Małgorzata Guzowska; Cónall Kelly; Alexandra Rodkina
We consider the Euler discretisation of a scalar linear test equation with positive solutions and show for both strong and weak approximations that the probability of positivity over any finite interval of simulation tends to unity as the step size approaches zero. Although a.s. positivity in an approximation is impossible to achieve, we develop for the strong (Maruyama) approximation an asymptotic estimate of the number of mesh points required for positivity as our tolerance of non-positive trajectories tends to zero, and examine the effectiveness of this estimate in the context of practical numerical simulation. We show how this analysis generalises to equations with a drift coefficient that may display a high level of nonlinearity, but which must be linearly bounded from below (i.e. when acting towards zero), and a linearly bounded diffusion coefficient. Finally, in the linear case we develop a refined asymptotic estimate that is more useful as an a priori guide to the number of mesh points required to produce positive approximations with a given probability.