In this paper we stochastically perturb the delay Lotka–Volterra model ẋ(t)= diag(x1(t), . . . , xn(t))[A(x(t)− x̄)+B(x(t − τ )− x̄)] into the stochastic delay differential equation (SDDE) dx(t)= diag(x1(t), . . . , xn(t)){[A(x(t)− x̄)+B(x(t − τ )− x̄)]dt + σ (x(t)− x̄)dw(t)}. The main aim is to reveal the effects of environmental noise on the delay Lotka–Volterra model. Our results can essentially ...

it is known that a stochastic dierential equation (sde) induces two probabilisticobjects, namely a diusion process and a stochastic ow. while the diusion process isdetermined by the innitesimal mean and variance given by the coecients of the sde,this is not the case for the stochastic ow induced by the sde. in order to characterize thestochastic ow uniquely the innitesimal covariance give...

Abstract: In this paper we show that there are applications that transform the movement of a pendulum into movements in R3. This can be done using Euler top system of differential equations. On the constant level surfaces, Euler top system reduces to the equation of a pendulum. Those properties are also considered in the case of system of differential equations with delay argument and in the fr...

Most of the existing results on stochastic stability use a single Lyapunov function, but we shall instead use multiple Lyapunov functions in this paper. We shall establish the sufficient condition, in terms of multiple Lyapunov functions, for the asymptotic behaviours of solutions of stochastic differential delay equations. Moreover, from them follow many effective criteria on stochastic asympt...

We consider optimal control problems for systems described by stochastic differential equations with delay. We prove two (sufficient) maximum principles for certain classes of such systems, one for ordinary stochastic delay control and one which also includes singular stochastic delay control. As an application we find explicitly the optimal consumption rate from an economic quantity described ...

Models consisting of linear, N-dimensional stochastic delay differential equations present a particular set of challenges for numerical simulation. While the user often seeks the probability density function of the solution, currently available methods rely on Monte Carlo sampling to generate sample paths, from which a density function must be estimated statistically. In the present work, we de...