Stability of equilibriums of stochastically perturbed delay differential neoclassical growth model

The known nonlinear delay differential neoclassical growth model is considered. It is assumed that this model is influenced by stochastic perturbations of the white noise type and these perturbations are directly proportional to the deviation of the system state from the zero or a positive equilibrium. Sufficient conditions for stability in probability of the positive equilibrium and for exponential mean square stability of the zero equilibrium are obtained. Numerical calculations and figures illustrate the obtained stability regions and behavior of stable and unstable solutions of the considered model. The proposed investigation procedure can be applied for arbitrary nonlinear stochastic delay differential equations with the order of nonlinearity higher than one.


Introduction.
The well known delay differential neoclassical growth model is often enough discussed in mathematical economics (see, for instance, [2,3,4,6,7]). Here it is supposed that this model is influenced by stochastic perturbations of the white noise type that are directly proportional to the deviation of the system state from the one of system equilibrium. It is shown how the stability theory for stochastic delay differential equations [10] can be applied to stability investigation of equilibriums of nonlinear models under stochastic perturbations. In particular, if γ = 1 then the equation (1.1) is well known Nicholson's blowflies equation [8] and (1.2) by the condition a > c has the explicit positive solution 1566

LEONID SHAIKHET
Below we will investigate stability of both equilibriums of the equation (1.1) under stochastic perturbations of the white noise type that are directly proportional to the deviation of the solution x(t) from the equilibrium point x * . By that the equation (1.1) takes the form of stochastic differential equation [5,10] where σ is a constant and w(t) is the standard Wiener process. In the case γ = 1 this problem was in detail investigated in [1,9,10], where, in particular, sufficient conditions for stability in probability both for the zero and the positive equilibriums were obtained. Below the case γ ≥ 1 is considered, sufficient conditions for stability in probability of the positive equilibrium and sufficient conditions for exponential mean square stability of the zero equilibrium are obtained.
Neglecting o(y), as a result we obtain the linear part of the equation (1.4) in the formż (1.6)

Auxiliary definitions and statements.
) .  mean square stable if for each ε > 0 there exists a δ > 0 such that E|y(t, ϕ)| 2 < ε, t ≥ 0, provided that ∥ϕ∥ 2 = sup s≤0 E|ϕ(s)| 2 < δ; asymptotically mean square stable if it is mean square stable and for each initial function ϕ the solution y(t) of the equation (1.4) satisfies the condition lim t→∞ E|y(t)| 2 = 0; exponentially mean square stable if it is mean square stable and there exists λ > 0 such that for each initial function ϕ there exists C > 0 (which may depend on ϕ) such that E|y(t, ϕ)| 2 ≤ Ce −λt for t > 0.

Lemma 2.3. [9, 10]
A necessary and sufficient condition for asymptotic mean square stability of the zero solution of the linear Ito stochastic differential equatioṅ (2.5) Let L be the generator [5,10] of the stochastic differential equation (1.4), y(t) be the solution of (1.4) in the time moment t, y t be the trajectory of the solution of (1.4) until the time moment t, E be the expectation.
Theorem 2.1. [10] Let there exist a functional V (t, ϕ) and λ > 0 such that for a solution y(t) of the equation (1.4) the following inequalities hold:
Note that via (1.5) the first inequality (3.1) is equivalent to the condition 1+µ > 0 that follows from the first inequality (2.4).
The red part of the stability region is placed between two straight lines: 1) c = K 1 a and 2) c = K 0 a, where K 0 is defined in (2.1) and Via Lemma 3.2 the region of exponential mean square stability (yellow) of the equation (3.3) zero equilibrium is placed above the straight line 3) c = K 0 a + p.
In      One can see that in contrast to stability in probability all trajectories converge to the exponentially mean square stable zero equilibrium for each initial function, even if the initial function is placed far enough from the zero (b 0 = 100).

Conclusion.
In the present paper, the nonlinear delay differential neoclassical growth model under stochastic perturbations is analyzed. Stability conditions for the zero and positive equilibriums of the considered model are obtained. Numerical simulations show a principal difference in solution behavior in the case of stable and unstable equilibriums and in the case of stable equilibriums for different types of stability: stability in probability and exponential mean square stability. The proposed research method can be applied to investigation of other stochastic nonlinear models with the order of nonlinearity higher than one.