Explicit form of global solution to stochastic logistic differential equation and related topics

This paper presents the explicit form of positive global solution to stochastically perturbed nonautonomous logistic equation dN(t) = N(t) [ (a(t)− b(t)N(t))dt+ α(t)dw(t) + ∫ R γ(t, z)ν̃(dt, dz) ] , N(0) = N0, where w(t) is the standard one-dimensional Wiener process, ν̃(t, A) = ν(t, A)− tΠ(A), ν(t, A) is the Poisson measure, which is independent on w(t), E[ν(t, A)] = tΠ(A), Π(A) is a finite measure on the Borel sets in R. If coefficients a(t), b(t), α(t) and γ(t, z) are continuous on t, T -periodic on t functions, a(t) > 0, b(t) > 0 and ∫ T 0 [ a(s)− α(s)− ∫ R γ(s, z) 1 + γ(s, z) Π(dz) ] ds > 0, then there exists unique, positive T -periodic solution to equation for E[1/N(t)].


Introduction
The construction of the logistic model and its properties are presented in M. Iannelli and A. Pugliese [1].A deterministic nonautonomous logistic equation has a form and models the number N of a single species whose members compete among themselves for a limit amount of food and living space.Here a(t) is the rate of growth and a(t)/b(t) is the carrying capacity at time t.In the paper by D. Jiang and N. Shi [2] it is considered the nonautonomous stochastic logistic differential equation 59 where w(t) is the standard one-dimensional Wiener process, N 0 > 0 is a random variable independent on w(t), a(t), b(t) and α(t) are bounded, continuous functions.The authors prove, that if a(t) > 0, b(t) > 0, then there exists a unique continuous, positive global solution N (t) to equation (1).
where N (t) is the solution to the equation (1) for any initial value In this paper, we consider the stochastic nonautonomous logistic differential equation of the form where w(t) is the standard one-dimensional Wiener process, ] ds > 0, we will show, that the equation for function E[1/N (t)] has a unique positive T -periodic solution.
Mainly we use the notations and approaches proposed in D. Jiang and N. Shi [2].If coefficient γ(t, z) ≡ 0, then our results are consistent with the corresponding results in D. Jiang and N. Shi [2].
The rest of this paper is organized as follows.In Section 2, we obtain the explicit representation of the unique global positive solution to equation (2).In Section 3, we derive the explicit form of unique positive T -periodic solution to the equation for E[1/N (t)] in the case of periodical coefficients, and in Section 4, we consider the stochastic nonautonomous logistic differential equation of the form where θ > 0 is an odd integer.

Explicit form of global solution
Let (Ω, F, P ) be a probability space and w(t), t ≥ 0 is a standard one-dimensional Wiener process on (Ω, F, P ), N 0 > 0 is a random variable on (Ω, F, P ), which is independent on w(t), and is a finite measure on the Borel sets in R. On the probability space (Ω, F, P ) we consider an increasing, right continuous family of complete sub-σ-algebras {F t } t≥0 , where The main result of this section is following

Theorem 1
Let a(t) > 0, b(t) > 0 and α(t) be a bounded continuous functions defined on [0, +∞).Assume that Π(R) < ∞ and γ(t, z) is continuous on t function and Then there exists a unique positive solution N (t) to equation (2) for any initial value N (0) = N 0 > 0, which is global and has a representation where Proof.The coefficients of the equation (2) are local Lipschitz continuous.Therefore for any initial random value N 0 > 0, which is independent on w(t) and ν(t, A), there exists a unique local solution N (t) on [0, τ e ), where sup t<τe |N (t)| = +∞ (cf.Theorem 6, p.246, [3]).We will derive the explicit form of this solution and we will see, that this solution is global. Let where Using the Ito formula, we derive the stochastic differential equation for the process x(t): Let N (t) = 1/x(t), then N (t) > 0 and N (t) is stochastically continuous process with trajectories, which are right continuous and have a left limit at every t almost surely.Under conditions of the theorem, N (t) will not explode in a finite time.

Remark 1
If b(t) < 0, then equation (2) has only the local solution } ds , 0 ≤ t < τ e , where explosion time defined by

Periodic solution to equation for E[1/N (t)]
From the Lemma (p.465, [3]) we have the following result. where Let us consider equation (2) with T -periodic on t coefficients.

Theorem 2
Let coefficients a(t), b(t), α(t) and γ(t, z) be continuous on t and T -periodic on t functions, a(t) > 0, b(t) > 0 and Then the equation where N (t) is a solution to equation (2), has a unique positive T -periodic solution and for solution N (t) to equation (2) with any initial value Proof.From (3) we have By Lemma 1 we obtain Therefore, using the independence of w(t), ν(t, A) and N 0 , we derive ] dτ } ds.
Let N (t) be a solution to equation (11), then by Ito formula applied to stochastic process N θ (t) we derive For stochastic process M (t) = N θ (t) we have an (2)-type equation It is easy to see that under conditions of Theorem 1 on coefficients a(t), b(t), α(t), γ(t, z) the same conditions are fulfilled for coefficients â(t), b(t), α(t), γ(t, z).Therefore we obtain the following results.