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

• Dmytro Borysenko Department of Integral and Differential Equations, Taras Shevchenko National University of Kyiv, Ukraine
• Oleksandr Borysenko Department of Probability Theory, Statistics and Actuarial Mathematics, Taras Shevchenko National University of Kyiv
Keywords: Nonautonomous Stochastic Logistic Differential Equation, Explicit Form, Global Solution, Periodical Solution

### Abstract

This paper presents the explicit form of positive global solution to stochastically perturbed nonautonomous logistic equation $$dN(t)=N(t)\left[(a(t)-b(t)N(t))dt+\alpha(t)dw(t)+\int_{\mathbb{R}}\gamma(t,z)\tilde\nu(dt,dz)\right],\ N(0)=N_0,$$ where $w(t)$ is the standard one-dimensional Wiener process, $\tilde\nu(t,A)=\nu(t,A)-t\Pi(A)$, $\nu(t,A)$ is the Poisson measure, which is independent on $w(t)$, $E[\nu(t,A)]=t\Pi(A)$, $\Pi(A)$ is a finite measure on the Borel sets in $\mathbb{R}$. If coefficients $a(t), b(t), \alpha(t)$ and $\gamma(t,z)$ are continuous on $t$, $T$-periodic on $t$ functions, $a(t)>0, b(t)>0$ and $$\int_{0}^{T}\left[a(s)-\alpha^2(s)-\int_{\mathbb{R}}\frac{\gamma^2(s,z)}{1+\gamma(s,z)}\Pi(dz)\right]ds>0,$$ then there exists unique, positive $T$-periodic solution to equation for $E[1/N(t)]$.

### Author Biography

Oleksandr Borysenko, Department of Probability Theory, Statistics and Actuarial Mathematics, Taras Shevchenko National University of Kyiv

### References

M.Iannelli, and A. Pugliese, An Introduction to Mathematical Population Dynamics, Springer, 2014.

D. Jiang, and N. Shi, A note on nonautonomous logistic equation with random perturbation, J. Math. Anal. Appl., 303, pp. 164 – 172, 2005.

I.I Gikhman, and A.V. Skorokhod, Stochastic Differential Equations and its Applications, Kiev, Naukova Dumka, 1982.(In Russian)

G. Sansone, Ordinary Differential Equations, v.1, Moscow, Inostrannaya Literatura, 1953.(In Russian)

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Published
2017-03-04
How to Cite
Borysenko, D., & Borysenko, O. (2017). Explicit form of global solution to stochastic logistic differential equation and related topics. Statistics, Optimization & Information Computing, 5(1), 58-64. https://doi.org/10.19139/soic.v5i1.262
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Section
Research Articles