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)

Statistics

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