1978 Volume 14 Issue 6 Pages 685-692
In this paper, we are concerned with the existence of a periodic solution of a weakly nonlinear stochastic system with periodic coefficients (Intensity of nonlinearity is expressed by a parameter ε>0). Here the term “periodic” means that every finite dimensional distribution function of the solution process is periodic in t.
We prove the existence of a periodic solution along two different approaches under the condition that the characteristic exponents of the corresponding linear system have negative real parts.
At first, we give a proof by actually constructing a periodic solution, where the successive approximation procedure is adopted. Techniques used there are similar to the ones of A. Ja. Dorogovcev 2) or of H. Bunke 3), but some properties of a Wiener process are incorporated in the proof as essential parts. At the same time, we give a sufficient condition in terms of the range of a parameter ε>0, under which there exists a periodic solution.
Next, we construct a periodic solution according to I. Vrkoc 4), but in this time, we exclusively deal with distribution functions of the process. First, we show that solutions of the original system are uniformly exponentially stable in the quadratic mean for sufficiently small ε>0. Under this stability condition, it is shown that distribution functions of the process which are sampled at every T-time, where T is a period of coefficients, converge. Let the limit distribution be F*. Then the distribution after one period of the solution process with the initial distribution F* is also given by F*. In this way, we can prove the existence of a periodic solution again.