Tymoshenko O.A.

The asymptotic behaviour of the solutions of stochastic differential equations

In this paper, we study the asymptotic behaviour of the solutions of some stochastic differential equations with the coefficients, dependent on time dη(t)=g(η(t))φ(t)dt + σ(η(t))θ(t)dw(t), η(0)≡b. Moreover, we define the conditions on the functions g, φ, σ, θ, under which the asymptotic behaviour of the solution η coincides with the solution μ of the differential equation dμ(t)=g(μ(t))φ(t)dt, μ(0)≡b.

The exact order of growth of stochastic differential equation solutions with time dependent coefficient of drift

We consider the behavior of solutions of stochastic differential equations with time dependent coefficient of drift and diffusion – dη(t) = g (η(t))ϕ(t)dt + + σ(η(t))θ(t)dw(t), η(0) ≡ b, where g and σ are positive continuous functions and θ and ϕ are continuous functions. In addition, we determine the conditions on g,ϕ,σ, θ functions, under which the exact order of increase η agrees with μ solution of the differential equation dμ(t) = g(μ(t))ϕ(t)dt , μ(0) ≡ b.