## Limit theorems for integral functionals of a process with instantaneous reflection.(English. Russian original)Zbl 0839.60037

Theory Probab. Math. Stat. 47, 101-105 (1993); translation from Teor. Jmovirn. Mat. Stat. 47, 99-104 (1992).
Let $$\xi (t)$$ be a solution of the stochastic differential equation $$d \xi (t) = a(\xi (t)) dt + \sigma (\xi (t)) dw(t)$$ with instantaneous reflection at the boundary $$x = 0$$, where $$a(x)$$ and $$\sigma (x)$$ satisfy a Lipschitz condition, $$a(0 +) = 0$$, $$w(t)$$ is a Wiener process. Two classes of equations are considered, namely equations of class $$K_i$$, for which $\lim_{x \to \infty} {f(x) \over \int^x_0 \bigl[ f'(v) \sigma^2 (v) \bigr]^{-1} dv} = 0, \quad 0 < f'(x) \sigma (x) \leq c,$ where $$f(x) = \int^x_0 \exp \{- 2 \int^u_0 {a(v) \over \sigma^2 (v)} dv\} du$$, and equations of class $$K_2$$, for which $$\lim_{x \to \infty} xa(x) = a_0$$, $$\lim_{x \to \infty} \sigma (x) = \sigma_0$$, $$2a_0 + \sigma^2_0 > 0$$. For both classes the limit behavior, as $$t \to \infty$$, of the functionals $$\int^t_0 g(\xi (s))ds$$ is investigated.

### MSC:

 60F17 Functional limit theorems; invariance principles 60G50 Sums of independent random variables; random walks 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)

### Keywords:

stochastic differential equation; Wiener process