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

Minbaeva, G. U.

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.

Reviewer: O.K.Zakusilo (Kiev)

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