A limit theorem for stochastic equations with the local time. (English. Ukrainian original) Zbl 0998.60058

Makhno, S. Ya.

Theory Probab. Math. Stat. 64, 123-127 (2002); translation from Teor. Jmovirn. Mat. Stat. 64, 106-109 (2001).

This paper deals with necessary and sufficient conditions for weak convergence as \(\varepsilon \to 0\) of solutions of the stochastic equation \[ \xi_{\varepsilon}(t)=x+b_{\varepsilon}L^{\xi_{\varepsilon}}(t,0)+ \int_0^t g_{\varepsilon}(\xi_{\varepsilon}(s)) ds+ \int_0^t\sigma_{\varepsilon}(\xi_{\varepsilon}(s)) dW(s), \] where \(W(s)\) is a standard Brownian motion, \(g_{\varepsilon}(\cdot)\) and \(\sigma_{\varepsilon}(\cdot)\) are non-random functions, \(L^{\xi_{\varepsilon}}(t,0)\) is a symmetric local time of the process \(\xi_{\varepsilon}\) in zero defined by Tanaka’s formula.

Reviewer: N.M.Zinchenko (Kyïv)

MSC:

60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)
60F17	Functional limit theorems; invariance principles

Keywords:

stochastic equation; local time; Tanaka’s formula; weak convergence