Let $A_\pm>0$, $\beta\in(0,1)$, and let $Z^{ (\alpha)}$ be a strictly $\alpha$-stable L\'evy process with the jump measure $\nu(\mathrm{ d} z)=(C_+\mathbb{ I}_{ (0,\infty)}(z)+ C_-\mathbb{ I}_{ (-\infty,0)}(z))|z|^{ -1-\alpha}\,\mathrm{ d} z$, $\alpha\in (1,2)$, $C_\pm\geq 0$, $C_++C_->0$. The selection problem for the model stochastic differential equation $\mathrm{ d} \bar X^\varepsilon=(A_+\mathbb{ I}_{ [0,\infty)}(\bar X^\varepsilon) - A_-\mathbb{ I}_{ (-\infty,0)}(\bar X^\varepsilon))|\bar X^\varepsilon|^\beta \,\mathrm{ d} t +\varepsilon \mathrm{ d} Z^{ (\alpha)}$ states that in the small noise limit $\varepsilon\to 0$, solutions $\bar X^\varepsilon$ converge weakly to the maximal or minimal solutions of the limiting non-Lipschitzian ordinary differential equation $\mathrm{ d} \bar x=(A_+\mathbb{ I}_{ [0,\infty)}(\bar x)- A_-\mathbb{ I}_{ (\infty,0)}(\bar x))|\bar x|^\beta \,\mathrm{ d} t$ with probabilities $\bar p_\pm=\bar p_\pm(\alpha,C_+/C_-,\beta, A_+/A_-)$, see [Pilipenko and Proske, Stat. Probab. Lett., 132:62-73, 2018]. In this paper we solve the generalized selection problem for the stochastic differential equation $\mathrm{ d} X^\varepsilon=a(X^\varepsilon)\,\mathrm{ d} t+\varepsilon b(X^\varepsilon)\,\mathrm{ d} Z$ whose dynamics in the vicinity of the origin in certain sense reminds of dynamics of the model equation. In particular we show that solutions $X^\varepsilon$ also converge to the maximal or minimal solutions of the limiting irregular ordinary differential equation $\mathrm{ d} x=a(x) \,\mathrm{ d} t$ with the same model selection probabilities $\bar p_\pm$. This means that for a large class of irregular stochastic differential equations, the selection dynamics is completely determined by four local parameters of the drift and the jump measure.
Categories: math.PR