Generalized selection problem with Lévy noise

2020/04/11 by Pavlyukevich, Ilya, Pilipenko, Andrey
#34A12 #34E10 #34F05 (Secondary) #60G51 #60H10 (Primary) 60F05 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2004.05421

Abstract

Let A_±>0, β∈(0,1), and let Z^(α) be a strictly α-stable Lévy process with the jump measure ν(d z)=(C₊\mathbbI_(0,∞)(z)+ C_-\mathbbI_(-∞,0)(z))|z|^-1-α d z, α∈ (1,2), C_±≥ 0, C₊+C_->0. The selection problem for the model stochastic differential equation d X^ε=(A₊\mathbbI_[0,∞)( X^ε) - A_-\mathbbI_(-∞,0)( X^ε))| X^ε|^β d t +ε d Z^(α) states that in the small noise limit ε→ 0, solutions X^ε converge weakly to the maximal or minimal solutions of the limiting non-Lipschitzian ordinary differential equation d x=(A₊\mathbbI_[0,∞)( x)- A_-\mathbbI_(∞,0)( x))| x|^β d t with probabilities p_±= p_±(α,C₊/C_-,β, 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 d X^ε=a(X^ε) d t+ε b(X^ε) 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^ε also converge to the maximal or minimal solutions of the limiting irregular ordinary differential equation d x=a(x) d t with the same model selection probabilities p_±. 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.

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