A limit theorem for singular stochastic differential equations

Abstract

We study the weak limits of solutions to SDEs \[dX_n(t)=a_n\bigl(X_n(t)\bigr)\,dt+dW(t),\] where the sequence $\{a_n\}$ converges in some sense to $(c_{-}1{\mathrm{l}}_{x<0}+c_{+}1{\mathrm{l}}_{x>0})/x+\gamma {\delta }_0$. Here ${\delta }_0$ is the Dirac delta function concentrated at zero. A limit of $\{X_n\}$ may be a Bessel process, a skew Bessel process, or a mixture of Bessel processes.

arXiv commentPublished at http://dx.doi.org/10.15559/16-VMSTA63 in the Modern Stochastics: Theory and Applications (https://www.i-journals.org/vtxpp/VMSTA) by VTeX (http://www.vtex.lt/)