SDEs with random and irregular coefficients

TL;DR

This paper proves the unique solvability of singular stochastic differential equations with random, irregular coefficients by employing backward stochastic Kolmogorov equations and a modified Zvonkin transformation under low regularity and Malliavin differentiability assumptions.

Contribution

It introduces a novel approach to solve singular SDEs with irregular coefficients using backward stochastic PDEs and a modified Zvonkin transformation, expanding the class of solvable equations.

Findings

Established unique solvability under low regularity conditions

Developed a method combining backward stochastic PDEs with Zvonkin transformation

Extended solvability results to more irregular coefficient scenarios

Abstract

We consider It\^o uniformly nondegenerate equations with random coefficients. When the coefficients satisfy some low regularity assumptions with respect to the spatial variables and Malliavin differentiability assumptions on the sample points, the unique solvability of singular SDEs is proved by solving backward stochastic Kolmogorov equations and utilizing a modified Zvonkin type transformation.