On strong solutions and explicit formulas for solutions of stochastic integral equations

TL;DR: In this article, a method for constructing strong solutions of one-dimensional Stochastic Differential Equations where the drift may be discontinuous and unbounded was developed. But this method is not based on a pathwise uniqueness argument.

TL;DR: In this article, the authors construct for every starting point weak solution to SDEs up to their explosion times including the following conditions: irreducibility and strong Feller property.

TL;DR: This paper presents a simple Stochastic Differential Equation with Discontinuous Drift that is applicable to Hybrid Autonomous Systems and demonstrates the power of the explicit specification of a discrete-time step.

TL;DR: In this article, the authors prove existence results for classical solutions to some general non-egenerate Cauchy problems which are natural generalizations of Isaacs equations, and extend these results by admitting local conditions for coefficients.

TL;DR: In this paper, a quasi-isometric transformation of a phase space that allows passing from a diffusion process with nonzero drift coefficient to a process without drift is presented, and strong solutions of stochastic differential equations with a "bad" drift coefficient are constructed.

TL;DR: In this paper, the authors prove the existence of a strong solution of a stochastic integral equation of the form, using the theory of differential equations of parabolic type, and prove the proof of these criteria is based on finding formulas expressing via multiple Stochastic integrals, formulas which in the case give an expression for, if is a strong solutions of the stochastically equation.