On the Law of the Minimum in a Class of Unidimensional SDEs

Abstract

We prove that the law of the minimum $${m := {\rm min}_{t\in[0,1]}\xi(t)}$$ of the solution $${\xi}$$ to a one-dimensional stochastic differential equation with good nonlinearity has continuous density with respect to the Lebesgue measure. As a byproduct of the procedure, we show that the sets $${\{x \in C([0,1]) : {\rm inf} x \geq r\}}$$ have finite perimeter with respect to the law $${\nu}$$ of the solution $${\xi({\cdot})}$$ in $${L^{2}(0,2)}$$ .