Variably skewed Brownian motion

Abstract

Given a standard Brownian motion B, we show that the equation X [subscript] t = x [subscript] 0 + B [subscript] t + [beta](L [to the power of X] [subscript] t ); t [is greater than or equal to] 0 ; has a unique strong solution X. Here L [to the power of X] is the symmetric local time of X at 0, and [beta] is a given differentiable function with [beta](0) = 0, -1 < [beta prime](.) < 1. (For linear [beta](.), the solution is the familiar skew Brownian motion).