A Note on One-dimensional Stochastic Differential Equations with Generalized Drift

Pathwise solvability of stochastic integral equations with generalized drift and non-smooth dispersion functions

“…The first claim of Theorem 4.4 is proved as in Theorem 4.48 in Engelbert and Schmidt [24]; see also Le Gall [48], Barlow and Perkins [3], Engelbert and Schmidt [23], and Blei and Engelbert [11,12]. The argument proceeds by the familiar Zvonkin [74] method of removal of drift; Stroock and Yor [64], Le Gall [48], and Engelbert and Schmidt [23] contain early usage of this technique in the context of stochastic integral equations with generalized drifts.…”

“…However, motivated by the pioneering work of Walsh [68] and Harrison and Shepp [38] on the "skew Brownian motion," several authors have studied SIEs without such a continuity assumption in quite some generality, beginning with Stroock and Yor [64], Le Gall [48], Barlow and Perkins [3], and Engelbert and Schmidt [23]. In the years since, Engelbert and Schmidt [24], Engelbert [22], Flandoli et al [32,33], Bass and Chen [4], Russo and Trutnau [60], and Blei and Engelbert [11,12] have provided deep existence and uniqueness results about such equations.…”

One-dimensional stochastic differential equations with generalized and singular drift

“…Engelbert and W. Schmidt [6] and [7] concerning equations of type (1.4) come into play. As in (1.6) and according to [5], Theorem 2.2 and Corollary 2.5 and 2.6, we always suppose ν({a}) < 1/2, a ∈ F − , which implies ν G ({a}) < 1/2, a ∈ G(F − ). On the one hand, we want to exclude the phenomenon of reflection in the points of F − , which is described by ν({a}) = 1/2.…”

“…In S. Blei and H.J. Engelbert [5] the reader can find a complete treatment of the features of Eq. (1.4) in the cases ν({x}) > 1/2 and ν({x}) = 1/2 for some x ∈ R.…”