A note on one-dimensional stochastic differential equations with generalized drift

Abstract

We consider one-dimensional stochastic differential equations with generalized drift which involve the local time L X of the solution process:where b is a measurable real function, B is a Wiener process and ν denotes a set function which is defined on the bounded Borel sets of the real line R such that it is a finite signed measure on B([−N, N ]) for every N ∈ N. This kind of equation is, in dependence of using the right, the left or the symmetric local time, usually studied under the atom condition ν({x}) < 1/2, ν({x}) > −1/2 and |ν({x})| < 1, respectively. This condition allows to reduce an equation with generalized drift to an equation without drift and to derive conditions on existence and uniqueness of solutions from results for equations without drift. The main aim of the present note is to treat the cases ν({x}) ≥ 1/2, ν({x}) ≤ −1/2 and |ν({x})| ≥ 1, respectively, for some x ∈ R, and we give a complete description of the features of equations with generalized drift and their solutions in these cases.

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]…”

“…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.…”

“…Since F has Lebesgue measure zero, the continuity properties of L X m imply the desired result. 5 Note that L X m (t, y+) = L X m (t, y), t < S X ∞ , y ∈ R, P-a.s.…”