Stefan Blei | Searcharxiv

arXiv2012

One-dimensional Stochastic Differential Equations with Generalized and Singular Drift

Introducing certain singularities, we generalize the class of one-dimensional stochastic differential equations with so-called generalized drift. Equations with generalized drift, well-known in the literature, possess a drift that is described by the semimartingale local time of the unknown process integrated with respect to a locally finite signed measure ν. The generalization which we deal with can be interpreted as allowing more general set functions ν, for example signed measures which are only σ-finite. However, we use a different approach to describe the singular drift. For the considered class of one-dimensional stochastic differential equations, we derive necessary and sufficient conditions for existence and uniqueness in law of solutions.

math.PR↗

arXiv2012

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

We consider one-dimensional stochastic differential equations with generalized drift which involve the local time $L^X$ of the solution process: X_t = X_0 + \int_0^t b(X_s) dB_s + \int_\mathbb{R} L^X(t,y) ν(dy), 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 $\mathbb{R}$ such that it is a finite signed measure on $\mathscr{B}([-N,N])$ for every $N \in \mathbb{N}$. 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 $ν({x}) \geq 1/2$, $ν({x}) \leq -1/2$ and $|ν({x})| \geq 1$, respectively, for some $x \in \mathbb{R}$, and we give a complete description of the features of equations with generalized drift and their solutions in these cases.

math.PR↗