On reflected solutions of stochastic equations driven by symmetric stable processes

Preprint series: 04-03 , Reports on Stochastics and Statistics

MSC:
60G44 Martingales with continuous parameter

Abstract: We study the one-dimensional stochastic differential equation (SDE) of the
form $X_{t}=x_{0}+\int_{0}^{t}b(X_{s-})dM_{s}+K_{t},\mbox{ }t\geq 0$, where $% b:[0,\infty )\rightarrow \mathbb{R}$ is a Borel measurable function, $% x_{0}\in \lbrack 0,\infty )$ is an arbitrary initial value, the process $X$
is nonnegative, $K$ is a right-continuous increasing process with $K_{0}=0$
and $M$ is a symmetric stable process of arbitrary stability index $0<\alpha \leq 2$ with $M_{0}=0$. The process $K$ satisfies the condition $% \int_{0}^{\infty }\mathbf{1}_{\{X_{t}\neq 0\}}dK_{t}=0$, that means that $K$
is a reflecting process for the solution $X$. For every $x_{0}\in \lbrack 0,\infty )$ we state conditions on $b$ for the existence of a reflected
solution $X$ with $X_{0}=x_{0}$. In particular, our results generalize the
results of W. M. Schmidt who considered the given SDE in the case of the
Brownian motion ($\alpha =2$).

Keywords: Symmetric stable processes, Skorohod reflection problem, integral functionals, stochastic stable integrals, existence of solutions