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

**MSC:**- 60H10 Stochastic ordinary differential equations [See also 34F05]
- 60J60 Diffusion processes [See also 58J65]
- 60J65 Brownian motion [See also 58J65]
- 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*

**Upload:** 2004-05-04

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