  The skew Brownian motion is a strong Markov process which behaves like a
Brownian motion until hitting zero and exhibits an asymmetry at zero. We
address the following question: what is a natural counterpart of the skew
Brownian motion in the situation that the noise is a stable L\'{e}vy process
with finite mean and infinite variance. We define a skew stable L\'{e}vy
process $X$ as the limit of a sequence of stable L\'{e}vy processes which are
perturbed at zero. We point out a formula for the resolvent of $X$ and show
that $X$ is a solution to a stochastic differential equation with a local time.
Also, we provide a representation of $X$ in terms of It\^{o}`s excursion
theory.
