  Since the quasiconvex risk measures is a bigger class than the well known
convex risk measures, the study of quasiconvex risk measures makes sense
especially in the financial markets with volatility. In this paper, we will
study the quasiconvex risk measures defined on a special space $L^{p(\cdot)}$
where the variable exponent $p(\cdot)$ is no longer a given real number like
the space $L^{p}$, but a random variable, which reflects the possible
volatility of the financial markets. The dual representation for this
quasiconvex risk measures will also provided.
