Dynamic risk measures with fluctuation of market volatility under Bochne-Lebesgue space

Starting from the global financial crisis to the more recent disruptions brought about by geopolitical tensions and public health crises, the volatility of risk in financial markets has increased significantly. This underscores the necessity for comprehensive risk measures capable of capturing the complexity and heightened fluctuations in market volatility. This need is crucial not only for new financial assets but also for the traditional financial market in the face of a rapidly changing financial environment and global landscape. In this paper, we consider the risk measures on a special space $L^{p(\cdot)}$, where the variable exponent $p(\cdot)$ is no longer a given real number as in the conventional risk measure space $L^{p}$, but rather a random variable reflecting potential fluctuations in volatility within financial markets. Through further development of axioms related to this class of risk measures, we also establish dual representations for them.