Before revision:  Furthermore, the obtained equivalent probability can be chosen as close to the real-world probability measure as we want (but might not be equal) {\it   .
After revision:  Furthermore, the obtained equivalent probability can be chosen as close to the real-world probability measure as we want (but might not be equal) . Without changing the underlying probability measure and under mild assumptions, our second contribution proves that the NUPBR is equivalent to the "{\it .