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) {\it " existence of the optimal portfolio. This constitutes an alternative to the first contribution, if one insists on working under the real-world probability. These two contributions lead naturally to new types of viability that we call weak and local viabilities .