Quasi-copulas with a given sub-diagonal section

As is well known, the Fréchet–Hoeffding bounds are the best possible for both copulas and quasi-copulas: for every (quasi-) copula QQ, max{x+y−1,0}≤Q(x,y)≤min{x,y}max{x+y−1,0}≤Q(x,y)≤min{x,y} for all x,y∈[0,1]x,y∈[0,1]. Sharper bounds hold when the (quasi-)copulas take prescribed values, e.g., along their diagonal or horizontal resp. vertical sections. Here we pursue two goals: first, we investigate construction methods for (quasi-)copulas with a given sub-diagonal section, i.e., with prescribed values along the straight line segment joining the points (x0,0)(x0,0) and (1,1−x0)(1,1−x0) for x0∈]0,1[. Then, we determine the best-possible bounds for sets of quasi-copulas with a given sub-diagonal section.