Koksma-Hlawka's Inequality

Page 19 of Niederreiter, his Theorem 2.11 states Koksma-Hlawka's inequality.

Theorem 4 (Koksma-Hlawka's Inequality)   If h has bounded (total) variation on the unit hyper cube in s dimensions, then for any points $x_1,...,x_N$, in said unit hypercube, we have

$$\vert\frac{1}{N} \sum_{i=1}^{N} h(x_i) - \int h(u) du \le V(h) D_{N}^{*}(x_1,...,x_N)$$ (2.13)

where V(h) is the Hardy-Krause Variation in s dimensions of the function h on the unit hypercube in s dimensions. Here the integral is over the unit hypercube in s dimensions.

The Koksma-Hlawka inequality extends the Koksma inequality to s dimensions. This is what applies to many practical problems in finance and insurance. However, we may not have that $V(f)$ is bounded, especially if the payoff is unbounded.