Under the risk-neutral measure by application of Ito:
dS^3_t = 3 \left[ (r + \sigma^2)S^3_t dt + \sigma S^3_t dW_t \right]
The risk-neutral drift is not the risk-free rate and hence $S_t^3 \; \forall t$ cannot be the price of a claim or any other tradable asset.
So basically along the same lines as your proof, but without calculating expectations etc. ...
The price of most (not all) bonds is quoted as a percentage of face value (par).
For most amortizing bonds that have already amortized, the percentage is of the face value now, after amortizations, not the initial face value.
(Bonds that are quoted / trading dirty / flat / on proceeds are different and I won't go there.)
Suppose, for concreteness, than some ...
It sounds to me that they just mean that each bound can be seen as a function of the parameter(s) in the parametrization and this function is Lipschitz continuous.
An example: Consider the XY-plane. Let $Y(x)$ be a function of $x$. This function can be seen as describing the upper bound of the area below the graph.
This function can then have the Lipschitz ...