Extend the Convex Envelope Algorithm to multidimensional superhedging

Develop an extension of the Convex Envelope Algorithm of Degano–Ferrando–González (2018) for computing the dynamic superhedging bounds \overline{U}_i F in trajectory-based market models to dimensions d ≥ 2, so that portfolios trading multiple assets can be used to compute the quantities \overline{U}_i F (and thus \overline{V}_i F and superhedging prices) on multidimensional conditional trajectory sets \mathcal{X}_{(X,i)}. The extension should retain the backward dynamic programming structure and deliver an algorithmic procedure analogous to the one-dimensional case that is implementable for d > 1.

Background

The paper constructs trajectory sets in a multidimensional setting (notably d=2 for two traded assets), but computes superhedging prices using a one-dimensional portfolio due to the available algorithmic machinery. In the one-dimensional case, the dynamic bounds \overline{U}_i F are computed via a convex envelope algorithm (as in Degano et al., 2018), which enables backward dynamic programming to obtain superhedging prices.

A full generalization of this computational procedure to higher dimensions would allow portfolios that trade multiple assets simultaneously to superhedge payoffs, aligning the computation with the multidimensional trajectory models the paper develops. The authors explicitly note that the key obstruction is extending the convex envelope algorithm to d > 1, and identify this as the reason they restrict pricing computations to one-dimensional portfolios despite working with multidimensional trajectories.