Geometric Martingale Benamou-Brenier transport and geometric Bass martingales
arxiv(2024)
Abstract
We introduce and study geometric Bass martingales. Bass martingales were
introduced in \cite{Ba83} and studied recently in a series of works, including
\cite{BaBeHuKa20,BaBeScTs23}, where they appear as solutions to the martingale
version of the Benamou-Brenier optimal transport formulation. These arithmetic,
as well as our novel geometric, Bass martingales are continuous martingale on
$[0,1]$ with prescribed initial and terminal distributions. An arithmetic Bass
martingale is the one closest to Brownian motion: its quadratic variation is as
close as possible to being linear in the averaged $L^2$ sense. Its geometric
counterpart we develop here, is the one closest to a geometric Brownian motion:
the quadratic variation of its logarithm is as close as possible to being
linear. By analogy between Bachelier and Black-Scholes models in mathematical
finance, the newly obtained geometric Bass martingales} have the potential to
be of more practical importance in a number of applications.
Our main contribution is to exhibit an explicit bijection between geometric
Bass martingales and their arithmetic counterparts. This allows us, in
particular, to translate fine properties of the latter into the new geometric
setting. We obtain an explicit representation for a geometric Bass martingale
for given initial and terminal marginals, we characterise it as a solution to
an SDE, and we show that geometric Brownian motion is the only process which is
both an arithmetic and a geometric Bass martingale. Finally, we deduce a dual
formulation for our geometric martingale Benamou-Brenier problem. Our main
proof is probabilistic in nature and uses a suitable change of measure, but we
also provide PDE arguments relying on the dual formulation of the problem,
which offer a rigorous proof under suitable regularity assumptions.
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