Optimal embeddings by unbiased shifts of Brownian motion

TL;DR

This paper demonstrates that a known solution to the unbiased embedding problem of Brownian motion minimizes the expected value of all nonnegative, concave functions of the shift time, across all such solutions.

Contribution

It proves that the existing unbiased embedding solution minimizes expected concave functionals of the shift time for all nonnegative, concave functions.

Findings

The solution minimizes ${ m E} \psi(T)$ for all nonnegative, concave functions $\psi$.

A new discrete concavity inequality is established, which may have independent applications.

The result extends the optimality properties of the embedding solution to a broad class of functionals.

Abstract

An unbiased shift of the two-sided Brownian motion $(B_t \colon t\in{\mathbb R})$ is a random time $T$ such that $(B_{T+t} \colon t\in{\mathbb R})$ is still a two-sided Brownian motion. Given a pair $\mu, \nu$ of orthogonal probability measures, an unbiased shift $T$ solves the embedding problem, if $B_0\sim\mu$ implies $B_{T}\sim\nu$. A solution to this problem was given by Last et al. (2014), based on earlier work of Bertoin and Le Jan (1992), and Holroyd and Liggett (2001). In this note we show that this solution minimises ${\mathbb E} \psi(T)$ over all nonnegative unbiased solutions $T$, simultaneously for all nonnegative, concave functions $\psi$. Our proof is based on a discrete concavity inequality that may be of independent interest.