Self-similar martingales derived from Root embedding

Abstract

Given a family $({\mu }_{\lambda },\lambda \ge 0)$ of integrable mean-zero probability measures such that, for every $\lambda \ge 0$, ${\mu }_{\lambda }$ is the image of ${\mu }_1$ under the homothety $y⟼\sqrt{\lambda }y$, we provide a necessary and sufficient condition on ${\mu }_1$ under which the Root embedding algorithm yields a self-similar martingale with one-dimensional marginals $({\mu }_{\lambda },\lambda \ge 0)$. Precisely, if ${\tau }_{\lambda }$ and $R_{\lambda }$ denote the Root solution to the Skorokhod embedding problem (SEP) and the Root regular barrier for ${\mu }_{\lambda }$ respectively, then this condition is equivalent to the property that $(R_{\lambda },\lambda \ge 0)$ is non-increasing in the sense of inclusion, which in turn is equivalent to the assertion that $({\tau }_{\lambda },\lambda \ge 0)$ is non-decreasing a.s. We show that there are many examples for which this result applies and we provide some numerical simulations to illustrate the monotonicity property of regular barriers $(R_{\lambda },\lambda \ge 0)$ in this case.

arXiv comment13 pages, 2 figures

References in corpus (1)

• On the Root solution to the Skorokhod embedding problem given full marginals