Schur–Weyl duality and the product of randomly-rotated symmetries by a unitary Brownian motion

In this paper, we introduce and study a unitary matrix-valued process which is closely related to the Hermitian matrix-Jacobi process. It is precisely defined as the product of a deterministic self-adjoint symmetry and a randomly-rotated one by a unitary Brownian motion. Using stochastic calculus and the action of the symmetric group on tensor powers, we derive an ordinary differential equation for the moments of its fixed-time marginals. Next, we derive an expression of these moments which involves a unitary bridge between our unitary process and another independent unitary Brownian motion. This bridge motivates and allows to write a second direct proof of the obtained moment expression.

Protection of historic stained glass

The process of deterioration of a stained glass window, more precisely of its component glass, lead and paint pigments, has in recent decades been researched thoroughly and is now well understood. The need for the protection in particular of the potash window glasses of the middle ages therefore is widely accepted but not the means of how to achieve this aim. Early attempts such as coatings applied on the glass or the sandwich process, known as Jacobi–Process, have failed for a variety of reasons and with disastrous consequences. Short of removing the windows to safe storage, the only response currently available is the creation of a museum like condition on site in the form of the introduction of an isothermal glazing system. Despite its unquestioned merits, the isothermal glazing system is still in dispute mainly because of its interference with a given historic setting. A variety of designs for the isothermal glazing will be discussed, all aimed at minimising their physical and visual impact on the building, making a strong case in favour of this important protective measure.

QUANTO PRICING IN STOCHASTIC CORRELATION MODELS

Correlation plays an important role in pricing multi-asset options. In this work we incorporate stochastic correlation into pricing quanto options which is one special and important type of multi-asset option. Motivated by the market observations that the correlations between financial quantities behave like a stochastic process, instead of using a constant correlation, we allow the asset price process and the exchange rate process to be stochastically correlated with a parameter which is driven either by an Ornstein–Uhlenbeck process or a bounded Jacobi process. We derive an exact quanto option pricing formula in the stochastic correlation model of the Ornstein–Uhlenbeck process and a highly accurate approximated pricing formula in the stochastic correlation model of the bounded Jacobi process, where correlation risk has been hedged. The comparison between prices using our pricing formula and the Monte-Carlo method are provided.

Inverse of the flow and moments of the free Jacobi process associated with one projection

This paper is a companion to a series of papers devoted to the study of the spectral distribution of the free Jacobi process associated with a single projection. Actually, we note that the flow derived in [N. Demni and T. Hmidi, Spectral distribution of the free Jacobi process associated with one projection, Colloq. Math. 137(2) (2014) 271–296] solves a radial Löwner equation and as such, the general theory of Löwner equations implies that it is univalent in some connected region in the open unit disc. We also prove that its inverse defines the Aleksandrov–Clark measure at [Formula: see text] of some Herglotz function which is absolutely-continuous with an essentially bounded density. As a by-product, we deduce that [Formula: see text] does not belong to the continuous singular spectrum of the unitary operator whose spectral dynamics are governed by the flow. Moreover, we use a previous result due to the first author in order to derive an explicit, yet complicated, expression of the moments of both the unitary and the free Jacobi processes. The paper is closed with some remarks on the boundary behavior of the flow’s inverse.