Kusuoka (2001) has obtained explicit representation theorems for comonotone risk measures and, more generally, for law invariant risk measures. These theorems pertain, like most of the previous literature, to the case of scalar-valued risks. Jouini, Meddeb, and Touzi (2004) and Burgert and...

S. Kusuoka [K 01, Theorem 4] gave an interesting dual characterizationof law invariant coherent risk measures, satisfying the Fatou property.The latter property was introduced by F. Delbaen [D 02]. In thepresent note we extend Kusuoka's characterization in two directions, thefirst one being...

We prove a general version of the super-replication theorem, which applies to Kabanov’s model of foreign exchange markets under proportional transaction costs. The market is described by a matrix-valued càdlàg bid-ask process $$(\Pi_t)_{t\in [0,T]}$$ evolving in continuous time. We propose a...

Every submartingale S of class D has a unique Doob–Meyer decomposition S=M+A, where M is a martingale and A is a predictable increasing process starting at 0.

Let $X$ be an ${\Bbb R}^d$-valued special semimartingale on a probability space $(\Omega , {\cal F} , ({\cal F} _t)_{0 \leq t \leq T} ,P)$ with canonical decomposition $X=X_0+M+A$. Denote by $G_T(\Theta )$ the space of all random variables $(\theta \cdot X)_T$, where $\theta $ is a predictable...

For portfolio optimisation under proportional transaction costs, we provide a duality theory for general cadlag price processes. In this setting, we prove the existence of a dual optimiser as well as a shadow price process in a generalised sense. This shadow price is defined via a "sandwiched"...

This article discusses static hedges for installment options, which are finding broad application in cases where the option‐buyer may reduce up‐front premium costs via early termination of an option. An installment option is a European option in which the premium, instead of being paid up...