Causal optimal transport and its links to enlargement of filtrations and continuous-time stochastic optimization

The martingale part in the semimartingale decomposition of a Brownian motion with respect to an enlargement of its filtration, is an anticipative mapping of the given Brownian motion. In analogy to optimal transport theory, we define causal transport plans in the context of enlargement of filtrations, as the Kantorovich counterparts of the aforementioned non-adapted mappings. We provide a necessary and sufficient condition for a Brownian motion to remain a semimartingale in an enlarged filtration, in terms of certain minimization problems over sets of causal transport plans. The latter are also used in order to give robust transport-based estimates for the value of having additional information, as well as model sensitivity with respect to the reference measure, for the classical stochastic optimization problems of utility maximization and optimal stopping

A trajectorial interpretation of Doob's martingale inequalities

We present a unified approach to Doob’s Lp maximal inequalities for 1 ≤ p < 1. The novelty of our method is that these martingale inequalities are obtained as consequences of elementary deterministic counterparts. The latter have a natural interpretation in terms of robust hedging. Moreover our deterministic inequalities lead to new versions of Doob’s maximal inequalities. These are best possible in the sense that equality is attained by properly chosen martingales

Short note on inf-convolution preserving the Fatou property

Monetary utility functions, Fatou property, Fenchel–Legendre transform, Convolution, D81,

A trajectorial interpretation of Doob's martingale inequalities

We present a unified approach to Doob's $L^p$ maximal inequalities for $1\leq p

Risk assessment for uncertain cash flows: model ambiguity, discounting ambiguity, and the role of bubbles

We study the risk assessment of uncertain cash flows in terms of dynamic convex risk measures for processes as introduced in Cheridito et al. (Electron. J. Probab. 11(3):57–106, 2006). These risk measures take into account not only the amounts but also the timing of a cash flow. We discuss their robust representation in terms of suitably penalised probability measures on the optional σ-field. This yields an explicit analysis both of model and discounting ambiguity. We focus on supermartingale criteria for time consistency. In particular, we show how “bubbles” may appear in the dynamic penalisation, and how they cause a breakdown of asymptotic safety of the risk assessment procedur