Risk measures and progressive enlargement of filtration: a BSDE approach

We consider dynamic risk measures induced by Backward Stochastic Differential Equations (BSDEs) in enlargement of filtration setting. On a fixed probability space, we are given a standard Brownian motion and a pair of random variables $(\tau, \zeta) \in (0,+\infty) \times E$, with $E \subset \mathbb{R}^m$, that enlarge the reference filtration, i.e., the one generated by the Brownian motion. These random variables can be interpreted financially as a default time and an associated mark. After introducing a BSDE driven by the Brownian motion and the random measure associated to $(\tau, \zeta)$, we define the dynamic risk measure $(\rho_t)_{t \in [0,T]}$, for a fixed time $T > 0$, induced by its solution. We prove that $(\rho_t)_{t \in [0,T]}$ can be decomposed in a pair of risk measures, acting before and after $\tau$ and we characterize its properties giving suitable assumptions on the driver of the BSDE. Furthermore, we prove an inequality satisfied by the penalty term associated to the robust representation of $(\rho_t)_{t \in [0,T]}$ and we discuss the dynamic entropic risk measure case, providing examples where it is possible to write explicitly its decomposition and simulate it numerically.Comment: 30 pages, 2 figure

The Recognition of Words in Pure Alexia and Hemianopic Alexia: a Neuropsychological Study of 6 Patients

During my PhD I investigated how shape and motion information are processed by the rat visual system, so as to establish how advanced is the representation of higher-order visual information in this species and, ultimately, to understand to what extent rats can present a valuable alternative to monkeys, as experimental models, in vision studies. Specifically, in my thesis work, I have investigated: 1) The possible visual strategies underlying shape recognition. 2) The ability of rat visual cortical areas to represent motion and shape information. My work contemplated two different, but complementary experimental approaches: psychophysical measurements of the rat\u2019s recognition ability and strategy, and in vivo extracellular recordings in anaesthetized animals passively exposed to various (static and moving) visual stimulation. The first approach implied training the rats to an invariant object recognition task, i.e. to tolerate different ranges of transformations in the object\u2019s appearance, and the application of an mage classification technique known as The Bubbles to reveal the visual strategy the animals were able, under different conditions of stimulus discriminability, to adopt in order to perform the task. The second approach involved electrophysiological exploration of different visual areas in the rat\u2019s cortex, in order to investigate putative functional hierarchies (or streams of processing) in the computation of motion and shape information. Results show, on one hand, that rats are able, under conditions of highly stimulus discriminability, to adopt a shape-based, view-invariant, multi-featural recognition strategy; on the other hand, the functional properties of neurons recorded from different visual areas suggest the presence of a putative shape-based, ventral-like stream of processing in the rat\u2019s visual cortex. The general purpose of my work is and has been the unveiling the neural mechanisms that make object recognition happen, with the goal of eventually 1) be able to relate my findings on rats to those on more visually-advanced species, such as human and non-human primates; and 2) collect enough biological data to support the artificial simulation of visual recognition processes, which still presents an important scientific challeng

On the penalty function and on continuity properties of risk measures

We discuss two issues about risk measures: we first point out an alternative interpretation of the penalty function in the dual representation of a risk measure; then we analyze the continuity properties of comonotone convex risk measures. In particular, due to the loss of convexity, local and global continuity are no more equivalent and many implications true for convex risk measures do not hold any more

Capital allocation rules and generalized collapse to the mean

In the context of capital allocation principles for (not necessarily coherent) risk measures, we derive - under mild conditions - some representation results as ''collapse to the mean" in a generalized sense. This approach is related to the well-known Gradient allocation and allows to extend a result of Kalkbrener (Theorem 4.3 in [27]) to a non-differentiable setting as well as to more general capital allocation rules and risk measures

The Term Structure of Sharpe Ratios and Arbitrage-Free Asset Pricing in Continuous Time

Recent empirical studies suggest a downward sloping term structure of Sharpe ratios. We present a theoretical framework in continuous time that can cope with such a non-flat forward curve of risk prices. The approach departs from an arbitrage-free and incomplete market setting when different pricing measures are possible. Involved pricing measures now depend on the time of evaluation or the maturity of payoffs. This results in a time inconsistent pricing scheme. The dynamics can be captured by a time-delayed backward stochastic Volterra integral equation, which to the best of our knowledge, has not yet been studied

Capital Allocation \ue0 La Aumann-Shapley for Non Differentiable Risk Measures

We study capital allocation rules satisfying suitable properties for convex and quasi-convex risk measures, by focusing in particular on a family of capital allocation rules based on the dual representation for risk measures and inspired by the Aumann\u2013Shapley allocation principle. These rules extend some well known methods of capital allocation for coherent and convex risk measures to the case of non-Gateauxdifferentiable risk measures. We also analyze the properties of the allocation principles here introduced and discuss their suitability in the quasi-convex context

Capital allocation for set-valued risk measures

We introduce the notion of set-valued Capital Allocation rule, and study Capital allocation principles for multivariate set-valued coherent and convex risk measures. We compare these rules with some of those mostly used for univariate (single-valued) risk measures

Fully-dynamic risk measures: horizon risk, time-consistency, and relations with BSDEs and BSVIEs

In a dynamic framework, we identify a new concept associated with the risk of assessing the financial exposure by a measure that is not adequate to the actual time horizon of the position. This will be called horizon risk. We clarify that dynamic risk measures are subject to horizon risk, so we propose to use the fully-dynamic version. To quantify horizon risk, we introduce h-longevity as an indicator. We investigate these notions together with other properties of risk measures as normalization, restriction property, and different formulations of time-consistency. We also consider these concepts for fully-dynamic risk measures generated by backward stochastic differential equations (BSDEs), backward stochastic Volterra integral equations (BSVIEs), and families of these. Within this study, we provide new results for BSVIEs such as a converse comparison theorem and the dual representation of the associated risk measures