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

Bounding inferences for large-scale continuous-time Markov chains : a new approach based on lumping and imprecise Markov chains

If the state space of a homogeneous continuous-time Markov chain is too large, making inferences becomes computationally infeasible. Fortunately, the state space of such a chain is usually too detailed for the inferences we are interested in, in the sense that a less detailed—smaller—state space suffices to unambiguously formalise the inference. However, in general this so-called lumped state space inhibits computing exact inferences because the corresponding dynamics are unknown and/or intractable to obtain. We address this issue by considering an imprecise continuous-time Markov chain. In this way, we are able to provide guaranteed lower and upper bounds for the inferences of interest, without suffering from the curse of dimensionality

PRICE-CONDITIONAL TECHNOLOGY

Economics theorists for years have considered the possibility that the direction of technical change is altered by changes in relative prices. Prices also have been identified as one of the determinants of technical change through innovation. This article extends the theory of the firm to cover situations in which the firm’s' technology set is conditional on expected prices. The basic idea is to distinguish between “"market prices,"” or the prices that guide the firm’s choices subject to the technology that is in place, and “"normal prices,"” the prices conditioning the choice of technology. A “"generalized”" price effect is obtained that included the traditional price effect as well as the technical change effect of price changes, and an example is presented.Demand and Price Analysis,

Martingale Problem under Nonlinear Expectations

We formulate and solve the martingale problem in a nonlinear expectation space. Unlike the classical work of Stroock and Varadhan (1969) where the linear operator in the associated PDE is naturally defined from the corresponding diffusion process, the main difficulty in the nonlinear setting is to identify an appropriate class of nonlinear operators for the associated fully nonlinear PDEs. Based on the analysis of the martingale problem, we introduce the notion of weak solution for stochastic differential equations under nonlinear expectations and obtain an existence theorem under the H\"older continuity condition of the coefficients. The approach to establish the existence of weak solutions generalizes the classical Girsanov transformation method in that it no longer requires the two (probability) measures to be absolutely continuous.Comment: The new version simplifies some proofs for the main theorems and generalizes some result

Approximating multivariate distributions with vines

In a series of papers, Bedford and Cooke used vine (or pair-copulae) as a graphical tool for representing complex high dimensional distributions in terms of bivariate and conditional bivariate distributions or copulae. In this paper, we show that how vines can be used to approximate any given multivariate distribution to any required degree of approximation. This paper is more about the approximation rather than optimal estimation methods. To maintain uniform approximation in the class of copulae used to build the corresponding vine we use minimum information approaches. We generalised the results found by Bedford and Cooke that if a minimal information copula satis¯es each of the (local) constraints (on moments, rank correlation, etc.), then the resulting joint distribution will be also minimally informative given those constraints, to all regular vines. We then apply our results to modelling a dataset of Norwegian financial data that was previously analysed in Aas et al. (2009)