Catálogo de publicaciones - libros

Some of the recent developments in the rapidly expanding field of Ambit Stochastics are here reviewed. After a brief recall of the framework of Ambit Stochastics, two topics are considered: (i) Methods of modelling and inference for volatility/intermittency processes and fields; (ii) Universal laws in turbulence and finance in relation to temporal processes. This review complements two other recent expositions.

First, we revisit basic theory of functional Itô/path-dependent calculus, using the formulation of calculus via regularization. Relations with the corresponding Banach space valued calculus are explored. The second part of the paper is devoted to the study of the Kolmogorov type equation associated with the so called window Brownian motion, called path-dependent heat equation, for which well-posedness at the level of strict solutions is established. Then, a notion of strong approximating solution, called strong-viscosity solution, is introduced which is supposed to be a substitution tool to the viscosity solution. For that kind of solution, we also prove existence and uniqueness.

For Hölder continuous functions (, ) and , we define nonlinear integral via fractional calculus. This nonlinear integral arises naturally in the Feynman-Kac formula for stochastic heat equations with random coefficients (Hu and Lê, Nonlinear Young integrals and differential systems in Hölder media. Trans. Am. Math. Soc. (in press)). We also define iterated nonlinear integrals.

In this paper we present a weak limit theorem for a numerical approximation of Brownian semi-stationary processes studied in []. In the original work of [] the authors propose to use Fourier transformation to embed a given one dimensional (Lévy) Brownian semi-stationary process into a two-parameter stochastic field. For the latter they use a simple iteration procedure and study the strong approximation error of the resulting numerical scheme given that the volatility process is fully observed. In this work we present the corresponding weak limit theorem for the setting, where the volatility/drift process needs to be numerically simulated. In particular, weak approximation errors for smooth test functions can be obtained from our asymptotic theory.

Relying on the method developed in [], we prove the existence of a density for two different examples of random fields indexed by . The first example consists of SPDEs with Lipschitz continuous coefficients driven by a Gaussian noise white in time and with a stationary spatial covariance, in the setting of []. The density exists on the set where the nonlinearity of the noise does not vanish. This complements the results in [] where is assumed to be bounded away from zero. The second example is an ambit field with a stochastic integral term having as integrator a Lévy basis of pure-jump, stable-like type.

We propose new notions of regular solutions and viscosity solutions for path-dependent second order partial differential equations. Making use of the martingale problem approach to path-dependent diffusion processes, we explicitly construct families of time-consistent dynamic risk measures on the set of càdlàg paths valued endowed with the Skorokhod topology. These risk measures are shown to have regularity properties. We prove then that these time-consistent dynamic risk measures provide viscosity supersolutions and viscosity subsolutions for path-dependent semi-linear second order partial differential equations.

Contingent Convertible Bonds, or , are contingent capital instruments which are converted into shares, or may suffer a principal write-down, if certain trigger event occurs. In this paper we discuss some approaches to the problem of pricing when its conversion and the other relevant credit events are triggered by the issuer’s share price. We introduce a new model of partial information which aims at enhancing the market trigger approach while remaining analytically tractable. We address also having the additional feature of being callable by the issuer at a series of pre-defined dates. These are thus exposed to a new source of risk—referred to as —since they have no fixed maturity, and the repayment of the principal may take place at the issuer’s convenience.

In this paper the effect of the choice of the model on partial hedging in incomplete markets in finance is estimated. In fact we compare the quadratic hedging strategies in a martingale setting for a claim when two models for the underlying stock price are considered. The first model is a geometric Lévy process in which the small jumps might have infinite activity. The second model is a geometric Lévy process where the small jumps are replaced by a Brownian motion which is appropriately scaled. The hedging strategies are related to solutions of backward stochastic differential equations with jumps which are driven by a Brownian motion and a Poisson random measure. We use this relation to prove that the strategies are robust towards the choice of the model for the market prices and to estimate the model risk.

We establish a stochastic maximum principle (SMP) for control problems of partially observed diffusions of mean-field type with risk-sensitive performance functionals.

According to theoretical arguments, a properly designed emission trading system should help reaching pollution reduction at low social burden based on the theoretical work of environmental economists, cap-and-trade systems are put into operations all over the world. However, the practice from emissions trading yields a real stress test for the underlying theory and reveals a number of its weak points. This paper aims to fill the gap between general welfare concepts underlying understanding of liberalized market and specific issues of real-world emission market operation. In our work, we present a novel technique to analyze emission market equilibrium in order to address diverse questions in the setting of market players. Our contribution significantly upgrades all existing models in this field, which neglect risk-aversion aspects at the cost of having a wide range of singularities in their conclusions, now resolved in our approach. Furthermore, we show both how the architecture of an environmental market can be optimized under the realistic assumption of risk-aversion.