Infinite-server systems with Hawkes arrivals and Hawkes services

Queueing Systems - This paper is devoted to the study of the number of customers in infinite-server systems driven by Hawkes processes. In these systems, the self-exciting arrival process is...     

Stochastic Control Methods: Hedging in a Market Described by Pure Jump Processes

This paper considers the asset price movements in a financial market with a risky asset and a bond. The dynamics of the risky asset, modeled by a marked point process, depend on a stochastic factor, modeled also by a marked point process. The possibility of common jump times with the price is allowed. The problem studied is to determine a strategy maximizing the expected value of a utility function of the hedging error. Two different approaches are considered: an Hamilton Jacobi Bellmann equation is studied for a simplified model and a contraction technique is introduced for a more general model.     

Markov Chain Model with Catastrophe to Determine Mean Time to Default of Credit Risky Assets

This article deals with the problem of probabilistic prediction of the time distance to default for a firm. To model the credit risk, the dynamics of an asset is described as a function of a homogeneous discrete time Markov chain subject to a catastrophe, the default. The behaviour of the Markov chain is investigated and the mean time to the default is expressed in a closed form. The methodology to estimate the parameters is given. Numerical results are provided to illustrate the applicability of the proposed model on real data and their analysis is discussed.     

Existence of Optimal Controls for Partially Observed Jump Processes

A partially observable control problem for an R ^d -valued jump process with counting observations is studied. The state and the observations may be strongly dependent and, in particular, the two processes may jump together. An equivalent separated problem is introduced and the existence of an optimal control for the separated problem is obtained in the class of relaxed and generalized controls. Equivalence between the initial problem and the relaxed generalized separated control problem is discussed.     

A Partially Observed Ultra-High-Frequency Data Model: Risk-Minimizing Hedging

An ultra-high-frequency data model on financial asset price movements is considered. This model allows us to relate the changes in price volatility and trading activity to news or information arrivals. The underlying event arrivals process is assumed to be unobserved by the market agents. Then, the study of the risk-minimizing hedging-strategies for derivatives under partial information bring us to a nonlinear filtering problem. Taking into account the weak form of market efficiency, under some Markovianity assumptions, classical filtering techniques are used to evaluate the risk-minimizing hedging-strategies.      

Probabilistic prediction of credit ratings: a filtering approach

In a financial market, to analyze the credit quality of firms, this note proposes a dynamical model. The population of firms is divided into a finite number of classes, depending on their credit status. The cardinality of the population can increase during the time, since new firms can enter in the market. Due to changes in credit quality and to the defaults, each firm can move from a class to another, or can go to the class of the defaulted firms. Different rating agencies are considered, each of them defines its own partition of the population. Aim of this note is to find the probabilistic prediction of the actual partition of the population, and of the conditional distribution of the distance to defaults. In a partial observing setting, this topic is discussed using stochastic filtering techniques.     

Credit Risk in an Economy with New Firms Arrivals

We propose a dynamic model to analyze the credit quality of firms. In the market in which they operate, the firms are divided into a finite number of classes representing their credit status. The cardinality of the population can increase, since new firms can enter the market and the partition is supposed to change over time, due to defaults and changes in credit quality, following a class of Markov processes. Some conditional probabilities related to default times are investigated and the role of occupation numbers is highlighted in this context. In a partial information setting at discrete time, we present a particle filtering technique to numerically compute by simulation the conditional distribution of the number of firms in the credit classes, given the information up to time t.     

Filtering on a Partially Observed Ultra-High-Frequency Data Model

A model for intraday stock price movements is considered. The jump-intensity of the logreturn process is a function of the whole history of a hidden marked point process. The aim is to find the conditional law of such intensity given the history of the logreturn process. Under a Markovianity assumption, related with the weak form of market efficiency, classical filtering techniques are used. The law of the jump-intensity, given the history of the logreturn price, is evaluated and a discussion on a particular case is performed.