What happens after a default: the conditional density approach

We present a general model for default time, making precise the role of the intensity process, and showing that this process allows for a knowledge of the conditional distribution of the default only "before the default". This lack of information is crucial while working in a multi-default setting. In a single default case, the knowledge of the intensity process does not allow to compute the price of defaultable claims, except in the case where immersion property is satisfied. We propose in this paper the density approach for default time. The density process will give a full characterization of the links between the default time and the reference filtration, in particular "after the default time". We also investigate the description of martingales in the full filtration in terms of martingales in the reference filtration, and the impact of Girsanov transformation on the density and intensity processes, and also on the immersion property

Coherent Price Systems and Uncertainty-Neutral Valuation

We consider fundamental questions of arbitrage pricing arising when the uncertainty model is given by a set of possible mutually singular probability measures. With a single probability model, essential equivalence between the absence of arbitrage and the existence of an equivalent martingale measure is a folk theorem, see Harrison and Kreps (1979). We establish a microeconomic foundation of sublinear price systems and present an extension result. In this context we introduce a prior dependent notion of marketed spaces and viable price systems. We associate this extension with a canonically altered concept of equivalent symmetric martingale measure sets, in a dynamic trading framework under absence of prior depending arbitrage. We prove the existence of such sets when volatility uncertainty is modeled by a stochastic differential equation, driven by Peng's G-Brownian motions

Asymptotics of Asynchronicity

In this article we focus on estimating the quadratic covariation of continuous semimartingales from discrete observations that take place at asynchronous observation times. The Hayashi-Yoshida estimator serves as synchronized realized covolatility for that we give our own distinct illustration based on an iterative synchronization algorithm. We consider high-frequency asymptotics and prove a feasible stable central limit theorem. The characteristics of non-synchronous observation schemes affecting the asymptotic variance are captured by a notion of asymptotic covariations of times. These are precisely illuminated and explicitly deduced for the important case of independent time-homogeneous Poisson sampling.Comment: technical report, 36 page

The filtering equations revisited

The problem of nonlinear filtering has engendered a surprising number of mathematical techniques for its treatment. A notable example is the change-of--probability-measure method originally introduced by Kallianpur and Striebel to derive the filtering equations and the Bayes-like formula that bears their names. More recent work, however, has generally preferred other methods. In this paper, we reconsider the change-of-measure approach to the derivation of the filtering equations and show that many of the technical conditions present in previous work can be relaxed. The filtering equations are established for general Markov signal processes that can be described by a martingale-problem formulation. Two specific applications are treated