arXiv subjects

Barbara Torti

Publications and source records attributed to Barbara Torti.

6 recordsLinked to original sources

arXiv2024

Thin-thick approach to martingale representations on progressively enlarged filtrations

We study the predictable representation property in the progressive enlargement F^\tau of a reference filtration F by a random time \tau. Our approach is based on the decomposition of any random time into two parts, one overlapping F-stopping times (thin part) and the other one that avoids F-stopping times (thick part). We assume that the F-thin part of \tau is nontrivial and prove a martingale representation theorem on F^\tau. We thus extend previous results dealing with F-avoiding random times. We collect some examples of application to the enlargement of the natural filtration of a L\'evy process.

math.PR↗

arXiv2021

Martingale representations in progressive enlargement by multivariate point processes

We show that all local martingales with respect to the initially enlarged natural filtration of a vector of multivariate point processes can be weakly represented up to the minimum among the explosion times of the components. We also prove that a strong representation holds if any multivariate point process of the vector has almost surely infinite explosion time and discrete mark's space. Then we provide a condition under which the components of the multidimensional local martingale driving the strong representation are pairwise orthogonal.

math.PR↗

arXiv2017

Martingale representation on enlarged filtrations: the role of the accessible jump times

We consider a filtration $\mathbb{G}$ obtained as enlargement of a filtration $\mathbb{F}$ by a filtration $\mathbb{H}$. We assume that all $\mathbb{F}$-local martingales are represented by a martingale $M$ and all $\mathbb{H}$-local martingales are represented by a martingale $N$. $M$ and $N$ are not necessarily quasi-left continuous processes and their jump times may overlap. We first analyze the contribution of the accessible jump times of $M$ and $N$ to the Jacod's dimension of the space of the $\mathcal{H}^1(\mathbb{G})$-martingales. Then we prove a new martingale representation theorem on $\mathbb{G}$.

math.PR↗

arXiv2016

Martingale representation in progressive enlargement by the reference filtration of a semimartingale: a note on the multidimensional case

Let X and Y be an m-dimensional F-semimartingale and an n-dimensional H-semimartingale respectively on the same probability space, both enjoying the strong predictable representation property. We propose a martingale representation result under the probability measure P for the square integrable G-martingales, where G is the union of F and H. As a first application we identify the biggest possible value of the multiplicity in the sense of Davis and Varaiya of the union of F1,..., Fd, where, fixed i in (1,...,d), Fi is the reference filtration of a real martingale Mi, which enjoys the Fi-predictable representation property. A second application falls into the framework of credit risk modeling and in particular into the study of the progressive enlargement of the market filtration by a default time. More precisely, when the risky asset price is a multidimensional semimartingale enjoying the strong predictable representation property and the default time satisfies the density hypothesis, we present a new proof of the analogous of the classical theorem of Kusuoka.

math.PR↗

arXiv2016

Enlargement of filtration and predictable representation property for semi-martingales

We present two examples of loss of the predictable representation property for semi-martingales by enlargement of the reference filtration. First of all we show that the predictable representation property for a square-integrable semi-martingale X does not transfer from the reference filtration F to a larger filtration G when the information starts growing up to a positive time. Then we study the case when F coincides with the natural filtration of X and G is obtained by adding the natural filtration of a second square-integrable semi-martingale, Y. We establish conditions under which the triplet (X,Y,[X,Y]) enjoys the predictable representation property with respect to G.

math.PR↗

arXiv2013

Rare Mutations in Evolutionary Dynamics

In this paper we study the effect of rare mutations, driven by a marked point process, on the evolutionary behavior of a population. We derive a Kolmogorov equation describing the expected values of the different frequencies and prove some rigorous analytical results about their behavior. Finally, in a simple case of two different quasispecies, we are able to prove that the rarity of mutations increases the survival opportunity of the low fitness species.

math.DS↗