arXiv subjects

Paolo Di Tella

Publications and source records attributed to Paolo Di Tella.

12 recordsLinked to original sources

arXiv2024

On moments of integrals with respect to Markov additive processes and of Markov modulated generalized Ornstein-Uhlenbeck processes

We establish sufficient conditions for the existence, and derive explicit formulas for the $κ$'th moments, $κ\geq 1$, of Markov modulated generalized Ornstein-Uhlenbeck processes as well as their stationary distributions. In particular, the running mean, the autocovariance function, and integer moments of the stationary distribution are derived in terms of the characteristics of the driving Markov additive process. Our derivations rely on new general results on moments of Markov additive processes and (multidimensional) integrals with respect to Markov additive processes.

math.PR↗

arXiv2023

Product formulas for multiple stochastic integrals associated with Lévy processes

In the present paper, we obtain an explicit product formula for products of multiple integrals w.r.t. a random measure associated with a Lévy process. As a building block, we use a representation formula for products of martingales from a compensated-covariation stable family. This enables us to consider Lévy processes with both jump and Gaussian part. It is well known that for multiple integrals w.r.t. the Brownian motion such product formulas exist without further integrability conditions on the kernels. However, if a jump part is present, this is, in general, false. Therefore, we provide here sufficient conditions on the kernels which allow us to establish product formulas. As an application, we obtain explicit expressions for the expectation of products of iterated integrals, as well as for the moments and the cumulants for stochastic integrals w.r.t. the random measure. Based on these expressions, we show a central limit theorem for the long time behaviour of a class of stochastic integrals. Finally, we provide methods to calculate the number of summands in the product formula.

math.PR↗

arXiv2021

Progressively Enlargement of Filtrations and Control Problems for Step Processes

In the present paper we address stochastic optimal control problems for a step process $(X,\mathbb{F})$ under a progressive enlargement of the filtration. The global information is obtained adding to the reference filtration $\mathbb{F}$ the point process $H=1_{[τ,+\infty)}$. Here $τ$ is a random time that can be regarded as the occurrence time of an external shock event. We study two classes of control problems, over $[0,T]$ and over the random horizon $[0,T \wedge τ]$. We solve these control problems following a dynamical approach based on a class of BSDEs driven by the jump measure $μ^ Z$ of the semimartingale $Z=(X,H)$, which is a step process with respect to the enlarged filtration $\mathbb G$. The BSDEs that we consider can be solved in $\mathbb{G}$ thanks to a martingale representation theorem which we also establish here. To solve the BSDEs and the control problems we need to ensure that $Z$ is quasi-left continuous in the enlarged filtration $\mathbb{G}$. Therefore, in addition to the $\mathbb{F}$-quasi left continuity of $X$, we assume some further conditions on $τ$: the {\it avoidance} of $\mathbb{F}$-stopping times and the {\it immersion} property, or alternatively {\it Jacod's absolutely continuity} hypothesis.

math.PR↗

arXiv2020

Martingale Representation in the Enlargement of the Filtration Generated by a Point Process

Let $X$ be a point process and let $\mathbb{X}$ denote the filtration generated by $X$. In this paper we study martingale representation theorems in the filtration $\mathbb{G}$ obtained as an initial and progressive enlargement of the filtration $\mathbb{X}$. The progressive enlargement is done here by means of a whole point process $H$. We do not require further assumptions on the point process $H$ nor on the dependence between $X$ and $H$. In particular, we recover the special case of the progressive enlargement by a random time $τ$.

math.PR↗

arXiv2020

Martingale Representation in Progressively Enlarged Lévy Filtrations

In this paper we obtain a martingale representation theorem in the progressive enlargement $\mathbb{G}$ by a random time $τ$ of the filtration $\mathbb{F}^L$ generated by a Lévy process $L$. The assumptions on the random time are that $\mathbb{F}^ L$ is immersed in $\mathbb{G}$ and that $τ$ avoids $\mathbb{F}^ L$ stopping times. We also study the multiplicity of a progressively enlarged filtration.

math.PR↗

arXiv2020

On the Propagation of the Weak Representation Property in Independently Enlarged Filtrations: The General Case

In this paper we investigate the propagation of the weak representation property (WRP) to an independently enlarged filtration. More precisely, we consider an $\mathbb{F}$-semimartingale $X$ possessing the WRP with respect to $\mathbb{F}$ and an $\mathbb{H}$-semimartingale $Y$ possessing the WRP with respect to $\mathbb{H}$. Assuming that $\mathbb{F}$ and $\mathbb{H}$ are independent, we show that the $\mathbb{G}$-semimartingale $Z=(X,Y)$ has the WRP with respect to $\mathbb{G}$, where $\mathbb{G}:=\mathbb{F}\vee\mathbb{H}$. In our setting, $X$ and $Y$ may have simultaneous jump-times. Furthermore, their jumps may charge predictable times. This generalizes all available results about the propagation of the WRP to independently enlarged filtrations.

math.PR↗

arXiv2019

BSDEs and log-utility maximization for Lévy processes

In this paper we establish the existence and the uniqueness of the solution of a special class of BSDEs for Lévy processes in the case of a Lipschitz generator of sublinear growth. We then study a related problem of logarithmic utility maximization of the terminal wealth in the filtration generated by an arbitrary Lévy process.

math.PR↗

arXiv2019

On The Weak Representation Property in Progressively Enlarged Filtrations with an Application to Exponential Utility Maximization

In this paper we show that the weak representation property of a semimartingale $X$ with respect to a filtration $\mathbb{F}$ is preserved in the progressive enlargement $\mathbb{G}$ by a random time $τ$ avoiding $\mathbb{F}$-stopping times and such that $\mathbb{F}$ is immersed in $\mathbb{G}$. As an application of this, we can solve an exponential utility maximization problem in the enlarged filtration $\mathbb{G}$ following the dynamical approach, based on suitable BSDEs, both over the fixed time horizon $[0,T]$, $T>0$, and over $[0,T\wedgeτ]$.

math.PR↗

arXiv2018

Product and Moment Formulas for Iterated Stochastic Integrals (associated with Lévy Processes)

In this paper, we obtain explicit product and moment formulas for products of iterated integrals generated by families of square integrable martingales associated with an arbitrary Lévy process. We propose a new approach applying the theory of compensated-covariation stable families of martingales. Our main tool is a representation formula for products of elements of a compensated-covariation stable family, which enables to consider Lévy processes, with both jumps and Gaussian part.

math.PR↗

arXiv2017

Semi-Static and Sparse Variance-Optimal Hedging

We consider hedging of a contingent claim by a 'semi-static' strategy composed of a dynamic position in one asset and static (buy-and-hold) positions in other assets. We give general representations of the optimal strategy and the hedging error under the criterion of variance-optimality and provide tractable formulas using Fourier-integration in case of the Heston model. We also consider the problem of optimally selecting a sparse semi-static hedging strategy, i.e. a strategy which only uses a small subset of available hedging assets. The developed methods are illustrated in an extended numerical example where we compute a sparse semi-static hedge for a variance swap using European options as static hedging assets.

q-fin.MF↗

arXiv2017

Semi-Static Variance-Optimal Hedging in Stochastic Volatility Models with Fourier Representation

In a financial market model, we consider the variance-optimal semi-static hedging of a given contingent claim, a generalization of the classic variance-optimal hedging. To obtain a tractable formula for the expected squared hedging error and the optimal hedging strategy, we use a Fourier approach in a general multidimensional semimartingale factor model. As a special case, we recover existing results for variance-optimal hedging in affine stochastic volatility models. We apply the theory to set up a variance-optimal semi-static hedging strategy for a variance swap in both the Heston and the 3/2-model, the latter of which is a non-affine stochastic volatility model.

math.PR↗

arXiv2015

The Chaotic Representation Property of Compensated-Covariation Stable Families of Martingales

In the present paper, we study the chaotic representation property for certain families of square integrable martingales. For this purpose, we introduce the notion of compensated-covariation stability of such families. The chaotic representation property will be defined using iterated integrals with respect to a given family of square integrable martingales having deterministic mutual predictable covariation. The main result of the present paper is: If $\mathscr{X}$ is a compensated-covariation stable family of square integrable martingales such that $\langle{X},{Y}\rangle$ is deterministic for all $X,Y\in\mathscr{X}$ and, furthermore, the system of monomials generated by $\mathscr{X}$ is total in $L^2(Ω,\mathscr{F}^\mathscr{X}_T,\mathbb{P})$, then $\mathscr{X}$ possesses the chaotic representation property. We shall apply this result to the case of Lévy processes. Relative to the filtration generated by a Lévy process, we construct families of martingales which possess the chaotic representation property. As an illustration of the general results, we will also discuss applications to continuous Gaussian families of martingales and independent families of compensated Poisson processes. We conclude the paper by giving, for the case of Lévy processes, several examples of concrete families $\mathscr{X}$ of martingales including Teugels martingales.

math.PR↗