Mounir Zili

Mounir Zili
University of Monastir, Faculty of sciences, Monastir · Mathematics

Phd

About

62
Publications
8,905
Reads
How we measure 'reads'
A 'read' is counted each time someone views a publication summary (such as the title, abstract, and list of authors), clicks on a figure, or views or downloads the full-text. Learn more
358
Citations
Additional affiliations
September 2008 - October 2012
Preparatory school to the military academies, Sousse, Tunisia
Position
  • Chair
April 2004 - October 2012
Preparatory school to the military academies, Sousse, Tunisia
Position
  • Chair
February 2002 - August 2008
Preparatory school to the military academies, Sousse, Tunisia
Position
  • Professor (Associate)

Publications

Publications (62)
Article
Full-text available
We investigated a novel stochastic fractional partial differential equation (FPDE) characterized by a mixed operator that integrated the standard Laplacian, the fractional Laplacian, and the gradient operator. The equation was driven by a random noise, which admitted a covariance measure structure with respect to the time variable and behaved as a...
Article
Full-text available
This work is a first step toward developing a stochastic calculus theory with respect to the generalized fractional Brownian motion, which a recently introduced Gaussian process is extending both fractional and sub-fractional Brownian motions. A Malliavin divergence operator and a stochastic symmetric integral with respect to this process are defin...
Book
Depuis trente ans, le développement des mathématiques financières a connu un véritable essor du fait de leurs applications à la modélisation, à la quantification et à la compréhension des phénomènes régissant les marchés financiers.Didactique et accessible Martingales et mathématiques financières en temps discret présente la théorie des martingales...
Article
We introduce a new stochastic heat equation with a mixed operator, which is a combination of the standard Laplacian, a fractional Laplacian and the gradient operator, driven by an additive Gaussian noise which is white in time and in space. We establish the existence of the solution and we study its behavior with respect to the time variable. In pa...
Chapter
This chapter presents the concept of conditional expectation. This is a very important concept that generalizes the concept of conditional probability, when conditioning occurs not only on a single event, but on an entire family of events. The chapter reviews the definition of conditional probability with respect to an event with strictly positive...
Chapter
This chapter defines the concept of a “martingale”, which is a family of discrete‐time stochastic processes that are especially important for applications in financial mathematics. It looks at the important concept of the transform of a martingale. The chapter introduces the Doob decomposition theorem. This theorem justifies the universality of mar...
Chapter
This chapter offers an introduction to an interesting example of stochastic process, simple symmetric walk, with minimum formalism. In most bibliographic references, random walks and the classical problem of the gambler's ruin are treated as applications of the theory of martingales or of Markov chains. The chapter defines the simple symmetric rand...
Chapter
This chapter reviews the basic concepts related to probability and random variables. It examines the chief definitions and properties of random variables and their distribution. The chapter also reviews the concept of a probability measure or probability distribution, and the concept of random variable, as well as the chief properties of these conc...
Chapter
This chapter offers solutions for a series of exercises and practical work in using the concept of martingales and discrete‐time financial market model discussed in this book. It is a useful reference for students at the master's or doctoral level who are specializing in applied mathematics or finance as well as teachers, researchers in the field o...
Chapter
This chapter looks at a second family of options called American options. The chief difference between these and European options lies in the date of exercise: American options can be exercised at any point of time between the initial date and the maturity date. The chapter presents the definition of American options and provides a recursive formul...
Chapter
This chapter examines a specific form of a financial contract or conditional asset called a European option . It introduces the concept of a complete financial market and its characterization in terms of a martingale. The chapter resolves the problem of the valuation and hedging of European options in a general framework. It aims to apply these res...
Chapter
This chapter discusses financial mathematics properly, and focuses on discrete‐time models. It introduces the concept of a financial asset, the concept of an investment strategy, and the concept of arbitration. This key concept allows us to establish a relationship with the martingale theory. The chapter also introduces a discrete‐time financial ma...
Preprint
Full-text available
We investigate a stochastic partial differential equation with second order elliptic operator in divergence form, having a piecewise constant diffusion coefficient, and driven by a space-time white noise. We introduce a notion of weak solution of this equation and prove its equivalence to the already known notion of mild solution.
Article
Full-text available
We introduce a fractional stochastic heat equation with second-order elliptic operator in divergence form, having a piecewise constant diffusion coefficient, and driven by an infinite-dimensional fractional Brownian motion. We characterize the fundamental solution of its deterministic part, and prove the existence and the uniqueness of its solution...
Preprint
Full-text available
We introduce a stochastic partial differential equation (SPDE) with elliptic operator in divergence form, with measurable and bounded coefficients and driven by space-time white noise. Such SPDEs could be used in mathematical modelling of diffusion phenomena in medium consisting of different kinds of materials and undergoing stochastic perturbation...
Article
We investigate a stochastic partial differential equation with second order elliptic operator in divergence form, having a piecewise constant diffusion coefficient, and driven by a space–time white noise. We introduce a notion of weak solution of this equation and prove its equivalence to the already known notion of mild solution.
Preprint
Full-text available
We introduce a fractional stochastic heat equation with second order elliptic operator in divergence form, having a piecewise constant diffusion coefficient, and driven by an infinite-dimensional fractional Brownian motion. We characterize the fundamental solution of its deterministic part, and prove the existence and the uniqueness of its solution...
Article
Full-text available
We introduce a stochastic partial differential equation (SPDE) with elliptic operator in divergence form, with measurable and bounded coefficients and driven by space-time white noise. Such SPDEs could be used in mathematical modelling of diffusion phenomena in medium consisting of different kinds of materials and undergoing stochastic perturbation...
Article
We expand the quartic variations in time and the quadratic variations in space of the solution to a stochastic partial differential equation with piecewise constant coefficients. Both expansions allow us to deduce an estimation method of the parameters appearing in the equation.
Article
Full-text available
We expand the quartic variations in time and the quadratic variations in space of the solution to a stochastic partial differential equation with piecewise constant coefficients. Both expansions allow us to deduce an estimation method of the parameters appearing in the equation.
Article
Full-text available
The generalized fractional Brownion motion (gfBm) is a new extension of both fractional and sub-fractional Brownian motions, introduced very recently. We show that this process could serve to obtain new models, better than those constructed from fractional and sub-fractional Brownian motions, permitting to take the level of correlation between the...
Article
Full-text available
We study a stochastic differential equation, the diffusion coefficient of which is a function of some adapted stochastic process. The various conditions for the existence and uniqueness of weak and strong solutions are presented. The drift parameter estimation in this model is investigated, and the strong consistency of the least squares and maximu...
Book
This book reflects upon the current trends arising in both the theoretical study and the practical modeling of phenomena that exhibit fractionality. There is a tendency to transition from one type of fractionality to several types presented simultaneously; thus, there is a need to construct models that include different types of fractionality. The...
Article
Full-text available
We introduce a new stochastic partial differential equation with second-order elliptic operator in divergence form, having a piecewise constant diffusion coefficient, and driven by a space–time white noise. Such equation could be used in mathematical modeling of diffusion phenomena in medium consisting of two kinds of materials and undergoing stoch...
Article
We study the asymptotic behavior of the spatial quadratic variation for the solution to the stochastic wave equation driven by additive space-time white noise. We prove that the sequence of its renormalized quadratic variations satisfies a central limit theorem (CLT for short). We obtain the rate of convergence for this CLT via the Stein–Malliavin...
Preprint
We introduce a new Gaussian process, a generalization of both fractional and subfractional Brownian motions, which could serve as a good model for a larger class of natural phenomena. We study its main stochastic properties and some increments characteristics. As an application, we deduce the properties of nonsemimartingality, H\"{o}lder continuity...
Article
Full-text available
We introduce a new Gaussian process, a generalization of both fractional and subfractional Brownian motions, which could serve as a good model for a larger class of natural phenomena. We study its main stochastic properties and some increments characteristics. As an application, we deduce the properties of nonsemimartingality, H\"{o}lder continuity...
Article
Full-text available
We consider a stochastic differential equation of the form \[dX_t=\theta a(t,X_t)\,dt+\sigma_1(t,X_t)\sigma_2(t,Y_t)\,dW_t\] with multiplicative stochastic volatility, where $Y$ is some adapted stochastic process. We prove existence--uniqueness results for weak and strong solutions of this equation under various conditions on the process $Y$ and th...
Preprint
Full-text available
We consider a stochastic differential equation of the form \[dX_t=\theta a(t,X_t)\,dt+\sigma_1(t,X_t)\sigma_2(t,Y_t)\,dW_t\] with multiplicative stochastic volatility, where $Y$ is some adapted stochastic process. We prove existence--uniqueness results for weak and strong solutions of this equation under various conditions on the process $Y$ and th...
Article
Full-text available
We consider a stochastic partial differential equation involving a second order differential operator whose drift is discontinuous. The equation is driven by a Gaussian noise which behaves as a Wiener process in space and the time covariance generates a signed measure. This class includes the Brownian motion, fractional Brownian motion and other re...
Article
In 1962 Lamperti introduced a transformation that associates to every non-trivial self-similar process a strictly stationary process. This transform has been widely studied for Gaussian processes and in particular for fractional Brownian motion. Our aim is to analyze various properties of the Lamperti transform of the fractional Brownian sheet. We...
Article
Full-text available
We introduce a new stochastic heat equation with a colored-white fractional noise, which behaves as a Wiener process in the spatial variable and as mixed sub-fractional Brownian motion in time. A necessary and sufficient condition for the existence of its solution is reported. We also analyze regularity properties of this equation, with respect to...
Article
Full-text available
We investigate the problem of estimation of the unknown drift parameter in the stochastic differential equations driven by fractional Brownian motion, with the coefficients supplying standard existence–uniqueness demands. We consider a particular case when the ratio of drift and diffusion coefficients is non-random, and establish the asymptotic str...
Article
A sub-mixed fractional Brownian motion (smfBm) is a linear combination of a Brownian motion and an independent sub-fractional Brownian motion with Hurst index 0 < H < 1. Its main properties are studied. They suggest that smfBm lies between the sub-fractional Brownian motion and the mixed fractional Brownian motion. We also determine the values of H...
Article
Full-text available
Let {S t H , t ≥ 0} be a linear combination of a Brownian motion and an independent sub-fractional Brownian motion with Hurst index 0 < H < 1. Its main properties are studied. They suggest that S H lies between the sub-fractional Brownian motion and the mixed fractional Brownian motion. We also determine the values of H for which S H is not a semi-...
Article
Full-text available
We study the linear heat equation driven by a random noise which admits a covariance measure structure with respect to the time variable and has a spatial covariance given by the Riesz kernel. We focus our attention on the particular case when the noise behaves as a bifractional Brownian motion in time.
Article
Full-text available
We study a second-order parabolic equation with divergence form elliptic operator, having piecewise constant diffusion coefficients with two points of discontinuity. Such partial differential equations appear in the modelization of diffusion phenomena in medium consisting of three kind of materials. Using probabilistic methods, we present an explic...
Article
Full-text available
A new extension of the sub-fractional Brownian motion, and thus of the Brownian motion, is introduced. It is a linear combination of a finite number of sub-fractional Brownian motions, that we have chosen to call the mixed sub-fractional Brownian motion. In this paper, we study some basic properties of this process, its non-Markovian and non-statio...
Article
Full-text available
We present an explicit series expansion of the sub-mixed fractional Brownian motion and study its rate of convergence. We show that the obtained expansion is rate-optimal in the sense that the expected uniform norm of the truncated series vanishes at optimal rate as the truncation point tends to infinity. As an application of this result, we presen...
Article
Full-text available
Let ${S_t^H, t \geq 0} $ be a linear combination of a Brownian motion and of an independent sub-fractional Brownian motion with Hurst index $0 < H < 1$. Its main properties are studied and it is shown that $S^H $ can be considered as an intermediate process between a sub-fractional Brownian motion and a mixed fractional Brownian motion. Finally, we...
Conference Paper
Full-text available
This special issue of Random Operators and Stochastic Equations contains some refereed submitted papers, among those presented at the International Conference on Stochastic Analysis and Applied Probability (SAAP 2010).
Conference Paper
Selected papers submitted by participants of the international Conference “Stochastic Analysis and Applied Probability 2010” ( www.saap2010.org ) make up the basis of this volume. The SAAP 2010 was held in Tunisia, from 7-9 October, 2010, and was organized by the “Applied Mathematics & Mathematical Physics” research unit of the preparatory institu...
Article
Full-text available
Paper presents a methodology for estimating the parameters of stochastic differential equation (SDE) driven by fractional Brownian motion (fBm). The main idea is connected with simulated maximum likelihood. To develop this methodology two important questions: generation the fBm sample paths with different Hurst parameter values and Hurst parameter...
Conference Paper
Full-text available
Portfolio Optimization Problem of Mertons' Market Driven by a Fractional Brownian Motion.
Article
Full-text available
We construct a fundamental solution of a second-order parabolic partial differential equation, with piecewise constant coefficients and admitting a generalized drift. We present a calculation method, which permits us to give an explicit expression of the solution usable in concrete applications.
Article
Full-text available
The aim of this article is to construct a fundamental solution of a partial differential equation with pieacewise constant coefficients and admitting a generalized drift. We present a calcul method, which permits us to give an explicit formulation of the solution usable in concrete applications.
Article
Full-text available
In this paper we prove the pathwise uniqueness of solutions of a stochastic differential equation with a singular drift whiçh depends on time. Our method is of probabilistic nature, and it is based on an Al-Hussaini and Elliott result.
Article
In this paper, we prove pathwise uniqueness of the solutions to a stochastic differential equation with time-dependent singular drift. Our method is probabilistic and based on Itô's-type formula by Al-Hussaini and Elliott [1]

Network

Cited By