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M. Hassani

Publications and source records attributed to M. Hassani.

12 recordsLinked to original sources
arXiv2024

Set-valued stochastic integrals in UMD spaces and applications

The purpose of this paper is to study certain set-valued integrals in UMD Banach spaces and provide a compatible form of the martingale representation theorem for set-valued martingales. Under specific conditions, these martingales can be expressed using revised set-valued stochastic integrals with respect to a real standard Brownian motion $W = (W_t)_{t\in[0,T ]}$. Moreover, we prove the existence of solutions to the following set-valued backward stochastic differential equation of the form $$ Y_t=\left(\xi+\int_t^TH_u du+\int_{[0,t]}^{\mathscr{R}}{Z}\, dW_u\right) \circleddash \int_{[0,T]}^{\mathscr{R}}{Z}\, dW_u\quad a.s.,\quad t\in [0,T], $$ where the right-hand side, of this equation, represents the Hukuhara difference of two quantities containing revised set-valued stochastic integrals, $\xi$ is a terminal set-valued function condition and $H$ is a set-valued function satisfying some suitable conditions.

math.PR↗
arXiv2023

Doubly Reflected BSDEs With Stochastic Quadratic Growth: Around The Predictable Obstacles

We prove the existence of maximal (and minimal) solution for one-dimensional generalized doubly reflected backward stochastic differential equation (RBSDE for short) with irregular barriers and stochastic quadratic growth, for which the solution $Y$ has to remain between two rcll barriers $L$ and $U$ on $[0; T[$, and its left limit $Y_-$ has to stay respectively above and below two predictable barriers $l$ and $u$ on $]0; T]$. This is done without assuming any $P$-integrability conditions and under weaker assumptions on the input data. In particular, we construct a maximal solution for such a RBSDE when the terminal condition $ξ$ is only ${\cal F}_T-$measurable and the driver $f$ is continuous with general growth with respect to the variable $y$ and stochastic quadratic growth with respect to the variable $z$. Our result is based on a (generalized) penalization method. This method allow us find an equivalent form to our original RBSDE where its solution has to remain between two new rcll reflecting barriers $\overline{Y}$ and $\underline{Y}$ which are, roughly speaking, the limit of the penalizing equations driven by the dominating conditions assumed on the coefficients. A standard and equivalent form to our initial RBSDE as well as a characterization of the solution $Y$ as a generalized Snell envelope of some given predictable process $l$ are also given.

math.PR↗
arXiv2023

Backward Stochastic Evolution Inclusions in UMD Banach Spaces

In this paper, we prove the existence of a mild $L^p$-solution for the backward stochastic evolution inclusion (BSEI for short) of the form \begin{align*}%\label{BSDI3} \begin{cases} dY_t+AY_tdt\in G(t,Y_t,Z_t)dt+Z_tdW_t,\quad t\in [0,T] Y_T =ξ, \end{cases} \end{align*} where $W=(W_t)_{t\in [0,T]}$ is a standard Brownian motion, $A$ is the generator of a $C_0$-semigroup on a UMD Banach space $E$, $ξ$ is a terminal condition from $L^p(Ω,\mathscr{F}_T;E)$, with $p>1$ and $G$ is a set-valued function satisfying some suitable conditions. The case when the processes with values in spaces that have martingale type $2$, has been also studied.

math.PR↗
arXiv2019

A Classification Of Cohomogeneity One Actions On The Minkowski Space $\mathbb{R}^{3,1}$

The aim of this paper is to classify cohomogeneity one isometric actions on the 4-dimensional Minkowski space $\mathbb{R}^{3,1}$, up to orbit equivalence. Representations, up to conjugacy, of the acting groups in $O(3,1)\ltimes \mathbb{R}^{3,1}$ are given in both cases, proper and non-proper actions. When the action is proper, the orbits and the orbit spaces are determined.

math.DG↗
arXiv2019

Continuity of the quantum Fisher information

In estimating an unknown parameter of a quantum state the quantum Fisher information (QFI) is a pivotal quantity, which depends on the state and its derivate with respect to the unknown parameter. We prove the continuity property for the QFI in the sense that two close states with close first derivatives have close QFIs. This property is completely general and irrespective of dynamics or how states acquire their parameter dependence and also the form of parameter dependence---indeed this continuity is basically a feature of the classical Fisher information that in the case of the QFI naturally carries over from the manifold of probability distributions onto the manifold of density matrices. We demonstrate that in the special case where the dependence of the states on the unknown parameter comes from one dynamical map (quantum channel), the continuity holds in its reduced form with respect to the initial states. In addition, we show that when one initial state evolves through two different quantum channels, the continuity relation applies in its general form. A situation in which such scenario can occur is an open-system metrology where one of the maps represents the ideal dynamics whereas the other map represents the real (noisy) dynamics. In the making of our main result, we also introduce a regularized representation for the symmetric logarithmic derivative which works for general states even with incomplete rank, and its features continuity similarly to the QFI.

quant-ph↗
arXiv2018

Wall effects on spatial correlations of non-affine strain in a 3D model glass

Effects of hard planar walls with a particle scale roughness on the spatial correlations of non-affine strain in amorphous solids are investigated via molecular dynamics simulations. When determined within layers parallel to the wall plane, normalized non-affine strain correlations are enhanced within layers closer to the wall. The amplitude of these correlations, on the other hand, is found to be suppressed by the wall. While the former is connected to the effects of a hard boundary on the continuum mechanics scale, the latter is attributed to molecular scale wall effects on the size of the region (nearest neighbor cage), explored by particles on intermediate times scales.

cond-mat.stat-mech↗
arXiv2013

Generalized BSDE With 2-Reflecting Barriers and Stochastic Quadratic Growth

We study the existence of a solution for a one-dimensional generalized backward stochastic differential equation with two reflecting barriers (GRBSDE for short) under assumptions on the input data which are weaker than that on the current literature. In particular, we construct a maximal solution for such a GRBSDE when the terminal condition ξis only F_T-measurable and the driver f is continuous with general growth with respect to the variable y and stochastic quadratic growth with respect to the variable z without assuming any P-integrability conditions. The work is suggested by the interest the results might have in Dynkin game problem and American game option.

math.PR↗
arXiv2011

Stochastic Quadratic BSDE With Two RCLL Obstacles

We study the problem of existence of solutions for generalized backward stochastic differential equation with two reflecting barriers (GRBSDE for short) under weaker assumptions on the data. Roughly speaking we show the existence of a maximal solution for GRBSDE when the terminal condition ξis F_T-measurable, the coefficient f is continuous with general growth with respect to the variable y and stochastic quadratic growth with respect to the variable z and the reflecting barriers L and U are just right continuous left limited. The result is proved without assuming any P-integrability conditions.

math.PR↗
arXiv2010

p-integrable solutions to multidimensional BSDEs and degenerate systems of PDEs with logarithmic nonlinearities

We study multidimensional backward stochastic differential equations (BSDEs) which cover the logarithmic nonlinearity u log u. More precisely, we establish the existence and uniqueness as well as the stability of p-integrable solutions (p > 1) to multidimensional BSDEs with a p-integrable terminal condition and a super-linear growth generator in the both variables y and z. This is done with a generator f(y, z) which can be neither locally monotone in the variable y nor locally Lipschitz in the variable z. Moreover, it is not uniformly continuous. As application, we establish the existence and uniqueness of Sobolev solutions to possibly degenerate systems of semilinear parabolic PDEs with super-linear growth generator and an p-integrable terminal data. Our result cover, for instance, certain (systems of) PDEs arising in physics.

math.PR↗
arXiv2008

General Existence Results for Reflected BSDE and BSDE

In this paper, we are concerned with the problem of existence of solutions for generalized reflected backward stochastic differential equations (GRBSDEs for short) and generalized backward stochastic differential equations (GBSDEs for short) when the generator $fds + gdA_s$ is continuous with general growth with respect to the variable $y$ and stochastic quadratic growth with respect to the variable $z$. We deal with the case of a bounded terminal condition $\xi$ and a bounded barrier $L$ as well as the case of unbounded ones. This is done by using the notion of generalized BSDEs with two reflecting barriers studied in \cite{EH}. The work is suggested by the interest the results might have in finance, control and game theory.

math.PR↗
arXiv2007

On a Result of Hardy and Ramanujan

In this paper, we introduce some explicit approximations for the summation $\sum_{k\leq n}Ω(k)$, where $Ω(k)$ is the total number of prime factors of $k$.

math.NT↗