arXiv subjects

Francesco Russo

Publications and source records attributed to Francesco Russo.

At least 91 records · Page 5Linked to original sources

arXiv2016

Infinite dimensional weak Dirichlet processes and convolution type processes

The present paper continues the study of infinite dimensional calculus via regularization, started by C. Di Girolami and the second named author, introducing the notion of weak Dirichlet process in this context. Such a process X, taking values in a Banach space H, is the sum of a local martingale and a suitable orthogonal process. The concept of weak Dirichlet process fits the notion of convolution type processes, a class including mild solutions for stochastic evolution equations on infinite dimensional Hilbert spaces and in particular of several classes of stochastic partial differential equations (SPDEs). In particular the mentioned decomposition appears to be a substitute of an It{ô}'s type formula applied to f (t, X(t)) where f : [0, T ] $\times$ H $\rightarrow$ R is a C 0,1 function and X a convolution type processes.

math.PR↗

arXiv2016

BSDEs, c{à}dl{à}g martingale problems and orthogonalisation under basis risk

The aim of this paper is to introduce a new formalism for the deterministic analysis associated with backward stochastic differential equations driven by general c{à}dl{à}g martingales. When the martingale is a standard Brownian motion, the natural deterministic analysis is provided by the solution of a semilinear PDE of parabolic type. A significant application concerns the hedging problem under basis risk of a contingent claim $g(X\_T,S\_T)$, where $S$ (resp. $X$) is an underlying price of a traded (resp. non-traded but observable) asset, via the celebrated F{ö}llmer-Schweizer decomposition. We revisit the case when the couple of price processes $(X,S)$ is a diffusion and we provide explicit expressions when $(X,S)$ is an exponential of additive processes.

math.PR↗

arXiv2015

Multidimensional stochastic differential equations with distributional drift

This paper investigates a time-dependent multidimensional stochastic differential equation with drift being a distribution in a suitable class of Sobolev spaces with negative derivation order. This is done through a careful analysis of the corresponding Kolmogorov equation whose coefficient is a distribution.

math.PR↗

arXiv2015

On countably skewed Brownian motion with accumulation point

In this work we connect the theory of Dirichlet forms and direct stochastic calculus to obtain strong existence and pathwise uniqueness for Brownian motion that is perturbed by a series of constant multiples of local times at a sequence of points that has exactly one accumulation point in $\mathbb{R}$. The considered process is identified as special distorted Brownian motion $X$ in dimension one and is studied thoroughly. Besides strong uniqueness, we present necessary and sufficient conditions for non-explosion, recurrence and positive recurrence as well as for $X$ to be semimartingale and possible applications to advection-diffusion in layered media.

math.PR↗

arXiv2015

Elliptic PDEs with distributional drift and backward SDEs driven by a c{à}dl{à}g martingale with random terminal time

We introduce a generalized notion of semilinear elliptic partial differential equations where the corresponding second order partial differential operator $L$ has a generalized drift. We investigate existence and uniqueness of generalized solutions of class $C^1$. The generator $L$ is associated with a Markov process $X$ which is the solution of a stochastic differential equation with distributional drift. If the semilinear PDE admits boundary conditions, its solution is naturally associated with a backward stochastic differential equation (BSDE) with random terminal time, where the forward process is $X$. Since $X$ is a weak solution of the forward SDE, the BSDE appears naturally to be driven by a martingale. In the paper we also discuss the uniqueness of a BSDE with random terminal time when the driving process is a general c{à}dl{à}g martingale.

math.PR↗

arXiv2015

Functional it{ô} versus banach space stochastic calculus and strict solutions of semilinear path-dependent equations

Functional Itô calculus was introduced in order to expand a functional $F(t, X\_{\cdot+t}, X\_t)$ depending on time $t$, past and present values of the process $X$. Another possibility to expand $F(t, X\_{\cdot+t}, X\_t)$ consists in considering the path $X\_{\cdot+t}=\{X\_{x+t},\,x\in[-T,0]\}$ as an element of the Banach space of continuous functions on $C([-T,0])$ and to use Banach space stochastic calculus. The aim of this paper is threefold. 1) To reformulate functional Itô calculus, separating time and past, making use of the regularization procedures which matches more naturally the notion of horizontal derivative which is one of the tools of that calculus. 2) To exploit this reformulation in order to discuss the (not obvious) relation between the functional and the Banach space approaches. 3) To study existence and uniqueness of smooth solutions to path-dependent partial differential equations which naturally arise in the study of functional Itô calculus. More precisely, we study a path-dependent equation of Kolmogorov type which is related to the window process of the solution to an Itô stochastic differential equation with path-dependent coefficients. We also study a semilinear version of that equation.

math.PR↗

arXiv2015

Probabilistic representation of a class of non conservative nonlinear Partial Differential Equations

We introduce a new class of nonlinear Stochastic Differential Equations in the sense of McKean, related to non conservative nonlinear Partial Differential equations (PDEs). We discuss existence and uniqueness pathwise and in law under various assumptions. We propose an original interacting particle system for which we discuss the propagation of chaos. To this system, we associate a random function which is proved to converge to a solution of a regularized version of PDE.

math.PR↗

arXiv2014

Calculus via regularizations in Banach spaces and Kolmogorov-type path-dependent equations

The paper reminds the basic ideas of stochastic calculus via regularizations in Banach spaces and its applications to the study of strict solutions of Kolmogorov path dependent equations associated with "windows" of diffusion processes. One makes the link between the Banach space approach and the so called functional stochastic calculus. When no strict solutions are available one describes the notion of strong-viscosity solution which alternative (in infinite dimension) to the classical notion of viscosity solution.

math.PR↗

arXiv2014

On dual defective manifolds

An embedded manifold is dual defective if its dual variety is not a hypersurface. Using the geometry of the variety of lines through a general point, we characterize scrolls among dual defective manifolds. This leads to an optimal bound for the dual defect, which improves results due to Ein. Among other things we also provide a short and easy proof of the famous Landman Parity Theorem for dual defective manifolds based on our approach to the subject. We also discuss our conjecture that every dual defective manifold with cyclic Picard group should be secant defective, of a very special type, namely a local quadratic entry locus variety.

math.AG↗

arXiv2014

On cubic hypersurfaces with vanishing hessian

If $X = V(f) \subset \mathbb P^N$ is a reduced complex hypersurface, the hessian of $f$ (or by abusing the terminology the hessian of $X$) is the determinant of the matrix of the second derivatives of the form $f$, that is the determinant of the hessian matrix of $f$. Hypersurfaces with vanishing hessian were studied systematically for the first time in the fundamental paper [GN], where Gordan and M. Noether analyze Hesse's claims in [Hesse1, Hesse2] according to which these hypersurfaces are necessarily cones. Of course cones have vanishing hessian. Clearly the claim is true if deg(X)=2 so that the first relevant case for the problem is that of cubic hypersurfaces. One immediately sees that $V(x_0x_3^2 + x_1x_3x_4 + x_2x_4^2)\subset \mathbb P^4$ is a cubic hypersurface with vanishing hessian but not a cone (for example one could check that the first partial derivatives of the equation are linearly independent). As firstly pointed out in [GN], the claim is true for $N\leq 3$ and in general false for every $N\geq 4$. Here we prove that for $N\leq 6$ an irreducible cubic hypersurface with vanishing hessian in $\mathbb P^N$ is either a cone or a scroll in linear spaces tangent to the dual of the image of the polar map of the hypersurface. We also provide canonical forms and a projective characterization of {\it Special Perazzo Cubic Hypersurfaces}, which, a posteriori, exhaust the class of cubic hypersurfaces with vanishing hessian, not cones, for $N\leq 6$. Finally we show by pertinent examples the technical difficulties arising for $N\geq 7$.

math.AG↗

arXiv2014

The stochastic porous media equation in $\R^d$

Existence and uniqueness of solutions to the stochastic porous media equation $dX-\Dψ(X) dt=XdW$ in $\rr^d$ are studied. Here, $W$ is a Wiener process, $ψ$ is a maximal monotone graph in $\rr\times\rr$ such that $ψ(r)\le C|r|^m$, $\ff r\in\rr$, $W$ is a coloured Wiener process. In this general case the dimension is restricted to $d\ge 3$, the main reason being the absence of a convenient multiplier result in the space $\calh=\{φ\in\mathcal{S}'(\rr^d);\ |ξ|(\calfφ)(ξ)\in L^2(\rr^d)\}$, for $d\le2$. When $ψ$ is Lipschitz, the well-posedness, however, holds for all dimensions on the classical Sobolev space $H^{-1}(\rr^d)$. If $ψ(r)r\geρ|r|^{m+1}$ and $m=\frac{d-2}{d+2}$, we prove the finite time extinction with strictly positive probability.

math.PR↗

arXiv2014

Gaussian and non-Gaussian processes of zero power variation, and related stochastic calculus

We consider a class of stochastic processes $X$ defined by $X\left( t\right) =\int_{0}^{T}G\left( t,s\right) dM\left( s\right) $ for $t\in\lbrack0,T]$, where $M$ is a square-integrable continuous martingale and $G$ is a deterministic kernel. Let $m$ be an odd integer. Under the assumption that the quadratic variation $\left[ M\right] $ of $M$ is differentiable with $\mathbf{E}\left[ \left\vert d\left[ M\right] (t)/dt\right\vert ^{m}\right] $ finite, it is shown that the $m$th power variation $$ \lim_{\varepsilon\rightarrow0}\varepsilon^{-1}\int_{0}^{T}ds\left( X\left( s+\varepsilon\right) -X\left( s\right) \right) ^{m} $$ exists and is zero when a quantity $δ^{2}\left( r\right) $ related to the variance of an increment of $M$ over a small interval of length $r$ satisfies $δ\left( r\right) =o\left( r^{1/(2m)}\right) $. When $M$ is the Wiener process, $X$ is Gaussian; the class then includes fractional Brownian motion and other Gaussian processes with or without stationary increments. When $X$ is Gaussian and has stationary increments, $δ$ is $X$'s univariate canonical metric, and the condition on $δ$ is proved to be necessary. In the non-stationary Gaussian case, when $m=3$, the symmetric (generalized Stratonovich) integral is defined, proved to exist, and its Itô formula is established for all functions of class $C^{6}$.

math.PR↗

arXiv2014

A stochastic Fokker-Planck equation and double probabilistic representation for the stochastic porous media type equation

The purpose of the present paper consists in proposing and discussing a double probabilistic representation for a porous media equation in the whole space perturbed by a multiplicative colored noise. For almost all random realizations $ω$, one associates a stochastic differential equation in law with random coefficients, driven by an independent Brownian motion. The key ingredient is a uniqueness lemma for a linear SPDE of Fokker-Planck type with measurable bounded (possibly degenerated) random coefficients.

math.PR↗

arXiv2014

A regularization approach to functional Itô calculus and strong-viscosity solutions to path-dependent PDEs

First, we revisit functional Itô/path-dependent calculus started by B. Dupire, R. Cont and D.-A. Fournié, using the formulation of calculus via regularization. Relations with the corresponding Banach space valued calculus introduced by C. Di Girolami and the second named author are explored. The second part of the paper is devoted to the study of the Kolmogorov type equation associated with the so called window Brownian motion, called path-dependent heat equation, for which well-posedness at the level of classical solutions is established. Then, a notion of strong approximating solution, called strong-viscosity solution, is introduced which is supposed to be a substitution tool to the viscosity solution. For that kind of solution, we also prove existence and uniqueness. The notion of strong-viscosity solution motivates the last part of the paper which is devoted to explore this new concept of solution for general semilinear PDEs in the finite dimensional case. We prove an equivalence result between the classical viscosity solution and the new one. The definition of strong-viscosity solution for semilinear PDEs is inspired by the notion of "good" solution, and it is based again on an approximating procedure.

math.PR↗

arXiv2013

The covariation for Banach space valued processes and applications

This article focuses on a new concept of quadratic variation for processes taking values in a Banach space $B$ and a corresponding covariation. This is more general than the classical one of Métivier and Pellaumail. Those notions are associated with some subspace $χ$ of the dual of the projective tensor product of $B$ with itself. We also introduce the notion of a convolution type process, which is a natural generalization of the Itô process and the concept of $\bar ν_0$-semimartingale, which is a natural extension of the classical notion of semimartingale. The framework is the stochastic calculus via regularization in Banach spaces. Two main applications are mentioned: one related to Clark-Ocone formula for finite quadratic variation processes; the second one concerns the probabilistic representation of a Hilbert valued partial differential equation of Kolmogorov type.

math.PR↗

arXiv2013

The ASTRI SST-2M Prototype: Camera and Electronics

ASTRI is a Flagship Project financed by the Italian Ministry of Education, University and Research, and led by INAF, the Italian National Institute of Astrophysics. The primary goal of the ASTRI project is the realization of an end-to-end prototype of a Small Size Telescope for the Cherenkov Telescope Array. The prototype, named ASTRI SST-2M, is based on a completely new double mirror optics design and will be equipped with a camera made of a matrix of SiPM detectors. Here we describe the ASTRI SST-2M camera concept: basic idea, detectors, electronics, current status and some results coming from experiments in lab.

astro-ph.IM↗

arXiv2013

Second Order PDEs with Dirichlet White Noise Boundary Condition

In this paper we study the Poisson and heat equations on bounded and unbounded domains with smooth boundary with random Dirichlet boundary conditions. The main novelty of this work is a convenient framework for the analysis of such equations excited by the white in time and/or space noise on the boundary. Our approach allows us to show the existence and uniqueness of weak solutions in the space of distributions. Then we prove that the solutions can be identified as smooth functions inside the domain, and finally the rate of their blow up at the boundary is estimated. A large class of noises including Wiener and fractional Wiener space time white noise, homogeneous noise and Lévy noise is considered.

math.PR↗

arXiv2013

BSDEs under partial information and financial applications

In this paper we provide existence and uniqueness results for the solution of BSDEs driven by a general square integrable martingale under partial information. We discuss some special cases where the solution to a BSDE under restricted information can be derived by that related to a problem of a BSDE under full information. In particular, we provide a suitable version of the Föllmer-Schweizer decomposition of a square integrable random variable working under partial information and we use this achievement to investigate the local risk-minimization approach for a semimartingale financial market model.

math.PR↗