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Papers by Francesco Russo

arXiv (Cornell University), Apr 28, 2021

arXiv (Cornell University), Jun 13, 2016

HAL (Le Centre pour la Communication Scientifique Directe), Apr 12, 2013

arXiv (Cornell University), Sep 13, 2017

arXiv (Cornell University), Jan 27, 2017

arXiv (Cornell University), Feb 3, 2012

Latin American Journal of Probability and Mathematical Statistics

Stochastic Analysis and Applications, 2019

Probability Theory and Related Fields, 1990

Journal of Functional Analysis, 2005

SIAM Journal on Control and Optimization

The paper is devoted to the construction of a probabilistic particle algorithm. This is related t... more The paper is devoted to the construction of a probabilistic particle algorithm. This is related to nonlin-ear forward Feynman-Kac type equation, which represents the solution of a nonconservative semilinear parabolic Partial Differential Equations (PDE). Illustrations of the efficiency of the algorithm are provided by numerical experiments.

Let (È s,x) (s,x)∈[0,T ]×E be a family of probability measures, where E is a Polish space, define... more Let (È s,x) (s,x)∈[0,T ]×E be a family of probability measures, where E is a Polish space, defined on the canonical probability space ([0, T ], E) of E-valued cadlag functions. We suppose that a martingale problem with respect to a time-inhomogeneous generator a is well-posed. We consider also an associated semilinear Pseudo-PDE for which we introduce a notion of so called decoupled mild solution and study the equivalence with the notion of martingale solution introduced in a companion paper. We also investigate well-posedness for decoupled mild solutions and their relations with a special class of BSDEs without driving martingale. The notion of decoupled mild solution is a good candidate to replace the notion of viscosity solution which is not always suitable when the map a is not a PDE operator. MSC 2010 Classification. 60H30; 60H10; 35S05; 60J35; 60J60; 60J75. KEY WORDS AND PHRASES. Martingale problem; pseudo-PDE; Markov processes; backward stochastic differential equation; decou...

Existence and uniqueness of solutions to the stochastic porous media equation dX −∆ψ(X)dt = XdW i... more Existence and uniqueness of solutions to the stochastic porous media equation dX −∆ψ(X)dt = XdW in R d are studied. Here, W is a Wiener process, ψ is a maximal monotone graph in R × R such that ψ(r) ≤ C|r| m , ∀r ∈ R. In this general case, the dimension is restricted to d ≥ 3, the main reason being the absence of a convenient multiplier result in the space H = {ϕ ∈ S (R d); |ξ|(F ϕ)(ξ) ∈ L 2 (R d)}, for d ≤ 2. When ψ is Lipschitz, the well-posedness, however, holds for all dimensions on the classical Sobolev space H −1 (R d). If ψ(r)r ≥ ρ|r| m+1 and m = d−2 d+2 , we prove the finite time extinction with strictly positive probability. Résumé NousétudionsNous´Nousétudions existence et unicité pour les solutions d'uné equation de milieux poreux dX − ∆ψ(X)dt = XdW dans R d. Ici W est un processus de Wiener, ψ est un graphe maximal monotone dans R × R tel que ψ(r) ≤ C|r| m , ∀r ∈ R. Dans ce contexte général, la dimension est restreintè a d ≥ 3, essentiellement compte tenu de l'ab...

We discuss a class of Backward Stochastic Differential Equations (BSDEs) with no driving martinga... more We discuss a class of Backward Stochastic Differential Equations (BSDEs) with no driving martingale. When the randomness of the driver depends on a general Markov process X, those BSDEs are denominated Markovian BSDEs and can be associated to a deterministic problem, called Pseudo-PDE which constitute the natural generalization of a parabolic semilinear PDE which naturally appears when the underlying filtration is Brownian. We consider two aspects of well-posedness for the Pseudo-PDEs: classical and martingale solutions. MSC 2010 Classification. 60H30; 60H10; 35S05; 60J35; 60J60; 60J75. KEY WORDS AND PHRASES. Martingale problem; pseudo-PDE; Markov processes; backward stochastic differential equation.

About Fokker-Planck equation with measurable coefficients and applications to the fast diffusion ... more About Fokker-Planck equation with measurable coefficients and applications to the fast diffusion equation

Abstract: We consider a new class of random partial dierential equation of parabolic type where t... more Abstract: We consider a new class of random partial dierential equation of parabolic type where the stochastic term is constituted by an irregular noisy drift, not necessarily Gaussian. We provide a suitable interpretation and we study existence. After freezing a realization of the drift (stochastic process), we study existence and uniqueness (in some suitable sense) of the associated parabolic equation and we investigate probabilistic interpretation.

The present paper continues the study of infinite dimensional calcu-lus via regularization, start... more The present paper continues the study of infinite dimensional calcu-lus 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 Hilbert space H, is the sum of a local martingale and a suitable orthogonal process. The new concept is shown to be useful in several contexts and directions. On one side, the mentioned decomposition appears to be a substitute of an Itô’s type formula applied to f(t,X(t)) where f: [0, T] ×H → R is a C0,1 function and, on the other side, the idea of weak Dirichlet process fits the widely used notion of mild solution for stochastic PDE. As a specific application, we provide a verification theorem for stochastic optimal con-trol problems whose state equation is an infinite dimensional stochastic evolution equation.