# Francesco RussoENSTA Institut Polytechnique de Paris · Unité de Mathématiques Appliquées

Francesco Russo

PhD

## About

179

Publications

15,474

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Introduction

I am involved in research in the following fields.
- Probabilistic representations of non-conservative PDEs via McKean type equations.
- Backward Stochastic Differential equations and Pseudo-Partial Differential Equations
- Stochastic differential equations with distributional drift.
- Stochastic calculus via regularization.
- Path-dependent partial differential equations
- Mathematical finance.

Additional affiliations

September 2010 - July 2015

Education

January 1983 - December 1987

September 1978 - January 1983

## Publications

Publications (179)

We examine the relation between a stochastic version of the rough path integral with the symmetric-Stratonovich integral in the sense of regularization. Under mild regularity conditions in the sense of Malliavin calculus, we establish equality between stochastic rough path and symmetric-Stratonovich integrals driven by a class of Gaussian processes...

In this paper we explain how the notion of "weak Dirichlet process" is the suitable generalization of the one of semimartingale with jumps. For such a process we provide a unique decomposition which is new also for semimartingales: in particular we introduce "characteristics" for weak Dirichlet processes. We also introduce a weak concept (in law) o...

Usually Fokker–Planck type partial differential equations (PDEs) are well-posed if the initial condition is specified. In this paper, alternatively, we consider the inverse problem which consists in prescribing final data: in particular we give sufficient conditions for uniqueness. In the second part of the paper we provide a probabilistic represen...

We propose a fully backward representation of semilinear PDEs with application to stochastic control. Based on this, we develop a fully backward Monte-Carlo scheme allowing to generate the regression grid, backwardly in time, as the value function is computed. This offers two key advantages in terms of computational efficiency and memory. First, th...

We investigate existence and uniqueness for a new 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 forward BSDEs and can be associated to a deter-ministic problem, called Pseudo-PDE which constitute the natural...

We establish the existence of smooth densities for solutions to a broad class of path-dependent SDEs under a Hörmander-type condition. The classical scheme based on the reduced Malliavin matrix turns out to be unavailable in the path-dependent context. We approach the problem by lifting the given n-dimensional path-dependent SDE into a suitable Lp-...

The paper investigates existence and uniqueness for a stochastic differential equation (SDE) with distributional drift depending on the law density of the solution. Those equations are known as McKean SDEs. The McKean SDE is interpreted in the sense of a suitable singular martingale problem. A key tool used in the investigation is the study of the...

We prove existence and uniqueness of Crandall-Lions viscosity solutions of Hamilton-Jacobi-Bellman equations in the space of continuous paths, associated to the optimal control of path-dependent SDEs. This seems the first uniqueness result in such a context. More precisely, similarly to the seminal paper of P.L. Lions, the proof of our core result,...

Calculus via regularizations and rough paths are two methods to approach stochastic integration and calculus close to pathwise calculus. The origin of rough paths theory is purely deterministic, calculus via regularization is based on deterministic techniques but there is still a probability in the background. The goal of this paper is to establish...

We propose a fully backward representation of semilinear PDEs with application to stochastic control. Based on this, we develop a fully backward Monte-Carlo scheme allowing to generate the regression grid, backwardly in time, as the value function is computed. This offers two key advantages in terms of computational efficiency and memory. First, th...

The paper presents the study on the existence and uniqueness (strong and in law) of a class of non-Markovian SDEs whose drift contains the derivative in the sense of distributions of a continuous function.

This paper presents a partial state of the art about the topic of representation of generalized Fokker-Planck Partial Differential Equations (PDEs) by solutions of McKean Feynman-Kac Equations (MFKEs) that generalize the notion of McKean Stochastic Differential Equations (MSDEs). While MSDEs can be related to non-linear Fokker-Planck PDEs, MFKEs ca...

In this paper, we focus on the so-called identification problem for a BSDE driven by a continuous local martingale and a possibly non-quasi-left-continuous random measure. Supposing that a solution [Formula: see text] of a BSDE is such that [Formula: see text] where [Formula: see text] is an underlying process and [Formula: see text] is a determini...

We establish the existence of smooth densities for solutions to a broad class of path-dependent SDEs under a H\"ormander-type condition. The classical scheme based on the reduced Malliavin matrix turns out to be unavailable in the path-dependent context. We approach the problem by lifting the given $n$-dimensional path-dependent SDE into a suitable...

In this paper, we present a Longstaff-Schwartz-type algorithm for the discretization method designed in Le\~ao, Ohashi and Russo [28]. In contrast to previous works, our methodology applies to optimal stopping problems for fully non-Markovian and non-semimartingale state processes. Based on statistical learning theory techniques, we provide overall...

We focus on a class of path-dependent problems which include path-dependent (possibly Integro) PDEs, and their representation via BSDEs driven by a cadlag martingale. For those equations we introduce the notion of decoupled mild solution for which, under general assumptions, we study existence and uniqueness and its representation via the afore men...

Usually Fokker-Planck type partial differential equations (PDEs) are well-posed if the initial condition is specified. In this paper, alternatively, we consider the inverse problem which consists in prescribing final data: in particular we give sufficient conditions for existence and uniqueness. In the second part of the paper we provide a probabil...

This paper investigates some one-dimensional path-dependent SDEs, which includes an irregular (distributional) drift b' depending on the present position. We treat essentially two cases: the first one concerns the case when the drift b' is the derivative of a continuous function, the second one when b ' is the derivative of a logarithmic or an Heav...

In this paper we focus on the so called identification problem for a backward SDE driven by a continuous local martingale and a possibly non quasi-left-continuous random measure. Supposing that a solution (Y, Z, U) of a backward SDE is such that $Y(t) = v(t, X(t))$ where X is an underlying process and v is a deterministic function, solving the iden...

This paper presents a partial state of the art about the topic of representation of generalized Fokker-Planck Partial Differential Equations (PDEs) by solutions of McKean Feynman-Kac Equations (MFKEs) that generalize the notion of McKean Stochastic Differential Equations (MSDEs). While MSDEs can be related to non-linear Fokker-Planck PDEs, MFKEs ca...

We address our interest to the development of a theory of viscosity solutions {\`a} la Crandall-Lions for path-dependent partial differential equations (PDEs), namely PDEs in the space of continuous paths C([0, T ]; R^d). Path-dependent PDEs can play a central role in the study of certain classes of optimal control problems, as for instance optimal...

In this paper we investigate BSDEs where the driver contains a distributional term (in the sense of generalised functions) and derive general Feynman-Kac formulae related to these BSDEs. We introduce an integral operator to give sense to the equation and then we show the existence of a strong solution employing results on a related PDE. Due to the...

We are interested in path-dependent semilinear PDEs, where the derivatives are of G{\^a}teaux type in specific directions k and b, being the kernel functions of a Volterra Gaussian process X. Under some conditions on k, b and the coefficients of the PDE, we prove existence and uniqueness of a decoupled mild solution, a notion introduced in a previo...

The aim of the present work is the introduction of a viscosity type solution, called strongviscosity solution emphasizing also a similarity with the existing notion of strong solution in the literature. It has the following peculiarities: it is a purely analytic object; it can be easily adapted to more general equations than classical partial diffe...

The paper introduces and investigates the natural extension to the path-dependent setup of the usual concept of canonical Markov class introduced by Dynkin and which is at the basis of the theory of Markov processes. That extension, indexed by starting paths rather than starting points, will be called path-dependent canonical class. Associated with...

We analyze the well-posedness of a so called McKean Feynman-Kac Equation (MFKE), which is a McKean type equation with a Feynman-Kac perturbation. We provide in particular weak and strong existence conditions as well as pathwise uniqueness conditions without strong regularity assumptions on the coefficients. One major tool to establish this result i...

An overview of my human and scientific relations with my PhD advisor, Srishti Dhar Chatterji.

Two generalizations of It{\^o} formula to infinite-dimensional spaces are
given. The first one, in Hilbert spaces, extends the classical one by taking
advantage of cancellations, when they occur in examples and it is applied to
the case of a group generator. The second one, based on the previous one and a
limit procedure, is an It{\^o} formula in a...

We are concerned with BSDEs where the driver contains a distributional term (in the sense of generalised functions). We introduce an integral operator to give sense to the equation and then we show the existence of a strong solution. Because of the irregularity of the driver, the Y-component of a couple (Y, Z) solving the BSDE is not necessarily a...

The paper introduces and investigates the natural extension to the path-dependent setup of the usual concept of canonical Markov class introduced by Dynkin and which is at the basis of the theory of Markov processes. That extension, indexed by starting paths rather than starting points will be called path-dependent canonical class. Associated with...

This paper investigates two existence theorems for the path-dependent heat equation, which is the Kolmogorov equation related to the window Brownian motion, considered as a C([--T, 0])-valued process. We concentrate on two general existence results of its classical solutions related to different classes of final conditions: the first one is given b...

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.

The purpose of the present note consists of first showing a uniqueness result for a stochastic Fokker-Planck equation under very general assumptions. In particular, the second order coefficients may be just measurable and degenerate. We also provide a proof for uniqueness of a stochastic porous media equation in a fairly large space.

This note develops shortly the theory of time-inhomogeneous additive functionals and is a useful support for the analysis of time-dependent Markov processes and related topics. It is a significant tool for the analysis of BSDEs in law. In particular we extend to a non-homogeneous setup some results concerning the quadratic variation and the angular...

We focus on a class of BSDEs driven by a càdlàg martingale and the corresponding Markovian BSDEs which arise when the randomness of the driver appears through a Markov process. To those BSDEs we associate a deterministic equation which, when the Markov process is a Brownian diffusion, is nothing else but a parabolic semi-linear PDE. We prove existe...

In this paper, we present an approximation scheme to solve optimal stopping problems based on fully non-Markovian reward continuous processes adapted to the filtration generated by the multi-dimensional Brownian motion. The approximations satisfy suitable variational inequalities which allow us to construct $\epsilon$-optimal stopping times and opt...

Let $(\mathbb{P}^{s,x})_{(s,x)\in[0,T]\times E}$ be a family of probability measures, where $E$ is a Polish space, defined on the canonical probability space ${\mathbb D}([0,T],E)$ 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 sem...

We discuss numerical aspects related to a new class of nonlinear Stochastic Differential Equations in the sense of McKean, which are supposed to represent non conservative nonlinear Partial Differential equations (PDEs). We propose an original interacting particle system for which we discuss the propagation of chaos. We consider a time-discretized...

A stochastic optimal control problem driven by an abstract evolution equation in a separable Hilbert space is considered. Thanks to the identification of the mild solution of the state equation as $\nu$-weak Dirichlet process, the value processes is proved to be a real weak Dirichlet process. The uniqueness of the corresponding decomposition is use...

This paper investigates existence results for path-dependent differential equations driven by a H{\"o}lder function where the integrals are understood in the Young sense. The two main results are proved via an application of Schauder theorem and the vector field is allowed to be unbounded. The H{\"o}lder function is typically the trajectory of a st...

First, we revisit basic theory of functional Itô/path-dependent calculus, using the formulation of calculus via regularization. Relations with the corresponding Banach space valued calculus 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...

We propose a nonlinear forward Feynman-Kac type equation, which represents the solution of a non-conservative semilinear parabolic Partial Differential Equations (PDE). We show in particular existence and uniqueness in the first part of the article. The second part is devoted to the construction of a probabilistic particle algorithm and the proof o...

The purpose of the present paper consists in proposing and discussing a doubly probabilistic representation for a stochastic porous media equation in the whole space R^1 perturbed by a multiplicative coloured noise. For almost all random realizations $\omega$, one associates a stochastic differential equation in law with random coefficients, driven...

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 we...

The aim of this paper is to introduce a new formalism for the deterministic analysis associated with backward stochastic differential equations driven by general càdlàg 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 a...

The paper reminds the basic ideas of stochastic calculus via regularizations in Banach spaces and its applications to the studyof 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 str...

This paper considers a forward BSDE driven by a random measure, when the
underlying forward process X is special semimartingale, or even more generally,
a special weak Dirichlet process. Given a solution (Y, Z, U), generally Y
appears to be of the type u(t, X\_t) where u is a deterministic function. In
this paper we identify Z and U in terms of u a...

This paper develops systematically stochastic calculus via regularization in
the case of jump processes. In particular one continues the analysis of
real-valued c\`adl\`ag weak Dirichlet processes with respect to a given
filtration. Such a process is the sum of a local martingale and an adapted
process A such that $[N, A] = 0$, for any continuous l...

In this work we connect the theory of symmetric 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 identifie...

Functional It\^o 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...

The aim of the present work is the introduction of a viscosity type solution,
called strong-viscosity solution to distinguish it from the classical one, with
the following peculiarities: it is a purely analytic object; it can be easily
adapted to more general equations than classical partial differential
equations. First, we introduce the notion of...

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.

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 cha...

This book presents in thirteen refereed survey articles an overview of modern activity in stochastic analysis, written by leading international experts. The topics addressed include stochastic fluid dynamics and regularization by noise of deterministic dynamical systems; stochastic partial differential equations driven by Gaussian or Lévy noise, in...

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 st...

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{\`a}dl{\`a}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 signi...

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 provid...

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 di...

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...

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 $\omega$, one associates a stochastic differential equation in law
with random coefficients, driven by an independe...

In this paper we provide Galtchouk-Kunita-Watanabe representation results in
the case where there are restrictions on the available information. This allows
to prove existence and uniqueness for linear backward stochastic differential
equations driven by a general c\`adl\`ag martingale under partial information.
Furthermore, we discuss an applicati...

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.

First, we revisit functional It\^o/path-dependent calculus started by B.
Dupire, R. Cont and D.-A. Fourni\'e, 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...

Nous étudions existence et unicité pour les solutions d'une équation de milieux poreux dX−Δψ(X)dt=XdWdX−Δψ(X)dt=XdW dans RdRd. Ici W est un processus de Wiener, ψ est un graphe maximal monotone dans R×RR×R tel que ψ(r)≤C|r|mψ(r)≤C|r|m, ∀r∈R∀r∈R. Dans ce contexte général, la dimension est restreinte à d≥3d≥3, essentiellement compte tenu de l'absence...

Existence and uniqueness of solutions to the stochastic porous media equation
$dX-\D\psi(X) dt=XdW$ in $\rr^d$ are studied. Here, $W$ is a Wiener process,
$\psi$ is a maximal monotone graph in $\rr\times\rr$ such that $\psi(r)\le
C|r|^m$, $\ff r\in\rr$, $W$ is a coloured Wiener process. In this general case
the dimension is restricted to $d\ge 3$,...

This paper focuses on the valuation and hedging of gas storage facilities,
using a spot-based valuation framework coupled with a financial hedging
strategy implemented with futures contracts. The first novelty consist in
proposing a model that unifies the dynamics of the futures curve and the spot
price, which accounts for the main stylized facts o...

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 specia...

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 existenc...

The purpose of this paper consists in proposing a generalized solution for a
porous media type equation on a half-line with Neumann boundary condition and
prove a probabilistic representation of this solution in terms of an associated
microscopic diffusion. The main idea is to construct a stochastic differential
equation with reflection which has a...

The object of this paper is a multi-dimensional generalized porous media equation (PDE) with not smooth and possibly discontinuous coefficient
$\beta $
, which is well-posed as an evolution problem in
$L^1(\mathbb R ^d)$
. This work continues the study related to the one-dimensional case by the same authors. One expects that a solution of the m...

For a large class of vanilla contingent claims, we establish an explicit
F\"ollmer-Schweizer decomposition when the underlying is an exponential of an
additive process. This allows to provide an efficient algorithm for solving the
mean variance hedging problem. Applications to models derived from the
electricity market are performed.

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 M\'etivier and Pellaumail. Those notions
are associated with some subspace $\chi$ of the dual of the projective tensor
product of $B$ with itself. We also intr...

This book presents refereed research or review articles presented at the 7th Seminar on Stochastic Analysis, Random Fields and Applications, which was held at the Centro Stefano Franscini (Monte Verità) in Ascona, Switzerland, in May 2011. The seminar mainly focused on:
• stochastic (partial) differential equations, especially with regard to jump p...

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 Hilbert space $H$, is the sum of a local
martingale and a suitable "orthogonal" process. The new...

We consider the discretized version of a (continuous-time) two-factor model introduced by Benth and coauthors for the electricity markets. For this model, the underlying is the exponent of a sum of independent random variables. We provide and test an algorithm, which is based on the celebrated Foellmer-Schweizer decomposition for solving the mean-v...

Given a process with independent increments $X$ (not necessarily a
martingale) and a large class of square integrable r.v. $H=f(X_T)$, $f$ being
the Fourier transform of a finite measure $\mu$, we provide explicit
Kunita-Watanabe and F\"ollmer-Schweizer decompositions. The representation is
expressed by means of two significant maps: the expectatio...

The object of this paper is the uniqueness for a $d$-dimensional
Fokker-Planck type equation with non-homogeneous (possibly degenerated)
measurable not necessarily bounded coefficients. We provide an application to
the probabilistic representation of the so called Barenblatt solution of the
fast diffusion equation which is the partial differential...

This paper concerns a class of Banach valued processes which have finite
quadratic variation. The notion introduced here generalizes the classical one,
of M\'etivier and Pellaumail which is quite restrictive. We make use of the
notion of $\chi$-covariation which is a generalized notion of covariation for
processes with values in two Banach spaces $...

This paper does not suppose a priori that the evolution of the price of a
financial asset is a semimartingale. Since possible strategies of investors are
self-financing, previous prices are forced to be finite quadratic variation
processes. The non-arbitrage property is not excluded if the class
$\mathcal{A}$ of admissible strategies is restricted....

This paper discusses a new notion of quadratic variation and covariation for
Banach space valued processes (not necessarily semimartingales) and related
It\^o formula. If $\X$ and $\Y$ take respectively values in Banach spaces
$B_{1}$ and $B_{2}$ and $\chi$ is a suitable subspace of the dual of the
projective tensor product of $B_{1}$ and $B_{2}$ (...

We develop a stochastic analysis for a Gaussian process $X$ with singular covariance by an intrinsic procedure focusing on several examples such as covariance measure structure processes, bifractional Brownian motion, processes with stationary increments. We introduce some new spaces associated with the self-reproducing kernel space and we define t...

The object of this paper is a one-dimensional generalized porous media equation (PDE) with possibly discontinuous coefficient $\beta$, which is well-posed as an evolution problem in $L^1(\mathbb{R})$. In some recent papers of Blanchard et alia and Barbu et alia, the solution was represented by the solution of a non-linear stochastic differential eq...

We provide a suitable framework for the concept of finite quadratic variation for processes with values in a separable Banach space $B$ using the language of stochastic calculus via regularizations, introduced in the case $B= \R$ by the second author and P. Vallois. To a real continuous process $X$ we associate the Banach valued process $X(\cdot)$,...

This paper develops some aspects of stochastic calculus via regularization to Banach valued processes. An original concept of $\chi$-quadratic variation is introduced, where $\chi$ is a subspace of the dual of a tensor product $B \otimes B$ where $B$ is the values space of some process $X$ process. Particular interest is devoted to the case when $B...

This paper considers the class of stochastic processes $X$ which are Volterra convolutions of a martingale $M$. When $M$ is Brownian motion, $X$ is Gaussian, and the class includes fractional Brownian motion and other Gaussian processes with or without homogeneous increments. Let $m$ be an odd integer. Under some technical conditions on the quadrat...

For a large class of vanilla contingent claims, we establish an explicit F\"ollmer-Schweizer decomposition when the underlying is a process with independent increments (PII) and an exponential of a PII process. This allows to provide an efficient algorithm for solving the mean variance hedging problem. Applications to models derived from the electr...

We consider a possibly degenerate porous media type equation over all of $\R^d$ with $d = 1$, with monotone discontinuous coefficients with linear growth and prove a probabilistic representation of its s