Paul Eric Chaudru de Raynal: Smoothing properties of degenerate Kolmogorov Equations of weak Hörmander type: application to regularization by stochastic drift
Abstract. In a seminal work of Zvonkin in 1974, it is shown that infinitesimal non-degenerate stochastic perturbation can restore uniqueness for differential equation. In other words, non-degenerate stochastic differential equations are well posed under a larger set of assumptions than ordinary differential systems. The proof of Zvonkin, which has been widely used since then, consists in exhibiting suitable smoothing properties on the Kolmogorov Partial Differential Equation (PDE) associated to the stochastic process.
In this talk, we consider the case where the noise acts macroscopically: it is added in the drift of the original system. We show that it still regularizes the system and that the Zvonkin Strategy leads us to study the smoothing effects of a degenerate Kolmogorov PDE of weak Hörmander type.
Diego Chamorro: Non Linear Singular Drifts and Fractional Operators: when Besov meets Morrey and Campanato
Abstract. We study the Hölder regularity properties of the solutions of a transport-diffusion equation with nonlinear singular drifts that satisfy a Besov stability property. We will see how this Besov information is relevant and how it allows to improve previous results. Moreover, in some particular cases we show that as the nonlinear drift becomes more regular, in the sense of Morrey-Campanato spaces, the additional Besov stability property will be less useful.
Gennaro Cibelli: Sharp estimates for Geman-Yor processes and applications to Arithmetic Average Asian options
Abstract. In this talk we will present optimal pointwise lower and upper bounds for the fundamental solution of a degenerate second order partial differential equation, related to Geman-Yor stochastic processes, which arises in models for option pricing theory in fi nance. For the achievement of the results we will take advantage of Stochastic Process Theory, Partial Differential Equations of Kolmogorov type and Optimal Control Theory.
In particular, the existence and uniqueness of a smooth fundamental solution is proven by using Malliavin Calculus. The lower bound is obtained by using repeatedly an invariant Harnack inequality and by solving an associated optimal control problem with quadratic cost. The upper one is obtained by the fact that the optimal cost satisfies a specific Hamilton-Jacobi-Bellman equation.
(This talk is based on a joint work with Polidoro S. and Rossi F. (ArXiv: 1610.07838).)
Chiara Guardasoni: Efficient Method for Barrier Option Evaluation
Abstract. We will illustrate an efficient method for the evaluation of barrier options provided that the fundamental solution of the partial differential model problem is known in an exact or approximated form.
The method is based on an integral representation of the exact solution. It has been tested in the classical Black-Scholes framework and then implemented for time-dependent interest rate, volatility and in the Heston stochastic volatility model.
Igor Honoré: Non-asymptotic Gaussian Estimates for the Recursive Approximation of the Invariant Measure of a Diffusion
Abstract. We obtain sharp non-asymptotic Gaussian concentration bounds for the difference between the invariant measure v of an ergodic Brownian diffusion process and the empirical distribution
of an approximating scheme with decreasing time step along a suitable class of (smooth enough) test functions f such that f-v(f) is a coboundary of the infinitesimal generator. We show that these bounds can still be improved when the (squared) Fröbenius norm of the diffusion coefficient also lies in this class. We apply these bounds to design computable non-asymptotic confidence intervals for the approximating scheme. As a theoretical application, we finally derive non-asymptotic deviation bounds for the almost sure Central Limit Theorem.
Elena Issoglio: FBSDEs with distributional coefficients
Abstract. In this talk I will present some recent results about systems of forward-backward stochastic differential equations (FBSDEs) where some of the coefficients are Schwartz distributions, in particular they are elements of a fractional Sobolev space of negative order (with regularity > -1/2). A notion of virtual solution is introduced in order to make sense of the singular integrals that appear in the FBSDE. One of the key tools we use is a theorem of existence, uniqueness and regularity of the solution of a PDEs with distributional coefficients – a singular PDE that plays the role of the Kolmogorov backward equation. In this setting, we investigate existence and uniqueness of a virtual solution for the singular FBSDE and we also show the validity of the so-called non-linear Feynman-Kac formula.
(This talk is based on a joint work with Shuai Jing (ArXiv:1605.01558).)
Valentin Konakov: Random walks in non homogeneous Poissonian environment
Abstract. We consider the moving particle process in Rd which is defined in the following way. There are two independent sequences (Tk) and (dk) of random variables. The variables Tk are non negative and form an increasing sequence, while variables dk form an i.i.d. sequence with common distribution concentrated on the unit sphere. The values dk are interpreted as the directions, and Tk as the moments of change of directions. A particle starts from zero and moves in the direction d1 up to the moment T1. It then changes direction to d2 and moves on within the time interval T2 minus T1, etc. The speed is constant at all sites. The position of the particle at time t is denoted by X(t). We suppose that the points (Tk) form a non homogeneous Poisson point process and we are interested in the global behavior of the process (X(t)), namely, we are looking for conditions under which the processes (Y(T,t), T is non negative), Y(T,t) is X(tT) normalized by B(T), t in (0, 1), weakly converges in C(0, 1) to some process Y when T tends to infinity. In the second part of the paper the process X(t) is considered as a Markov chain. We construct diffusion approximations for this process and investigate their accuracy. The main tool in this part is the parametrix method.
Alberto Lanconelli : Nash estimates and upper bounds for non-homogeneous Kolmogorov equations
Abstract. We prove a Gaussian upper bound for the fundamental solutions of a class of ultra-parabolic equations in divergence form. The bound is independent on the smoothness of the coefficients and generalizes some classical results by Nash, Aronson and Davies. The class considered has relevant applications in the theory of stochastic processes, in physics and in mathematical finance.
Stéphane Menozzi: L^p Estimates For Degenerate Non-Local Kolmogorov Operators
Abstract. We provide L^p estimates for degenerate Kolmogorov operators satisfying a kind of weak Hörmander condition. Using mixed analytical and probabilistic techniques we manage to enter the Coifman and Weiss setting for singular integrals. The corresponding martingale problem will be discussed as well.
(This talk is based on a joint work with L. Huang and E. Priola).
Enzo Orsingher : Parabolic, Hyperbolic and Fractional equations arising in the field of stochastic processes
Abstract. First of all we recall some classical results of probabilistic potential theory and some extensions to Non-Euclidean Spaces.
In the treatment of finite velocity random motions with a finite number of possible directions naturally arise hyperbolic equations of order equal to the number of possible directions. We will give some examples of random motions where equations of higher order emerge and how hyper-Bessel functions play a central role in their analysis.
We will also consider finite-velocity random motions where also hyperbolic equations with non-constant coefficients arise including the Euler-Poisson-Darboux equation.
The last part of the talk is devoted to fractional counterparts of the equations analysed before and include space-time fractional equations whose solutions can be interpreted as the law of stable-processes at random times obtained as inverses of combinations of stable subordinators.
Andrea Pascucci: Local densities and the Taylor formula of implied volatility
Abstract. In a model driven by a multi-dimensional local diffusion, we study the behavior of implied volatility {\sigma} and its derivatives with respect to log-strike k and maturity T near expiry and at the money. We recover explicit limits of these derivatives for (T,k) approaching the origin within the parabolic region |x-k|^2 < {\lambda} T, with x denoting the spot log-price of the underlying asset and where {\lambda} is a positive and arbitrarily large constant. Such limits yield the exact Taylor formula for implied volatility within the parabola |x-k|^2 < {\lambda} T. In order to include important models of interest in mathematical finance, e.g. Heston, CEV, SABR, the analysis is carried out under the assumption that the infinitesimal generator of the diffusion is only locally elliptic..
Enrico Priola: Well-posedness of semilinear stochastic wave equations with Hölder continuous coefficients
Abstract. We prove that semilinear stochastic abstract wave equations, including wave and plate equations, are well-posed in the strong sense with an α-Hölder continuous drift coefficient, if α є (2/3,1).
The uniqueness may fail for the corresponding deterministic PDE and well-posedness is restored by adding an external random forcing of white noise type.This shows a kind of regularization by noise for the semilinear wave equation. To prove the result we introduce an approach based on backward stochastic differential equations. We also establish regularizing properties of the transition semigroup associated to the stochastic wave equation by using control theoretic results.
(This talk is based on a joint work with Federica Masiero.)
Alexandre Veretennikov: On Poisson equations with potentials in the whole space
Abstract. In a series of papers by the author – and others – Poisson equation in the whole space was studied for so called ergodic generators L corresponding to homogeneous Markov diffusions in finite-dimensional Euclidean spaces. Solving such an equation is one of the main tools for the diffusion approximation method in the theory of stochastic averaging and homogenisation. In this talk a similar equation with a potential will be considered, firstly because it is natural for PDEs, and secondly with a hope that it may be also useful for some extensions related to homogenization and averaging.