Seminari del Dipartimento di Matematica

nell'ambito della serie: SEMINARI DI PROBABILITÀ E STATISTICA MATEMATICA

probabilità

In this talk we will discuss the validity of Harnack inequalities for linear evolution equations modelled after the Kolmogorov operator. The main focus will be on nondivergence form equations with non-smooth coefficients, and on the absence of an analogue of the ABP maximum principle and of the Krylov-Safonov theory in this setting. In particular, we will highlight as the (very general) crucial ingredient some quantitative point-to-measure estimate for nonnegative subsolutions. With this perspective in mind, we will show a potential theory approach (established in a 2019 joint work with F. Abedin) which allows to prove invariant Harnack inequalities for Kolmogorov-type operators with coefficients satisfying either a Cordes-Landis assumption or a continuity hypothesis.

Maggio

06

2021

Aernout van Enter

One-sided versus two-sided stochastic processes

nell'ambito della serie: SEMINARI DI PROBABILITÀ E STATISTICA MATEMATICA

probabilità

Stochastic processes can be parametrised by time (such as occurs in Markov chains), in which case conditioning is one-sided (on the past) or by one-dimensional space (which is the case, for example, for one-dimensional Markov fields), where conditioning is two-sided (on the right and on the left). I will discuss some examples, in particular generalising this distinction to g-measures versus Gibbs measures, where, instead of a Markovian dependence, the weaker property of continuity (in the product topology) is considered. In particular I will discuss when the two descriptions (one-sided or two-sided) produce the same objects and when they are different. We show moreover the role one-dimensional entropic repulsion plays in this setting. Based on joint work with R. Bissacot, E. Endo and A. Le Ny, and S. Shlosman

A Kolmogorov PDE with non-linear drift: analysis and applications to stochastic filtering

nell'ambito della serie: SEMINARI DI PROBABILITÀ E STATISTICA MATEMATICA

probabilità

BACKWARD SDES, MARTINGALE PROBLEMS AND APPLICATIONS TO MATHEMATICAL FINANCE

nell'ambito della serie: SEMINARI DI PROBABILITÀ E STATISTICA MATEMATICA

probabilità

The aim of this talk consists in introducing a formalism for the deterministic analysis associated with backward stochastic differential equations driven by general càdlàg martingales, coupled with a forward process. When the martingale is a standard Brownian motion, the natural deterministic analysis is provided by the solution u of a semilinear PDE of parabolic type coupled with a function v which is associated with the gradient ∇u, when u is of class C1 in space. When u is only a viscosity solution of the PDE, the link associating v to u is not completely clear: sometimes in the literature it is called the identification problem. We introduce in particular the notion of a decoupled mild solution of a PDE, a IPDE, a path-dependent PDE or more generally a deterministic problem associated with a BSDE. The idea is to introduce a suitable analysis to investigate the equivalent of the identification problem first in a general Markovian setting with a class of examples. An interesting application concerns the hedging problem under basis risk of a contingent claim g(XT,ST ), where S (resp. X) is an underlying price of a traded (resp. non-traded but observable) asset, via the celebrated Föllmer-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. Extensions to non-Markovian (path-dependent) cases are discussed.

On stochastic differential equations

nell'ambito della serie: SEMINARI DI PROBABILITÀ E STATISTICA MATEMATICA

probabilità

Thermodynamic limit and phase transitions in non-cooperative games: some mean-ﬁeld examples

nell'ambito della serie: SEMINARI DI PROBABILITÀ E STATISTICA MATEMATICA

probabilità

In stochastic dynamics inspired by Statistical Mechanics the interaction between diﬀerent particles, or agents, is usually expressed as a given function of their states. The behavior of the system, in the limit of inﬁnitely many particles (thermodynamic limit), may change dramatically by small changes in the parameters of the model: when this occurs we say there is a phase transition. In many applications the interaction cannot be given a priori but it is rather a result of agents’ strategy, aimed at optimizing a given performance. Using the simplest models of this nature, mean ﬁeld games, we illustrate some examples of phase transitions, pointing to diﬃculties in the proof of the thermodynamic limit.

Revolving random walks in 2D: analysis of type and winding number.

nell'ambito della serie: SEMINARI DI PROBABILITÀ E STATISTICA MATEMATICA

probabilità

We consider simple random walks on two directed versions of the $\mathbb{Z}^2$ lattice; one characteristic feature of these random walks is that they are bound to revolve, according to the orientation prescribed by the edges. The first model was studied by Campanino and Petritis(‘03) and shown to be transient; the other one appeared recently in a paper by Menshikov et al. (’17), where the authors conjectured its recurrence. We shall indeed confirm this conjecture: our proof is done by considering a continuous analogue of the random walk and then applying the Lyapunov function criteria. On the other hand, we obtain a local limit theorem for the return probabilities of the first random walk, which in particular gives a new proof of transience. Finally, we deduce some results related to the winding number for both random walks. This results are joint work with Y.Peres (Microsoft Research, Redmond) and Y.Hu (University of Washington).

Novembre

15

2018

Gianluca Bosi, Università di Bologna

Revolving random walks in 2D: analysis of type and winding number.

nell'ambito della serie: SEMINARI DI PROBABILITÀ E STATISTICA MATEMATICA

probabilità

We consider simple random walks on two directed versions of the $\mathbb{Z}^2$ lattice; one characteristic feature of these random walks is that they are bound to revolve, according to the orientation prescribed by the edges. The first model was studied by Campanino and Petritis(‘03) and shown to be transient; the other one appeared recently in a paper by Menshikov et al. (’17), where the authors conjectured its recurrence. We shall indeed confirm this conjecture: our proof is done by considering a continuous analogue of the random walk and then applying the Lyapunov function criteria. On the other hand, we obtain a local limit theorem for the return probabilities of the first random walk, which in particular gives a new proof of transience. Finally, we deduce some results related to the winding number for both random walks. This results are joint work with Y.Peres (Microsoft Research, Redmond) and Y.Hu (University of Washington).

Novembre

07

2018

Claudio Macci, Dipartimento di Matematica, Università di Roma Tor Vergata

Asymptotic behavior of some hitting probabilities for sums of IID Gaussian random sets

nell'ambito della serie: SEMINARI DI PROBABILITÀ E STATISTICA MATEMATICA

probabilità

We consider sequences of partial sums of iid Gaussian random sets (with respect to the Minkowski sum) and we study the asymptotic behavior of some hitting probabilities (of suitable sets of $R^d$) for these partial sums. We also illustrate the use of the importance sampling for the estimation of these hitting probabilities by Monte Carlo simulations. We obtain the analog of well-known results for level crossing probabilities of random walks, and we refer to a version of the classical Cramér's Theorem in large deviations for random compact sets existing in the literature. Joint work with Barbara Pacchiarotti.

Settembre

27

2018

Pietro Rigo - Università di Pavia

CLASSICAL VERSUS COHERENT CONDITIONAL PROBABILITIES

nell'ambito della serie: SEMINARI DI PROBABILITÀ E STATISTICA MATEMATICA

probabilità

The standard (Kolmogorovian) notion of conditional probability is compared with the (de Finettian) notion based on the coherence principle. Each notion has both merits and drawbacks, and the talk aims to highlight them with special attention to the connection points. Various results, both old and new, are discussed and some open problems are mentioned. The talk is split into four parts. (i) Classical (Kolmogorovian) conditional probabilities; (ii) Disintegrability; (iii) Coherent (de Finettian) conditional probabilities; (iv) Some statistical implications.

Febbraio

28

2018

Davide Dardari, Dipartimento di Ingegneria dell'Energia Elettrica e dell'Informazione "Guglielmo Marconi"

Probabilità e Informazione

nell'ambito della serie: SEMINARI DI PROBABILITÀ E STATISTICA MATEMATICA

probabilità

Gennaio

24

2018

Alberto Lanconelli, Università degli Studi di Bari

On Wong - Zakai approximation theorems for Ito stochastic differential equations

nell'ambito della serie: SEMINARI DI PROBABILITÀ E STATISTICA MATEMATICA

probabilità

Matematica in finanza

nell'ambito della serie: SEMINARI DI PROBABILITÀ E STATISTICA MATEMATICA

finanza matematica

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