Oxford-Man Institute

This talk is based on a joint work with Céline Labart. We are interested in this paper in the numerical simulation of solutions to Backward Stochastic Differential Equations. There are several existing methods to handle this problem and one of the main difficulty is always to compute conditional expectations.

Even though our approach can also be applied in the case of the dynamic programmation equation, our starting point is the use of Picard's iterations that we write in a forward way

In order to compute the conditional expectations, we use Wiener Chaos expansions of the underlying random variables. From a practical point of view, we keep only a finite number of terms in the expansions and we get explicit formulas.

We will present numerical experiments and results on the error analysis.

Random wavelet series were introduced in the mid 90s as simple and flexible models that allow to take into account observed statistics of wavelet coefficients in signal and image processing. One of their most interesting properties is that they supply random processes whose pointwise regularity jumps form point to point in a very erratic way, thus supplying examples of multifractal processes.

Interest in such models has been renewed recently under the spur of new applications coming from widely different fields; e.g.

-in functional analysis, they allow to derive the regularity properties of ``generic'' functions in a given function space (in the sense of

prevalence)

-they offer toy examples on which one can check the accuracy of numerical algorithms that allow to derive the multifractal parameters associated with signals and images.

We will give an overview of these properties, and we will focus on recent extensions whose sample paths are not locally bounded, and offer models for signals which share this property.

Free probability theory has been introduced by Voiculescu in the 80's for the study of the von Neumann algebras of the free groups. It consists in an algebraic setting of non commutative probability, where one encodes "non commutative random variables" in abstract (non commutative) algebras endowed with linear forms (which satisfies properties in order to play the role of the expectation). In this context, Voiculescu introduce the notion of freeness which is the analogue of the classical independence.

A decade later, he realized that a family of independent random matrices invariant in law by conjugation by unitary matrices are asymptotically free. This phenomenon is called asymptotic freeness. It had a deep impact in operator algebra and probability and has been generalized in many directions. A simple particular case of Voiculescu's theorem gives an estimate, for N large, of the spectrum of an N by N Hermitian matrix H_N = A_N + 1/\sqrt N X_N, where A_N is a given deterministic Hermitian matrix and X_N has independent gaussian standard sub-diagonal entries.

Nevertheless, it turns out that asymptotic freeness does not hold in certain situations, e.g. when the entries of X_N as above have heavy-tails. To infer the spectrum of a larger class of matrices, we go further into Voiculescu's approach and introduce the distributions of traffics and their free product. This notion of distribution is richer than Voiculescu's notion of distribution of non commutative random variables and it generalizes the notion of law of a random graph. The notion of freeness for traffics is an intriguing mixing between the classical independence and Voiculescu's notion of freeness. We prove an asymptotic freeness theorem in that context for independent random matrices invariant in law by conjugation by permutation matrices.

The purpose of this talk is to give an introductory presentation of these notions.

Convergence of the Bayes posterior measure is considered in canonical statistical settings (like density estimation or nonparametric regression) where observations sit on a geometrical object such as a compact manifold, or more generally on a compact metric space verifying some conditions.

A natural geometric prior based on randomly rescaled solutions of the heat equation is considered. Upper and lower bound posterior contraction rates are derived.

Jump processes, and Lévy processes in particular, are notoriously difficult to simulate. The task becomes even harder if the process is stopped when it crosses a certain boundary, which happens in applications to barrier option pricing or structural credit risk models. In this talk, I will present novel adaptive discretization

schemes for the simulation of stopped Lévy processes, which are several orders of magnitude faster than the traditional approaches based on uniform discretization, and provide an explicit control of the bias. The schemes are based on sharp asymptotic estimates for the exit probability and work by recursively adding discretization dates in the parts of the trajectory which are close to the boundary, until a specified error tolerance is met.

This is a joint work with Jose Figueroa-Lopez (Purdue).

In this talk we consider two questions about conformally invariant random curves known as Schramm-Loewner evolutions (SLE). The first question is about the "boundary zig-zags", i.e. the probabilities for a chordal SLE to pass through small neighborhoods of given boundary points in a given order. The second question is that of obtaining explicit descriptions of "multiple SLE pure geometries", i.e. those extremal multiple SLE probability measures which can not be expressed as non-trivial convex combinations of other multiple SLEs. For both problems one needs to find solutions of a system of partial differential equations with asymptotics conditions written recursively in terms of solution of the same problem with a smaller number of variables. We present a general correspondence, which translates these problems to linear systems of equations in finite dimensional representations of the quantum group U_q(sl_2), and we then explicitly solve these systems. The talk is based on joint works with Eveliina Peltola (Helsinki), and with Niko Jokela (Santiago de Compostela) and Matti Järvinen (Crete).

We will first review the return to equilibrium of the Ising model when a small external field is applied. The relaxation time is extremely long and can be estimated as the time needed to create critical droplets of the stable phase which will invade the whole system. We will then discuss the impact of disorder on this metastable behavior and show that for Ising model with random interactions (dilution of the couplings) the relaxation time is much faster as the disorder acts as a catalyst. In the last part of the talk, we will focus on the droplet growth and study a toy model describing interface motion in disordered media.

Abstract: The aim of this lecture is to give a general introduction to the theory of interacting particle methods and an overview of its applications to numerical finance. We survey the main techniques and results on interacting particle systems and explain how they can be applied to deal with a variety of financial numerical problems such as: pricing complex path dependent European options, computing sensitivities, American option pricing or solving numerically partially observed control problems.

In many models of Applied Probability, the distributional limits of recursively defined quantities satisfy distributional identities that are reminiscent of equations of stability. Therefore, there is an interest in generalized concepts of equations of stability.

One extension of this concept is that of random variables ``stable by random weighted mean'' (this notion is due to Liu).

A random variable $X$ taking values in $\mathbb{R}^d$ is called ``stable by random weighted mean'' if it satisfies a recursive distributional equation of the following type:

\begin{equation} \tag{1} \label{eq:1}

X ~\stackrel{\mathcal{D}}{=}~ C + \sum_{j \geq 1} T_j X_j.

\end{equation}

Here, ``$\stackrel{\mathcal{D}}{=}$'' denotes equality of the corresponding distributions, $(C,T_1,T_2,\ldots)$ is a given sequence of real-valued random variables,

and $X_1, X_2, \ldots$ denotes a sequence of i.i.d.\;copies of the random variable $X$ that are independent of $(C,T_1,T_2,\ldots)$.

The distributions $P$ on $\mathbb{R}^d$ such that \eqref{eq:1} holds when $X$ has distribution $P$ are called fixed points of the smoothing transform

(associated with $(C,T_1,T_2,\ldots)$).

A particularly prominent instance of \eqref{eq:1} is the {\texttt Quicksort} equation, where $T_1 = 1-T_2 = U \sim \mathrm{Unif}(0,1)$, $T_j = 0$ for all $j \geq 3$ and $C = g(U)$ for some function $g$.

In this talk, I start with the {\texttt Quicksort} algorithm to motivate the study of \eqref{eq:1}.

Then, I consider the problem of characterizing the set of all solutions to \eqref{eq:1}

in a very general context.

Special emphasis is put on \emph{endogenous} solutions to \eqref{eq:1} since they play an important role in the given setting.

Abstract: Non-geometric rough paths arise
when one encounters stochastic integrals for which the the classical
integration by parts formula does not hold. We will introduce two notions of
non-geometric rough paths - one old (branched rough paths) and one new (quasi
geometric rough paths). The former (due to Gubinelli) assumes one knows nothing
about products of integrals, instead those products must be postulated as new
components of the rough path. The latter assumes one knows a bit about
products, namely that they satisfy a natural generalisation of the
"Ito" integration by parts formula. We will show why they are both
reasonable frameworks for a large class of integrals. Moreover, we will show
that Ito's formula can be derived in either framework and that this derivation
is completely algebraic. Finally, we will show that both types of non-geometric
rough path can be re-written as geometric rough paths living above an extended
version of the original path. This means that every non-geometric rough
differential equation can be re-written as a geometric rough differential
equation, hence generalising the Ito-Stratonovich correction formula.