Past Stochastic Analysis Seminar

When cast in a Bayesian setting, the solution to an inverse problem is given as a distribution over the space where the quantity of interest lives. When the quantity of interest is in principle a field then the discretization is very high-dimensional. Formulating algorithms which are defined in function space yields dimension-independent algorithms, which overcome the so-called curse of dimensionality. These algorithms are still often too expensive to implement in practice but can be effectively used offline and on toy-models in order to benchmark the ability of inexpensive approximate alternatives to quantify uncertainty in very high-dimensional problems. Inspired by the recent development of pCN and other function-space samplers [1], and also the recent independent development of Riemann manifold methods [2] and stochastic Newton methods [3], we propose a class of algorithms [4,5] which combine the benefits of both, yielding various dimension-independent and likelihood-informed (DILI) sampling algorithms. These algorithms can be effective at sampling from very high-dimensional posterior distributions.

[1] S.L. Cotter, G.O. Roberts, A.M. Stuart, D. White. "MCMC methods for functions: modifying old algorithms to make them faster," Statistical Science (2013).

[2] M. Girolami, B. Calderhead. "Riemann manifold Langevin and Hamiltonian Monte Carlo methods," Journal of the Royal Statistical Society: Series B (Statistical Methodology) 73 (2), 123–214 (2011).

[3] J. Martin, L. Wilcox, C. Burstedde, O. Ghattas. "A stochastic newton mcmc method for large-scale statistical inverse problems with application to seismic inversion," SIAM Journal on Scientific Computing 34(3), 1460–1487 (2012).

[4] K. J. H. Law. "Proposals Which Speed Up Function-Space MCMC," Journal of Computational and Applied Mathematics, in press (2013). http://dx.doi.org/10.1016/j.cam.2013.07.026

[5] T. Cui, K.J.H. Law, Y. Marzouk. Dimension-independent, likelihood- informed samplers for Bayesian inverse problems. In preparation.

Abstract : The Bayesian approach is a possible way to build estimators in statistical models. It consists in attributing a probability measure -the prior- to the unknown parameters of the model. The estimator is then the posterior distribution, which is a conditional distribution given the information contained in the data.

The Bernstein-von Mises theorem in parametric models states that under mild regularity conditions, the posterior distribution for the finite-dimensional model parameter is asymptotically Gaussian with `optimal' centering and variance.

In this talk I will discuss recent advances in the understanding of posterior distributions in nonparametric models, that is when the unknown parameter is infinite-dimensional, focusing on a concept of nonparametric Bernstein-von Mises theorem.

This talk develops some aspects of stochastic calculus via regularization for processes with values in a general Banach space B.

A new concept of quadratic variation which depends on a particular subspace is introduced.

An Itô formula and stability results for processes admitting this kind of quadratic variation are presented.

Particular interest is devoted to the case when B is the space of real continuous functions defined on [-T,0], T>0 and the process is the window process X(•) associated with a continuous real process X which, at time t, it takes into account the past of the process.

If X is a finite quadratic variation process (for instance Dirichlet, weak Dirichlet), it is possible to represent a large class of path-dependent random variable h as a real number plus a real forward integral in a semiexplicite form.

This representation result of h makes use of a functional solving an infinite dimensional partial differential equation.

This decomposition generalizes, in some cases, the Clark-Ocone formula which is true when X is the standard Brownian motion W. Some stability results will be given explicitly.

This is a joint work with Francesco Russo (ENSTA ParisTech Paris)."

Abstract: The expected signature is often viewed as a direct analogue of the Laplace transform, and as such it has been asked whether, under certain conditions, it may determine the law of a random signature. In this talk we first introduce a meaningful topology on the space of (geometric) rough paths which allows us to study it as a well-defined probability space. With the help of compact symplectic Lie groups, we then define a set of characteristic functions and show that two random variables in this space are equal in law if and only if they agree on each characteristic function. We finally show that under very general boundedness conditions, the value of each characteristic function is completely determined by the expected signature, giving an affirmative answer to the aforementioned question in many cases. In particular, we demonstrate that the Stratonovich signature is completely determined in law by its expected signature, and show how a similar technique can be used to demonstrate convergence in law of random signatures.

Background material: http://arxiv.org/abs/1307.3580

A review of a valuation strategy to price American-style option contracts in a “limited information” framework is discussed where sequential Monte Carlo (SMC) techniques, as presented in Doucet, de Freitas, and Gordon’s text Sequential Monte Carlo Methods in Practice, and the least–squares Monte Carlo (LSM) approach of Longstaff and Schwartz (Review of Financial Studies 14:113-147, 2001), are used as part of the valuation methodology. We utilize a risk–neutralized version of a mean-reverting model to model the volatility process. We assume that volatility is a latent stochastic process, and we capture information about it using “summary vectors” based on sequential Monte Carlo posterior filtering distributions. Of primary interest in this work is an empirical assessment of American options governed by a stochastic volatility model where the focus is on the market price of volatility risk (or the volatility risk premium). We discuss statistical modeling of the market price of volatility risk as our current evidence reveals interesting nuances about the volatility risk premium, and we hypothesize that switching models or more sophisticated time-series models could be of value to understand the empirical observations we found on the market price of volatility risk. Prior studies have shown that the magnitude of the volatility risk premium changes markedly when an American index option (NYSE Arca Oil Index Options) is in its expiration month relative to prior months, or that the magnitude varies across equities. Our objective is to study if useful information can be extracted from the volatility risk premium process, and how this information can better inform holders of American options when making decisions under uncertainty.

Key words: American options, stochastic volatility, volatility risk, sequential, Monte Carlo, risk premium, decisions, uncertainty

Disclaimer: The views expressed in this abstract (and the paper that will accompany it) are solely those of the authors and do not, in any way, reflect the opinions of the Office of the Comptroller of the Currency (OCC).

The talk will recall the results of three preprints, first two authored by my former student Mickael Launay, and the final coauthored by Mickael and myself. All three works are available on arXiv. At this point it is not clear that they will ever get published (or submitted for review) but hopefully this does not make their contents less interesting. This class of interacting urn processes was introduced in Launay's thesis, in an attempt to model more realistically the memory sharing that occurs in food trail pheromone marking or in similar collective learning phenomena. An interesting critical behavior occurs already in the case of exponential reinforcement. No prior knowledge of strong urns will be assumed, and I will try to explain the reason behind the phase transition.

The coalescing Brownian flow on $\R$ is a process which was introduced by Arratia (1979) and Toth and Werner (1997), and which formally corresponds to starting coalescing Brownian motions from every space-time point. We provide a new state space and topology for this process and obtain an invariance principle for coalescing random walks. The invariance principle holds under a finite variance assumption and is thus optimal. In a series of previous works, this question was studied under a different topology, and a moment of order $3-\eps$ was necessary for the convergence to hold. Our proof relies crucially on recent work of Schramm and Smirnov on scaling limits of critical percolation in the plane. Our approach is sufficiently simple that we can handle substantially more complicated coalescing flows with little extra work -- in particular similar results are obtained in the case of coalescing Brownian motions on the Sierpinski gasket. This is the first such result where the limiting paths do not enjoy the non-crossing property.

Joint work with Christophe Garban (Lyon) and Arnab Sen (Minnesota).

With F. Johansson Viklund (Columbia) and A. Turner (Lancaster), we have studied a regularized version of the Hastings-Levitov model of random Laplacian growth. In addition to the usual feedback parameter $\alpha>0$, this regularized version of the growth process features a smoothing parameter $\sigma>0$.

We prove convergence of random clusters, in the limit as the size of the individual aggregating particles tends to zero, to deterministic limits, provided the smoothing parameter does not tend to zero too fast. We also study scalings limit of the harmonic measure flow on the boundary, and show that it can be described in terms of stopped Brownian webs on the circle. In contrast to the case $\alpha=0$, the flow does not always collapse into a single Brownian motion, which can be interpreted as a random number of infinite branches being present in the clusters.

Abstract: The boundary Harnack principle states that the ratio of any two functions, which are positive and harmonic on a domain, is bounded near some part of the boundary where both functions vanish. A given domain may or may not have this property, depending on the geometry of its boundary and the underlying metric measure space.

In this talk, we will consider a scale-invariant boundary Harnack principle on domains that are inner uniform. This has applications such as two-sided bounds on the Dirichlet heat kernel, or the identification of the Martin boundary and the topological boundary for bounded inner uniform domains.

The inner uniformity provides a large class of domains which may have very rough boundary as long as there are no cusps. Aikawa and Ancona proved the scale-invariant boundary Harnack principle on inner uniform domains in Euclidean space. Gyrya and Saloff-Coste gave a proof in the setting of non-fractal strictly local Dirichlet spaces that satisfy a parabolic Harnack inequality.

I will present a scale-invariant boundary Harnack principle for inner uniform domains in metric measure Dirichlet spaces that satisfy a parabolic Harnack inequality. This result applies to fractal spaces.

''Regression analysis aims to use observational data from multiple observations to develop a functional relationship relating explanatory variables to response variables, which is important for much of modern statistics, and econometrics, and also the field of machine learning. In this paper, we consider the special case where the explanatory variable is a stream of information, and the response is also potentially a stream. We provide an approach based on identifying carefully chosen features of the stream which allows linear regression to be used to characterise the functional relationship between explanatory variables and the conditional distribution of the response; the methods used to develop and justify this approach, such as the signature of a stream and the shue product of tensors, are standard tools in the theory of rough paths and seem appropriate in this context of regression as well and provide a surprisingly unified and non-parametric approach.''

(joint work with Aurélien Alfonsi and Arturo Kohatsu-Higa)

We are interested in the Wasserstein distance on the space of continuous sample-paths equipped with the supremum norm between the laws of a uniformly elliptic one-dimensional diffusion process and its continuous-time Euler scheme with N steps. This distance controls the discretization biais for a large class of path-dependent payoffs.

Its convergence rate to 0 is clearly intermediate between -the rate -1/2 of the strong error estimation obtained when coupling the stochastic differential equation and its Euler scheme with the same Brownian motion -and the rate -1 of the weak error estimation obtained when comparing the expectations of the same function of the diffusion and its Euler scheme at the terminal time.

For uniformly elliptic one-dimensional stochastic differential equations, we prove that this rate is not worse than -2/3.

Abstract: In 2010, Erschler, Tóth and Werner introduced the so-called Stuck Walks, which are a class of self-interacting random walks on Z for which there is competition between repulsion at small scale and attraction at large scale. They proved that, for any positive integer L, if the relevant parameter belongs to a certain interval, then such random walks localize on L + 2 sites with positive probability. They also conjectured that it is the almost sure behaviour. We settle this conjecture partially, proving that the walk localizes on L + 2 or L + 3 sites almost surely, under the same assumptions.

In this talk, we consider the setting: a random realization of an evolving dynamical system, and explain how, using notions common in the theory of rough paths, such as the signature, and shuffle product, one can provide a new united approach to the fundamental problem of predicting the conditional distribution of the near future given the past. We will explain how the problem can be reduced to a linear regression and least squaresanalysis. The approach is clean and systematic and provides a clear gradation of finite dimensional approximations. The approach is also non-parametric and very general but still presents itself in computationally tractable and flexible restricted forms for concrete problems. Popular techniques in time series analysis such as GARCH can be seen to be restricted special cases of our approach but it is not clear they are always the best or most informative choices. Some numerical examples will be shown in order to compare our approach and standard time series models.

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.

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).

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.

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.

We relate the expected signature to the Fourier transform of n-point functions, first studied by O. Schramm, and subsequently
by J. Cardy and Simmon, D. Belyaev and J. Viklund. We also prove that the signatures determine the paths in the complement of a Chordal SLE null set. In the end, we will also discuss an idea on how to extend the uniqueness of signatures result by Hambly and Lyons (2006) to paths with finite 1<p<2variations.

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.