Titles & Abstracts

• Yacine Aït-Sahalia (Princeton University)

  Estimating the degree of activity of jumps in high frequency financial data

  We define a generalized index of jump activity, propose estimators of that index for a discretely sampled process, and derive the estimators' properties. These estimators are applicable despite the presence of Brownian volatility in the process, which makes it more challenging to infer the characteristics of the small, infinite activity, jumps. When the method is applied to high frequency stock returns, we find evidence of infinitely active jumps in the data and estimate their index of activity. The full paper (joint work with Jean Jacod) is here.

• Anestis Antoniadis (Universite Joseph Fourier, Grenoble)

  Smoothing non equispaced heavy noisy data with wavelet kernels

  We consider a nonparametric noisy data model $Y_k = f(x_k) + \epsilon_k, \quad k=1, \dots, n,$ where the unknown signal $f: \ [0,1] \rightarrow \RR$ is assumed to belong to a wide range of function classes, including discontinuous functions and the $\epsilon_k's$ are independent identically distributed noises with zero median. The distribution of the noise is assumed to be unknown and satisfying some weak conditions. Possible noise distributions may have heavy tails, so that, for example, no moments of the noise exist. The design points are assumed to be deterministic points, not necessarily equispaced within the interval $[0,1]$. We first use local medians to construct certain variables $Z_k$ structured as a Gaussian nonparametric regression, but the resulting data being not equispaced, we apply a wavelet block penalizing procedure adapted to non equidistant designs to construct an estimator of the regression function. Under mild assumptions on the design it is shown that our estimator simultaneously attains the optimal rate of convergence over a wide range of Besov classes, without prior knowledge of the smoothness of the underlying functions or prior knowledge of the error distribution.

• Sara Biagini (University of Pisa)

  The Relaxed Investor and the Relaxed Utility Maximization Problem

  For utility functions U finite only on the positive real line, Kramkov and Schachermayer showed that under a condition on U, the well- known Reasonable Asymptotic Elasticity, the associated utility maximization problem has a (unique) optimal solution, independently of the probabilistic model. What about the *relaxed* investor, whose utility does not satisfy RAE? This has been also addressed by Kramkov and Schachermayer, but the optimal solution is characterized only for sufficiently small initial endowments. Under a sufficient (and basically necessary) joint condition on the probabilistic model and the utility, we show by relaxation and duality techniques that the maximization problem admits solution for any initial endowment. However, a singular part may pop up, that is the optimal investment may have a component which is concentrated on a set of probability zero. This singular part may fail to be unique. (joint work with P. Guasoni)

• Erhan Çinlar (Princeton University)

  Jump-Diffusions

  Markovian motions that combine jumps with diffusions are getting renewed interest in financial mathematics and in the theory of Markov processes. Such processes are considered difficult, because their generators are integro-differential operators. We take a decomposition approach that concentrates on the jumps. Embedded at a sequence of appropriately chosen jump times, there is a Markov chain (discrete-time, continuous space) that decomposes the original process into a sequence of diffusions. Then, the original resolvent can be written as the potential operator of that Markov chain acting on the resolvent of a diffusion. Similar decompositions are possible for hitting distributions and the transition semigroup. The net effect is to reduce the original jump-diffusion into a nested sequence of Markov chains.

• Albina Danilova (Oxford University)

  Stock Market Insider Trading in Continuous Time with Imperfect Dynamic Information

  This paper studies the equilibrium pricing of asset shares in the presence of dynamic private information. The market consists of a risk-neutral informed agent who observes the firm value, noise traders, and competitive market makers who set share prices using the total order flow as a noisy signal of the insider's information. I provide a characterization of all optimal strategies, and prove existence of both markovian and non markovian equilibria by deriving closed form solutions for the optimal order process of the informed trader and the optimal pricing rule of the market maker. The consideration of non markovian equilibrium is relevant since market maker might decide to re-weight past information after recieving a new signal. Also, I show that a) there is a unique markovian equilibrium price process which allows the insider to trade undetected, and that b) the presence of an insider increases the market informational efficiency, in particular for times close to dividend payment.

• Ingrid Daubechies (Princeton University)

  Sparse and Stable Markowitz Portfolios

  We consider the problem of portfolio selection within the classical Markowitz mean-variance framework, reformulated as a constrained least-squares regression problem. We propose to add to the objective function a penalty proportional to the sum of the absolute values of the portfolio weights. This penalty regularizes (stabilizes) the optimization problem, encourages sparse portfolios (i.e. portfolios with only few active positions), and allows to account for transaction costs. Our approach recovers as special cases the no-short-positions portfolios, but does allow for short positions in limited number. We implement this methodology on two benchmark data sets constructed by Fama and French. Using only a modest amount of training data, we construct portfolios whose out-of-sample performance, as measured by Sharpe ratio, is consistently and significantly better than that of the naive evenly-weighted portfolio which constitutes, as shown in recent literature, a very tough benchmark.

• Jean-Pierre Fouque (University of California at Santa Barbara)

  An Introduction to Time Reversal and Applications

  I will explain what is a physical time-reversal experiment as conducted, for instance, by Mathias Fink and his group in Paris in the context of ultrasounds. Refocusing, multipathing, superresolution, and aperture enhancement will be discussed as well as applications in detection, destruction, imaging, communications,... Finally, scaling regimes will be introduced and some recent mathematical results in the case of randomly layered media will be presented.

• Damir Filipovic (Vienna Institute of Finance)

  Dynamic CDO Term Structure Modelling

  This paper provides a unifying approach for valuing contingent claims on a portfolio of credits, such as collateralized debt obligations (CDOs). We introduce the defaultable (T; x)-bonds, which pay one if the aggregated loss process in the underlying pool of the CDO has not exceeded x at maturity T, and zero else. Necessary and suffient conditions on the stochastic term structure movements for the absence of arbitrage are given. Moreover, we show that any exogenous specification of the forward rates and spreads volatility curve actually yields a consistent loss process and thus an arbitrage-free family of (T; x)-bond prices. For the sake of analytical and computational efficiency we then develop a tractable class of affine term structure models.

• Leonard Gross (Cornell University)

  From Brownian motion to Lie's third theorem

  The Brownian motion into a compact Lie group is ergodic with respect to translations by the Cameron-Martin subgroup. The same is true for the pinned versions. The proof of these theorems has led to a non-commutative analog of the Fock space, the natural isomorphisms long associated with its representations, a spectral gap for Schrodinger operators over loop groups, major steps toward a Hodge-deRham theorem over loop spaces, heat kernel analysis for hypoelliptic heat kernels over Lie groups, and insights into geometric quantization. The past fifteen years of evolution of these ideas will be surveyed and an application of the methods to a new proof of Lie's third theorem will be sketched.

• Elliott Lieb (Princeton University)

  Four decades of "stability of matter" and analytic inequalities

  I will briefly review the progress that has been made on the problem of stability of matter since its inception in 1967 with the work of Dyson and Lenard. Each bit of progress usually required a novel inequality in functional analysis or potential theory and my intention is to try to list some of the more interesting ones, including some very recent developments.

• Mike Ludkovski (University of Michigan)

  Optimal stopping under partial information

  We study the numerical solution of nonlinear partially observed optimal stopping problems. The system state is taken to be a nonlinear multi-dimensional diffusion and drives the drift of the observation process, which is another multi-dimensional diffusion with correlated noise. Alternatively, discrete observations can also be considered. Such models are of interest in mathematical finance in connection with pricing of American options with stochastic volatility, optimal investment under partial information and pricing commodity derivatives in convenience yield models. We propose a new approximate numerical algorithm that is based on particle filters and regression Monte Carlo methods. The algorithm maintains a continuous state-space and yields an integrated approach to the filtering and control sub-problems. Our approach is entirely simulation-based and therefore allows for a robust implementation with respect to model specification. We carry out the error analysis of our scheme and illustrate with several computational examples.

• Terry Lyons (Oxford University)

  New uses for Ito's map

  It is an interesting exercise to compute the iterated integrals of Brownian motion and to calculate the expectations (of polynomial functions of these integrals). Recent work on constructing discrete measures on path space, which give the same value as Wiener measure to certain of these expectations, has led to promising new numerical algorithms for solving 2nd order parabolic PDEs in moderate dimensions. Old work of Krylov associated finitely additive signed measures to certain constant coefficient PDEs of higher order. Recent work with Levin allows us to identify the relevant expectations of iterated integrals in this case, leaving many interesting open questions and possible numerical algorithms for solving high dimensional elliptic PDEs.

• Stanislav Molchanov (University of North Carolina at Charlotte)

  Critical behavior of the homopolymers

  My talk will present the results on the behavior of the long homogeneous polymer chains (homopolymers) near critical temperature, in other words, the description of the phase transition from the globular to the diffusive states. The theory includes the analysis of the critical indexes and the renormalization groups for the corresponding Gibb's measures.

• David Nualart (University of Kansas)

  Central limit theorem for functionals of Gaussian processes

  The aim of this talk is to present some recent advances in the central limit theorem for functionals of Gaussian processes. It turns out that in the case of a normalized sequence of multiple stochastic integrals of a fixed order, the convergence in distribution to a normal law is equivalent to the convergence of the moments of order fourth. An equivalent condition is the convergence in mean square of the square norm of the derivatives in the sense of Malliavin calculus to a constant, which is equal to the order to the multiple stochastic integrals. These results can be extended to the case of multidimensional sequences and they can be used to provide useful criteria for the central limit theorem to hold for general sequences of square integrable functionals of the underlying Gaussian process. We plan to discuss some applications of these criteria to the analysis of the asymptotic behavior of quantities such as the weighted p-variations, and self-intersection local time of the fractional Brownian motion.

• Anastasia Papavasiliou (Warwick University)

  Parameter Estimation for Rough Differential Equations

  I will describe a moment-matching method for estimating parameters of a differential equation driven by rough paths, when the expected signature of the rough path is known. The motivation comes from multiscale simulations.

• Boris Rozovsky (Brown University)

  On Elliptic PDEs with Random Coefficients

  I will discuss boundary problems for elliptic PDEs with highly variable random coefficients. A typical example of such equation is given by the following Dirichlet problem:
  $(K(x) \cdot u_x(X))_X + f(x) = 0, x\in (a,b), u_{|\partial G}=0$ (1)
  where $K(x) = \bar{K}(x) + \varepsilon \dot{W}(x),$ $\bar{K}(x)$ is a uniformly positive bounded deterministic function, and $\dot{W}(x)$ is a spatial white noise. This example is related to problems of identification and validation of mathematical models with uncertain parameters.
  Mathematical theory of equations with highly variable coefficients presents a number of challenging problems. For one, equation (1) is not elliptic in the sense of the classical theory. Nevertheless, it is possible to show that this equation and more general equations of this type allow interpretation that make them statistically well posed. Solutions to these equations are not square integrable and shall be considered in weighted spaces.
  It will be shown that a weighted version of Malliavin calculus is an effective tool in studying elliptic equations with highly variable coefficients. We will also discuss weighted Wiener chaos approximations of the solutions and related statistical moments.

• Nizar Touzi (Ecole Polytechnique)

  Stochastic Target Problems with Controlled Loss

  We first consider the target reachability problem with unbounded controls. The derivation of the dynamic programming equation in this context needs special care. This result is used crucially in order to solve the problem of target reachability with given probability of success, and more generally with given loss. The main idea is to convert the latter problem into a stochastic target problem by introducing an additional controlled state, whose control process turns out to be unbounded. In particular, this method provides a direct PDE approach to the Follmer and Leukert problem of quantile hedging.

• Frederi Viens (Purdue University)

  Polymers in Gaussian environments: Lyapunov exponents, memory length, and superdiffusivity

  Rene Carmona and collaborators pioneered work on the asymptotic behavior of the stochastic Anderson model in the early 1990's: the techniques and results of these works are still at the center of current preoccupations on stochastic PDEs, polymers, and random media. In this talk, we will review these works, and other recent results on the almost-sure Lyapunov exponents for the stochastic Anderson model, in discrete and continuous space. We will relate these exponents to the model's role as the partition function of a polymer's Gibbs measure in a Gaussian environment, explaining the connection with weak and strong disorder, with questions of space and time memory length, and with the property of superdiffusivity.