# Research Seminars

## Volatility estimation under one-sided errors with applications to limit order books

Speaker(s):
Markus Bibinger (Universität Mannheim)
Date:
Thursday, December 17, 2015 - 4:15pm
Location:
HU Berlin, Rudower Chaussee 25, Room 1.115

We consider a semi-martingale which forms a stochastic boundary, where we have discrete observations with one-sided errors. A rate-optimal estimator of the quadratic variation is constructed. The problem is embedded in a Poisson point process framework, which reveals an interesting connection to the theory of Brownian excursion areas. We derive n 1=3 as optimal (minimax) convergence rate in a high-frequency framework with n observations (in mean).

## Concentration bounds and asympotic distribution for the empirical spectral projectors of sample covariance operators

Speaker(s):
Karim Lounici (Georgia Institute of Technology, Atlanta)
Date:
Wednesday, December 9, 2015 - 10:00am
Location:
Paul-Drude-Institut für Festkörperelektronik, Hausvogteiplatz 5-7, 10117 Berlin, EG Raum 007

Let $X,X_1,\dots, X_n$ be i.i.d.

## Backward Stochastic Partial Differential Equations in Hölder Spaces

Speaker(s):
Wenning Wei (Fudan University)
Date:
Thursday, December 3, 2015 - 5:15pm
Location:
HU Berlin, Rudower Chaussee 25, Room 1.115

My talk is concerned with solution in Hölder spaces for linear and semi-linear backward stochastic partial differential equations (BSPDEs) of super-parabolic type. The pair of unknown functional variables are viewed as deterministic time-space functionals, but take values in Banach spaces of random (vector) variables or processes. We define suitable functional Hölder spaces for them and give some inequalities among these Hölder norms. The existence, uniqueness as well as the regularity of solutions are proved for BSPDEs, which contain new assertions even on deterministic PDEs.

## Generalized Dynkin games and game options in an imperfect market with default

Speaker(s):
Roxana Dumitrescu (HU Berlin)
Date:
Thursday, December 3, 2015 - 4:15pm
Location:
HU Berlin, Rudower Chaussee 25, Room 1.115

In the first part of the talk, we introduce a generalized Dynkin game problem with non linear conditional expectation induced by a Backward Stochastic Differential Equation (BSDE) with jumps. Under Mokobodski's condition, we establish the existence of a value function for this game. This value can be characterized via a doubly reflected BSDE. Using this characterization, we provide some new results on these equations, such as comparison theorems and a priori estimates. When the obstacles are left upper semicontinuous along stopping times, we prove the existence of a saddle point.

## Non-asymptotic upper bounds for the reconstruction error of PCA

Speaker(s):
Martin Wahl (HU Berlin)
Date:
Wednesday, December 2, 2015 - 10:00am
Location:
WIAS, Erhard-Schmidt-Saal, Mohrenstraße 39, 10117 Berlin

Principal component analysis (PCA) is a standard tool for dimension reduction. Classical objects of interest are the principal subspaces and their empirical counterparts. In this talk, we focus on the reconstruction error of PCA, and prove a non-asymptotic upper bound for the corresponding excess risk. This bound unifies and improves several upper bounds which were previously obtained by empirical process theory. Moreover, the bound reveals that the excess risk differs considerably from the usual subspace distance based on canonical angles.

## On the Optimality of Averaging in Distributed Statistical Learning

Speaker(s):
Jonathan Rosenblatt (Ben-Gourion University Negev)
Date:
Wednesday, November 25, 2015 - 10:00am
Location:
WIAS, Erhard-Schmidt-Saal, Mohrenstraße 39, 10117 Berlin

A common approach to statistical learning on big data is to randomly split it among m machines and calculate the parameter of interest by averaging their m individual estimates. Focusing on empirical risk minimization, or equivalently M-estimation, we study the statistical error incurred by this strategy. We consider two asymptotic settings: one where the number of samples per machine n \to inf but the number of parameters p is fixed, and a second high-dimensional regime where both p, n \to inf with p/n \to \kappa.

## Optimal market making

Speaker(s):
Olivier Guéant (ENSAE ParisTech)
Date:
Thursday, November 19, 2015 - 5:15pm
Location:
HU Berlin, Rudower Chaussee 25, Room 1.115

Market makers provide liquidity to other market participants: they propose prices at which they stand ready to buy and sell a wide variety of assets. Market makers face a complex dynamical optimization problem. They need to propose bid and offer/ask prices in an optimal way for making money out of the difference between these two prices (their bid-ask spread), while mitigating the risk associated with price changes. In practice, market makers indeed seldom buy and sell simultaneously. Therefore, they hold long or short inventories and are exposed to market risk.

## On optimal transport under the causality constraint

Speaker(s):
Julio Backhoff (Universität Wien)
Date:
Thursday, November 19, 2015 - 4:15pm
Location:
HU Berlin, Rudower Chaussee 25, Room 1.115

In this talk we shall examine causal transports and the associated optimal transportation problem under the causality constraint (Pc) introduced by Rémi Lasalle. Loosely speaking, causal transports are a relaxation of adapted processes in the same sense as Kantorovich transport plans are the extension of Monge-type transport maps. We will establish a simple primal-dual picture of both (Pc) and the so-called bicausal transportation problem (whereby causality runs in both directions) in euclidean space or equiv. for discrete-time processes.

## The Emergence of Delta-Vega Hedging in the Black-Scholes Model

Speaker(s):
Johannes Muhle-Karbe (ETH Zürich)
Date:
Thursday, November 5, 2015 - 5:15pm
Location:
HU Berlin, Rudower Chaussee 25, Room 1.115

We study option pricing and hedging with uncertainty about a Black-Scholes reference model. For dynamic trading in the underlying asset and a liquidly traded vanilla option, delta-vega hedging is asymptotically optimal in the limit for small uncertainty aversion. The corresponding price corrections are determined by a number of second-order greeks, namely the option’s gamma, vanna, and volga.

(Joint work with Sebastian Herrmann)

## Fast low-rank estimation by projected gradient descent: Statistical and algorithmic guarantees

Speaker(s):
Martin Wainwright (University Berkeley)
Date:
Wednesday, October 28, 2015 - 10:00am
Location:
WIAS, Erhard-Schmidt-Saal, Mohrenstraße 39, 10117 Berlin

Optimization problems with rank constraints arise in many applications, including matrix regression, structured PCA, matrix completion and matrix decomposition problems. An attractive heuristic for solving such problems is to factorize the low-rank matrix, and to run projected gradient descent on the nonconvex problem in the lower dimensional factorized space. We provide a general set of conditions under which projected gradient descent, when given a suitable initialization, converges geometrically to a statistically useful solution.