Das Institut für Mathematische Wirtschaftsforschung veranstaltet im Rahmen des Bielefeld Stochastic Afternoon regelmäßig Seminare zum Thema Finanzmathematik. Das Programm des aktuellen Semesters finden Sie hier.
Mathieu Rosenbaum (Ecole Polytechnique, Paris)
Titel: Optimal make-take fees for market making regulation
Abstract: We consider an exchange who wishes to set suitable make-take fees to attract liquidity on its platform. Using a principal-agent approach, we are able to describe in quasi-explicit form the optimal contract to propose to a market maker. This contract depends essentially on the market maker inventory trajectory and on the volatility of the asset. We also provide the optimal quotes that should be displayed by the market maker. The simplicity of our formulas allows us to analyze in details the effects of optimal contracting with an exchange, compared to a situation without contract. We show in particular that it leads to higher quality liquidity and lower trading costs for investors. This is joint work with Omar El Euch, Thibaut Mastrolia and Nizar Touzi.
Sören Christensen (University of Hamburg)
Titel: Nonparametric learning in stochastic control - exploration vs. exploitation
Abstract: One of the fundamental assumptions in stochastic control of continuous time processes is that the dynamics of the underlying (diffusion) process is known. This is, however, usually obviously not fulfilled in practice. On the other hand, over the last decades, a rich theory for nonparametric estimation of the drift (and volatility) for continuous time processes has been developed. The aim of this talk is to make a first (small) step to bringing together techniques from stochastic control with methods from statistics for stochastic processes to find a way to both learn the dynamics of the underlying process and control good at the same time. To this end, we study a toy example motivated from optimal harvesting, mathematically described as an impulse control problem. One of the problems that immediately arises is an Exploration vs. Exploitation behavior as is well known in Machine Learning. We propose a way to deal with this issue and analyse the proposed strategy asymptotically.
Rama Cont (University of Oxford)
Titel: Rough calculus: pathwise integration and non-anticipative calculus for functionals of irregular paths
Abstract: We construct a pathwise calculus which extends the Ito calculus to smooth functionals of continuous paths with regularity defined in terms of the p-th variation along a sequence of time partitions for arbitrary large p >0. We show pointwise convergence of appropriately defined compensated Riemann sums. The corresponding pathwise integral satisfies a change of variable formula and an isometry formula. Results for functions are extended to regular path-dependent functionals using the concept of vertical derivative of a functional. Finally, we obtain a “signal plus noise” decomposition for regular functionals of paths with strictly increasing p-th variation. Our results apply to sample paths of semimartingales as well as fractional Brownian motions with arbitrary Hurst parameter H>0.
Based on joint work with: Anna Ananova (Oxford) and Nicholas Perkowski (Berlin).
References:
A Ananova, R Cont (2017) Pathwise integration with respect to paths of finite quadratic variation, Journal de Mathematiques Pures et Appliquees, Volume 107, No 6, June 2017, 737-757.
R Cont, N Perkowski (2019) Pathwise integration and change of variable formulas for continuous paths with arbitrary regularity, Transactions of the American Mathematical Society (Series B), Volume 6, 161-186.
Marco Maggis (University of Milan)
Titel: A general framework for quasisure functional analysis
Abstract: Motivated by the economic phenomenon of Knightian uncertainty, working with a measurable structure where a set of probability measures replaces the single reference probability has become a common model assumption in mathematical finance and economic theory. More precisely, this replacement has led to the development of robust finance, and the techniques are often referred to as “quasisure analysis”.
We propose a general framework which allows the study of financial and economic models exploiting functional analytical methods. The abstract discussion will be followed through a main significant example, namely the usual volatility uncertainty financial market model, which perfectly fits our construction. This talk is based on a joint work with F. Liebrich and G. Svindland.
Markus Fischer (University of Padova)
Titel: On the convergence problem in mean field games: a two state model without uniqueness
Abstract: Mean field games are limit models for symmetric non-cooperative dynamic N-player games as the number of players N tends to infinity. The notion of solution usually adopted for the prelimit models is that of a Nash equilibrium. The convergence problem consists in making the passage to the limit rigorous. For Nash equilibria in Markov feedback strategies, Cardaliaguet, Delarue, Lasry, and Lions (2015) established convergence under the condition that the so-called master equation possesses a unique (regular) solution. This implies uniqueness of solutions for the mean field game. Here, we consider a simple two-state mean field game that exhibits multiple solutions. We show that the (uniquely determined) Markov feedback Nash equilibria of the associated N-player games select, as N tends to infinity, a particular solution of the mean field game. That solution can be characterized in different ways, as the unique entropy solution of the master equation interpreted as a scalar conservation law, but also as the optimizer of an associated deterministic control problem. This is based on a joint work with Alekos Cecchin, Paolo Dai Pra, and Guglielmo Pelino.
Jan Palczewski (University of Leeds)
Titel: Zero-sum stopping games with asymmetric information
Abstract: We study the value of a zero-sum stopping game in which the payoff functional depends on the underlying process and on an additional randomness (with finitely many states) which is known to one player but unknown to the other. Such asymmetry of information arises naturally in insider trading when one of the counterparties knows an announcement before it is publicly released, e.g., central bank's interest rates decision or company earnings/business plans. In the context of game options this splits the pricing problem into the phase before announcement (asymmetric information) and after announcement (full information); the value of the latter exists and forms the terminal payoff of the asymmetric phase.
The above game does not have a value if both players use pure stopping times as the informed player's actions would reveal too much of his excess knowledge. The informed player manages the trade-off between releasing information and stopping optimally employing randomised stopping times. We reformulate the stopping game as a zero-sum game between a stopper (the uninformed player) and a singular controller (the informed player). We prove existence of the value of the latter game for a large class of underlying strong Markov processes including multi-variate diffusions and Feller processes. The main tools are approximations by smooth singular controls and by discrete-time games.